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DIPOG-2.1 User Guide Direct Problems for Optical Gratings

1. 0 if Smk gt 0 Indeed these coefficients A describe magnitude and phase shift of the propagating plane waves More precisely the modulus A n is the amplitude of the nth reflected resp trans mitted wave mode and arg A A the phase shift The terms with n U lead to evanescent waves only The optical efficiencies of the grating are defined by z a Pr A e e a 4 e y t neU U m neU 24 Bg AG which is the ratio of energy of the incident wave entailed to the nth propagating mode Note that these efficiencies of propagating modes exist for non absorbing materials i e for Smk 0 If the transverse component has been computed approximately then the Rayleigh coefficients can be obtained by a discretized Fourier series expansion applied to the FEM solution restricted to cf and 2 3 Formula yields the efficiencies 2 2 The classical TM problem The case of TM polarization is quite similar to TE Indeed this time the vector of the magnetic field H shows in the direction of the grooves i e in the direction of the z axis 16 Analogously to formula 2 1 given in the last subsection for the incident electric field we get Hircident a y z t Hircident y Y z exp iwt 2 5 EgV nt Honing y z va exp ikt sinf z ik cos y k w m n Jio for the z component of the incident magnetic field Hi dert Note t
2. if Grating data is profiles 3 gt N_mat n 1 with n nmb_curves from the file GEOMETRIES profiles_par c if Grating data is pin gt N_mat 3 if Grating data is cpin gt N_mat 4 if Grating data is cpin2 gt N_mat 4 125 if Grating data is stack k gt N_mat k 1 if Grating data is bOX gt N_mat number indicated as second integer in second lind after line with code word bO0X if Grating data is rough_mls gt N_mat nlatnidtnlbt2 if width of additional layer above and below is positive N_mat nlatnidtnlb 1 if instead of the value for the width of additional layer above and below a no is given in the input file Number of materials 4 FERRER PETITE TET TEEPE TTT TEETH ee EEE TE ETE TERE ETE HE ATE ERE E EEE Optical indices of grating materials This is c times square root of mu times epsilon If meaningful then the refractive indices should be ordered according to the location from above to below If an input file namel inp is used then the optical index of a subdomain with the material index j is just the j th optical index following below If grating is lAmellar then first material is cover material last material is substrate and all other materials are ordered from left to right and inside the columns from below to above For technical
3. 0 004059 0 131487 0 003041 0 001482 0 009015 0 008718 0 007285 0 148711 2 035991 Transmission efficiencies and coefficients n O theta 26 95 n 1 theta 50 41 n 1 theta 7 80 0 439727 0 251543 0 521995 0 681800 0 061566 0 302391 145 ooo oO I l 2 1 3 O DOOH Ze 55 Ti 462309 lt 117800 236279 757185 164372 761614 730540 002606 034668 268177 066021 589973 162601 0 115618 0 050452 0 160324 1 609071 0 781807 1 754536 n 2 theta 10 48 0 021529 n 3 theta 29 96 0 071286 n 4 theta 54 77 0 002095 Transmitted energy 97 964009 INFO OF SOLUTION LEVEL 3 degrees of freedom 12628 stepsize of discr 0 19522 numb of nonzero entr 92896 rate of nonzero entr 0 058254 per cent memory for pardiso 12659 kB Reflection efficiencies and coefficients n 0 n 1 n 2 n 3 theta theta theta theta 65 00 15 74 21 33 87 07 Reflected energy C 0 012070 0 003512 0 002098 0 079551 2 179773 0 139848 0 014030 0 002527 0 082779 Transmission efficiencies and coefficients 0 127814 0 568916 0 299329 0 125325 0 111135 0 069144 2 1 970304 047639 002377 159453 l o OOH 27 975753 52 762242 12619565 2 015636 1 414798 1 032234 n O theta 26 95 0 498922 n 1 theta 650 41 0 613317 n
4. DIPOG 2 1 User Guide Direct Problems for Optical Gratings over Triangular Grids A Rathsfeld Weierstraf Institut f r Angewandte Analysis und Stochastik Leibniz Institut im Forschungsverbund Berlin e V Mohrenstr 39 D 10117 Berlin Germany rathsfeld wias berlin de if NS Weierstrass Institute for Applied Analysis and Stochastics Abstract This guide describes how to use the programs FEM_CHECK GFEM_CHECK FEM GFEM FEM_PLOT GFEM_PLOT FEM_FULLINFO GFEM_FULLINFO and OPTIMIZE of the package DIPOG 2 1 The package is a collection of finite element FEM programs to determine the efficiencies of the diffraction of light by a periodic grating structure It is based on the software package PDELIB and solves the classical case of TE and TM polarization and the case of conical diffraction The code provides a conventional FEM and a generalized FEM called GFEM The latter is the variational approach of the conventional FEM combined with a new trial space Other routines of DIPOG 2 1 determine optimal gratings of certain grating classes minimizing objective functionals depending on the efficiencies We note that the DIPOG 2 1 programs require the installation of the previ ous version DIPOG 1 3 or DIPOG 1 5 of the grid generator TRIANGLE 1 4 of the graphical user interface package FLTK and of the equations solver PARDISC Additionally some of them need the graphical package openGL or the MESA emulation
5. with i 0 1 2 as long as level 2 ix3 is less or equal to 8 The initial solution of the computation on level 2 is the input initial solution The initial solution of the computation on level 2 ix3 for i gt 0 is the final solution of the level 2 i 1 3 Number af HEHEHE EH HAAR RR AEE RRR aaa OBJECTIVE FUNCTIONAL AA E A TE AE TE E A E A E E E TE EEE PE E E AE E A EE T PE PEPER TE LESS TE A TE E PP EEE AE EET PP HEE E HEHE EE HE HE BE HE HE TETEE HERE HE reflected value w_lin_ene_re energy transmitted w_lin_ene_tr energy total w_lin_ene_to energy n_lin_re reflected gt w_lin_re efficiency ass J o_lin_re j 1 j n_lin_tr gt transmitted gt w_lin_tr efficiency see J o_lin_tr j 1 j n_qua_re reflected 2 gt w_qua_re efficiency c_qua_re 55 j o_qua_re kl H j 1 j n_qua_tr gt transmitted 2 gt w_qua_tr efficiency c_qua_ttr J 183 HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHH RH HR HH HH j o guatr j j 1 j n_lin_1_re sss reflected gt w_lin_i_re efficiency_1i gt j o_lin_i_re j 1 j H lin 1 tf ss transmitted gt w_lin_i_tr efficiency_1 asa j o_lin_1_tr j 1 j n_qua_i_re i reflected 2 gt w_qua_t_re efficiency c_qua_i_re j o_qua_i_re j jt j n_
6. p4 Tm a dat pa Te p T 7 To indicate that terms of this type are to be included instead of the corresponding terms depending on one polarization only one easily adds a TE TM before the input of the number of terms cf Section 10 1 3 Similarly if the type of polarization is TE TM if the illumination is classical angle of incidence 0 and if the output data is presented in the TE TM form cf Section 2 3 then the objective functional might contain terms with the phase difference between the classical TE and TM case In other words one might need terms like Son pa TE wet Lo ee whe p Pe wT ey 106 Such terms are indicated by adding a CL TE TM before the input of the number of terms cf Section 10 1 3 10 1 2 Optimization via OPTIMIZE In order to solve the above optimization problem the user changes to the subdirectory OPTIM Here he finds the executable OPTIMIZE the input file conical Dat with the control parameters of the generalized FEM cf the description of the similar file generalized Dat in Section a as well as the example input file example dat cf the enclosed data file in Section At first the user creates his own input file e g name dat containing all the data a tlie grating geometry of the illumination conditions the initial values of the parameterq to be optimized the data of the objective function
7. 0 49973 0 42367 62 77211 1 20 54 135 99 0 26115 0 05446 0 32171 0 20006 17 87244 1 128 80 161 77 0 12492 0 19255 0 13873 0 20804 14 07546 2 158 51 138 27 0 08036 0 05007 153 0 10289 0 07817 2 25198 Transmitted energy 96 97198 Reflection efficiencies and coefficients Order Phi Theta E2 H z Efficiency 0 06129 0 00846 0 09079 0 04955 1 45263 0 02946 0 01247 0 02120 0 02318 0 20498 0 05922 0 00385 0 01123 0 00823 0 02441 O 47 00 30 00 1 128 80 27 98 2 158 51 86 74 Reflected energy 1 68202 Transmission efficiencies and coefficients Order Phi Theta EZ H z Efficiency 0 38108 0 19799 0 58998 0 31418 66 49400 0 22472 0 00453 0 37512 0 09299 15 48188 0 07243 0 23512 0 06585 0 25190 15 85666 0 05169 0 00215 0 03000 0 03203 0 48544 O 47 00 160 53 1 20 54 135 99 A 12880 161 77 2 1568 51 138 27 PP FO LO ON LO OO ON ES Transmitted energy 98 31798 Reflection efficiencies and coefficients Order Phi Theta E Z 154 H_z Efficiency 06068 0 01649 07617 0 03097 1 07145 02951 0 00963 01652 0 00861 0 13367 04706 0 03015 00604 0 01967 0 02330 O 47 00 30 00 1 128 80 27 98 2 158 51 86 74 Reflected energy 1 22842 Transmission efficiencies and coefficients Order Phi Theta E Z H_z Efficiency 0 38965 0 12856 0 61617 0 20649 61 81559 0 25045 0 03050
8. J M Melenk and I Babuska The partition of unity method Basic theory and appli cations Comp Methods Appl Mech Eng 139 pp 289 314 1998 For the solver of the linear system of equations we refer to 22 O Schenk K Gartner and W Fichtner Efficient Sparse LU Factorization with Left Right Looking Strategy on Shared Memory Multiprocessors BIT Vol 40 158 176 2000 O Schenk and K Gartner Solving unsymmetric sparse systems of linear equations with PARDISO Journal of Future Generation Computer Systems 20 pp 475 487 2004 O Schenk and K Grtner On fast factorization pivoting methods for symmetric indef inite systems Elec Trans Numer Anal 23 pp 158 179 2006 3 Geometry Input 3 1 Geometrical data in input file name dat Computation starts with the change of the working directory to directory CLASSICAL for classical diffraction or to CONICAL conical diffraction and by calling an executable e g FEM or GFEM followed by the data file name dat as argument of the executable Here name dat contains all information on the grating and the light cd DIPOG 2 1 CLASSICAL FEM name dat or cd DIPOG 2 1 CONICAL GFEM name dat On the screen there will appear the output data of the computation and the name of an additional output file where the output data is stored Mainly the geometrical information of the input data in name dat is fixed by the lines Length factor of ad
9. cf the last section The profile is vertically shifted by h in wm Note that for technical reasons the blaze angles a and must be less than 90 50 If the defining two lines are lineg 1 binary i do dy dy eine dy lines f then we get a BINARY GRATING PROFILE with 2 teeth with height do with transition points d d2 do and without coating layer In other words before the shift the grating function is zero between 0 and d between dz and d3 between dg _2 and dzi 1 and between dy and the period d The grating function is d between d and dz between ds and d4 between d2 _3 and d2 _2 and between dz _ and dy _ The profile is vertically shifted by h um Note that i lt 6 0 lt do O lt di lt d2 lt lt dy lt d with d the period For 2 equal to 1 or 2 either d 0 or dz d is allowed If the defining two lines are linez2 j 1 trapezoid d d d3 a line hy then we get asymmetric TRAPEZOIDAL GRATING profile cf Section 3 5 with the trapezoid starting at x d4 ending at x dz and with the angle a and the height d3 The profile is vertically shifted by hy wm Note that we require d3 gt 0 0 lt d lt dy lt d and 0 lt a lt 90 with d the period The height must be sufficiently small such that the trapezoid does not degenerate If the defining two lines are linezj 1 profile ccode lines j h then we get a profile curve DETERMINED By A SMOO
10. nameG inp without tag inp Recall that this is the 92 P k P P J N 1 Figure 30 Location of interface in yellow area A geometry input file of the grating which is to be extended cf Section 3 2 and the example in Section 12 1 The fifth class requires the following ni_geom_param 6 integer parameters i geom_param l index 5 of the grating class i_geom_param 2 number of materials which must be one plus the number of different materials indicated in nameG inp if i geom_param 3 gt 0 and which is exactly the number of different materials indicated in nameG inp if i geom_param 3 0 i_geom_param 3 number N of interior corners in the polygonal interface non negative integer i_geom_param 4 index of end point P1 as a grid point in nameG inp resp dummy if i geom_param 3 0 i_geom_param 5 index of end point P2 as a grid point in nameG inp resp dummy if i_geom_param 3 0 i_geom_param 6 index of convex domain A which is divided by the new poly gonal interface as a subdomain in nameG inp resp dummy if i geom_param 3 0 After dividing the convex domain A by the new interface the first subdomain on the right of 93 the polygonal interface running from P1 to P2 inherits the material index i_geom_param 6 and the index of the second subdomain is set to i geom_param 2 1 Before the subdivision i geom_param 2 1 was the material index of the domain adjacent to the
11. sjrj from sjlj s u The partial derivative Of Or turns into Of Or 1 s Of Or Choosing the right scaling factors sj the partial derivatives Of Or can be made to be almost of the same size and the iteration converges well The gradients printed after calling OPTIMIZE with flag f may be helpful to find the scalar factors sj For the notation from Section recall that the variables r are those parameters d_geom_param j which are not fixed Ty setting dl_geom_param j du_geom_param j The corresponding scaling factors s are the values d_geom scal j In the case of fixed real parameters d_geom_param j with dl_geom_param j du_geom_param j the scaling factor d_geom_scal j must be set to one 10 3 2 Conjugate gradient method with projection Suppose the optimization problem is to find a local minimum i e to find an admissible vector Top in RY such that f rop lt f r holds at least for any admissible r RY close to Topt Here a vector r RY is called admissible if the coordinates r of r satisfy l lt ri lt uj and if the constraint conditiong gt Gm r lt 0 are fulfilled for any m 1 M The functionals f and gm are supposed to be continuously differentiable and possibly non linear The conjugate gradient methods consists of the following Steps 1 3 25Obviously for the classes in the Sections 10 2 2 10 2 3 10 2 5 10 2 5 10 2 6 and 10 2 7 7 there are no additional con
12. 0 c_ene_tr no line if w_ene_tr 0 QUADRATIC TERMS TOTAL ENERGY w_qua_ene_to 0 c_ene_to no line if w_ene_to 0 HEHEHE RRA RARER RRR RH HEHEHEHEH OPTIMIAZAAT TON ALGORITHM HEHE AH HRA Aa AE Raa Data for optimization algorithm maximal number of iterations This is usually a positive number However if the level is varying incremental input of level then the maximal number of iterations can be chosen in dependence of the level For m different number of levels the corresponding input consists of m 1 lines The first line contains LEV and is followed by m lines each containing a positive number of maximal iterations E g for the level input I 2 8 3 se LEV 5 T 13 29 means maximal 5 iterations for level 2 maximal 7 iterations for level 5 and maximal 13 iterations for level 8 indicator ind_opt of method ind_opt 1 conjugate gradient method projection onto feasibility set ind_opt 2 interior point method ind_opt 3 augmented Lagrangian method ind_opt 4 simulated annealing ind_opt 5 Newton type method number of integer parameters ni_opt vector i_opt of integer parameters each number in a separate line ni_opt numbers number of real parameters nd_opt vector d_opt of real parameters each number in a separate line nd_opt numbers number of string parameters ns_opt vector s_opt of string parameters each string HHHHHHHHHHHHHHHHHHHHHHHH HHH HH HH
13. 0 0250000 eff_2 tra 0 27 0614650 2 15000 0000000 psh_1 tra 1 135 6202900 _ 2 50 0000000 psh_2 tra 0 69 1695300 _ 2 pshi Cocagk He ue 2 sin 2 Pix psh_i k c 360 functional value data for optimization max n of iterations 100 method of optimization interior point method initial parameter of op 0 10000000000000001 reduct fact for param 0 50000000000000000 const in Armijo criter 0 00100000000000000 bound of stepsize factor 0 90000000000000002 threshold for gradient 0 00000000000001000 204 threshold of rel diff 0 00001000000000000 max n of almost same res 3 level of discr 3 stepsize of discr 0 03978873577297385 times period number of nodes 3935 degrees of freedom 7870 number of iterations 81 max 100 per call number of eval grad 230 norm of red gradient 0 63985931836072452 call of inter_pnt_me only once stopping criterion Warning 3 times the same gradient norm optimal value of objective functional 0 00006906968592636 optimal set of parameters param 3 1 39654044740417649 param 4 0 02205203290024821 param 5 1 29279672856900452 param 6 0 08879797922972749 param 7 1 16142455471499551 param 8 0 06526282507629259 param 11 0 07701248245801905 param 12 0 13277015218125693 date 9 Aug 2005 10 30 13 Thank you for choosing OPTIMIZE Bye bye 205 13 Copyright Responsible programmer
14. 1 theta 7 80 0 134396 n 2 theta 10 48 C 0 040241 n 3 theta 29 96 0 038098 n 4 theta 54 77 l 0 101705 Transmitted energy 97 820227 INFO OF SOLUTION LEVEL 4 degrees of freedom 50133 stepsize of discr 0 09156 numb of nonzero entr 369931 rate of nonzero entr memory for pardiso 0 014719 per cent 59346 kB Reflection efficiencies and coefficients n 0 1 B Il theta theta 65 00 15 74 0 012847 0 000916 146 0 143535 0 015911 e_ 2 e_ 3 e_ 4 e_ 0 e_ e_ e_ 3 e_ 0 e_ a e e_ 2 e_ 3 e_ 4 e_ 0 e_ i 2 076732 0 057844 n 3 theta 87 Reflected energy Transmission efficiencies n O theta 26 n 1 theta 50 n 1 theta T3 n 2 theta 10 n 3 theta 29 n 4 theta 54 Transmitted energy OT 0 088879 0 060381 2 274312 and coefficients 1 2 3 4 Ris 52 13 020839 168650 788186 degrees of freedom stepsize of discr numb of nonzero entr rate of nonzero entr memory for pardiso 95 0 502017 0 095119 41 0 636562 0 542658 80 0 165801 0 295567 48 0 064145 0 115134 96 0 059586 0 088603 TI 0 049270 0 095506 97 725688 200431 0 04493 1481809 0 003689 per cent 278626 kB Reflection efficiencies and coefficients n O theta 65 n 1 theta 15 n 2 theta 21 n 3 theta Sfx Reflected energy Transmission eff
15. 20 indicate that the number of quadratic efficiency terms with non zero weight factor is two i e there are exactly two terms in the sum gt w e c of 10 2 The corresponding orders n of the transmitted plane wave mode are 1 and 0 the weight factors w_ 2 and Wo 3 and the prescribed efficiency values c_ 10 and cy 20 In other words the term J wz le c in turns into 2 e2 10 3 e9 20 The lines QUADRATIC TERMS FIRST TE or S TRANSMITTED EFFICIENCY n_qua_1_tr 1 w_qua_l _tr 2 F o qua ltr 1 equa ltr 10 indicate that the number of first typd quadratic efficiency terms with non zero weight factor is one and the sum gt gt wh eb c 7 turns into 2 et 10 Moreover the values c cht GF c and c may depend on the wave lengths on the angles of incidence 0 and and on the polarization type tm If e g the wave length runs through the values 7 1 2 3 jy then the 7 x n_qua_tr 6 different values can be fixed e g by n QUADRATIC TERMS TRANSMITTED EFFICIENCY n_qua_tr 2 F w_qua_tr WAL a o gqua tr 1 0 2 First type efficiency means efficiency of the TE part of the mode if the type of the output is set to TE T second variant of output and the S part of the mode if the type of the output is set to Jones third variant of output in Section 2 3 80 F equat WAL 10 15 20 25
16. Bry yz I neZ on 0 in BE k an 72 Re Bt gt 0 Smp gt 0 resp E z y z SA exp i lanz b y 72 neZ H xz y z ya exp il ilanz Bau 72l l neZ Now there are three variants of output data The first computes the third components i e the z components of the Rayleigh coefficients 7 220 2 Ja Pn and the efficiencies aa kt c EE t Ste 2 9 1 c Bs te Ps SPP 20 gt X Figure 4 Coordinate system based on x z plane of the nth reflected resp transmitted wave mode The second output variant computes the TE and TM part of the total wave i e if s stands for the direction perpendicular to the y axis and to the direction of propagation of the nth reflected resp transmitted plane wave mode s7 an 87 y x 0 1 0 an 87 y x 0 1 0 then the output coefficients are the scalar products A s _ ap beg n non 1 e Va e gt aq bcp n an 87 7 B 7n a a b c m 2 The efficiencies of the second output are the total efficiencies e of 2 9 and the efficiencies am e 2 l E nt 4 Pn J s l Py Bt si p B n i e the efficiencies of the projection of the nth reflected resp transmitted wave mode to third variant computes the S and
17. Parameter HHEHHHHHHHHHHHHHHHHHHHHHHEHHHHHHHHEHHEHH RHE HHRHHRHRAE HEHEHE Angle of incident wave in degrees theta HHHHHHHHHHHHHHH HHH HH HH HH Eol If type of polarization is pol then the incident light beam takes the direction sin theta cos phi cos theta cos phi sin phi with the restriction 90 lt phi theta lt 90 If type of polarization is TE TM TP then the incident light beam takes the direction sin theta cos phi cos theta sin theta sin phi with the restriction 0 lt theta lt 90 Either add a single value e g 45 Either add more values by e g ce V Oy OS The last means that computation is to be done for the angles from the Vector of length 5 SOB Pg ENGR 2 a GE aA G aa and A ETO T Or add e g e TA5 56 277 The last means that computation is to be done for the angles 45 i 2 with i 0 1 2 and with angle 45 i 2 less or equal to 56 132 Note that either the wave length or the angle of incident wave theta must be single valued Angle 30 HEHEHE RRA R RANNE P PPPH RRR aaa Angle of incident wave in degrees phi If type of polarization is pol then the incident light beam takes the direction sin theta cos phi cos theta cos phi sin phi with the restriction 90 lt phi theta lt 90 If type of polarization is TE Z TM TIP then the incident light beam ta
18. c_ lambdal phim2 mode2 c_ lambdai phim2 moden_ c_ lambda2 phii mode1 c_ lambda2 phil mode2 c_ lambda2 phil moden_ c_ lambda2 phi2 mode1 c_ lambda2 phi2 mode2 187 HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHH RHR azz lambda2 phi2 Clambdam1 phi1 lambdam1 phil Clambdam1 phii Clambdam1 phi2 lambdam1 phi2 lambdam1 phi2 lambda1 thetal lambdal thetal lambdal thetal lambda1 theta2 phil lambdal theta2 phii lambda1 theta2 lambdal theta2 lambdal theta2 moden_ lambda2 phim2 mode1 lambda2 phim2 mode2 lambda2 phim2 moden_ model mode2 moden_ modei mode2 moden_ Clambdam1 phim2 mode1 lambdam1 phim2 mode2 lambdam1 phim2 moden_ lambdal thetai phiil lambdal thetai phil lambda1 theta1 phil lambdai theta1 phi2 lambdai theta1 phi2 lambdai theta1 phi2 lambda1 theta2 phil lambdai theta2 phi2 lambdai theta2 phi2 lambdai theta2 phi2 P modei mode2 moden_ modei mode2 moden_ phim3 mode1 phim3 mode2 phim3 moden_ modei mode2 moden_ modei mode2 moden_ 2 phim3 mode1 phim3 mode2 phim3 moden_ lambdai thetam2 phi1 mode1 lambdal thetam2 phi1 mode2 lambdal thetam2 phi1 moden_ 188 HHHHHHHHHHHH
19. is to use the graphical user interface program TGUI in the subdirectory GEOMETRIES Just call TGUI and draw one period of the cross section of the grating Note that TGUI is equipped with a complete help system For the special DIPOG 2 1 format we mention the following The dimensions of the cross section details including the period horizontal dimension of the grating are measured in nano meters Typically the vertical and horizontal diameters are about 1000 nano meters Note however that the geometry will be scaled to period 1 for the input format and the actual period is fixed in the data file name dat cf Sects 3 1 and 5 1 The cross section domain is bounded by two lateral sides as well as an upper and lower boundary curve connecting the upper resp lower points of the lateral sides Possibly this domain is split into several material parts by polygonal interfaces If the upper or the lower curve is a horizontal straight line segment then DIPOG can add additional upper and lower rectangular strips to the geometry However these strips are described in the data file name dat cf Sect and not by the geometry input format namel inp The lateral sides of the cross section domain must form straight line segments in exactly vertical direction The segment indicator of the segments of these sides must be 3 whereas the segments forming the upper and lower boundaries get indicator 2 and 1 respectively 33 The indices
20. refractive index of k th layer beneath stack Re nl_k gt O Im nl_k gt 0 k 1 i_geom_param 4 d_geom_param nd_geom_param 4 i d_geom_param nd_geom_param 3 refractive index of substrate material resp material of adjacent lower coatingstrip Re n_su gt 0 Im n_su 0 174 HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHH azz te ee fase ee PERE k th trapez h_k k 2 f hel Se v pete Ses o ees r 4 0 0 posa 0 posb 0 d 0 pee Se eee ee Sa ee a eee hl_k k th layer k 2 SSS Ss Ss SSS ee SS SS SSeS sees eS SSS SSS Sa SS SS SS se The height hel of the extra layer beside the bridge may be any non negative number h_1 lt hel lt h_2 is not supposed h_k d_geom_param 5 k 4 height of k th trapezoid in stack in micro meter h_k gt 0 b_k d d_geom_param 5 k 3 ratio of distance of right upper corner from the left boundary line of the period and period d 0 lt b_k d lt 1 a_k b_k d_geom_param 5 k 2 ratio of distance of left upper corner from the left boundary line of the period and distance of right upper corner from the left boundary line of the period 0 lt a_k b_k lt 1 If not both of the parameters b_k d and a_k b_k are fixed then we require a_k b_k gt 0O for b_k d 1 k 1 i_geom_param 3 posb d d_geom_param 5 i_geom_param 3 1 ratio of distance of right lower corner from the left boundary line of the period and period d 0
21. 0 w_lin_1i_tr n_lin_1_tr numbers in n_lin_i_tr lines o_lin_i_tr n_lin_i_tr numbers in n_lin_1i_tr lines LINEAR TERMS SECOND TM or P TRANSMITTED EFFICIENCY n_lin_2_tr 0 w_lin_2_tr n_lin_2_tr numbers in n_lin_2_tr lines o_lin_2_ tr n_lin_2_tr numbers in n_lin_2_tr lines QUADRATIC TERMS REFLECTED EFFICIENCY n_qua_re 0 w_qua_re n_qua_re numbers in n_qua_re lines o_qua_re n_qua_re numbers in n_qua_re lines c_qua_re n_qua_re numbers in n_qua_re lines QUADRATIC TERMS FIRST TE or S REFLECTED EFFICIENCY n_qua_i_re 194 1 w_qua_1_re n_qua_i_re 0 03 o_qua_1_re n_qua_i_re 0 c_qua_1_re n_qua_i_re 51 603605 QUADRATIC TERMS SECOND n_qua_2_re 0 w_qua_2_re n_qua_2_re o_qua_2_re n_qua_2_re numbers in n_qua_i_re numbers in n_qua_i_re numbers in n_qua_i_re lines lines lines TM or P REFLECTED EFFICIENCY numbers in n_qua_2_re numbers in n_qua_2_re lines lines c_qua_2_re n_qua_2_re numbers in n_qua_2_re lines QUADRATIC n_qua_tr 0 w_qua_tr o_qua_tr c_qua_tr QUADRATIC n_qua_i_tr 0 w_qua_1_tr n_qua_1_tr o_qua_1_tr n_qua_1_tr c_qua_1_tr n_qua_1_tr QUADRATIC TERMS SECOND n_qua_2_tr O w_qua_2_tr n_qua_2_tr o_qua_2_tr m gua 2 tr TERMS TRANSMITTED EFFICIENCY numbers in n_qua_1_tr numbers in n_qua_1_tr numbers in n_qua_1_tr TM or P TRANSMITTED numbers in n_qua_2_tr numbers i
22. 1 Additionally a coating layer is attached located between the first curve fe 1 t fy t O lt t lt 1 united with the the two straight line segments z 0 21 lt lt Lmin and z 0 1 min lt lt x2 and a second curve fx 2 t fy 2 t O lt t lt 1 The last connects 31 then this can be accomplished by calling the executable GEN_CPIN2 from the subdirectory GEOMETRIES More precisely suppose min and the profile lines f2 j t fy j t 0 lt t lt 1 j 1 2 are given by the c code in the file GEOMETRIES cpin2 c Then GEN_CPIN2 name 0 05 creates the file name inp of the desired profile grating where the profile curves are ap proximated by a polygonal line with a stepsize equal to 0 05 times the length of period cf Figure 19 where Lmin 0 2 fe 1 t Emin 1 22min t fy 1 t 0 5 sin rt and fel 2 t 0 5 min 1 Smin t fyl 2 t 0 8 sin rt In some applications the user might think that meshes graded towards the corner points of the polygonal interfaces should improve the convergence behaviour of the FEM solution Such graded meshes can be generated by adding clouds of points to the neighbourhood of the corners If a geometry is given by the input file name inp then an advanced input file name inp including point clouds at the corners can be generated by the command POINT_CLOUD_INP name 30 5 100 Here the last argument 100 is the angle in degrees such
23. 1 1 What is DIPOG 2 1 and Dipog 1 5 J ooa aaa eed aw eed ees 1 2 Programming language and used packages o oo aaa a 1 3 Get executables comment lines in input files a a a aa aa 1 4 Structure of the package ooa a a OBA Ree RE ORS 1 5 Environment variables oaoa a a a a a 0 0000 a a a ee Diffraction Problems for Gratings 2 1 The classical TE problem 4 aaa 62406 eee hoe Re RES 2 2 The classical TM problem ous lt x 4 lt 0 2h 6 4 9 eo eee eee be SG 2 3 Conical problems aac 4h ha dS Ode 2 SA EAS RES AK DE ERED ES 2 4 References 2 2 0 000 0 ee A On Geometry Input 3 1 Geometrical data in input file name dat a a aaa 3 2 How to get an input file namel inp aooaa aa tae ee ae 3 4 Input file namel inp by TGUL 2 ed ee ewe ee ewe ee Be ele deine heh thas EE PE ee ee ee e oe ae ee ee ee Input of Refractive Indices Computation of Efficiencies Using FEM in CLASSICAL 5 1 How to get an input file name2 dat 2 0 020 0 04 5 2 Simple calculation with minimal output o a a a 5 3 Check before computation more infos and plots Computation of Efficiencies Using GFEM in CLASSICAL Computation of Efficiencies Using FEM GFEM in CONICAL Plot a Graph with the Efficiencies Parameter Test for GFEM 0 Optimization Tools eee owed esate maid dias ae in oe op tee ete 6 ao es 494 oe eee 8 de Ae AG Aah eee eee ee aes
24. 180 to s equal to the unit vector in the direction opposite to the z coordinate 73 consists in finding a parameter set fop for which the value f ropt is less or equal to f r for all r in the set of the admissible parameter sets of the class The arithmetic expression for f r looks a little bit complicated and is defined as in ko l me yD uim 10 2 j l k 1 l 1 m 1 tot Ee altel Sikim tet 0 e Uet J Dret J we J wy tent n n n rr 1 at 2 25 2 2 Wn En X Wn En X Wn En F Wy le cr n n n n L l 1 2 2 22 2 2 Ts 72 Wh lel G J Wh le Ce T Wp len c n raS 3 x n n Su len dy Sow ax a ey wt e ct E n n 23 g abt att A n n re O de al 2 2 2 2 2 oe ler Sele eee He les n n x 2 p C ik sin n n x 2 where the w w w wE gt 0 wbt gt 0 w gt 0 w gt 0 ew c cdt c c c and are constants fixed by the user Moreover these constants can be chosen in dependence on Aj k and tm Note that the phase shift is an angle between 180 and 180 Since the angles 180 and 180 correspond to the same phase shift we have replaced the quadratic term p amp by its periodic variant p Choosing w gt 0 i e including phase shift terms can be dangerous Namely if the corresponding efficienc
25. 5 Threshold acc gt 0 iterate is treated as a boundary point if its distance to the boundary is less than ace cf Step 2 and choose its value about the expected accuracy of the final solution 6 Threshold ste gt 0 stop if the change in the iterative solution is less than Este cf Step 2 2 and choose e g Este 0 1 Eace 7 Threshold Egra gt 0 stop if gradient is less than Egra cf Step 2 and choose its value about the discretization error of gradient calculation or less 8 Threshold Enorm gt 0 stop if relative change of the norm of the gradient is less than Enorm for the last Nnorm steps cf Equation and choose e g Enorm 0 01 103 The parameters i_opt 1 and d_opt i i 1 2 3 4 5 8 should be chosen as recommended For the error d_opt 7 gra of the gradient computation a first estimate can be obtained using the command OPTIMIZE g name dat On the other hand the user can choose niter_max larger than necessary and the positive parameters d_opt i i 6 7 smaller than recommended In the worst case a large number of unnecessary iteration steps with very small changes in the iterative solutions are performed at the end of the optimization procedure Even these redundant iteration steps can be interrupted Indeed the user can invoke the optimization by the command OPTIMIZE g name dat In this case the actual iterative solutions are printed on the screen and pushing the Ctrl C keys
26. BaTiO3_palik BeO_llnl_cxro Be_palik BN llnl_cxro C10H804 CaF2_llnl_cxro CH2CC12_llnl_cxro CH2 Co304_llnl_cxro Co Co_palik Cr203_llnl_cxro Cr3C2 IMD Cr CsI_Ilnl_cxro Cu20_llnl_cxro Cu4 i_llnl_cxro IMD Cu CuO_palik c ZnSe_palik d C_Ilnl_cxro d C_windt 56 a Al203 a C_Ilnl_cxro Ac a C_windt91 Ag20 Ag_IInl_cxro Ag windt A1203 Al 3Ga 7As_palik Al_Inl_cxro AIN AlSb palik a SiC_kortright a SiH a Si a SiO2_palik a Si_windt91 Au_hagemann IMD Au Au windt B4C_llnl_cxro BaTiO3 e_palik Be_llnl_cxro BeO B_llnl_cxro BN C5H802_llnl cxro CaF2 CH2CC12 Co203_llnl_cxro Co304 CoO_lIlnl_cxro CoSi2_llnl_cxro Cr203 Cr3C2_windt CrO_llnl_cxro CsI Cu20 Cu4Si CuO_llnl_cxro Cu_palik c ZnS d C Fe203_llnl_cxro Fe203 Fe_lln _cxro FeO GaAs GaP _llnl_cxro GaSb_llnl_cxro g C_llnl_cxro Ge H20 Hf Hf te_lynch Hg _Ilnl_cxro HgTe h ZnS e_palik InAs InP_palik Ir_Iln _cxro Ir_windt ir ZnS KBr_palik K LiF palik LiNbO3 Li_palik IMD MgF2 Mg MgO_palik Mn _Ilnl_cxro MnO2 MoC_llnl_cxro Mo Mo03_llnl cxro MoS 2_llnl_cxro MoSi2 Mo_windt91 NaClL_palik Na Nb205 NbO2_IIn _cxro NbO Nb_windt Ni 93V 07_IInl_cxro Ni Ni palik Fe304_llnl_cxro Fe Fe_palik GaAs_palik GaP GaSb g C Ge_palik H2O_palik HfO2_IInl_cxro Hf tm_tynch Hg HgTe_palik h ZnS InAs_palik InSb Ir ir ZnSe ir ZnS_palik KCl K_palik LiINbO3 e LiNbO3_palik MgF2 e MegF2_palik MgO_llnl_cxro
27. In some applications with a large number of geometrical parameters only a small num ber of parameters can be chosen freely and the other depend on these free parameter through explicit formulas If the jth real parameter d_geom_param j is a function of the parameters d_geom_param k l 1 L then the input of the bounds dl_geom_param j and du_geom_param j and the initial value d_ geom_param j is to be replaced by adding the dependency function More precisely instead of the numbers for du_geom_param j and d_geom_param j the word Dep indicates that the corresponding parameter depends on other parameters The number for dl_geom_param j must be replaced by the string Dep followed by the dependency function This function is to be written as a c code where the argument d_geom_param k is denoted by pk For instance the dependency d_geom_param 7 d_geom_param 8 d_geom_param 3 d_geom_param 2 together with the values d_geom_param 1 0 5 bounded to the prescribed interval 0 1 0 9 and d_geom_param 2 0 3 from the interval 0 2 0 4 is indicated by the input lines Lower bounds dl geom param 0 1 85 0 2 Dep p7 p8 p2 Upper bounds du_geom_param 0 9 0 4 Dep Number ni geom param of integer parameters Integer parameters i_geom_param Number ns_geom_param of name parameters 0 Parameter names s_geom_param AY YY I YY A AH AAE A AAE A A AAAA i A A a e e a a A a e i AA TN
28. N_mat k times m plus 2 if Grating data is polygon filet gt N_mat 2 if Grating data is polygon2 filet file2 gt N_mat 3 if Grating data is profile gt N_mat 2 if Grating data is profile 2 3 33 gt N_mat 2 if Grating data is profile 0 125 sin 2 M_PI t gt N mat 2 if Grating data is profile 0 5 0 5 cos M_PI 1 t 0 25 sin M_PI t gt N_mat 2 if Grating data is profiles gt N_mat nti with n nmb_curves from the file GEOMETRIES profiles c if Grating data is profiles gt N_mat n 1 with n nmb_curves from the file GEOMETRIES profiles_par c eg Eg if Grating data is pin gt N_mat 3 if Grating data is cpin gt N_mat 4 if Grating data is stack k gt N_mat k 1 if Grating data is bOX gt N_mat number indicated as second integer in second lind after line with code word bO0X if Grating data is rough_mls gt N_mat nlatnidtnlbt2 if width of additional layer above and below is positive N_mat nlatnidtnlbt 1 if instead of the value for the width of additional layer above and below a no is given in the input file Number of materials 2 TEEPE ESE PEPE TET EET EE EEE TE TE EHHHHEH Optical indices of grating materials This is c times square root of mu times epsilon If mea
29. Number of materials Must include in its number the two materials of the regions immediately above and below the grating structure Number of materials 4 HEHEHE RARER aa RH HHS Minimal angle of subdivision triangles 20 000000 HEHEHE RRR RRR aH EEE EEEE EEE A HHE Upper bound for mesh size 0 500000 FERRET PETTITT TEEPE TTT TEE AHRR RRRA R RRRA EEE TE PETE TERE TEAL HE EATER TEPER E HH Width of additional strip above and below Automatic choice of small width 112 if this value is 0 For no additional strip add no H For additional strip below but no additional strip above add no_up 0 2 For additional strip above but no additional strip below add no_lo 0 2 lf no nocup or nolo is added then the input for the Number of materials must not contain the materials of the excluded strips Width QO 200000 HERE E EE EPPS EERE THEE EEE EE Ee Eee Grid points a HHHHHHHHHHH HHH HH HHH PERERA EE Be EEE ETT TEEPE PEELE EEE EE ETE TE TE PETE EEE points of triangulation which is part of the domain for the FEM gt x components between O and 1 gt triangles should be disjoint gt union of triangles should be a simply connected domain gt union of triangles should connect the lines x 0 x 1 gt union of triangles should be bounded by two vertical lines and by two piecewise linear functions in x gt diameter
30. fctx 2 0 fcty 2 0 with x2 0 fctx 2 1 fcty 2 1 with O lt x1 lt xmin lt 1 xmin lt x2 Moreover fctx 2 t fcty 2 t 0 lt t lt 1 is a simple open arc above fctx 1 t fcty 1 t 0 lt t lt 1 such that O lt ictx 2 t lt 1 O lt t lt i The functions fctx 1 fetes fcty 1 and fcty 2 and the parameter xmin are defined by the code of the file GEOMETRIES cpin2 c stack 3 profile t 0 2 sin 2 M_PI t 2 profile 0 2 sin 2 M_PI t i profile t 0 Q 23 STACK GRATING i e a stack of 3 profile curves shifted by 2 1 0 micro meter in vertical direction For more details see the description in the USERGUIDE E POX 0 1 ade 13 3 2 46 0 6 1 0 4 cos 2 M_PI t 0 4 sin 2 M_PI t 0 8 a 0 15 0 89 2 BOX GRATING i e a box geometry of size 0 period 2 x 1 1 given in micro m period is supposed to be set to 2 in previous input lines number 0 1 behind word BOX is a mesh size factor 0 2 x 1 1 is divided by 2 curves into 3 different material parts curves are given by the c code in following 2 times two lines first code line is x component of first BREA AA AN 123 HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHH RR azz curve second code is y component of first curve third code is x component of second curve etc last three input lines define material points material with index one is located in part of box separated by above curves and
31. is zero cf Section 2 3 Then adding a CL TE TM before the input of n_phs_l_re resp n_phs_1_tr the terms are replaced by 10 6 This looks like n phs 1 tr CL TE TM 2 In general the objective functional may depend on 999 different values of efficiencies phase shifts energies If more than 999 efficiencies phase shifts energies are needed the environ ment variable NMB_OF_DATA must be set to a number larger than the number of required efficiencies phase shifts energies Finally the file name dat has to fix the scheme of numerical optimization together with its control parameters Generally this looks like Maximal number of iterations 10000 Indicator for optimization method 1 ni opt il i_opt 3 nd_opt 5 7 dopt L 0 001 le 3 le 2 le 2 Scaling parameters d_geom_scal 1 ie 1 Here the indicator 1 is the index of the numerical method The number and the meaning of the other parameters depend on the indicator value They will be explained in Section In the example from above the numerical method of index 1 requires one integer parameter i opt 1 and five real control parameters d_opt 1 d opt 2 d_opt 3 d_opt 4 and d_opt 5 Beside the control parameters the successful run of the local optimization routines depends on the scaling of the parameters d_geom_param j by the scaling factors d_geom_scal j j 1 nd_geom_param cf Section 10 3 1 If an optimization over more than one
32. of first polygon second polygon must be on left 0 QO 0 QO 0 136 HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHH Raza OO hand side of first one to one correspondence of the corners on the two polygons between first and last point of second polygon quadrilateral between corresponding segments on the left of first polygon these quadrilaterals must be disjoint last line should be End e g profile gt GRATING DETERMINED BY A SMOOTH PARAMETRIC CURVE i e grating determined by profile line given as fctx t fcty t 0 lt t lt 1 where the functions t gt fctx t and tl gt fcty t are defined by the c code of the file GEOMETRIES profile c e g profile_par 2 3 1 0 1 5 0 2 0 3 E gt GRATING DETERMINED BY A SMOOTH PARAMETRIC CURVE WITH PARAMETERS i e grating determined by profile line given as fctx t fcty t 0 lt t lt 1 where the functions t gt fctx t and tl gt fcty t are defined by the c code of GEOMETRIES profile_par c the last code uses 2 integer parameters and 3 real parameters named IPARaM1 IPARaM2 RPARaM1 RPARaM2 RPARaM3 the integer parameters take the values 1 and 0 following the first line of the calling sequence and the real parameters take the values 1 5 0 2 and 0 3 following the integer parameter values Any number of parameters is possible
33. period z 0 0 lt x lt 1 a material part is attached which is located between period x 0 0 lt x lt 1 and period f f O lt t lt 1 Here fo t fy t O lt t lt 1 is a simple open arc connecting f 0 fy 0 min 9 with f 1 fy 1 1 min 0 such that O lt Lmin lt 0 5 is a fixed number such that 0 lt felt lt 1 0 lt lt 1 and such that 0 lt f 0 lt lt 1 The functions fs fy and the parameter Zin are defined by the code in GEOMETRIES pin c If the program does not find the file GEOMETRIES pin c then it takes the file pin c of the current working directory Grating data cpin indicates a COATED PIN GRATING DETERMINED By Two PARAMETRIC CURVES cf Fig ure 18 where tia 0 2 a 02 08 ALI tant in t 2 0 5 sin t and f 2 t fy 2 t 0 lt t lt 1 is the polygonal curve connecting the four points f 1 a1 fy 1 a1 f2 1 a1 fy 1 0 5 0 1 f2 1 a2 fy 1 0 5 0 1 and fe 1 a2 fy 1 a2 ie over a flat grating with surface period 7 0 0 lt a lt 1 a material part is attached which is located between period z 0 0 lt a lt 1 and period fe 1 t f t OS Ge 1 Here a1 f 1 t 0 lt t lt 1 is a simple open arc connecting f 1 0 fy 1 0 Emin 0 with fa 1 1 f 1 1 1 Zmin 0 such that 0 lt 2min lt 0 5 is a fixed number such that 0 lt f 1 t lt 1
34. resp optimize 1 Then he enters 76 Obj funct col isols Grads arrs Lev 3 0 9 T k3 ae l 500 0 8 E E SSS ag a 450 i d 0 6 g 400 ao o EA 350 0 44 0 3 300 0 2 Yin 250 0 1 E 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 7th param Figure 27 Matlab plot of an objective functional and its gradient field kill 10 PID with PID replaced by the PID number of the process The flag 10 of the kill command raises the signal SIGUSR1 cf e g the manual page signal 7 of the linux system If the optimization catches this signal then the iteration is interrupted For the gradient based numerical optimization schemes cf Sections 10 3 2H10 3 4 the successful computation of the gradient is essential Therefore a visualization should help to control the gradient approximation In case of inaccurate gradients the discretization level must be refined The call of OPTIMIZE together with the flag p results in a Matlab code file MLplot m for a plot of the objective function and the gradient field over a two dimensional parameter set cf Figure 27 The code can be started calling MLplot from Matlab In the case of higher dimensional optimization problems the user can choose any pair of two parameters and fix the others in the input file name dat
35. with 3 columns each divided into 4 rectangular lay ers first column with x coordinate in 0 wm lt x lt 0 2 um second column with 0 2 wm lt x lt 0 6 um third column with 0 6 wm lt x lt period period given above whole grating with y coordinate s t 0 2 um lt y lt 1 0 um first column first layer with 0 2 wm lt y lt 0 um second with 0 um lt y lt 0 5 um third with 0 5 wm lt y lt 0 7 um and fourth with 0 7 um lt y lt 1 um second column first layer with 0 2 m lt y lt 0 0 um second with 0 0 pm lt y lt 0 50 um third with 0 50 um lt y lt 0 90 um and fourth with 0 90 wm lt y lt 1 um third column first layer with 0 2 pm lt y lt 0 00 um second with 0 00 wm lt y lt 0 500 um third with 0 500 u m lt y lt 0 900 um and fourth with 0 900 wm lt y lt 1 um Grating data 1Amellar 1 1 0 2 0 8 indicates a SIMPLE LAYER special case of lamellar grating with layer material s t y coordinate satisfies 0 2 wm lt y lt 0 8 um Grating data Figure 9 Multi trapezoidal grating 37 Figure 10 Lamellar grating polygon filel indicates a GRATING DETERMINED BY A POLYGONAL LINE cf Figure defined by the data in the file with name GEOMETRIES filel The z and y coordinates of the points in GEOMETRIES filel are supposed to be scaled such that the period is one in GEOMETRIES filel in each line beginning without there should be the z and y coordinate of one of the consecutive corner points first point
36. 0 41925 0 03884 18 86494 0 02063 0 25490 0 01257 0 27967 17 53158 0 05192 0 01214 0 04725 0 02301 0 55947 O 47 00 160 53 1 20 54 135 99 1 128 80 161 77 2 158 51 138 27 CO OO BOR Pe FO FO FO Transmitted energy 98 77158 Reflection efficiencies and coefficients Order Phi Theta E Z H_z Efficiency 0 47 00 30 00 0 06175 0 01688 0 07173 0 03108 1 02091 1 128 80 27 98 0 02812 0 00775 C 0 01579 0 00606 0 11595 2 158 51 86 74 0 04494 0 03733 0 00386 0 01787 0 02462 Reflected energy 1 16148 Transmission efficiencies and coefficients Order Phi Theta E Z H_z Efficiency 0 47 00 160 53 0 39161 0 10975 155 0 61958 0 17733 60 76400 0 25177 0 03911 0 42754 0 02212 19 38467 0 00565 0 25896 0 00395 0 28489 18 04542 0 05587 0 01457 0 05139 0 02021 0 64442 1 20 54 135 99 1 128 80 161 07 2 158 51 138 27 LOO PO LOE LOR a Transmitted energy 98 83852 date 10 Feb 2003 10 06 25 Thank you for choosing CONICAL Bye bye 12 8 Data file example dat of OPTIMIZE in OPTIM x makefile HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHE HHH HHH HHH HEH HHHHHHHHHHHHHHHHHH HHH HA HEHE TE E HE BE HEE HE HEE EE EE H EXAMPLE tee ton HAE HE EHETE HEHE HE HEHE HE TEHE HE HE HE HE HHH HA HEHE TE E HE AE HEHE AE HEHE EERE EE all lines beginning with are comments HHHH HH HH H HOH
37. 01 accuracy 72 10 Optimization Tools 10 1 Optimizing using OPTIMIZE in OPTIM 10 1 1 The optimization problem Optimizing an optical grating means the following First the user has to fix the period of the grating the refractive indices of the cover and substrate material one or more wave lengths of light A 7 1 jy one or more angles of incidence 0p k 1 k and dj l 1 ly one or two types of polarization tm TE TM m 1 m lt 2 Beside all these he has to fix a class of gratings cf the subsequent Section 10 2 Such a class is a certain grating geometry made of several materials The gratings of the class can be described by a small number of real parameters r i 1 N which are either geometrical parameters like lengths widths and heights or the real and or imaginary parts of the refractive indices Together with the class of gratings the user fixes the number of parameters He restricts the admissibility feasibility domain of real parameters by adding upper bounds u and lower bounds l To each set of parameters r r with li lt r lt u i 1 N there exists a unique grating Searching for an optimal grating is equivalent to finding the optimal set of parameters r The optimality of a grating resp set of parameters however is measured by an objective functional f r which is an arithmetic expression of the following data of the grating determined by the paramet
38. 28 1831 27 6783 27 4263 57 5905 52 7520 52 7661 17 3500 14 6183 13 7469 S O O N 0 gt w z Ko N Ko 0 H as O N pa N 149 1 04 2 36 conv ord 1 27 8 12 1 86 conv ord 0 39 0 79 1 99 conv ord 2 47 6 29 0 11 conv ord 1 78 1 26 1 91 11 VALUE ORDER 4 4 value extrap error conv ord 4 0 818 0 4156 0 2627 l 0 247 1 7545 1 0762 0 195 1 0322 1 2853 0 3539 0 89 0 092 0 7882 0 6637 0 1098 1 57 0 045 0 7333 OTi73 0 0549 2 15 4 12 VALUE TRANSMITTED ENERGY 4 4 h l value extrap error conv ord 0 818 93 4264 4 2479 0
39. 347 97 9640 0 2896 0 195 97 8202 97 8246 0 1459 4 98 0 092 97 7257 97 5442 0 0513 0 60 0 045 97 7000 97 6905 0 0257 1 88 4 END date 10 Feb 2003 09 56 18 Thank you for choosing dpogtr Bye bye 12 6 Output file example res of GFEM in CLASSICAL date 10 Feb 2003 10 13 19 KKK KK K kK kK FK FK FK FK FK K K K K 2 FK FK FK FK K FK K K K K K K FK FK FK FK FK FK K OK K DK 2 gg 2g K 2 K FK FK FK FK FK K K K K 2 K FK FK FK K FK FK K K K K K K 2K 2K 2K FK K K K K GDPOGTR x 150 OK gt KK KK K FK FK FK FK FK K K 2 K 2 K FK FK FK FK K K K K K K K 2g 2 2K FK FK FK 2k K K DK gt KK K K K FK FK 2 FK FK K K K K FK FK FK FK FK 2K K K K K K K K 2K 2K 2K 2K FK FK K 2K K Reflection efficiencies and coefficients n O theta n 1 theta n 2 theta n 3 theta Reflected energy 65 15 21 ST 00 74 33 07 k QO OF 0 Q 013025 000363 000035 091483 2 308647 OQ QO O Q 144735 016237 001347 053904 Transmission efficiencies and coefficients n O theta n 1 theta n 1 theta n 2 theta n 3 theta n 4 theta 26 50 T 10 29 54 Transmitted energy 95 41 80 48 96 TI ma Ar rN a i 0 QO 176698 0 0 0 0 502786 644183 071550 063904 03272
40. 4 4 d_opt 3 4 d_opt 4 d opt 5 HHHHHHHHHHHHHHHHHH HHHHHHHHHHH HHH HHH HHH HOH OF the algorithm manipulates over several temperatures each of this is obtained by cooling the previous by the factor fact if c_fact 1 then a logarithmic cooling scheme is used if c_fact 111 then a rational cooling scheme is used e g 0 95 stopping threshold eps_stop algorithm stops if the difference of the objective function of the solution from the previous temperature step differs from that of the current temperature by a value less than eps_stop no stopping rule for eps_stop 0 e g 0 initial value rho_ini of neighbourhood radius where the function transition searches a new iterate should satisfy O lt rho_ini e g expected accuracy of final solution reduction factor rho_fact of radius of neighbourhood for function transition the neighbourhood s diameter for this stochastic choice is reduced by this factor after each temperature step value should satisfy O lt rho_fact lt 1 e g 1 NEWTON METHOD PROJECTION ind_opt 5 hi_opt 2 i opt 1 i_opt 2 nd_opt 3 dopt i number n_norm of same gradient norms after which the algorithm stops e g 3 maximal number of iteration for which an increase of the functional is accepted e g 5 threshold eps_acc if difference of c
41. A TERETE HE T E E TEBE EEE TE ETE AE AE ESTEE E AE TE EEE TE H AE AE PEE EE E E HE E EEE E EHE E E HE EE E HEBE E HEEE E HE AEE HE STOCHASTIC ERROR ANALYSIS HHEHHHHHHHHHHHHHHHHHHAHHEHHHHHHHHHEHHHHHHHHH RHEE HRRE HEHEHE No further input is required However if the next line starts with SD then Suppose that prescribed input values for the efficiencies E and or the phase shifts P corresponding to the efficiency value denoted by E have a normally distributed error of H 201 expectation 0 and standard deviation sigma E and sigma P respectively Programm computes standard deviations of reconstructed values correlation factors The following three varaiants can be chosen 1 Suppose that standard deviation sigma E and sigma P are given as sigma E u E sigma P u E 360 100 The function u of variable E should be given in the next input line as a c code formula without blanks preceded by SD 2 Suppose that standard deviation sigma E and sigma P are equal to the deviation of the given measured values from those computed for the reconstructed grating Input line should be SD dev 3 Suppose that standard deviation sigma E and sigma P are to be equal to the reciprocal square root of the weight for the corresponding term in the objective functional Input line should be SD orig No error analysis is provided if the c code is not defined in th
42. KK K 2 K FK FK FK FK FK K K 2K 2K 2K K FK FK FK 2K K K K K K K K 2K 2K 2K 2K FK FK K 2K K CONICAL DK 2 gg 2g K 2 K FK FK FK FK FK K K K K FK FK FK FK FK 2K K FK K K K K K K 2K 2K 2K FK FK K K K DK gt KK K K K FK FK FK FK FK K K K 2 2K K FK FK FK 2K 2 K FK K K K K 2K 2K 2K 2K FK FK K 2K K Reflection efficiencies and coefficients Order Phi Theta E Z H_z Efficiency 0 47 00 30 00 0 19657 0 16355 0 05050 0 15697 9 25813 0 01482 0 03567 0 06213 0 13130 2 30360 0 07634 0 09663 1 128 80 27 98 4 2 158 51 86 74 152 0 12963 0 05033 0 22670 Reflected energy 11 78843 Transmission efficiencies and coefficients Order Phi Theta E_Z H_z Efficiency 0 47 00 160 53 0 18113 0 45695 0 14912 0 68850 80 24261 1 20 54 135 99 0 07497 0 15146 0 05129 0 15348 5 32346 1 128 80 161 77 0 05523 0 08440 0 02083 0 07834 2 29025 2 158 51 138 27 0 03830 0 01953 0 01062 0 03930 0 35525 Transmitted energy 88 21157 Reflection efficiencies and coefficients Order Phi Theta E_Z H_z Efficiency 0 00901 0 00844 0 13596 0 02289 1 91613 0 00200 08454 0 04245 0 01373 0 93218 0 06824 0 05857 C 0 12368 0 17972 O 47 00 30 00 1 128 80 27 98 O 2 158 51 86 74 0 06300 Reflected energy 3 02802 Transmission efficiencies and coefficients Order Phi Theta E_z H_z Efficiency 47 00 160 53 0 31129 0 27182
43. Mn304_llnl_exro Mn Mo2C _llnl cxro MoC Mo0O2_Ilnl_cxro Mo03 Mos2 MoSi2_windt Mo_windt92 NaF Na_palik Nb_llnl_cxro NbO2 Nb_palik Ni 8Cr 2_IIn _cxro Ni 93V 07 NiO_Ilnl_cxro Os_Ilnl_cxro 57 Fe304 FeO llnl cxro GaAsllnl_cxro GaAs_windt GaP _palik GaSb_palik Ge_llnl_cxro H20 llnl_cxro Hf_lnl_cxro HfO2 Hf_windt Hg _palik h ZnS e h ZnS_palik IMD InP InSb_palik Ir_palik ir ZnSe_palik KBr KCl_palik LiF LiNbO3 e_palik Li MegF2 e_palik Mg_llnl_cxro MgO Mn304 Mn0O2_Ilnl_cxro Mo2C Mo_llnl_cxro Mo02 Mo palik MoSi2_llnl_cxro Mo windt88 IMD NaCl NaF_palik Nb205_llnl_cxro Nb NbO_llnl_cxro Nb_weaver Ni 8Cr 2 Ni llnl cxro NiO Os lynch Os Os_palik PbSe_palik PbTe Pd Pd_palik Pt_lnl_cxro PtO2 Pt_palik Re207_lInl_cxro Re ReO3_llnl_cxro Re tm_lynch Rh203 Rh_palik Ru_cox RuO2_lln l_cxro RuO4 Ru te_weaver Ru_windt88 amp Sc203 ScN_llnl cxro Se e_palik Si 25Ge 75 Si3N4_llnl_cxro Si3N4 windt Si 5Ge 5_palik2 Si 8Ge 2_palik2 SiC_osantowski SiC_yanagihara SiO2 e SiO2 SiO Si_windt SnQO2_Ilnl_cxro SnO SrTiO3 Ta205 Ta_llnl_cxro TaN Ta_windt Te ThF4 palik TiC palik OsO2_Ilnl_cxro Os_windt PbS PbTe_palik PdO_llnl_cxro Pd_weaver Pt PtO_llnl_cxro Pt_weaver Re207 ReOQ2_llnl_cxro ReO3 Re_windt Rh_llnl_cxro Rh_weaver Ru_llnl_cxro RuO2 RuSi_lln _cxro Ru tm_weaver Ru_windt92 Sc_llnl_cxro ScN Se Si 25Ge 75_palik1 Si3 N4 Si 5Ge 5 Si 8Ge 2 S
44. Number ni_geom_param of integer parameters 3 Integer params i_geom_param ni_geom_param numbers in ni_geom_param lines 1 2 2 Number ns_geom_param of name parameters 0 Parameter names s_geom_param Each in one line i e ns_geom_param lines Parameters Or Oo Fe HHEHHHHHHHHHHHHHHHHHAHHHEEHRHHHHHHREHHEHH HHH HHRHRHHRE RHEE HHH HR HHH HH HH H HH OF Parameters d_geom_param of initial grating d_geom_param nd_geom_param numbers in nd_geom_param lines With dl_geom_param i lt d_geom_param i lt du_geom_param i for all i If initial solution is to be sought by a deterministic search algorithm then add the line no n_0 n_1 n_2 where n_0 is the refinement level for the FEM computation n_1 is the number of maximal subdivision points per dimension and n_2 is an indicator If n_2 1 then the minimum is improved by computing the local minimum of the linear Taylor polynomial around each mesh point Parameters 0 07 0 04 1 0 Lad 0 A E HE T E AE TE RESETS E A E E TE BEET TEESE E E A AE TE T AE TE E EET TE PEPE PE EEE AE EET PP EEE HE ETE HEE TETEE BE LEVEL OF DISGCGRETIZATILOUON 182 HHHHHHHHHHHHHHHEHHHEHHEHEHEHEHEHEHHHHHHHEHEEEEEHEEHEEHHHHHHHHHHHHHHHHHHHHH HH Number of levels Lev Computation is performed on this level Alternatively an incremental input is possible Eg meee dT ganr The last means that computation is to be done for the levels 2 i 3
45. RRR RR RAE RRHH MARRET EA EEEE aaa Period of grating in micro m i HEHEHE RRR RRR RAE E EEEE EEEE E H EEHEHE Grating data Either this should be e g namel if namel inp is the input file with the geometry data in sub directory GEOMETRIES Alternatively a path for the location of the file can be added before the name This must contain at least one slash E g for a file in the current working directory write namel Or this could be a stack of profiles given by the code word stack and many more lines cf Userguide Or this could be e g echellea R 0 3 0 03 0 04 gt ECHELLE GRATING TYPE A right angled triangle with hypotenuse parallel to the direction of the periodicity right interior angle gt 45 degrees with depth of 0 3 micro meter and with coated layers of height 0 03 micro meter resp 0 04 micro meter over the first resp second part of the grating measured in direction perpendicular to echelle profile height greater or equal to zero e g echellea L 0 3 0 03 0 04 gt ECHELLE GRATING TYPE A right angled triangle with hypotenuse parallel to the direction of the periodicity left interior angle gt 45 degrees with parameters like above e g echellea A 60 0 03 0 04 gt ECHELLE GRATING TYPE A right angled triangle with hypotenuse parallel to the direction of the periodicity with left interior angle Alpha 60 degrees
46. angle larger than the left To get the flipped grating with left blaze angle larger than the right the input should be A 90 and D 0 29 f trapezoid 60 0 6 3 0 2 0 1 0 10 05 gt TRAPEZOIDAL GRATING trapezoid with the basis parallel to the direction of the periodicity with angle of 60 degrees angle enclosed by basis and the sides with a base of length 0 6 micro meter consisting of 3 material layers of heights 0 2 0 1 and 0 1 micro meter respectively and with a coated layer of height 0 05 micro meter greater or equal to zero c e g mtrapezoid 3 0 005 0 015 0 005 80 90 80 0 01 7 0 05 0 075 0 05 0 05 0 075 0 075 0 05 0 150 3 0 4 0 5 0 6 0 75 0 9 7 gt MULTITRAPEZOIDAL GRATING finite number of symmetric trapezoidal bridges located side by side each bridge consists of 3 trapezoidal layers one over each other hights in micro meter of these layers from above to below are 0 005 0 015 and 0 005 sidewall angles in degrees of these layers from above to below are 80 90 80 height in micro meter at which the lateral width of the bridge is given is 0 01 number of trapezoidal bridges per period is 7 lateral widths in micro meter of these bridges are 0 05 0 075 0 05 0 05 0 075 0 075 and 0 05 x coordinates in micro meter of the mid points of the bottom lines of these bridges are 119 HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHH RHR azz e g e g gt e
47. coated pin grating determined by the simple non intersecting pro file lines given as fs j t fuj t 0 lt t lt 1 j 1 2 is needed then this can be accomplished by calling the executable GEN_CPIN from the subdirectory GEOMETRIES More precisely suppose Lin G1 2 and the profile lines f2 j t fy t 0 lt t lt 1 j 1 2 are given by the c code in the file GEOMETRIES cpin c Then GEN_CPIN name 0 06 creates the file name inp of the desired profile grating where the profile curves are ap proximated by a polygonal line with a stepsize equal to 0 06 times the length of period cf Figure 18 where Zmin 0 2 ay 0 2 a 0 8 F214 Emin 1 2 min t fy 1 t 0 5 sin 7t and f 2 t fy 2 t 0 lt t lt 1 is the polygonal curve connect ing the four points f 1 a1 fy 1 a1 fe 1 a1 fy 1 0 5 0 1 fr 1 a2 fy 1 0 5 0 1 and fe 1 a2 fy 1 a2 If an input file for a coated pin grating of type 2 determined by the simple non intersecting profile lines given as fr j t fy 7 t 0 lt t lt 1 j 1 2 is needed 2T e over a flat grating with surface x 0 0 lt x lt 1 a material part is attached which is located between z 0 0 lt a lt 1 and f t fy t 0 lt t lt 1 Here fr t fy t 0 lt t lt 1 is a simple open arc connecting f2 0 fy 0 min 0 with fe 1 fy 1 1 tmin 0 such that 0 lt min lt 0 5 is a fixed number such
48. containing 1 0 8 material with index two is in part of box separated by above curves and containing 1 0 etc Note that the curves must be simple not self intersecting contained in the box allowed intersection points of two different curves are end points only material areas separated by curves must be such that area with first index contains strip beneath upper boundary side of box and that area with last index contains strip above lower boundary side e g rough_mls name gt MULTILAYER SYSTEM WITH ROUGH INTERFACES i e nla layers nmld times nld layers nlb layers This system is described by the file name which is contained in the directory GEQMETRIES alternatively the name must contain the path of the file This input file name contains the following ordered data each in a separate line comment lines begin with dummy file name width of additional layer above and below the structure must be positive or could be no for no additional layer period of grating per number nla of layers above multilayer system number nlb of layers below multilayer system number nmld of multilayers inbetween number nld of layers in each multilayer number ncorn of corner points in each polygonal approximation interfaces randomly generated polygon number nrand of standard Gaussian distributed random numbers needed to ge
49. corner points first point with 29 x coordinate 0 last point with x coordinate 1 same y coordinate for first and last point all x coordinates between 0 and 1 at least two different y coordinates last line should be End then the command GEN_POLYGON name filel creates the file name inp of the desired polygonal grating If an input file for a grating determined by two polygonal profile lines is needed cf Fig ure 12 then this can be accomplished by calling the executable GEN_POLYGON2 from the subdirectory GEOMETRIES More precisely if the file GEOMETRIES filel contains the corner points of a polygonal profile line in GEOMETRIES filel in each line beginning without 4 there should be the z and y coordinate of one of the consecutive corner points first point with x coordinate 0 last point with x coordinate 1 same y coordinate for first and last point all x coordinates between 0 and 1 at least two different y coordinates last line should be End and if the file GEOMETRIES file2 contains the corner points of a sec ond polygonal profile line in GEOMETRIES file2 in each line beginning without there should be the z and y coordinate of one of the consecutive corner points first and last point must be corner of first polygon second polygon must be on left hand side of first one to one correspondence of the corners on the two polygons between first and last point of second p
50. f 1 0 fy 1 0 Lmin 0 with f2 1 1 fy 1 1 min 0 such that 0 lt min lt 0 5 is a fixed number such that 0 lt f 1 t lt 1 0 lt t lt 1 and such that 0 lt f 1 t 0 lt t lt 1 Additionally a coating layer is attached located between the first curve period f 1 fy 1 t 0 lt t lt 1 united with the the two straight line segments period 7 0 1 lt lt Lmin and period 2 0 1 2min lt lt 2 and a second curve period f 2 t POs tea 0 lt t lt 1 The last connects the first point period x1 0 period f 2 0 fy 2 0 with the last point period 2 0 period fa 2 1 fy 2 1 Moreover f2 2 t fy 2 t 0O lt t lt 1 44 Figure 17 Pin grating determined by parametric curve is a simple open arc above f2 1 t fy 1 0 lt t lt 1 such that 0 lt f 2 t lt 1 0 lt t lt 1 The functions f 1 fr 2 fy 1 and f 2 and the parameter min are defined by the code of the file GEOMETRIES cpin2 c If the program does not find the file GEOMETRIES cpin2 c then it takes the file cpin2 c of the current working directory Period of grating in micro meter 2 Grating data bOX 0 1 1 1 23 12 6 0 6 1 0 4 cos 2 M_PI t 0 4 sin 2 M_PI t 1 0 8 T i 1 0 8 indicates a Box GRATING cf Figure 20 where the box is 0 period 2 x 1 1 given in wm The number 0 1 behind the cod
51. for TM polarization Here the intensity I is defined as J Ren E where n is the refractive index and where 64 O Point Degree of freedom Function Interpolation polyn 0 1 of grid values Function FEM s of bou Function value 1 0 T7 values Figure 25 Trial basis function over a single grid triangle E is the electric field vector If the package Matlab is installed at your computer you can call these files from Matlab to see the plots Finally if you use the command GFEM_MOVIE name2 then the input file name2 dat is required to contain a single wave length and a single state of polarization only The program produces a Matlab file fct TE m for TE polarization and fet TM m for TM polarization If this is called from Matlab a movie of the time dependent third component of the electric field for TE polarization and of the magnetic field for TM is shown Moreover a movie file fct_TE avi resp fet TM avi is created 6 Computation of Efficiencies Using GFEM in CLAS SICAL The same computation from the last section can be performed by generalized FEM cf the result file enclosed in point 12 6 The latter is nothing else than the variational approach of the conventional FEM combined with a new trial space for the approximation of the unknown solution The trial space is defined over the triangular FEM partition and the trial functions are piecewise approximate solutions of the Helmholtz equat
52. for a corresponding file GEOMETRIES profile_par c e g profile 0 125 sin 2 M_PI t gt GRATING DETERMINED BY A SIMPLE SMOOTH FUNCTION i e grating determined by sine profile line given as t fcty t 0 lt t lt 1 where the function t gt fcty t is defined by the c code fcty t 0 125 sin 2 M_PI t do not use any blank space in the c code e g profile 0 5 0 5 cos M_PI 1 t 0 25 sin M_PI t gt GRATING DETERMINED BY A SIMPLE SMOOTH PARAMETRIC CURVE i e grating determined by ellipsoidal profile line given as fctx t fcty t 0 lt t lt 1 where the functions t gt fcty t and t gt fcty t are defined by the c codes fctx t 0 5 0 5 cos M_PI 1 t and fcty t 0 25 sin M_PI t respectively do not use any blank space in the c codes e g profiles gt GRATING DETERMINED BY SMOOTH PARAMETRIC CURVES i e grating determined by profile lines given as fctx j t fcty j t 0 lt t lt 1 j 1 n nmb_curves where the functions t gt fctx j t and 137 HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHH RR za Oz E E gt fcty j t are defined by the c code of the file GEOMETRIES profiles c e g profiles_par 1 2 3 0 5 0 50 ee gt GRATING DETERMINED BY SMOOTH PARAMETRIC CURVES WITH
53. grating in y direction 133 Must be a positive real number Length 1 HERE E EEE EE EERE EE EE Ee Eee AAA EEE EEEE HH H Length of additional shift of grating geometry in micro m This is shift into the y direction i e the direction perpendicular to the grating surface pointing into the cover material Must be a real number Length 0 HEHEHE NYAAH NARR RRR R AARE AA REEERE RRR AAA Period of grating in micro meter i TERR BETTE TETETET ETT PETE TET TEETH eee EE TEE TE PEE RRRA HE ETE ERE TEE R EHHH Grating data Either this should be e g namel if namel inp is the input file with the geometry data in sub directory GEOMETRIES Alternatively a path for the location of the file can be added before the name This must contain at least one slash E g for a file in the current working directory write namel Or this could be a stack of profiles given by the code word stack and many more lines cf Userguide Or this could be e g echellea R 0 3 0 03 0 04 gt ECHELLE GRATING TYPE A right angled triangle with hypotenuse parallel to the direction of the periodicity right interior angle gt 45 degrees with depth of 0 3 micro meter and with coated layers of height 0 03 micro meter resp 0 04 micro meter over the first resp second part of the grating measured in direction perpendicular to echelle profile height greater or equal to ze
54. here at perk i i i l 1 5 ARN a i E 2 ka i 1 5 1 0 5 0 0 1 ve 08 aa 0 y 05 06 0 5 0 9 1 Figure 24 Imaginary part of transverse component of magnetic field Output of executable GFEM PLOT in CLASSICAL via GNUPLOT Decrease mouse sensitivity increase control parameter Zoom out i Zoom in This help B Toggle background color black white d Dump actual picture to ppm file look for ppm F Toggle rendering volume frame bounding box drawing I Change number of isolines increase control parameter by a factor O Toggle Ortho D Print actual picture using ppm dump R Reset to internal default S Save actual state look for rndstate V Start Stop video recording a Switch to GUI c Toggle remembered lists g Toggle Gouraud flat shading h This help 63 i Toggle isoline mode decrease control parameter by a factor 1 Toggle level surface mode m Toggle model display when moving p Dump actual picture to eps file look for eps Mode control Quit Restore last saved state Toggle vscale for plane sections toggle wireframe mode Show x orthogonal plane section Show y orthogonal plane section Show z orthogonal plane section prev toggle state change mode next toggle state change mode left move left up move up right move right down move down Backspace Enter user control mode 38 3 fo To continue the computation of the
55. huge number of isoline curves surrounding a point indicate a singularity point Many isolines located close and parallel to the interface lines indicate surface layers For these cases graded meshes can enhance the approximation of the FEM and GFEM A mesh grading towards a corner point can be enforced if additional points close to the first point 71 0 f2 2 0 fy 2 0 with the last point x2 0 f2 2 1 fy 2 1 Moreover fr 2 t fy 2 t O lt t lt 1 is a simple open arc above f 1 t fy 1 0 lt t lt 1 such that 0 lt fr 2 t lt lt 1 0 lt t lt 1 The functions f 1 fr 2 fy 1 and f 2 and the parameter min are defined by the code of the file GEOMETRIES cpin2 c 32 the corners are introduced in the input file namel inp We recommend to approach the corner by these additional points from one two or three directions at distances of size 0 1 2 1 2 These points need not to be included into the list of triangle corners The mesh generator however will include the points into the finite element mesh such that a grading towards the corner is achieved Note that mesh gradings toward points does not increase the overall number of mesh points essentially A mesh grading towards an interface line can be enforced if a tiny additional layer on one side of the interface is introduced by the geometry input file namel inp This layer must have a different index of material O
56. i e depth period times sin Alpha times cos Alpha and other parameters like above e g echelleb 60 0 05 gt ECHELLE GRATING TYPE B right angled triangle with one of the legs parallel to the direction of the periodicity with angle 60 angle enclosed HHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHH 118 HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHH HHH HH HH 28 by hypotenuse and by the leg parallel to the period and with a coated layer of height 0 05 micro meter measured in direction perpendicular to echelle profile height greater or equal to zero lt echelle L 60 R 30 0 05 0 1 gt GENERAL ECHELLE GRATING with left blaze angle 60 degrees with right blaze angle 30 degrees with coated layer over left blaze side of height 0 05 micro meter measured perpendicular to profile height gt 0 and with coated layer over right blaze side of height 0 1 micro meter measured perpendicular to profile height gt 0 must be O if previous height is 0 Instead of the two inputs T 60 and R 30 one can choose also the inputs A 90 for an apex angle of 90 degrees or D 0 2 for a depth of grating 0 2 micro meter Moreover any combination of two inputs of the types CA 90 2 5 EL 11057 K 90 and D 0 29 1s accepted However the choice A 90 and D 0 2 might be ambiguous By definition it fixes an echelle grating with right blaze
57. i_geom_param 2 2 is larger than one In this case the upper and lower bounds for the internal parameters d_geom_param 1 with 1 5 k 3 1 5 i_geom_param 2 9 are set to 0 9 and 0 1 respectively The upper and lower bounds for the internal parameters d_geom_param 1 with 1 5 k 2 1 5 i_geom_param 2 8 are set to 0 9 and O respectively HETE AE E HE HEHE HE PEPE TEE H HHHHHHHHHHHHHHHHHHHHHHHHHHH HHH HH HH OH FH CLASS 5 i_geom_param 1 5 input grating to be extended Grating of input file name inp extended by a new polygonal interface curve and with refractive indices included into the set of optimization parameters new polygonal interface The new interface connects two given interface points P1 and P2 These two points are located at the boundary of a domain of a fixed material and are grid points of the file name inp This domain is convex and the periodic boundary lines 0 y y real and dperiod y y real must not intersect its interior The new interface curve divides this domain into two This new polygonal interface curve is a polygonal function over the straight line segment connecting the two points In other words if U is the convex domain fixed by a given material index and if P1 and P2 are two given grid points at its boundary then the j th corner j 1 m of the polygonal curve connecting P1 and P2 is chosen at the straight line segment P P1i P2 P1 j m 1 t n
58. in U i dl_geom_param j lt t lt du_geom_param j ii P tepsilon n in U iii P epsilon n in U Fo epsilon d_geom_param 2 i_geom_param 2 i_geom_param 3 1 where n is the unit vector normal at the segment P1 P2 pointing to the left side of P1 P2 Number of string parameters ns_geom_param 1 amp s_geom_param 1 1 buffer_size name inp name of input file for DIPOG 2 1 carrying the geometry information 170 HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHH Raza OO without the extension by the new polygonal interface curve Number of integer parameters ni_geom_param 6 i_geom_param 1 5 i_geom_param 2 number of materials should be number of materials in input file name inp plus one if i_geom_param 3 is positive and should be number of materials in input file name inp if i_geom_param 3 zero i_geom_param 3 number of interior knots in the polygonal interface should be non negative If this is zero then the convex domain is not divided into two and only the refractive indices are optimized i_geom_param 4 index of the grid and interface point P1 dummy if i_geom_param 3 0 i_geom_param 5 index of the grid and interface point P2 dummy if i_geom_param 3 0 i_geom_param 6 index of convex domain which is divided by the new polygonal interface should be between 2 and i_geom_param 2 1 and is a dummy if i_geom_param 3 0 After dividing the convex domain the first subdomain right of the polyg
59. initial solution fini RY from the data file The temperature must be positive If t 0 then an automatic choice of temperature is provided Set the first iterate ro to fin and set the first values of the optimal solution fop to Tini 2 Initialization of iteration Set the index j of the iteration step to zero Set the temperature t to tini and the value of the neighbourhood radius 0 to QOini 106 3 Steps of iteration If the index j of the iteration is larger than the prescribed maximal number niter_max then stop the iteration and go to Step 4 Also if 0 lt r rj 1 lt stop with a prescribed threshold stop stop the iteration and go to Step 4 Else get a new admissible iterate rj by a random search in the neighbourhood of r of radius o interpreted as a transition of the state in a cooling step If f rjzi lt f r then accept rj and if f rj41 gt f r holds then accept rj41 With a probability of exp f rj4i f r t In the case that rj41 is not accepted set rj rj If f rj 1 lt f ropt then mark the solution setting ro rj41 Cool the temperature by multiplying t with the prescribed cooling factor Cfact Decrease the radius of the neighbourhood o by multiplying 0 with the prescribed neighbourhood reduction factor Ofact Increase index j by one and repeat Step 3 4 Restarts If J is greater than the prescribed integer Nrest then generate at random a new starting s
60. introducing a small layer beside the interface filled with cover material The diameters of the triangles of the FEM grid will change smoothly from the maximal value to the width of this small layer The width of this small layer is the number defined by the environment variable WI_GRA_LAY 2 Diffraction Problems for Gratings 2 1 The classical TE problem Consider an ideal optical grating cf the cross section in Figure 1 We choose the co ordinate system such that the z axis shows in the direction of the grooves and that the y axis is orthogonal to the plane of the grooves The width of the grooves in x direction is the periodicity d of the grating The refractive index of the cover material is n that of the substrate under the grating surface structure n The grating part consists of several materials with indices n Above and below the grating structure there may exist some coated layers with different refractive index cf the indices n and n for one upper and one lower coating layer in Figure i We suppose that a plane wave is incident from above with a direction located in the x y plane i e in the plane perpendicular to the grooves and under the incident angle 6 The wave length of the light in air is A and we consider the case of TE polarization where the electric field vector is parallel to the grooves i e it shows in the z direction Hence if jug is the magnetic permeability of vacuum and c the 14 speed of
61. level is performed then the maximal number of iterations can be chosen in dependence on the level For instance if the input of the level 82 is the incremental I 1 5 2 then there are three different levels 1 3 and 5 The input Maximal number of iterations LEV 10 20 10000 would restrict the number of level 1 iteration to 10 that of level 3 to 20 and that of level 5 to 10000 10 1 4 The locality of the solution A warning The parameter set fop rj i 1 N for which f rop lt f r for all r in the fixed class of admissible parameters is called the global optimal solution Since the objective functional is continuous and since the class of parameters is compact the existence of rope is guaranteed However the optimal solution is in the general case not unique Moreover the topography of the graph of the objective function is often quite complex Usually there exist a lot of local minima Note that a set of parameters fioc is called local minimum if for all r close to fioc the value f fioe is less or equal to f r Sometimes there exists even a submanifold in the class of admissible parameter sets consisting of local minima Unfortunately the final result of the numerical optimization schemes is often a local minimum instead of global optimum Indeed the gradient based methods cf Sections 10 3 2 and are so called local methods which are designed to determine a local minimum To find a global minimum t
62. light then the transverse z coordinate of the electric field is given as Epemi ny at EPA yz exp iwt w gt 2 1 ae 1 incident spt os opt oa E x y z Jat exp ik sind x ik cos0 y kv w po o n The light is diffracted by the grating structure Beside some evanescent part the diffracted light splits into a finite number of reflected and transmitted TE polarized plane wave modes the propagation directions of which are independent of the grating geometry and the grating materials The problem is to determine the amplitude and the phase of the reflected and transmitted modes Note that the normalization factor 1 Vnt in the second line of has been introduced to obtain an incident light wave with a fixed intensity length of Poynting vector independent of the cover material If the cover material is air then nt 1 and the wave is normalized such that the amplitude of the electric field vector is of unit length Of course the values of the efficiencies and phase shifts of the reflected and transmitted plane wave modes are independent of this normalization factor The subsequent Rayleigh coefficients however depend on this scaling Using Maxwell s equations it can be shown that the transverse component E satisfies the scalar Helmholtz equation A k F 0 in any domain of the cross section plane with constant material as well as some transmission conditions on the interfaces between materials of different ref
63. ls O T BOX GRATING i e a box geometry of size 0 period 2 x 1 1 given in micro m period is supposed to be set to 2 in previous input lines number 0 1 behind word BOX is a mesh size factor 0 2 x 1 1 is divided by 2 curves into 3 different material parts curves are given by the c code in following 2 times two lines first code line is x component of first curve second code is y component of first curve third code is x component of second curve etc last three input lines define material points material with index one is located in part of box separated by above curves and containing 1 0 8 material with index two is in part of box separated by above curves and containing 1 0 etc Note that the curves must be simple not self intersecting contained in the box allowed intersection points of two different curves are end points only material areas separated by curves must be such that area with first index contains strip beneath upper boundary side of box and that area with last index contains strip above lower boundary side rough mls name MULTILAYER SYSTEM WITH ROUGH INTERFACES i e nla layers nmld times nld layers nlb layers This system is described by the file name which is contained in the directory GEQMETRIES alternatively the name must contain the path of the file This input file name contains the following ordered data each in a separate l
64. of openGL together with GLTOOLS 2 4 or alternatively the package GNUPLOT Examples of data and output files are enclosed 1 Don t read the complete user guide Don t you have anything better to do If you have to compute the efficiencies phase sifts and energies of the waves diffracted by gratings then go to the directory DIPOG 2 1 GUI and start the program DIPOG 2 1 GUI which is self explanatory Alternatively for gratings with more complex input data you should change to the directory DIPOG 2 1 CLASSICAL or DIPOG 2 1 CONICAL Read the data file example dat Change it according to your requirements Run one of the executables with example dat as argument Read the results Plots with efficiency curves can be produced via the executables in the directory DIPOG 2 1 RESULTS If you still have a question come back to this user guide and read the corresponding part only Good luck In case you have to optimize a grating read the Sections and 10 3 2 of this user guide Then go to the directory DIPOG 2 1 OPTIM and read the data file example dat Change it according to your requirements Run the executable OPTIMIZE with example dat as argument Read the results If you still have a question come back to this user guide and read the corresponding part only Good luck 2PARDISO itself requires some routines from LAPACK and some BLAS routines Contents 1 Introductory Remarks and the Structure of the Package N
65. of the material parts can be fixed to any positive integer Suppose that Nmat is the maximal index Then the materials will be determined in a list of Nmat refractive indices optical indices given in the data file name dat of DIPOG cf Sect 5 1p Note that index 1 is reserved for the refractive index of the material adjacent to the cross section domain from the upper side either cover material or lowest upper additional strip and index Nmat for the refractive index of the material adjacent to the cross section domain from the lower side either substrate material or highest lower additional strip The domain for the FEM computation is a rectangle which is a slight extension of the just mentioned cross section domain into the upper and lower direction The additional strips are possibly not contained in the FEM domain since they are treated via boundary operators Even if the cross section domain is rectangular small strips are added in order to enable a fast treatment of the boundary operators However in case of classical TE polarization and if the upper resp lower boundary curves are horizontal straight lines of the cross section domain the extension to the FEM domain by small upper resp lower strips can be suppressed cf Sect 3 2 To this end the indicators of the boundary segments must be changed from 2 resp 1 to 5 resp 4 Moreover since the fast treatment of the boundary operators requires uniform partitions the node points o
66. real parameters nd_geom_param i_geom_param 3 4 k 1 i_geom_param 3 d_geom_param k function value of profile curve at Period k i_geom_param 3 1 162 in other words polygonal grating has i_geom_param 3 2 corners at the points 0 0 x_k y_k k 1 i_geom_param 3 Period 0 where x_k Period k i_geom_param 3 1 y_k d_geom_param k k i_geom_param 3 1 d_geom_param k real part of refractive index of cover material k i_geom_param 3 2 d_geom_param k imaginary part of refractive index of cover material k i_geom_param 3 3 d_geom_param k real part of refractive index of substrate material k i_geom_param 3 4 d_geom_param k imaginary part of refractive index of substrate material Following parameters must be fixed by setting upper bound lower bound HHHHHHHHHHHH HH HHH HHH HHH H FH d_geom_param k k i_geom_param 3 1 i_geom_param 3 4 HHHHHHHHHHH HAH CLASS 2 i_geom_param 1 2 polygonal profile curve no proper constraints Polygonal grating defined by profile curve which is piecewise linear between the knots Warning iterative solutions are tested for selfintersection but the constraints expressed by no selfintersection is not included into the optimization Number of string parameters ns_geom_param 0 Number of integer parameters ni_geom_param 3 i_geom_param 1 2 i_geom_param 2 2 i_geom_param 3 number of interior knots with x coordinate in 0 Period Numb
67. reasons the index of the material adjacent to the upper line of the grating structure must coincide with that of the material in the adjacent upper coated layer resp in the adjacent superstrate Similarly the index of the materials adjacent to the lower line of the grating structure must coincide with that of the material in the adjacent lower coated layer resp in the adjacent substrate N_mat numbers are needed ptical indices 2 FT 5 10 ie oO 2 3 1 20 HEHEHE HERR AR Aa RRR HaHa Number of levels Lev In each refinement step the step size of the mesh is halved HHH HH HH HHH OH OF HHHHHHH HHH HH HH HHH OH Number of refinement steps is Lev Alternatively one can prescribe an bound for the maximal error of the efficiencies E g the input e 1 means that the level for the computation is the smallest positive integer such that all efficiencies are computed with an estimated error less than 1 per cent Number 3 FERRETTI RARUA ANNAAHAAA HRR RARR EE TEE TEETER TEAL HE TET ERE PETER EHHH H End AA E HE TE E AE TE BERETTA TEETER EEE AP ESTE HE E AE T HE PEPER EEE EEE TEE EEE PEE EEE E BEETS EEE PETE EEE 126 12 3 Data file generalized Dat for CLASSICAL resp cal Dat in CONICAL makefile HHEHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEHHEHHHHHEHHEHHH HEH HRB AAE Ee generalized Dat FEE TET EEE HH all lines beginning with are comments HHH HH HF HHH HHH HF AA E
68. suppose P and Q are the common corner points of the two polygonal curves and that R R Ri are the consecutive corner points between P and Q on the first polygonal line and R R3 R those on the second polygonal Then the code requires m n and that the coating area between the two polygonal lines is the disjoint union of the triangle PR R the quadrilaterals R R R3R o Ron 1 Rin Rin RGm 1 and the triangle R QRh 30 creates the file name inp of the desired profile grating where the profile curves are ap proximated by a polygonal line with a stepsize equal to 0 06 times the length of period cf Figure 16 where n 3 flj t t fyl t sin 2zt f 2 sin 2at 0 5 and f 8 t sin 27t 1 If an input file for a pin grating determined by a simple non intersecting profile line given as f t fy t O lt t lt 1 is needed then this can be accomplished by call ing the executable GEN PIN from the subdirectory GEOMETRIES More precisely suppose Lmin and the profile line fs t fy t 0 lt t lt 1 are given by the c code in the file GEOMETRIES pin c Then GEN_PIN name 0 06 creates the file name inp of the desired profile grating where the profile curves are ap proximated by a polygonal line with a stepsize equal to 0 06 times the length of period cf Figure 17 where Emin 0 2 felt Emin 1 22min t and fy t 0 5 sin rt If an input file for a
69. that name inp contains point clouds at all interface corners of name inp with angles less than 100 The third argument 5 is the number of layers in the point clouds i e the cloud points are located at circular curves around the corner with radii 0 5 o 0 5 o 0 55 The positive real is chosen as large as possible such that the point clouds do not intersect other point clouds or corner points Finally the second argument 30 is the number of points at each circular curve In other words the cloud points are the intersections of the circular curves with rays through the corner points such that the angle between two neighbour rays is about 360 30 3 3 Graded FEM mesh generated through geometry input file The geometry is read from the input file namel inp and the mesh generator will create a finite element partition with given refinement level The generator tries to compute regular almost uniform triangles Over the generated finite element mesh the trial functions and approximate solutions are determined Sometimes the field solution of the problem is difficult to approximate by the trial func tions defined over regular meshes For instance the field solutions may have singular points at corners and surface layers close to the interfaces If the corresponding field components by the executables GFEM_PLOT resp FEM_PLOT are visualised and if the the isoline mode is switched on cf Subsection 5 3 then a
70. that 0 lt f t lt 1 0 lt t lt 1 and such that 0 lt f t 0 lt t lt 1 13T e over a flat grating with surface x 0 0 lt x lt 1 a material part is attached which is located between x 0 0 lt x lt 1 and f 1 t fy 1 t 0 lt t lt 1 Here f 1 t fy 1 t 0 lt t lt 1 isa simple open arc connecting fz 1 0 fy 1 0 amin 0 with fell 1 fy 1 1 1 min 0 such that 0 lt amp min lt 0 5 is a fixed number such that 0 lt f 1 t lt 1 0 lt t lt 1 and such that 0 lt f 1 t 0 lt t lt 1 Additionally a coating layer is attached located between the first curve f2 1 t fy 1 0 lt t lt 1 and a second curve f2 2 t fy 2 t 0 lt t lt 1 The last connects the point f 1 a1 fy 1 a1 f 2 0 fy 2 0 with fe 1 a2 fy a2 B fa 2 1 fy 2 1 Moreover f2 2 fy 2 t 0 lt t lt 1 is a simple open arc above f 1 t fy 1 t 0 lt t lt 1 such that 0 lt f 2 t lt 1 0 lt t lt 1 147 over a flat grating with surface z 0 0 lt x lt 1 a material part is attached which is located between the line z 0 0 lt x lt 1 and fa 1 t fy 1 t O lt t lt 1 Here f 1 t fy 1 t 0 lt t lt 1 is a simple open arc connecting f 1 0 fy 1 0 min 0 with f2 1 1 fy 1 1 amp min 0 such that 0 lt amin lt 0 5 is a fixed number such that 0 lt f 1 t lt 1 0 lt t lt 1 and such that 0 lt f 1 t 0 lt t lt
71. the angle of direction D inside this inclined plane i e the angle between D and the y axis To change between the xy system and the xz system the following formulae are useful Pe ae sin6 COSPyz Gey arcsin sind sindz Ory arcsin y1 sin 6 sin Qz zz arccos cosy C08e xy 8 In other words 7 6 and are the spherical coordinates of k k t and 6 is not the angle enclosed by k k and the positive y axis but the angle enclosed by k k and the negative y axis 18 gt lt gt X Figure 2 Coordinate system based on x y plane arcsin a if Osy gt 0 1 cos O ey cos Pry r arcsin aes gt else 4 1 C08 Ony COS bay Though the user can choose his favourite spherical coordinate system for the input of the direction of incidence the output of the directions for the reflected and transmitted modes are presented in the xz system Clearly the fields E and H must be orthogonal Moreover the two vectors are uniquely determined by the normalization condition by Maxwell s equation and by the polarization type prescribing the polarization direction Here the normalization condition Fine 2 p 2 Z RBire Hive n z T nt z k z Eo z means that the incident light wave has a fixed intensity length of Poynting vector inde pendent of the cover material If the cover material is air then the wave is normalized such that the amplit
72. the integer parameter takes the value 3 following the first line of the calling sequence and the real parameters take the values 0 5 and 0 50 following the integer parameter values Any number of parameters is possible for a corresponding file GEOMETRIES profiles_par c ep pin gt PIN GRATING DETERMINED BY PARAMETRIC CURVE i e over a flat grating with surface x 0 0 lt x lt 1 a material part is attached which is located between x 0 0 lt x lt 1 and fctx t fcty t 0 lt t lt 1 Here fctx t fcty t 0 lt t lt 1 is a simple open arc connecting fctx 0 fcty 0 xmin 0O with fctx 1 fcty 1 1 xmin 0O such that O lt xmin lt 0 5 is a fixed number such that 0 lt fctx t lt 1 0 lt t lt 1 and such that 0 lt fcty t 0 lt t lt 1 The functions fctx fcty and the parameter xmin are defined by the code in f GEOMETRIES pin c exes C epim gt COATED PIN GRATING DETERMINED BY TWO PARAMETRIC CURVES i e over a flat grating with surface x 0 0 lt x lt 1 a material part is attached which is located between x 0 0 lt x lt 1 and fctx 1 t fcty 1 t 0 lt t lt 1 Here fctx 1 t fcty 1 t 0 lt t lt 1 is a simple open arc connecting fctx 1 0 fcty 1 0 xmin 0 with fctx 1 1 fcty 1 1 1 xmin 0 such that 0 lt xmin lt 0 5 is a fixed number such that 0 lt fctx 1 t lt 1 O lt t lt 1 and such that 0 lt fcty 1 t O lt t lt 1 Additionaly a coating la
73. the next Step 3 Else determine the next iterate r 41 as the optimal vector in RY for which the Lagrangian r L r ml mu attains its minimum This optimization problem without any restriction is solved by the non linear conjugate gradient method inner iteration and by choosing r as the initial solution In particular in each inner iteration a new search direction is sought cf and replace the projection P by the identity and a line search is performed in this direction cf Steps 2 1 2 2 in Section 10 3 2 For the line search a fixed parameter Amz for the maximal stepsize factor and a constant 104 c for the Armijo criterion are needed Moreover the maximal number of inner conjugate gradient iteration is bounded by the prescribed number liter_max The inner iteration stops even earlier if the norm of the gradient of the Lagrangian is less than the prescribed threshold gra times Cear or if the stepsize in the line search is less than Ege After the conjugate gradient iteration is finished define new multipliers mlj41 ml 1 Z and mujyi mu 1 1 by mt max 0 mj e t rj T i i gt muj i max fo muj eliri ui If the norm difference mlj41 ml muj41 mu of old and new multipliers is less than the prescribed positive threshold Emu and if V rj41 mlj41 muj41 is less than the prescribed gradient threshold then sto
74. the user can stop the iteration whenever he observes that the iterative solutions do not improve 10 3 4 Method of augmented Lagrangian Suppose the optimization problem is to find a local minimum i e to find an admissible vector Top in R such that f ropt lt f r holds at least for any admissible r R close to Topix Here a vector r R is called admissible if the coordinates r of r satisfy l lt rj lt uj The functional f is continuously differentiable and possibly non linear Moreover f r is supposed to be defined for all r RY To prepare the method of augmented Lagrangian some definitions are needed The La grangian multipliers are denoted by ml RY and mu RY and the augmented Lagrangian is defined by N 1 Lot ml mu Ceal f r T D max 0 mu olri z uj pe mui i 1 1 35 X max 0 ml e l ri mi 10 17 i 1 Here o stands for a prescribed positive real parameter and Cea is a fixed calibration factor Using this notation the method of augmented Lagrangian consists of the following Steps 1 3 1 Initialization Set the index j of the iteration to zero and choose ro from the initial values given by the user in the input file Choose the initial multipliers muo 0 and mlo 0 Increase j by one 2 Step of iteration If the number of iterations j is larger than the prescribed number niter_max then stop the iteration and go to
75. tr ja X lg X Mm X n_qua_tr and kg x ly X m X n_qua_tr values respectively the dependence on wave length plus 0 plus on wave length plus 0 plus polarization type on wave length plus plus polarization type and on plus plus polarization type respectively is managed Replacing WAL by WTP POL and adding ja x kg x lg X m xX n qua tr input numbers the dependence on wave length plus 0 plus plus polarization type is obtained As mentioned in Section 10 1 1 adding a TE TM before the input of n_phs_j_re resp n_phs_j_tr the terms can be replaced by 10 5 E g for the second type reflected phase shifts the corresponding input can look like n_phs_2_re TE TM 2 In the case of the terms 10 5 an input of the weights w_phs_j_re resp w_phs_j_tr and of 22The values for c and w7 respectively are given in a nested loop over the wave length angle 0 angle polarization type and the index of the mode The innermost loop is over the n_qua_tr modes The next is over the m polarization types the next over the l angles and the next over the kg angles 0 Finally the outermost loop is over the j values of the wave length 81 the prescribed values c_phs_j_re resp c_phs_j_tr beginning with a sequence containing POL is not possible since the phase shifts for different polarization are included into one term only Finally suppose that input and output are of type TE TM and that the angle of incidence
76. type A is needed right angled triangle with hypotenuse parallel to the direction of the periodicity cf Figure 6 then this can be accomplished by calling the executable GEN_ECHELLEA from the subdirectory GEOMETRIES More precisely the command GEN_ECHELLEA name R 0 3 0 03 0 04 creates the file name inp of the desired echelle profile grating with right blaze angle greater than 45 with a depth triangle height of 0 3 times period of the grating and with coated layers of height 0 03 resp 0 04 times period over the first resp second part of the grating measured in direction perpendicular to the echelle profile height greater or equal to zero If the input letter R is replaced by an L then the left blaze angle is greater than Tf this formal assumption is not fulfilled then a rectangular upper resp lower coating strip does not make sense and without coated layers a thicker artificial strip can be added without problem 10 Indeed the corner points at the upper resp lower boundary lines of the FEM domain must be part of a uniform grid This is required by the fast boundary element discretization 27 45 degrees Moreover if the input letter R is replaced by an A and the following input number 0 3 by 60 then the left blaze angle is 60 If an input file for an echelle grating of type B is needed right angled triangle with one of the legs parallel to the direction of the periodicity cf Figure 7 then this can be a
77. user can invoke the optimiza tion by the command OPTIMIZE g name dat In this case the actual iterative solutions are printed on the screen and pushing the Ctrl C keys the user can stop the iteration whenever he observes that the iterative solutions do not improve 10 3 3 Interior point method Suppose the optimization problem is to find a local minimum i e to find an admissible vector Top in RY such that f ropt lt f r holds at least for any admissible r RY close to Topt Here a vector r R is called admissible if the coordinates r of r satisfy l lt r lt uj The functional f is continuously differentiable and possibly non linear To prepare the interior point method some definitions are needed The slack variables sl and su as well the dual slack variables dl and du are given by al sh si ri li gt 0 dl dhi dl gt 0 10 13 su su su u i ri gt 0 du du du gt 0 10 14 101 Introducing the operator F r sl su dl du gt F r sl su dl du R Y by du dl V f r r I sl F r sl su dl du u r su sl dl 01 su du Oil the necessary Karush Kuhn Tucker condition of a locally optimal solution imply that there exist dual slack variables dl and du such that Fo r sl su dl du 0 sl gt 0 su gt 0 dl gt 0 du gt 0 Finally the interior point method defines iterative solutions r as the approxima
78. wave number as in the classical TE and TM case The direction k k must be prescribed by the user of DIPOG 2 1 This can be characterized by two parameters namely by the angles 6 and which are the spherical coordinates of a 3 7 We emphasize that and are the spherical coordinates of a 3 7 and not those of the normalized wave vector k k a vP Contrary to this the angles 6 and of the reflected and transmitted plane wave modes are exactly the spherical coordinates of the normalized wave vectors Unfortunately this traditional notation is a little bit confusing Either we use the spherical coordinate system with the x y plane as basis plane xy system or the spherical coordinates based on the x z plane xz system In the zy system we define the direction D a 8 7 as cf Figure 2 Dy sindry COSPgy COSA my COBP ny 4 singzy Here sy 90 90 is the angle of inclination of the plane containing the direction Dey and the z axis from the y z plane Angle E 90 90 is the angle of direction Dry inside this inclined plane i e the angle between D and the orthogonal projection of Dzy to the x y plane For the xz system the direction D a 5 7 is given by cf Figure 3 D sind cos z COSsO SINO yz sings Here is the angle of inclination of the plane containing the direction D and the y axis from the x y plane Angle 6 0 90 is
79. which are to be plotted executable PLOT_PS produces ps file of two dimensional graph of data argument name res and indices of modes the efficiencies of which are to be plotted executable PLOT MATLAB produces Matlab file of three dimensional graph of data works only for classical illumination argument name res and indices of modes the efficiencies of which are to be plotted executable PLOT_GNUPLOT produces gnuplot ps file of three dimensional graph of data works only for classical illumination argument 11 MAKES name res and indices of modes the efficiencies of which are to be plotted header MHEAD note that MHEAD is to be adapted to your computer system before installation executable MAKEHOME for another user produces new version of six subdirectories GEOMETRIES CLASSICAL CONICAL and RESULTS together with all data and example files and links to executables body of makefile makefile_all and more There exist subdirectories with technical files grid tri dpogtr edpogtr conical conical2 optim dipog 2 1 gui refr_ind_data results programs and input files for grating and grid data programs and input file for the FEM computation in the classical case programs and input file for the GFEM computation in the classical case programs and input file for the FEM computation in the conical case programs and input files for the GFEM computation in the con
80. with the Efficiencies If a result file name3 res is produced containing the values for several wave lengths or incident angles then one can have a look at the two dimensional graph of the efficiencies depending on the wave length or incident angle Make sure to be in the subdirectory RESULTS where the result file e g name3 res exists Then enter the command PLOT_DISPLAY name3 1 2 3 4 Here 1 2 3 and 3 stand for the efficiency energy to be plot E g setting 1 equal to R 1 means efficiency of reflected mode of order 1 setting 2 equal to T 0 means efficiency of transmitted mode of order 0 setting 3 equal to RE means total reflected energy and setting 4 equal to TE means total transmitted energy The number of efficiency energy can vary between one and nine Now a graph with the efficiencies energies pops up on the screen cf Figure 26 To interrupt the presentation of the picture press Enter Return Alternatively one can enter the command PLOT_PS name3 1 2 3 4 Everything is like in the previous case However instead of showing the graph on the screen a ps file cf Figure is produced The name of the ps file will be printed on the screen In the case of classical diffraction and for result files with varying wave lengths or and varying angles of incidence one can enter the command PLOT_MATLAB name3 1 68 Everything is like in the previous case However instead of creating a two dimens
81. with x coordinate 0 last point with x coordinate 1 same y coordinate for first and last point all x coordinates between 0 and 1 at least two different y coordinates last line should be End If the program does not find the file GEOMETRIES filel then it takes the file filel of the cur rent working directory Grating data polygon filel file2 indicates a COATED GRATING DETERMINED BY POLYGONAL LINES cf Figure 12 i e the grating profile line is defined by the data in the file with name GEOMETRIES file1 in GEOMETRIES filel in each line beginning without there should be the z and y coordinate of one of the consecutive corner points first point with x coordinate 0 last point with x coordinate 1 same y coordinate for first and last point all x coordinates be tween 0 and 1 at least two different y coordinates last line should be End and the coated layer is enclosed between the polygonal line of GEOMETRIES filel and the polygonal line of the file with name GEOMETRIES file2 in GEOMETRIES file2 in each line beginning without there should be the z and y coordinate of one of the consecutive corner points first and last point must be corner of first polygon second polygon must be on left hand 38 Figure 11 Grating determined by a polygonal line side of first one to one correspondence of the corners on the two polygons between first and last point of s
82. x lt d If necessary additional coating layers beneath the grating structure are allowed The class has no name parameters ns_geom_param 0 and ni_geom_param 2 integer parameters The value igeom_param 2 is the number of different materials which is equal to the number N of trapezoids plus two two for substrate and cover material The num ber of real parameters nd_geom_param is equal to 5 i geom_param 2 4 5N 6 These parameters include five reals for each trapezoid two reals for the location of the stack of trapezoids and four reals for the refractive indices of the substrate and cover material In particular cf Figure 28 if k is a positive integer less or equal to N then the parameter d_geom_param 5k 4 h gt 0 is the height of the kth trapezoid measured in um The pa rameter d_geom_param 5k 3 b d 0 1 is the ratio of the x coordinate of the right upper corner of the Ath trapezoid over the period d of the grating The real d_geom_param 5k 2 23 Unfortunately the optimization is interrupted if the width of the first coating layer adjacent to the stack of trapezoids is less than the width of the additional strip automatically added to the FEM domain cf Section 3 2 The last width is the sum of two numbers The first number is the minimum of the height of the lowest trapezoid and of half of the width of the first coating layer The second number is the period multiplied by the minimum of the meshsize 0 5 to the pow
83. 0 and R 30 one can choose also the inputs A 90 for an apex angle of 90 degrees or D 0 2 for a depth of grating 0 2 micro meter Moreover any combination of two inputs of the types PE ROO OE AO ROO and DO 22 is accepted However the choice A 90 and D 0 2 might be ambiguous By definition it fixes an echelle grating with right blaze angle larger than the left To get the flipped grating with left blaze angle larger than the right the input should be A 90 and D 0 2 e g trapezoid 60 0 6 3 0 2 0 1 0 1 0 05 gt TRAPEZOIDAL GRATING trapezoid with the basis parallel to the direction of the periodicity with angle of 60 degrees angle enclosed by basis and the sides with a base of length 0 6 micro meter consisting of 3 material layers of heights 0 2 0 1 and 0 1 micro meter respectively and with a coated layer of height 0 05 micro meter greater or equal to zero e g mtrapezoid 3 0 005 0 015 0 005 80 90 80 0 01 7 0 05 0 075 0 05 0 05 0 075 0 075 0 05 0415 O13 Oat 0 5 0 6 OTS Q9 7 gt MULTITRAPEZOIDAL GRATING finite number of symmetric trapezoidal bridges located side by side each bridge consists of 3 trapezoidal layers one over each other hights in micro meter of these layers from above to below are 0 005 0 015 and 0 005 sidewall angles in degrees of these layers from above to below are 80 90 80 height in
84. 0 97 691353 QO O O O 0 O 085087 532792 293270 111089 080675 097400 oo oOo oO l 1 2 3 i 2 3 4 OOON 20 x 52 13 031272 085740 720512 Reflection efficiencies and coefficients n O theta n 1 theta n 2 theta n 3 theta Reflected energy 65 15 21 87 00 74 33 07 0 0 0 0 013043 000365 000044 091547 2 309310 O Q 0 QO 144752 016240 001384 053854 Transmission efficiencies and coefficients n O theta n 1 theta n 1 theta n 2 theta n 3 theta n 4 theta 26 50 T 10 29 54 95 41 80 48 96 TI Pe PN OS A 0 0 502838 QO 176861 0 0 0 644282 071650 063954 032481 151 0 O O 0 0 O 084941 532607 293228 111036 080553 097406 eil 2 3 4 5 3 Oo 0 2a 62 13 031559 084372 719528 s1117 060075 000400 136395 424602 687989 741238 112318 060098 000423 136472 427416 682703 745111 Transmitted energy 97 690690 date 10 Feb 2003 10 30 53 E i Thank you for choosing gdpogtr Bye bye 12 7 Output file example res of FEM in CONICAL date 10 Feb 2003 10 06 22 KKK KK K K K FK FK FK FK FK K K 2 2 2K K FK FK FK K K K K K K K K FK FK 2K FK FK FK FK K K DK KK
85. 0 Number of integer parameters ni_geom_param 5 i_geom_param 1 6 indicator of EUV bridge i_geom_param 2 number of different materials number of trapezoids in stack bridge number of non stop layers beneath stack so far no non stop layers are allowed 3 if height of additional layer gt 0 i_geom_param 2 number of different materials number of trapezoids in stack bridge number of non stop layers beneath stack so far no non stop layers are allowed 2 if height of additional layer 0 i_geom_param 3 number of trapezoids in stack i_geom_param 4 number of non stop layers beneath stack i_geom_param 5 index of trapezoid in stack through which the upper line of the extra layer beside the stack goes 1 lt i_geom_param 5 lt i_geom_param 3 Number of real parameters nd_geom_param 5 i_geom_param 3 3 i_geom_param 4 12 Refractive indices n_co n_3 N nu2 f f 4 f nel Peers see eases nel n_1 nl_i 178 HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HH aaa OH OH hel 4 nil_2 4 nl3 4 n_su n_co d_geom_param nd_geom_param 6 ixd_
86. 0 lt t lt 1 and such that 0 lt fy 1 t 0 lt t lt 1 Additionally a coating layer is attached located between the first curve period f 1 t fy 1 t O lt t lt 1 and a second curve period f 2 t fy 2 t 0 lt t lt 1 The last connects the 43 Figure 16 Grating determined by smooth parametric curves first point period f 1 a1 fy 1 a1 fr 2 0 fy 2 0 with the last point petiod fa 1 a2 fy 1 2 period fe 2 t perio LOD ROI Moroen LOD RON 9 t lt 1 is a simple open arc above f 1 t fy O lt t lt 1 sucht that 0 lt f 2 t lt 1 0 lt t lt 1 The functions f 1 f 2 f 1 and f 2 and the parameters a1 aa and min are defined by the code of the file GEOMETRIES cpin c If the program does not find the file GEOMETRIES cpin c then it takes the file cpin c of the current working directory Grating data cpin2 indicates a COATED PIN GRATING DETERMINED By TWO PARAMETRIC CURVES TYPE 2 cf Figure 19 where Lmin 0 2 foll t ten 1 22min t Jy l t 0 5 sin rt and f 2 t 0 5 Emin 1 Emin t fy 2 t 0 8 sin 7t i e over a flat grating with surface period z 0 0 lt a lt 1 a material part is attached which is located between the line period x 0 0 lt x lt 1 and period f2 1 t f t 0 lt t lt 1 Here fa 1 t fy 1 t 0 lt t lt 1 is a simple open arc connecting
87. 1 c_ poltyp2 mode2 c_ poltyp2 moden_ 190 HHHHHHHHHHHHHH HHH HHH HOH if the values do not depend on lambda theta and phi In case it depends on lambda theta or phi the input starts with WAL POL for WAL ATH POL for ATH APH POL for APH W T POL for W T W P POL for W P T P POL for T P WIP POL for WIP Then there follows the doubled number of lines with values of c_ This corresponds to a loop over lambda theta phi as usually In the inner of the loop for fixed lambda theta and phi first the n_ values for the first polarization type poltypl are listed in the usual order Then the n_ values for the second polarization type poltyp2 follow If a value c_ or a value w_ depends on the wavelengths or and the angles or and the polarization type then the corresponding value w_ or a value c_ must be given in the same way In other words if e g W_ is constant but c_ depends on m values of wavelength then the input for w_ must start with the line WAL and after wards the constant value w_ must be repeated m times in m separate lines ETRE PE TEPER REPEL HE EE EEE HE HHHHHHHHHHHHHHHHHHHHH HHH HHH HHH FH If the type of polarization and coordinate system for the incoming wave vector is TE TM
88. 196 HHHHHHHHHHHHH HH HHH HHH HOF in a separate line nd_opt numbers vector d_geom_scal of positive scaling parameters d_geom_scal j j 1 nd_geom_param Indeed if partial derivatives of objective functional with respect to some parameter coordinate d_geom_param j are much larger than the others then this d_geom_param j together with the bounding dl_geom_param j du_geom_param j must be scaled d_geom_param j d_geom_param j d_geom_scal j dl_geom_param j du_geom_param j d_geom_scal j du_geom_param j dl_geom_param j d_geom_scal j Without scaling the iterative procedure reduces the large components of the gradient vector upto the discretization error and an optimization in the gradient directions of the remaining components is hindered by the relatively large discretization error of the gradient components which had formerly been large All scaling factors d_geom_scal j for the d_geom_param j fixed by setting dl_geom_param j du_geom_param j must be set to one ETRE PETE TERETE AEE EE HHHHHHHHHHHHHH HH HH HH HH HH H OH CONJUGATE GRADIENT METHOD PROJECTION ind_opt 1 ni_opt 1 i_opt 1 number n_norm of same gradient norms after which the algorithm stops e g 3 nd_opt 5 d_opt 1 maximal stepsize factor alpha_max in line search e g 1 d_opt 2 constant c_1 in Armijo stopping criterion for line search in conjugate gradient for conjugate gradient e g 0 001 d_o
89. 2 1 CLASSICAL cp example dat name2 dat Change name2 dat in the editor according to your requirements You will find the nec essary information as comments in the file name2 dat Indeed each line beginning with is a comment Comment lines can be added and deleted without any trouble 5 2 Simple calculation with minimal output Now enter the command FEM name2 The program is running and produces an output on the screen Additionally a result file is produced compare the similar file enclosed in point the name of which is announced on the screen You will find all Rayleigh coefficients the efficiencies and energies on both the screen and in this file Note that the result file has the tag res and is located in the subdirectory RESULTS If a lot of data is produced then computer programs should have an easy access to the data To enhance readability by computer a second output file can be produced setting a switch in name2 dat to yes The name of this second file is the same as that of the first but with tag erg instead of tag res The file is normally located in the subdirectory RESULTS The name name3 of the result files RESULTS name3 res resp RESULTS name3 erg is indicated by the name2 dat lines Name of output file name3 However if the code is started from a directory different from CLASSICAL or if the output file should be written into a different directo
90. 2i_geom_param 2 1 2i_geom_param 2 i geom_param 2 i_geom_param 3 1 must be satisfied For j 2i geom_param 2 k with 1 lt k lt i geom_param 3 the user defined upper and lower bounds du_geom_param j and dl_geom_param j will be corrected in order to guarantee that the corresponding interior corner points Q remain inside the convex domain A More precisely if the numbers dl dl_geom_param j and du du_geom_param j are the old user defined values then the new internally corrected values dlgz dl_geom_param j and dug du_geom_param j are given by cf Figure k du gt min du duo duo sup fn gt 0 P Nl P gt P hv a E k dl max dl dlo dlo int fh lt 0 Pit qih rite db e dgeom_param 2i geo _param 2 i geom param 3 1 i 10 9 94 EE 1 000 i 0 000 EE 1 350 i 0 100 EE 1 250 i 0 150 EE 1 150 i 0 100 EE 1 500 i 0 000 Figure 31 General grating without and with additional interface 10 2 7 Bridge composed of trapezoids under light in the EUV range The sixth class i_geom_param 1 6 is a flat grating with a bridge in form of a stack of trapezoids put on the flat interface cf Figure 32 The trapezoids are located with their parallel sides in the direction of the x axis and one trapezoid is placed over the other such that the upper resp lower sides of the two adjacent trapezoids coincide All these trapezoids do not exceed the period z y 0 lt x lt d If necess
91. 30 30 In this case the expression 7 Wp nln Ay cn Az turns into 2 e410 3 e An 15 2 e Oa 20 3 e5 On 95 2 e As 30 3 e5 As 35 Note that in our example though only the values c depend on the wave length the input of the corresponding weight numbers is done in the same mode In general the weights w and the corresponding prescribed values c must always be given in the same mode In other words if one of the two is constant and the other depends on an entity then the constant values must be repeated to get the same input form Replacing WAL by ATH APH or POL and the following j x n_qua_tr values for c and w by kg x n_qua_tr l X n_qua_tr and m xX n_qua_tr values respectively the user can define the values c and w depending on the angles and and on the polarization type respectively If the user replaces WAL by W T W P T P WAL POL ATH POL and APH POL and the following j x n qua tr values by j x kg x n qua tr ja x l x n qua tr ko X ly x n_qua_tr j X Mm X n_qua_tr kg X Mm x n_qua_tr and l x m X n_qua_tr values respectively then the user can fix the dependence on wave length plus 0 on wave length plus on 0 plus on wave length plus polarization type on 0 plus polarization type and on plus polarization type respectively Choosing WTP W T POL W P POL and T P POL for WAL as well as j x kg xX l x n qua tr ja X k X m X n_qua
92. 5 indicates an ECHELLE GRATING TYPE B right angled triangle with one of the legs par allel to the direction of the periodicity cf Figure fr with angle 60 angle enclosed by hypotenuse and by the leg parallel to the period and with a coated layer of height 0 05 wm measured in direction perpendicular to echelle profile height greater or equal to zero Grating data echelle L 100 R 30 0 05 0 1 indicates a GENERAL ECHELLE GRATING with a left blaze angle of 100 with a right blaze angle of 30 with a coated layer over the left blaze side of height 0 05 um measured in direction perpendicular to the echelle profile height greater or equal to zero and with a coated layer over the right blaze side of height 0 1 um measured in direction perpendicular to the echelle profile height greater or equal to zero must be zero if the previous height is zero Instead of the two inputs L 100 and R 30 one can choose also the inputs A 90 for an apex angle of 90 or D 0 2 for a depth of the grating equal to 0 2 um Moreover any combination of two inputs of the types A 90 L 110 R 90 and D 0 2 is accepted However the choice A 90 and D 0 2 might be ambiguous By definition it fixes an echelle grating with right blaze angle larger than the left To get the flipped grating with left blaze angle larger than the right the input should be A 90 and D 0 2 Figure 6 corres
93. 5 1 0 1 5 0 Upper bounds du_geom_param 0 5 0 5 1 0 1 5 0 Number ni_geom_param of integer parameters 3 Integer parameters i geom param 1 2 2 Number ns geom param of name parameters 0 Parameter names s geom param FATTER TE TETRA TA TT TA TA PE TT a Parameters d_geom_param of initial grating 0 07 0 04 1 78 0 1 5 0 Thus the parameter set of the grating geometry contains nd_geom_param 6 real param eters ni_geom_param 3 integers and ns_geom_param 0 names character strings The integer and name parameters are given in the lines following their number The first integer parameter i_geom_param 1 is the index of the grating class The meaning of all the other parameters depends on the class and will be explained in Section In general the real parameters d_geom_param i are the parameters r subject to the optimization The upper and lower bounds u and l are the numbers du_geom_param i and dl_geom_param i respectively Clearly the choice du_geom_param i dl_geom_param i would fix the pa rameter and only those d_geom_param i with du_geom_param i greater than the value dl_geom_param i are optimized Note that the geometry description includes geometry pa rameters like heights widths and lengths as well as material parameters like the refractive indices In the following input lines of name dat the level of discretization is given Level equal to l means that the FEM grid for
94. 8 More precisely the command GEN MTRAPEZOID namel name2 creates the file namel inp of the desired multitrapezoidal profile grating with data taken from the input file name2 INP In particular this contains lines with all lengths relative to period all angles in degrees number m of layers in one bridge m heights of the layers from above to below m sidewall angles from above to below height at which lateral width is given number n of bridges n lateral widths of bridges n x coordinates of midpoints of bridges If an input file for a lamellar grating is needed rectangular grating consisting of several materials placed in rectangular subdomains cf Figure 10 then this can be accomplished by calling the executable GEN_LAMELLAR from the subdirectory GEOMETRIES More precisely if the file GEOMETRIES lamellar INP contains the numbers each number in a separate line 3 4 0 2 0 6 0 2 1 0 0 0 5 0 70 0 0 50 0 900 00 0 500 and 0 900 then the command GEN_LAMELLAR name lamellar INP creates the file name inp of the desired lamellar profile grating with 3 columns each divided into 4 rectangular layers first column with x coordinate in 0 lt x lt 0 2 second column with 0 2 lt x lt 0 6 third column with 0 6 lt x lt 1 all coordinates are normalized with respect to the period period corresponds to x 1 whole grating with y coordinate s t 0 2 lt y
95. 999999999 d_opt 3 0 d_opt 4 2 d opt 5 1 With these parameters the objective functional is computed at niter max randomly chosen different parameter sets and the minimum of these values is determined Normally such a random search yields better results than a deterministic search at the points of a regular mesh of the parameter domain 10 3 6 Newton type method with projection Suppose the optimization problem is to find a local minimum i e to find an admissible vector Top in RY such that f ropt lt f r holds at least for any admissible r RY close to Topt Here a vector r R is called admissible if the coordinates r of r satisfy l lt r lt ui Moreover suppose the functional f is supposed to be composed of quadratic terms only cf 10 2 In this case f takes the form f r lle 8r where maps the admissible parameter sets r into the space RY with M gt N where c R and where is the Euclidean norm Clearly the components of r are just the scaled efficiencies energies or phase shift values for the grating determined by the parameter set r and the c is the vector of the scaled prescribed values in 10 2 The scaling factors are the square roots of the positive weights in 10 2 Denoting the Fr chet derivative of at r by V r we observe c ro O r roe r O r VO r fr r V r roe rt c Or row 7 VEE VO r VO r e r Thi
96. A Rathsfeld The programs are part of the package DIPOG Direct and Inverse Problems for Optical Gratings The programs require codes written by J R Shewchuk triangulation code TRIANGLE O Schenk K Gartner direct solver PARDISO R W Freund N M Nachtigal qmr solver B Spitzak and others FLTK for graphical user interface The programs are based on codes written by K Gartner direct solver cgs solver R Schlundt gmres solver J Ehlert simplex method J Fuhrmann T Koprucki H Langmach PDELIB adaption of GLTOOLS F Huth M Uhle TGUI graphical user interface for TRIANGLE T Arnold some routines for optimization B Kleemann G Schmidt A Rathsfeld adaption to the grating diffraction problem generalized FEM Owner of program Weierstrass Institute for Applied Analysis and Stochastics D 10117 Berlin Mohrenstr 39 Germany part of Forschungsverbund Berlin e V Wissenschaftsgemeinschaft Gottfried Wilhelm Leibniz e V References see Sections 2 4 and Acknowledgements The author gratefully acknowledges the support of the German Ministry of Education Research and Technology under Grant No 03 ELM3B5 K k k k k ee K k Copyright 2007 x 2K OOK OK OK OK OK OK OK OK OK OK OK OK OOK OK 206
97. E E BEE TERETE EERE SPEER HE HEE EEE PEE EPP EEE HE E EEE HE eg input file for GFEM located in directory CLASSICAL contains constants for numerical method in program GFEM HHH H H HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEHHEHHEHHE HEH Recommendation for n_DOF and n_LFEM a mild accuracy requirements and wave numbers not too large n_DOF 1 3 7 n_LFEM 2 n_DOF 1 b challenging accuracy requirements or large wave numbers n_DOF 3 with n_LFEM 31 OF n_DOF 7 with n_LFEM 127 or n_DOF 15 with n_LFEM 511 HHHHH HH HHH HH H OF Recommendation for n_UPA Take the first level 1_0 cf the last input in name dat which is the upper bound for all levels to be computed and cf the levels indicated in the result files name2 res such that the next level results in about four times the number of grid points Then if you wish to compute on level 1_0 1_1 set the HHH HHHH H 127 coni maximum level of computation last input in name dat to 1 0 and choose n_UPA as 2 to the power 1_1 H H HH ERRUR RREREEA AAA AAAA H A EEEE EEA AE E E EEEE E EEEE EAEE n_DOF Additional degrees of freedom on each triangle side Indeed trial functions on each subdivision triangle are approximate solutions of pde s t restriction to triangle sides coincides with Lagrange interpolation polynomials on triangle side Dirichlet s problem Her
98. ER 3 4 VALUE ORDER 3 h value extrap error 0 818 2 reZ 2 6240 0 347 0 2682 0 1350 0 195 0 1595 0 1545 0 0263 0 092 0 1397 0 1353 0 0065 0 045 0 1364 0 1358 0 0032 5 VALUE REFLECTED ENERGY h l value l extrap l error 0 818 6 5736 4 2479 0 347 2 0360 0 2896 0 195 2 1798 2 1754 0 1459 0 092 2 2743 2 4558 0 0513 0 045 2 3000 2 3095 0 0257 6 VALUE ORDER 0 h l value l extrap l error 22 1644 27 0660 27 9758 27 5337 27 4473 ORDER 1 56 7616 55 5900 52 7622 52 7521 52 7150 ORDER 1 4 o aM M o a a M M 9 2569 13 4623 ORDER 2 H
99. GFEM_PLOT from the directory DIPOG 2 1 CLASSICAL is used then this integer will be chosen as the number of iterations to improve the minimal eigenvalue in the estimate of the condition number of the linear system of equations The standard value is 10 EFF_PLO If this is set to yes then the efficiencies for grating corresponding to the approximate solution are plotted and compared to the values prescribed in the objective functional EFFRES If this is set then the efficiencies for the grating corresponding to the optimal solution are added to the result file name res EPS_OUT_VAL On screen and in result files only those numbers efficiencies energies Rayleigh coefficients phase shifts are printed which are greater than 5 10 However this threshold 5 107 can be changed to any positive number setting the environment variable EPS_OUT_VAL Moreover any value independent of its size is printed if EPS_OUT_VAL equals minus one EUV_SWA_90 Performing an optimization in the class of EUV bridges i e integer parameter i_geom_param 1 6 the user can restrict the search to bridges with sidewall angle less or equal to 90 by setting EUV_SWA_90 to yes This changes the meaning of the parameters cf Section 10 2 7 GET_COND_NUMB If this is set then the executables GFEM GFEM_FULLINFO and 13 GFEM_PLOT in the directory DIPOG 2 1 CLASSICAL will print estimates for the condition numbers instead of the memory requir
100. H RR azz k 5 i_geom_param 2 8 posa posb d_geom_param k ratio of distance of left lower corner from the left boundary line of the period and distance of right lower corner from the left boundary line of the period 0 lt posa posb lt 1 If not both of the parameters posb d and posa posb are fixed then we require posa posb gt 0 for posb d 1 k 5 i_geom_param 2 7 n_co d_geom_param k i d_geom_param k 1 refractive index of cover material Re n_co gt 0 Im n_co gt 0 k 5 i_geom_param 2 5 n_su d_geom_param k i d_geom_param k 1 refractive index of substrate material resp material of adjacent lower coating strip Re n_su gt 0 Im n_su 0 Following parameters must be fixed by setting upper bound lower bound d_geom_param k k 5 i_geom_param 2 7 5 i_geom_param 2 4 Refractive indices n_co ER E EA E n_3 EOE a X m2 i faa n_1 ae ie ae ee E n_su 167 HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHH Ea aaa OO b_k RE E Aver iets E E acy ae ee eee a aan E EEEE gt a_k Siew dace pii gt AEE T al l pee Se ae nena X i h_k k th trapez k 2 v ri post SsS Sea a sess EE prenia 0 0 posa 0 posb 0 d 0 The presented parameters of CLASS 4 are the internal parameters The class can be determined by external parameters too External geometry parameters d_k Sh iia al ich ea E ets ah anand ea E E ws
101. HHH HHH HH H HOF BEEEEEPE E E HE BESET ERT E E E A TE TETE STEEL HE HEPES TE E A AE PEE GE PEE TD AEE E EHE HE HE HE EE PET Ae input file for local optimization using GFEM for conical diffraction ERR RRRRRAAANGHHAAAANNAA HRR R RHH EEE EHH EE HHHH HEH NAME OF OUTPUT FILE HEHEHE RRR RRA aE RRR AE Raa Name of the output file 156 The tag res will be added The file will be written in the RESULTS directory Alternatively a path for the location of the file can be added before the name This must contain at least one slash E g for a file name res in the current working directory write name Name example HHEHHHHHHHHHHHHHHHHHEHHHEEHRHHHAHHEEH HHH HHH H RHEE HRRE RHEE RH ARR GRATING ILLUMINATION TERED ESET PETE PTET EEE ETT TE Number of coating layers over the grating N_co_ov The grating cross section consists of a rectangular area parallel to the axes This inhomogeneous part is determined by a triangular grid and can have already a few layers of coatings involved Beneath and above this rectangular structure there might be additional coated layers of rectangular shape These kind of layers are called coating layers over the grating and coating layers beneath the grating respectively Alternatively a a multilayer system input format is possible E g the in
102. HHHHHHHHHHHHHHHEHHEHHEH HEHEHE HY HEHHHHHHHHHAHH HH example dat HEHHHHHHHHHHHHAAH all lines beginning with are comments HHH H HH HF HHHH H HOH HHEHHHHHHHHHHHHHHHHHHHHHEEHHHHA HEHEHE EHHEH AHHH EHE RHA RHEE ERR H HH input file for FEM GFEM located in directory CLASSICAL HHEHHHHHHHHHHHHHHHHHHHHHEEHEHHHAHHHEHRAEEHHEH HEHEHE HREH HHH HRH RHEE HH Name of the output file H The tag res will be added File will be written into directory RESULTS Alternatively a path for the location of the file can be added before the name This must contain at least one slash E g for a file name res in the current working directory write name Name example HEHEHE RARER Ra RRR RAH Should there be an additional output file in the old style of DIPOG 1 3 resp an eps file for FEM_CHECK H Add no if not needed 114 Add yes if needed The name will be the same as the standard output file given above but with the tag erg instead of res Add phaseshifts if no additional output is needed but if phase shifts are preferred instead of Rayleigh coefficients yes or no or phaseshifts yes HHEHHHHHHHHHHHHHHHHHHHHHEEHEHHHHHHHHHRHEHHRHEHHHHHHHEHHHH HEHE HRHRH RHEE HH Number of coatings over the grating N_co_ov The grating cross section consists o
103. HHHHHHHHHHHHHHHHHHHH HHH HR RHR Ra Oz c_ lambdam1 theta1 phi2 mode1t c_ lambdam1 theta1 phi2 mode2 c_ lambdam1 thetal phi2 moden_ c_ lambdam1 theta1 phim3 mode1 c_ lambdam1 theta1 phim3 mode2 c_ lambdam1 theta1 phim3 moden_ c_ lambdam1 theta2 phi1 mode1 c_ lambdam1 theta2 phi1 mode2 c_ lambdam1 theta2 phi1 moden_ c_ lambdam1 theta2 phi2 mode1 c_ lambdam1 theta2 phi2 mode2 c_ lambdam1 theta2 phi2 moden_ c_ lambdam1 theta2 phim3 mode1 c_ lambdam1 theta2 phim3 mode2 c_ lambdam1 theta2 phim3 moden_ c_ lambdam1 thetam2 phi1 mode1 c_ lambdam1 thetam2 phi1 mode2 c_ lambdam1 thetam2 phi1 moden_ c_ lambdam1 thetam2 phi2 mode1 c_ lambdam1 thetam2 phi2 mode2 c_ lambdam1 thetam2 phi2 moden_ c_ lambdam1 thetam2 phim3 mode1 c_ lambdam1 thetam2 phim3 mode2 c_ lambdam1 thetam2 phim3 moden_ If additionally the type of polarization runs over two types in this case input type of polarization must be TE TM first type poltypel TE and second type poltype2 TM then the value of the objective function is replaced by the additional sum over the polarization types In this case the values c_ may depend on the type of polarization too The input of these values is e POE c_ poltyp1 mode1 c_ poltyp1 mode2 c_ poltyp1 moden_ c_ poltyp2 mode
104. HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHH RHR azz Clambdat lambdal lambdal lambdat lambdal lambdat lambda2 lambda2 lambda2 lambda2 lambda2 lambda2 lambda2 lambda2 lambda2 Lambda2 Lambda2 Lambda2 Lambda2 Lambda2 Lambda2 Lambda2 Lambda2 lambda2 Lambda2 Lambda2 Lambda2 Lambda2 Lambda2 Lambda2 Lambda2 lambda2 lambda2 thetam2 phi2 thetam2 phi2 thetam2 phi2 modei mode2 moden_ thetam2 phim3 mode1 thetam2 phim3 mode2 thetam2 phim3 moden_ thetal phiil thetal phil thetal phil thetal phi2 thetal phi2 thetal phi2 modei mode2 moden_ modei mode2 moden_ thetal phim3 model thetal phim3 mode2 thetal phim3 moden_ theta2 phil theta2 phil theta2 phil theta2 phi2 theta2 phi2 theta2 phi2 modei mode2 moden_ modei mode2 moden_ theta2 phim3 mode1 theta2 phim3 mode2 theta2 phim3 moden_ thetam2 phil thetam2 phil thetam2 phil thetam2 phi2 thetam2 phi2 thetam2 phi2 modei mode2 moden_ modei mode2 moden_ 2 thetam2 phim3 mode1 thetam2 phim3 mode2 thetam2 phim3 moden_ lambdam1 theta1 phi1 mode1 lambdam1 theta1 phil mode2 lambdam1 theta1 phi1 moden_ 189 HHHHHHHHHHHHHHHHHHHHHHH
105. KK GFEM GFEM GFEM K kK K kK 0 012 8 kk KKK K kK GFEM GFEM GFEM GFEM K kK 0 012 7 0 022 6 0 022 6 kKK KKK KKK 0 012 7 GFEM GFEM K kK 0 011 6 0 050 4 0 050 4 KKK KKK 0 011 6 0 022 5 GFEM 7 15 K kK 0 025 4 0 050 3 0 050 3 KKK KK 0 011 5 0 025 4 Relative mesh size h refinement levels necessary to reach 0 1 accuracy METHOD k 3 15 k 6 30 k 12 60 k 25 20 k 50 39 d 1 d 2 d 4 d 8 d 16 FEM 0 013 9 0 013 9 kk kK kk GFEM 1 3 0 023 7 0 023 7 kK ek eK GFEM 1 7 0 044 6 0 044 6 kK kek eK Grem 1 15 0 088 5 0 044 6 0 012 8 xxx kik GFEM 3 7 0 022 6 0 022 6 ke kk eK GFEM 3 15 0 044 5 0 044 5 kK kk aK Grem 3 31 0 100 4 0 100 4 0 012 7 xxx krs GFEM 3 63 0 201 3 0 100 4 0 022 6 0 012 7 xxx GFEM 7 15 0 022 5 0 022 5 kK kk xe Grem 7 63 0 100 3 0 100 3 0 011 6 xxx ksk GFEM 7 127 0 262 1 0 162 2 0 022 5 0 011 6 xxx GFEM 7 255 0 050 4 0 022 5 0 011 6 GFEM 15 31 0 025 4 kK kk ek Grem 15 127 0 081 2 0 011 5 xxx kk GFEM 15 255 0 025 4 0 011 5 xxx GFEM 15 511 0 025 4 0 011 5 Table 3 Relative mesh size h refinement levels necessary to reach 0
106. LINFO and GFEM_FULLINFO for calculation with additional information case of classical diffraction argument name dat executable GFEM_MOVIE creates movie of z coordinate of fields depending on time case of classical diffraction movie in form of Matlab file argument name dat input files name dat non geometrical data of the gratings data file conical Dat data for the GFEM executables FEM and GFEM for simple calculation case of conical diffraction argument name dat executables FEM_CHECK GFEM_CHECK for check of input case of conical diffraction exists only with openGL argument name dat executables FEM_PLOT and GFEM_PLOT for calculation with plots of resulting fields case of conical diffraction exists only with openGL or GNUPLOT argument name dat executables FEM_FULLINFO and GFEM_FULLINFO for calculation with additional information case of conical diffraction argument name dat input files name dat data of the gratings and optimization data file conical Dat data for the generalized finite elements executable OPTIMIZE using various flags this does all the work check data check gradients plot gradients optimize grating plot solution of optimization executable CONVTEST runs a convergence tests with initial solution set to corners of box domain defining the constraints input file is the same as for O
107. NICAL OPTIM Its name must begin with the letter u must not have a as a second letter and may consist of no more than five letters like e g user The file consists of at most 1000 lines each with three real numbers the first is the wave length in micro meter the second the real part of the corresponding optical index and the third the imaginary part of the index At the end of each line a comment beginning with the sign can be added Also the lines beginning with are comments Optical indices with negative real or imaginary parts are not admitted If the refractive index is given by a number then its value is independent for all com putations invoked by the input file However if the wavelength is varying in accordance with the input file then a refractive index independent of the wavelength is not realistic Hence for varying wavelength the input of refractive indices through user defined tables resp through the above mentioned code words is mandatory In this case an input by numbers is not accepted As seen in the example presented at the beginning of this section first the index of the cover material is given Then the indices of the materials of upper coated layers follow These are rectangular layers over the whole period and their number and widths are given in extra lines before the indices not presented in the example lines from above If the number of coated layers is zero then no lines with optical indices are nee
108. NUPLOT is available and if you use the command FEM_PLOT name2 then you have the same results as in point Additionally you will see pictures of the real part the imaginary part cf Figure efx an openGL picture and Figure for a GNUPLOT picture and the square modulus of the solution z component of electric field for TE polarization z component of magnetic field for TM polarization Note that the square modulus is proportional to the energy intensity distribution of the wave Moreover similar pictures for the fields above and below the coated grating area will be plot Program stops at each picture To control the graphical facilities of GLTOOLS in FEM_PLOT use Backspace Enter user control mode tab toggle state change mode Return Quit user control mode Space Mode control Increase mouse sensitivity decrease control parameter 62 Ma Field 22279 2 5 Imag P ATES 25 AHL berili DEEA i CLES Batata FEEF Atf A 2 EE LE EP REE eth the fea eae Hayate LAR aan I FE rar 1 kyl 1 LLLI TTE E Ee VS Te PIRATES EET ii A Tea APR L Ped RLU Deedee pee I KESLER b EEEE Kee TERA NIU PEPER PRO CCAD EPL TAT ay Sagoo oc ie a ees te ae TI LTI EG ya ra Lg LELETEI Le Ek O5 TMU ONIA ETRE h RE P TET N RENN 0 w Au OL Ret oaae N PERTE DEL 1I PLLNCELL amp MERU PAR SCRAP ILLLER tf hA AT AEAN 1 Ai Eta gipit L ity pees fi 0 5 Pui l HEA ean seat FEEL LEER LEE DAANA 1 Map pe L at Ed Lot
109. P parts of the electric field i e the components of and the direction of propagation of the nth reflected resp transmitted plane wave mode coefficients are gt ap bcq n A si _ apy beg corresponding to the TE and TM parts the component with electric resp magnetic field polarized in s direction Finally the the Jones vector If s is defined as above and if p gt is the direction orthogonal to s p Qn 87 7 x s Qn 6 7 x s then the S and P parts of the Rayleigh ETAO T 21 Riot 2 Bist GEES The efficiencies of the third output are the total efficiencies e of 2 9 and the efficiencies n corresponding to the S and P parts i e the efficiencies of the projection of the nth reflected resp transmitted wave mode to the component with electric field polarized in s resp p direction a Ae 2 fi n T abot 2 4 References For more details see the following publications and the references therein G Bao D C Dobson and J A Cox Mathematical studies in rigorous grating theory J Opt Soc Amer A 12 pp 1029 1042 1995 J Elschner and G Schmidt Diffraction in periodic structures and optimal design of binary gratings I Direct problems and gradient formulas Math Meth Appl Sci 21 pp 1297 1342 1998 J Elschner and G Schmidt The numerica
110. PARAMETERS i e grating determined by profile lines given as fctx j t fcty j t 0 lt t lt 1 j 1 n nmb_curves where the functions t gt fctx j t and t gt fcty j t are defined by the c code of the file GEOMETRIES profiles_par c the last code uses 1 integer parameter and 2 real parameters named IPARaM1 RPARaM1 RPARaM2 the integer parameter takes the value 3 following the first line of the calling sequence and the real parameters take the values 0 5 and 0 50 following the integer parameter values Any number of parameters is possible for a corresponding file GEOMETRIES profiles_par c e g pin gt PIN GRATING DETERMINED BY PARAMETRIC CURVE i e over a flat grating with surface x 0 0 lt x lt 1 a material part is attached which is located between x 0 0 lt x lt 1 and fctx t fcty t 0 lt t lt 1 Here fctx t fcty t 0 lt t lt 1 is a simple open arc connecting fctx 0 fcty 0 xmin 0 with fctx 1 fcty 1 1 xmin 0 such that O lt xmin lt 0 5 is a fixed number such that 0 lt fctx t lt 1 O lt t lt 1 and such that O lt fcty t O lt t lt i The functions fctx fety and the parameter xmin are defined by the code in GEOMETRIES pin c e g cpin gt COATED PIN GRATING DETERMINED BY TWO PARAMETRIC CURVES i e over a flat grating with surface x 0 0 lt x lt 1 a material part is attached
111. PTIMIZE but the initial solution of input file must be the exact solution executable OPTIM20PTIM 10 GUI RESULTS extracts new data input file for OPTIMIZE from old input file with reduced number of data in objective functional executable SHOWMEAS creates plots of data used in objective functional input is data input file for OPTIMIZE executable CLASSIC20PTIM extracts test data input file for OPTIMIZE from input and result file of GFEM in CLASSICAL executable SENSITIVITY extracts optimal sets of data for quadratic objective functional in OPTIMIZE input files are the data file for OPTIMIZE and files with Jacobians generated by OPTIMIZE executable OPTIM2JACOBIAN alternative executable to produce files with Jacobians for SENSITIVITY executable NOISETEST executable to test the dependency of the optimization result on noisy data input files name dat non geometrical data of the gratings like in CLASSICAL and CONICAL data files generlized Dat and conical Dat data for the generalized finite elements like in CLASSICAL and CONICAL executable DIPOG 2 1 GUI graphical user interface to replace the executables of CLASSICAL and CONICAL result files name res and name erg produced by executables in CLASSICAL and CONICAL executable PLOT_DISPLAY produces two dimensional graph of data on the screen argument name res and indices of modes the efficiencies of
112. Parameters d_geom_param of initial grating 0 5 0 3 Dep To avoid contradictions the arguments d_geom_param k of a dependency function must be free parameters If possible one should try to present rational dependency in the standard form Dep fi f2 fs where each f stands for an expression including the variables pj constants and the op erations and x but no blanks and no brackets The code realizes that the dependency is rational and generates code for the derivatives If the dependency is not rational or not in the standard form then the derivatives are approximated by difference formulas The number of integer parameters is larger or equal to two The first igeom_param 1 is an index between one and six indicating that the grating belongs to one of the six grating classes i geom_param 1 1 Profile grating determined by a polygonal profile function i geom_param 1 2 Profile grating determined by general polygonal profile curve I i geom_param 1 3 Profile grating determined by general polygonal profile curve IT i geom_param 1 4 Stack of trapezoids i geom_param 1 5 General grating with polygonal interface to be optimized i geom_param 1 6 Bridge composed of trapezoids under light in the EUV range The second number i geom_param 2 is the number of different materials contained in the grating including the cover and substrate material In general the code will generat
113. QUADRATIC TERMS FIRST TE or S REFLECTED EFFICIENCY Max_Like a b 0 1 Here the symbol a stands for the initial guess of the variance factor a and b for the fixed back ground noise level b The weight numbers following after the first line must be listed like the weights of quadratic efficiency terms Their values however must be zero or one A one indicates that the term with the corresponding efficiency is included into the summation for the objective functional f and a zero that it is not 10 2 Classes of gratings which can be optimized 10 2 1 General parameters For the description of the geometry nd geom param real ni geom param integer and ns_geom_param name parameters character string with less than 250 characters are needed 84 These are stored in the vectors d_geom_param j j 1 nd_geom_param i geom_param j j 1 ni geom_param s_geom_param j j 1 ns_geom_param The integer and name parameters are fixed by the user The vector of real parameters will be optimized Therefore initial values of these d_geom_param j j 1 nd_geom_param are to be fixed by the user Moreover the real parameters are restricted to intervals The upper bounds du_geom_param j and lower bounds dl_geom_param j of the intervals are given by the user The user can even fix a real parameter to a constant value setting the corresponding upper bound equal to the lower Some of the real parameters are the real or imagi
114. Stacie nae z rf h_k k th trapezoid k 2 angle a_k 168 HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHH RH HHH HOH Number of real parameters nd_geom_param 5 i_geom_param 2 4 k 1 i_geom_param 2 2 h_k d_geom_param 5 k 4 height of k th trapezoid in stack in micro meter h_k gt 0O d_k d_geom_param 5 k 3 length of the upper side of k th trapezoid in stack in micro meter a_k d_geom_param 5 k 2 angle at the right lower corner of k th trapezoid in stack in degrees n_k d_geom_param 5 k 1 i d_geom_param 5 k refractive index of k th trapezoid in stack Re n_k gt O Im n_k gt 0 k 5 i_geom_param 2 9 posa d_geom_param k distance of left lower stack corner from left starting point of the period in micro meter k 5 i_geom_param 2 8 posb d_geom_param k distance of right lower stack corner from left starting point of the period in micro meter k 5 i_geom_param 2 7 n_co d_geom_param k i d_geom_param k 1 refractive index of cover material Re n_co gt 0 Im n_co gt 0 k 5 i_geom_param 2 5 n_su d_geom_param k i d_geom_param k 1 refractive index of substrate material resp material of adjacent lower coating Strip Re n_su gt 0 Im n_su 0 169 HHH HH HH HHH HH HF OF Switch from external to internal parameter happens automatically if one of the parameters d_geom_param 5 k 2 k 1
115. TERMINED BY A SIMPLE SMOOTH FUNCTION cf Figure 14 i e grating determined by a profile line given as period t f t 0 lt t lt 1 where the function t f t is defined by the c code f t 0 125sin 27t do not use any blank space in the c code Grating data profile 0 5 0 5 x cos M_PI 1 t 0 25 sin M_PI x t indicates a GRATING DETERMINED BY A SIMPLE SMOOTH PARAMETRIC CURVE cf Fig ure 15 i e grating determined by ellipsoidal profile line given as period f t fy t 0 lt t lt 1 where the functions t f t and t f t are defined by the c codes fet 0 5 0 5cos a 1 t and fy t 0 25sin at respectively no blank space in c code Grating data profiles indicates a GRATING DETERMINED BY SMOOTH PARAMETRIC CURVES i e grating de termined by n non intersecting and periodic profile lines given from above to below as period felj t ft 0 lt t lt 1 j 1 n where n and the functions tr f j t and t f j t are defined by the c code of the file GEOMETRIES profiles c cf Figure 16 where n 3 20 0 t fy l t sin 2rt 7 2 0 sin 2rt 0 5 and f 8 t sin 27t 1 If the program does not find the file GEOMETRIES profiles c then it takes the file profiles c of the current working directory Grating data 41 Figure 14 Grating determined by a simple smooth function profiles_par 1 2 3 0 5 0 50 in
116. TH SIMPLE CURVE defined by fr j t t and f j t ccode The profile is vertically shifted by h wm Note that ccode is an expression of the parameter argument t 0 lt t lt 1 written in the c programming language This expression will appear in the code as fct ccode Even an if case is possible E g substituting the code O if t lt 0 3 fct t else if t lt 0 6 fct 0 6 t else fcet 0 into fcet ccode leads to the meaningful code fet 0 if t lt 0 3 fet t else if t lt 0 6 fct 0 6 t else fet 0 The code must be simple since the program can read no more than 399 symbols per input line If the defining two lines are lineg _1 profile ccode ccode2 linea hy then we get a profile curve DETERMINED By A SMOOTH SIMPLE PARAMETRIC CURVE defined by f j ccode and f j t ccode2 The profile is vertically shifted by h um The same remarks as in the previous mode apply If the defining two lines are 51 lineg 1 profilei ccode line hy dy d PEN di then we get a profile curve DETERMINED By A NON SMOOTH SIMPLE CURVE defined by f j t t and fy j t ccode The curve has i 1 lt i lt 9 corners with the parameter arguments d dz di such that 0 lt dy lt d2 lt lt di The profile is vertically shifted by h um The same remarks as in the previous mode apply If the defining two lines are lineg 1 profilei ccode ccodez line hi dy d Pe di th
117. Type of polarization and coordinate system for incoming wave vector Either TE means that incident electric field is perpendicular to wave vector and to normal of grating plane 131 plane of grating grooves and incoming wave vector is presented in xz system as H sin theta cos phi cos theta sin theta sin phi Either TM means that incident magnetic field is perpendicular to wave vector and to normal of grating plane and incoming wave vector is presented in xz system as sin theta cos phi cos theta sin theta sin phi Either TE TM means TE and TM two calculations Either TP means polarized electro magnetic field and incoming wave vector is presented in xz system as sin theta cos phi cos theta sin theta sin phi Or polt means polarized electro magnetic field and incoming wave vector is presented in xy system as sin theta cos phi cos theta cos phi sin phi Type TE TM TERETE EE ESE PEPE PETE TERETE TETETE HE ANER RRRA eee EEE HTT HEH Parameter of polarization HH HH H as If type of polarization is pol or TP then this is the angle in degrees between x axis axis in plane of grating grooves which is perpendicular to grooves and projection of electric field vector onto x z plane of grating grooves Needed only if polarization is of type pol or TP Else no entry and no line
118. _co_ov gt 0 Else no entry and no line For a multilayer system the indices of the n3 layers in the groups must be given only once I e for a multilayer system ni n3 n4 input lines are needed Optical indices TERETE IEEE ESE PEPE PETE TEEPE THE YHHH EE ee eee EEE ETE EH 158 Optical indices of the materials of the lower coating layers This is c times square root of mu times epsilon H N_co_be entries Needed only if N_co_be gt 0 Else no entry and no line Optical indices EHRRRNPPHHHHRHHRAHHN YHHH H NRR HHR RAAN AA EPERE EHENR ARRERA RAAREAR AHAHHH Optical index of substrate material This is c times square root of mu times epsilon Optical index TeS FE 0 PHRRRRPPHHHHGHHRAHHNYHHH EE eee ETE RRRHHARHAR RHH HHHHEH Type of output results Either TE TM results in terms of TE and TM part of Wave Either Jones results in terms of Jones vector representation Dr 3 Com results in terms of the component in the z axis that is in the direction of the grooves For more details cf Section 2 3 in USERGUIDE ps Type HEHEHE HRA AEH RRR aaa Type of polarization and coordinate system for incoming wave vector Either TE means that incident electric field is perpendicular to wave vector and to normal of grating plane plane of grating grooves and incoming wave vector is presented in xz system as sin theta cos phi cos theta
119. _geom_param 2 2 i_geom_param 3 nkn number of interior knots with x coordinate in 0 Period Number of real parameters nd_geom_param 2 i_geom_param 3 5 k 1 i_geom_param 3 P_k d_geom_param 2 m 1 d_geom_param 2 m corner point of profile curve in other words polygonal grating has i_geom_param 3 2 corners at the points P_O 0 0 P_m x_m y_m m 1 nks P_ nks 1 Period 0 where x_m d_geom_param 2 m 1 y_m d_geom_param 2 m nks i_geom_param 3 k 2 i_geom_param 3 1 d_geom_param k real part of refractive index of cover material HHHHHHHHHHHHHHHHHHHHHHHH HHH HHH HH 164 k 2 i_geom_param 3 2 d_geom_param k imaginary part of refractive index of cover material k 2 i_geom_param 3 3 d_geom_param k real part of refractive index of substrate material k 2 i_geom_param 3 4 d_geom_param k imaginary part of refractive index of substrate material k 2 i_geom_param 3 5 d_geom_param k small threshold EPSilon appearing in the constraint conditions for the feasible sets of parameters 0 0001 lt EPSilon lt 0 5 Constraints first nks 1 conditions two consecutive points not too close to each other next nks nks 1 conditions each point is not to close to each side of polygonal which does not contain the point point not in ellipse around side with small half axis about square root of EPSilon times side length last condition no intersection of non neighbour lines more precisely a i 1 2
120. admissible for our optimization are gratings with a polygonal profile separating the cover and substrate material special classes of multilayered trapezoidal gratings fixed grating structures where a part of a polygonal interface is to be optimized Note that for the classical case Dipog 1 5 computes optimal binary gratings for given efficiency sequences or for prescribed energy restrictions For Dipog 1 5 we refer to the German Benutzer Handbuch http www wias berlin de software DIPOG 1 2 Programming language and used packages All programs are written in fortran c or c language and based on the UNIX system The programs require the previous version DIPOG 1 3 or DIPOG 1 5 the grid generator TRIANGLE 1 4 the graphical user package FLTK and the linear equations solver PARDISO together with some LAPACK and BLAS routines For good visualization the package openGL or at least 3Contrary to the original meaning of binary several layers are admitted the MESA emulation of openGL is needed together with the auxiliary package GLTOOLS A minor visualization is possible with the program package GNUPLOT In emergency case the computations run also without any visualization i e without openGL and GNUPLOT For non comercial use the Levenberg Marquardt algorithm levmar 2 2 by Manolis Lourakis can be applied for optimization The package DIPOG 2 1 can use the refractive indices from the tables of the program package IMD cf Secti
121. alue of regularization parameter for Gaub Newton equation d_opt 2 Threshold gt 0 iteration stops if gradient norm satisfies the estimate J7e lt 1 with J the Jacobian of d_opt 3 Threshold 2 gt 0 iteration stops if correction Ap of iterative solution p satisfies Ap lt 2 Ap 3 with Ap the value of the previous step d_opt 4 Threshold 3 gt 0 iteration stops if least square deviation satisfies e 2 lt 3 To work with default parameters set d_opt 1 1 10 4 References For the computation of the gradients of the objective functional see J Elschner and G Schmidt Diffraction in periodic structures and optimal design of binary gratings I Direct problems and gradient formulas Math Meth Appl Sci 21 pp 1297 1342 1998 J Elschner and G Schmidt Conical diffraction by periodic structures Variation of interfaces and gradient formulas Math Nachr 252 pp 24 42 2003 For the numerical optimization methods see e g the following books and article Ch Gro mann and J Terno Numerik der Optimierung Teubner Studienbticher der Mathematik Teubner Stuttgart 1997 augmented Lagrangian method F Jarre and J Stoer Optimierung Springer Verlag New York Berlin Heidelberg 2004 interior point method P J M van Laarhoven and E H L Aarts Simulated annealing Theory and 110 Applications D Reidel Publishing Company Mathematics and Applic
122. am nd_geom_param and controls the strength of the required boundedness condition Setting upper bound equal to lower bound the user must fix the refractive indices of the substrate and the cover material i e the equality du_geom_param j dl_geom_param j must be satisfied for all indices 7 nd_geom_param 6 ud_geom_param 3 Clearly the real parts of all refractive indices must be positive and the imaginary parts non negative The imaginary part of the cover material must vanish Like the refractive indices of sub strate and cover material the last three parameters must be fixed setting upper bound equal to lower bound If the user wants to restrict the optimization to gratings with sidewall angles a lt 90 and 6 lt 90 cf Figure 32 then he must set the environment variable EUV_SWA_90 to yes In order to obtain box constraints this choice requires a change in the meaning of the parameters d_geom_param i for i 5 j 1 2 andi 5 j 1 3 with j 1 2 N More precisely b d_geom_param 5 j 1 2 a j 1 d_geom_param 5 j 1 4 3 a ji j 1 N i geom param 3 jmaji where the numbers a and b are defined in Figure 82 Clearly 0 lt a lt 90 and 0 lt Bk lt 90 imply the estimate 0 lt d geom param i lt 1 for i 5 j 1 2 andi 5 j 1 3 with j 1 2 N Therefore the corresponding lower and upper bounds must satisfy the relation 0 lt dl_geom_param i lt du_ge
123. and that of the method 20 Tf no good initial solution is known then the user can try to find one by a deterministic search algorithm over a tensor product grid of parameter points In other words the initial solution of parameter values is the parameter point of the grid at which the objective functional takes its minimum over the finite grid Instead of a list of initial values the user writes a line like no njey Nnmp O into the input file Here nies is the discretization level of the FEM method at which the search is to be accomplished The second number Nnmb is the maximal number of grid points in one dimension This is attained at least in the direction of the longest side of the box domain defined by the upper and lower parameter bounds If the user adds the input line no ney Nnmp 1 then the minimum search over the grid points is improved by replacing the value of the objective functional at each grid point by the minimum of the linear Taylor approximation taken over a small neighbourhood of the grid point 75 of optimization The best way to do this is to copy example dat to name dat and to change the data to meet the requirements of the user Numerous comments contained in this file will make it easy to get the right input for some more details cf the subsequent Section L0 1 3 Then the user enters the command OPTIMIZE name and the optimization starts All methods of optimization are iterative The number o
124. angles need to be computed only once In other words reducing the level by one and changing nypa from 1 to 2 turns GFEM into a competitive method even if WLrEm 1 is large Similarly the level can be reduced by 2 or 3 and nypa can be set to 4 or 8 So GFEM with larger npor and ny_rey outperforms the GFEM indicated by bold letters For TM polarization and the same grating and light scenario we get similar results E g in the case of k 12 60 i e d 4 and GFEM 3 31 we get an error of 1 0 1 and 0 01 choosing the refinement levels 4 6 and 6 respectively For k 25 20 i e d 8 and GFEM 7 127 we get an error of 1 0 1 and 0 01 choosing the refinement levels 4 4 and 5 respectively 7Note that the mesh size shown on the screen or in the result files name res after calling the program FEM_FULLINFO and GFEM_FULLINFO are just the h 8Fastest method means the one with the smallest complexity estimate 6 1 70 METHOD n d 16 FEM KKK GFEM GFEM K kK K kK 0 012 8 GFEM GFEM GFEM K kK 0 012 7 0 012 7 0 044 5 GFEM GFEM GFEM K kK 0 022 5 0 022 5 0 022 5 GFEM 7 15 K kK 0 011 5 0 025 4 0 025 4 Table 1 Relative mesh size h refinement levels necessary to reach 1 accuracy METHOD ke 25 20 d 8 k 50 39 d 16 FEM KKK K
125. aram 3 1 2i geom_param 3 5 In order to introduce the additional restrictions the corner points are denoted by P 0 0 P a5 y j 1 N and Pyii d 0 Excluding self intersection means to guarantee distance between neighbour corner points is positive each corner point is outside of an ellipsoidal neighbourhood of each polygonal side not containing the corner no intersection of non adjacent sides of polygonal curve Using the threshold parameter d_geom_param k the above restrictions can be refor mulated as P Fea P P Pj Pm B Pm m 1 ed i 0 N 10 7 E Pm Pml Mm 1 N41 j 0 N 1 j m 1 m 0 m i 1 N 1 m il gt 1 2 2 pee at N Peal Note that the first two types of restrictions are included in the choice of the new search direction for the conjugate gradient algorithm cf Section 10 3 2 The last is automatically fulfilled for a new iterative solution if this is close to the last iterate 10 2 5 Stack of trapezoids The fourth class i_geom_param 1 4 is a flat grating with a stack of trapezoids put on the flat interface cf Figure 28 and the example in Figure 29 The trapezoids are located with their parallel sides in the direction of the x axis and one trapezoid is placed over the other such that the upper resp lower sides of the two adjacent trapezoids coincide All these trapezoids do not exceed the period z y 0 lt
126. ary additional coating layers beneath the grating structure are allowed Beside the bridge an extra layer can be added The class has no name parameters ns_geom_param 0 and ni_geom_param 5 integer parameters The value i_geom_param 2 is the number of different materials which is equal to the number N of trapezoids plus two two for substrate and cover material and eventually plus one if an extra side layer is added The number i_geom_param 3 is the number N of trapezoids The fourth integer parameter igeom_param 4 is reserved for the number M of lower layers the refractive indices or widths of which are included into the set of optimization parameters Finally igeom_param 5 is the index of the trapezoid in the bridge intersected by the upper boundary line of the extra layer beside the bridge If there is no extra layer beside the bridge then igeom_param 5 1 and the height of the layer cf the subsequent parameter d_geom_param nd_geom_param 9 is zero The number of real parameters nd_geom_param is equal to 5N 3M 12 These parameters include five reals for each trapezoid two reals for the location of the stack of trapezoids three reals for each layer beneath the bridge and four reals for the refractive indices of the substrate and cover material In particular cf Figure 32 if k is a positive integer less or equal to N then the parameter d_geom_param 5k 4 h gt 0 is the height of 24 Tf the indexing should be confusing th
127. ations Member of the Kluwer Academic Publishing Group Dodrecht Boston Lancaster Tokyo 1988 J Nocedal and S J Wright Numerical optimization Springer Verlag New York Berlin Heidelberg Springer Series in Operation Research 2000 conjugate gradient method E M Dr ge R M Al Assaad and D M Byrne Mathematical analysis of inverse scat terometry Proc of SPIE 4689 pp 151 162 2002 Newton type method 11 The graphical user interface program DIPOG 2 1 GUI The graphical user interface program DIPOG 2 1 GUI can be called e g from the directory GUI Using this all the executables of the directories CLASSICAL CONICAL and RESULTS can be invoked All necessary information is provided by the interface We emphasize however the following Via the interface program the user can read a data file name dat cf Sects 5 1 and 7 and change its input data slightly In any case a new data file of the same type name dat but with different name will be produced storing the actual input data Of course the com putational results are written on the screen and into a result file name res cf Sect 5 2 If the input data is long then the original programs without graphical user interface are to be preferred since lots of data are easier to handle with data files and editor Therefore the input of DIPOG 2 1 GUI is restricted No more than nine upper and lower additional layers are admitted cf Sect 3 2 The grating par
128. ber of non stop layers beneath stack so far no non stop layers are allowed 2 if height of additional layer 0 i_geom_param 3 number of trapezoids in stack i_geom_param 4 number of non stop layers beneath stack i_geom_param 5 index of trapezoid in stack through which the upper line of the extra layer beside the stack goes 1 lt i_geom_param 5 lt i_geom_param 3 Number of real parameters nd_geom_param 5 i_geom_param 3 3 i_geom_param 4 12 Refractive indices 173 HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHH RH Razz nel nlk N 1 el a sS SsSsSSs nel n_1 4 4 Hl ae ww a oe a a a ee a oe a a a a a a a a a a a a a i a a a a a nl_2 ee we a a a a a a a ee ee a a a a a a a a a a a a a a a a a a a a i ii ii i i nl_3 ee a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a ee i i n_su d_geom_param nd_geom_param 6 ixd_geom_param nd_geom_param 5 refractive index of cover material Re n_co gt 0 Im n_co gt 0 d_geom_param 5 k 1 i d_geom_param 5 k refractive index of k th trapezoid in stack Re n_k gt O Im n_k gt 0 k 1 i_geom_param 3 d_geom_param nd_geom_param 8 i d_geom_param nd_geom_param 7 refractive index of extra layer material Re nel gt O Im nel gt 0 d_geom_param 5 i_geom_param 3 1 3 k ixd_geom_param 5 i_geom_param 3 2 3 k
129. between the implemented sub classes of gratings If the real parameter is the real resp imaginary part of a refractive index then its value can be fixed by an input string like Re Cr or Im Ag The same string must be the input for the corresponding lower and upper bounds cf Userguide If a real parameter depends on other free parameters then this dependency can be indicated as follows Write Dep on the input place for d_geom_param i and du_geom_param i On the input place of dl_geom_param i write Dep followed by the dependency in c code For instance the simple dependency expression d_geom_param i d_geom_param 11 d_geom_param 7 is indicated by the line cf Userguide Dep p11 p7 If possible try to present the dependency in the form Dep f1 2 3 with f1 f2 and f3 as terms including the variables d_geom_param i constants and the operations and but without any blank and bracket SETHE EEE EE HEHEHEHE a3 HHH HH HH HH H HH H OF CLASS 1 i_geom_param 1 1 polygonal function profile Polygonal grating defined by profile function which is piecewise linear between the knots over a uniform partition of the interval 0 Period Number of string parameters ns_geom_param 0 Number of integer parameters ni_geom_param 3 i_geom_param 1 1 i_geom_param 2 2 i_geom_param 3 number of interior knots in the uniform partition of 0 Period Number of
130. bination of two inputs of the types A 120 L 30 R 110 and D 0 4 is accepted However the choice A 120 and D 0 2 is ambiguous By definition it fixes an echelle grating with right blaze angle larger than the left To get the flipped grating with left blaze angle larger than the right the input should be A 120 and D 0 2 Figure 6 corresponds to a call of GEN_ ECHELLE with the parameter arguments A 90 D 0 3 0 03 0 04 If an input file for a trapezoidal grating is needed isosceles trapezoid with the basis parallel to the direction of the periodicity cf Figure s then this can be accomplished by calling the executable GEN_TRAPEZOID from the subdirectory GEOMETRIES More precisely the command GEN_TRAPEZOID name 60 0 6 3 0 2 0 1 0 1 0 05 creates the file name inp of the desired trapezoidal profile grating with angle 60 angle enclosed by basis and the sides with a basis of length 0 6 times period of the grating consisting of 3 material layers of heights 0 2 times period 0 1 times period and 0 1 times period respectively and with a coated layer of height 0 05 times period greater or equal to zero If an input file for a grating with several trapezoids one beside the other is needed sym metric trapezoids with the basis parallel to the direction of the periodicity then this can be accomplished by calling the executable GEN_MTRAPEZOID from the subdirectory GEOMETRIES 2
131. bstrate material must be the same as that of the grating material with index n Adding the widths of the upper and lower coated layers the user must not forget about the rectangular strips included already in the grating structure The width of the additional rectangular strips adjacent to the upper and lower boundary lines can be chosen automatically by adding the width input zero in the namel inp file More precisely adding a zero for the width the width is set to min 0 05 upper bound of meshsize x period 3 10 which approximates zero for the meshsize tending to zero The new rectangular strips are borrowed from the adjacent coated layer resp from the substrate or cover material i e the widths of the adjacent coated layers are reduced by the width of the strip In other words the automatic choice of the widths of the additional strips requires that the expression in 3 10 is less than the widths of the adjacent coated layers Now suppose that the width of the additional strips is chosen automatically and that there exist upper coated layers above the grating geometry fixed by the namel inp file The natural starting point of the additional upper strip is the point of the grating geometry with the highest y coordinate which belongs to an area occupied by a material different from that of the adjacent upper coated layer For technical reasons the grating geometry without the additionally added layers must not contain a stri
132. by setting upper and lower bounds to equal values This way any two dimensional section of the graph of the objective function and the gradient field can be displayed 10 1 3 The input file for the optimization An input file name dat cf the enclosed data file in Section 12 8 in the subdirectory OPTIM is needed To get this change the directory to OPTIM copy one of the existing files with tag dat e g the file example dat and call it name dat cd DIPOG 2 1 CLASSICAL 77 cp example dat name2 dat Change name dat in the editor according to your requirements You will find the nec essary information as comments in the file name dat Indeed each line beginning with 4 is a comment Comment lines can be added and deleted without any trouble The first input of name dat is the name name of the result file name2 res with out the tag res which will be written into the subdirectory RESULTS This is done by the lines Name of output file name3 These lines are followed by several inputs concerning the grating data which are similar to those in the input files for the conical diffraction cf the file of Section and compare Section 6 Of course the geometry description is different since now the geometry of the grating is to be optimized The geometry is described e g by the lines Number nd_geom_param of real parameters 6 Lower bounds dl geom param 0 5 0
133. ccomplished by calling the executable GEN_ECHELLEB from the subdirectory GEOMETRIES More precisely the command GEN_ECHELLEB name 60 0 05 creates the file name inp of the desired echelle profile grating with angle 60 angle enclosed by hypotenuse and by the leg parallel to the period and with a coated layer of height 0 05 times period of the grating measured in direction perpendicular to echelle profile height greater or equal to zero If an input file for a general echelle grating is needed then this can be accomplished by calling the executable GEN_ECHELLE from the subdirectory GEOMETRIES More precisely the command GEN_ECHELLE name A 120 L 30 0 05 0 1 creates the file name inp of the desired echelle profile grating with an apex angle of 120 with a left blaze angle of 30 with a coated layer over the left blaze side of height 0 05 times length of period of the grating measured in direction perpendicular to echelle profile height greater or equal to zero and with a coated layer over the right blaze side of height 0 1 times length of period of the grating measured in direction perpendicular to echelle profile height greater or equal to zero must be zero if previous height is zero Instead of the two inputs A 120 and L 30 one can choose also the inputs R 110 for a right blaze angle of 110 or D 0 4 for a depth of the grating equal to 0 4 times length of period of the grating Any com
134. computations using high discretization levels the algorithm may diverge However if the initial solution is sufficiently close to a local minimum then convergence is guaranteed 109 10 3 7 Levenberg Marquardt method Suppose the objective functional is the least squares sum of deviation terms for efficiencies and phase shifts cf 10 2 Moreover suppose the optimization problem is to find a local minimum i e to find an admissible vector fop in RY such that f rop lt f r holds at least for any admissible r R close to Top Here a vector r RY is called admissible if the coordinates r of r satisfy the box constraints l lt r lt u The objective funvtional f takes the form f r llelz e e amp r where maps the admissible parameter sets r into the space R with M gt N where c R and where is the Euclidean norm Clearly the components of r are just the scaled efficiencies or phase shift values for the grating determined by the parameter set r and the c is the vector of the scaled prescribed values in 10 2 The scaling factors are the square roots of the positive weights in 10 2 The Levenberg Marquardt method ind_opt 6 can easily be applied to find a numerical solution The method requires no integer input values ni opt 0 The number of real parameters is nd_opt 4 These parameters are d_opt 1 Factor u gt 0 for initial v
135. computed with an estimated error less than 1 per cent Number 1 A E HE T E AE AE E A E A E E A E E A A H E AE E A AE E PE AE TE E TE TE E AE E A AE TE E H EE E E A DE E E BE RE HEE HE HEHE EE HE HE AE HE E TETEE EREHE End AA TE TERED BEER ETT E E E A TE TETE AE AE TE EE TE TE HE AE TE H EE TE E AE AE AE AEE E PEE AE EEE E EHE HE HE HE EE E HEBE E HEEE Ae 12 5 Output file example res of FEM FULLINFO in CLASSI CAL 9K 2 gg K 2 2K K FK FK FK FK 3K K K K K K FK FK FK FK K FK K FK K K K K K FK 2K FK FK K K K OK 2 2g K K 2 2K 2g FK FK FK FK 3K K K K K FK FK FK FK FK 2K FK K K K K 2K K FK FK 2K FK FK K 2K K x DPOGTR x FKK K K K K 2K 2g FK FK FK FK FK K K K 2K 2 FK FK FK 2K K FK FK FK K K 2K K FK FK 2K FK FK K K K 2K 2 2g K K 2 2K 2g FK FK FK FK FK K K K K 2K FK FK FK FK K FK K FK K K K K FK FK 2K 2K 2K FK K K date 10 Feb 2003 09 55 17 Program solves Helmholtz equation for optical grating and TE TM polarization boundary conditions periodic with respect to x non local condition on x y y y_max and x y y y_min 142 method FEM partitioning based on Shewchuk s Triangle matrix assembly pdelib solver pardiso code generated with DCONV Name of input file without extension example Comments This is a fantasy grid for the test of gen_polyx Number of materials 4 Minimal angle of subdivision triangles 20 000000 Upper bound for mesh size 0 500000 Width
136. cond the real part of the corresponding optical index third the imaginary part of the corresponding index Optical index 1 0 i 0 FERRET PETE HARUAR PEPER PTET TET EA ARRA RAHENA PETER TEE HE ETE REEERE AHHH H Optical indices of the materials of the upper coatings H HOH HHHHH HHH HH HH HH OH OF Needed only if N_co_ov gt 0 Else no number no line For a multilayer system the indices of the n3 layers in the groups must be given only once I e for a multilayer system ni n3 n4 input lines are needed Optical indices 116 1 1 i 0 1 2 i 0 FERRET TITTIES TTT eee EE TEEPE PEEL TEE HE EAE ERE PEPER E HHHH Optical indices of the materials of the lower coatings Needed only if N_co_be gt 0 Else no number no line Optical indices 253 i 0 2 2 i 0 2 1 1 0 PHR RRRPPHHHHGHHRAHH PPHAH ARERR AARAA EEE EEA EEEE EEE HEHH Optical index of substrate material 2 0 i 0 HEHEHE HAAR AR RAH RRR aaa Angle of incident wave in degrees theta Either add a single value e g 45 angle 45 i 2 less or equal to 56 Note that either the wave length or the angle of incident wave must be single valued Angle of incident wave 65 HEHEHE AAAA AANA AARRE E E RRR HAAS Type of polarization Either TE TM or TE TM Type T M TEEPE EE ITE TETETETETE TERETE PE PTETE TE AHRR ERRAR RARAN EEE TEEPE PETE RE TEE HE ALT TE ER PETER EHHH H Length factor of ad
137. convenient is probably the following variant of external real parame ters using the angles and side lengths of the trapezoids The program recognizes that the parameter set is of this second type if there is an angle greater than one degree on a place where the input value of the internal variable must be a ratio less than one If the input is of the external type then the data is transformed into the internal parameter type The optimization is performed and the final result is transformed back to the external type Let us define the external type of parameter input For a positive k lt N the parameter d_geom_param 5k 4 h gt 0 is as before the height of the Ath trapezoid measured in um The parameter d_geom_param 5k 3 b a is the length of the upper side of the kth trapezoid measured in um The real d_geom_param 5k 2 is the interior angle a in degrees at the right lower corner of the kth trapezoid The last parameters of the kth trapezoid form the refractive index ng of the material i e the refractive index is n d_geom_param 5k 1 i d_geom_param 5kj The next parameter d_geom_param 5N 1 ap is the x coordinate in um of the left lower corner of the first trapezoid The next real parameter d_geom_param 5N 2 bo is the x coordinate in um of the right lower corner of the first trapezoid Finally the refractive indices of the cover and substrate material are given as in the internal parameter setting In the case of a switch
138. d d_geom_param k k nd_geom_param 6 nd_geom_param i e refractive index of cover and substrate material and the constants of the penalty terms if d_geom_param nd_geom_param 9 hel h_m 0 then d_geom_param nd_geom_param 9 i e degenerated height hel d_geom_param 5 k 4 k 1 2 i_geom_param 5 1 176 i e heights of trapezoids beneath upper boundary line of extra layer H if d_geom_param nd_geom_param 9 hel h_m 1 then d_geom_param nd_geom_param 9 i e degenerated height hel d_geom_param 5 k 4 k 1 2 i_geom_param 5 i e heights of trapezoids beneath upper H boundary line of extra layer Fe CLASS 6 i_geom_param 1 6 EUV bridge with environment Stack of several trapezoids bridge in grating with sidewall angles less or equal to 90 degrees and with refractive indices included into the set of optimization parameters Non stop layers of different heights beneath the stack with refractive indices included into the set of optimization parameters In other words Some of the lower additional layers can be added through the set of optimization parameters Fixed further layers can be added in the GRATING ILLUMINATION part of this input file Extra layer beside stack height gt zero If upper line of this layers contains a corner of the trapezoids in the stack then the height of the extra layer and all trapezoid heights of trapezoid beneath this line must be fixed by setting upp
139. d be a function of the prescribed efficiency value e g w_qua_1_tr 1 utu u f c_qua_1_tr f E sqrt E E 1le 2 Then the input is as follows n_qua_i_tr 2 w_qua_1_tr unc fct sqrt E E 1e 2 QO 1 o_qua_1_tr 1 0 c_qua_i_re 10 13 7 HHHHHHHHHHH HHH HHH HHH HOH OF Note that the two input values for the w_qua_1_tr 193 are dummy values The final values will be computed as w_qua_i_tr 1 u u u sqrt c_qua_1_re c_qua_1_re le 2 with c_qua_1_re 10 or c_qua_i_re 13 if the dummy input for w_qua_1_tr is positive and it will be set to zero if w_qua_i_tr is less or equal to zero HHH HH HH OF HHHHHHHHHHHHHHHHHHHHHHHH LINEAR TERMS REFLECTED EFFICIENCY w_ene_lin_re 0 w_ene_lin_tr 0 w_ene_lin_to 0 n_lin_re 0 w_lin_re n_lin_re numbers in n_lin_re lines o_lin_re n_lin_re numbers in n_lin_re lines LINEAR TERMS FIRST TE or S REFLECTED EFFICIENCY n_lin_i_re 0 w_lin_1_re n_lin_1_re numbers in n_lin_1_re lines o_lin_i_re n_lin_1_re numbers in n_lin_1_re lines LINEAR TERMS SECOND TM or P REFLECTED EFFICIENCY n_lin_2_re 0 w_lin_2_re n_lin_2_re numbers in n_lin_2_re lines o_lin_2_re n_lin_2_re numbers in n_lin_2_re lines LINEAR TERMS TRANSMITTED EFFICIENCY n_lin_tr 0 w_lin_tr n_lin_tr numbers in n_lin_tr lines o_lin_tr n_lin_tr numbers in n_lin_tr lines LINEAR TERMS FIRST TE or S TRANSMITTED EFFICIENCY n_lin_1i_tr
140. d_geom_param 2 minimal angle for sidewall angles of trapezoids except the uppermost trapezoid phi_max d_geom_param nd_geom_param 1 maximal angle for sidewall angles of trapezoids except the uppermost trapezoid phi_fac d_geom_param nd_geom_param factor of penalty term In other words to exclude solutions with too large or too small sidewall angles phi a penalty term of the following form is added to the objective functional for each sidewall angle phi l 2 phi_fac lt max 0 phi phi_max l 2 max O phi_min phi gt Following parameters must be fixed by setting upper bound lower bound d_geom_param k k nd_geom_param 6 nd_geom_param i e refractive index of cover and substrate material and the constants of the penalty terms if d_geom_param nd_geom_param 9 hel h_m 0 then d_geom_param nd_geom_param 9 i e degenerated height hel d_geom_param 5 k 4 k 1 2 i_geom_param 5 1 i e heights of trapezoids beneath upper boundary line of extra layer if d_geom_param nd_geom_param 9 hel h_m 1 then d_geom_param nd_geom_param 9 i e degenerated height hel 181 H H HH d_geom_param 5 k 4 k 1 2 i_geom_param 5 i e heights of trapezoids beneath upper boundary line of extra layer HAEE HEHE AE HE HE AE HEHE EEHEEHE E Number nd_geom_param of real parameters 6 Lower bounds dl_geom_param nd_geom_param numbers in nd_geom_param lines 0 5 0 5 20
141. ded Next the indices of the materials of lower coated layers and that of the substrate follow The indices of the materials in the area between upper and lower coatings resp between cover and substrate material if no rectangular coatings exist are the last refractive indices of the 6 N 59 input files These indices are listed from above to below if possible In some cases the ordering is indicated in the description of the geometrical part or the indices have to be in accordance with the numbering of the material parts in the file name inp In any case the first index of the grating materials is to be the same as that of the adjacent last upper coating layer resp of the cover if there does not exist any rectangular upper coating and the last index of the grating materials is to be the same as that of the adjacent first lower coating layer resp of the substrate if there does not exist any rectangular lower coating The input of refractive indices can be checked using the executables FEM_CHECK resp GFEM CHECK cf Sect 5 3 5 Computation of Efficiencies Using FEM in CLAS SICAL 5 1 How to get an input file name2 dat First an input file name2 dat cf the enclosed data file in 12 2 in the subdirectory CLASSICAL is needed To get this change the directory to CLASSICAL copy one of the existing files with tag dat e g the file example dat and call it name2 dat cd DIPOG
142. dicates a GRATING DETERMINED BY SMOOTH PARAMETRIC CURVES WITH PARAM ETERS i e grating determined by n non intersecting and periodic profile lines given from above to below as period f j fy 7 O lt t lt 1 j 1 n where n and the functions t gt f j t and t gt f j t are defined by the c code included in the file GEOMETRIES profiles_par c The last code uses 1 integer parameter and 2 real parameters named IPARaM1 RPARaM1 and RPARaM2 The integer parameter takes the value 3 following the first line of the calling sequence and the real parameters take the values 0 5 and 1 following the integer parameter values cf Figure 16 where m 2 JA ht i fy 1 t sin 27t fy 2 t sin 2rt 0 5 and fy 3 t sin 2rt 0 5 0 50 parame ter 3 is the number n of boundary and interface curves parameter 0 5 is the width of the first layer and parameter 0 50 that of the second Note that any number of parameters is possible for a corresponding file GEOMETRIES profiles par c If the program does not find the file GEOMETRIES profiles_par c then it takes the file profiles_par c of the current working directory Grating data pin indicates a PIN GRATING DETERMINED By PARAMETRIC CuRVE cf Figure 17 where Emin 0 2 falt fmin 1 22min t and f t 0 5 sin at ie over a flat grating 42 Figure 15 Grating determined by a simple smooth parametric curve with surface
143. dices of the grating must be that of the superstrate resp that of the adjacent upper layer If the width of the additional layers is positive then the last of the subsequent refractive indices of the grating must be that of AT Figure 20 Box grating with two curves the substrate resp that of the adjacent lower layer Grating data rough_mls k name indicates a MULTILAYER SYSTEM WITH ROUGH INTERFACES k TIMES This is just like above However k random realizations are computed and the output values are replaced by the mean values over the k computations Standard deviations are added too 48 Figure 21 Rough multi layer grating 49 3 6 Stack grating by code words For the stack grating there appear the following code words in the geometry input of the name dat file Grating data stack k line lines lines Here k is the number of profile curves in the stack These profile curves are defined by the following 2k code word lines Each profile curve is represented by two of the lines They are listed from above to below No intersection points of these curves are allowed With the exception of pin curves the j th curve j 1 k takes the form f 7 t fy 7 0 lt t lt 1 with the first end point such that f j 0 0 with the second end point such that fe j 1 1 and with 0 lt fe j t lt Lier 0 lt lt 1 If the defining two lines are linez2j 1 echellea R d lines hy the
144. dinate 1 same y coordinate for first and last point all x coordinates between 0 and 1 at least two different y coordinates last line should be End polygon2 filel file2 COATED GRATING DETERMINED BY POLYGONAL LINES i e grating profile line is defined by the data in the file with name GEOMETRIES file1 in GEOMETRIES filel in each line beginning without there should be the x and y coordinate of one of the consecutive corner points first point with x coordinate 0 last point with x coordinate 1 same y coordinate for first and last point all x coordinates between 0 and 1 at least two different y coordinates last line should be End and the coated layer is enclosed between the polygonal line of GEOMETRIES file1 and the polygonal line of the file with name GEOMETRIES file2 in GEOMETRIES file2 0 OF 0 0 O 0 0 120 HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHH RR aa in each line beginning without there should be the x and y coordinate of one of the consecutive corner points first and last point must be corner of first polygon second polygon must be on left hand side of first one to one correspondence of the corners on the two polygons between first and last point of second polygon quadrilateral between corresponding segments on the left of first polygon these quadrilaterals must be disjoint las
145. ditional shift of grating geometry This is shift into the x direction 4 This is length of shift relative to period Q Stretching factor for grating in y direction 1 Length of additional shift of grating in micro meter This is shift in y direction 0 Period of grating in micro meter E Grating data namel Here namel refers either to a file namel inp with geometrical data located in the subdirectory GEOMETRIES or to some special code words to fix the geometry of the grating We describe how to get the file namel inp in point 3 2 and the alternative code words in the subsequent point 3 5 of this section 23 As mentioned above the computation starts in the directory CLASSICAL and is based upon a geometry input file GEOMETRIES namel inp indicated in name dat However if the code is started from a directory different from CLASSICAL or if the geometry data file is located in a different directory with the path path1 then the file is to be specified by adding its path in name dat as Grating data path1 namel In particular for a geometry input file in the current working directory use Grating data namel Note that the geometry data in namel inp should be given relative to the period which is specified in the data file name dat of directory CLASSICAL resp CONICAL All data of namel inp will later be multiplied by the given leng
146. ditional shift of grating geometry This is shift into the x direction i e the direction of the period to the right This is length of shift relative to period i e the grating structure given by subsequent input will be shifted by factor times the period given in subsequent input However only the Rayleigh numbers and efficiencies will be computed according to the shift The field vectors in the plots are drawn without shift and the graphics of the executable with tag _CHECK is drawn without shift Must be a real number between O and 1 Either add more values by e g w N 5 63 64 65 69 TOs As The last means that computation is to be done for the angles from the Vector of length 5 Geay Bat EEA nega and TAO Or add e g I 45 56 2 The last means that computation is to be done for the angles 45 i 2 with i 0 1 2 and with i HHH HHH HHH H OF 117 Length 0 FERRET TT RARUA ANNRAAAAA HRR RRRRR RARAN HHHRR ERRAN R HARRARI E HHHH Stretching factor for grating in y direction Must be a positive real number Length 1 FERRET TET RAAUA AENAAAAAA HRR AR EEE EEE TEETER TEE EEE ERE EEE PE H Length of additional shift of grating geometry in micro m This is shift into the y direction i e the direction perpendicular to the grating surface pointing into the cover material Must be a real number Length 0 HEHEHE
147. e geometry input files of the form described in Section The upper bound for the mesh size is fixed by the input lines 86 Upper bound for mesh size 0 5 contained in the generated input files However this automatically chosen value 0 5 can be changed by the user setting the environment variable BND_MESH_SIZE to the desired value before calling the optimization routines 10 2 2 Profile grating determined by a polygonal profile function The first class i_geom_param 1 1 is a profile grating without coatings defined by a profile function which is the cross section of the interface between cover and substrate material Thus the number of materials is i geom_param 2 2 The profile is supposed to be the graph of a piecewise linear function The class has no name parameters ns_geom_param 0 and ni_geom_param 3 integer parameters The integer value igeom_param 3 is the number N of knots of the profile curve in the interior of the period i e with x coordinate strictly between zero and period d Hence the grating profile is the polygonal curve connecting the points 0 0 h y1 2h y2 peau Nh yn d 0 where h d N 1 The number of real input parameters nd_geom_param is equal to i_geom_param 3 4 and the first igeom_param 3 parameters define the profile curve by y d_geom_param j j 1 igeom_param 3 The last four parameters describe the materials More precisely the refractive indices of the cover and substrate materia
148. e interpolation is taken over uniform grid with n_DOF 2 interpolation knots Value should satisfy 0 lt n_DOF lt 100 value 3 HEHEHE RRR ARE E EEEE EEEE n_LFEM Approximate solution determined by FEM over subdivision triangle where additional uniform FEM partition on each small triangle is chosen such that the step size is side length divided by n_LFEM 1 If n_LFEM 1 n_DOF 0 conventional FEM method If n_DOF n_LFEM conventional FEM method with elimination of interior nodes of grid triangle i e real mesh size is mesh size shown in result file divided by n_DOF 1 If n_DOF lt n_LFEM method resembles p method or PUM Value should satisfy 1 lt n_LFEM lt 2048 and n_LFEM 1 must be a multiple of n_DOF 1 value 63 TEPER PETE TEE TETETET PETE APAA TELE EEE ER BBB Be PETE ETE TEEPE PEPE EE EEE n_UPA This is for additional uniform partition of all primary grid triangles into n_UPA n_UPA equal subdomains i e original side of grid triangle is split into n_UPA sides of uniform partition subtriangles HHH HH HH HH HH HF OF Value should satisfy 1 lt n_UPA lt 128 value al TERETE PETRIE PETE EEE EB EE EEH that s it HEHEHE RRR A aE RRHH ARRETE 128 12 4 Data file example dat for CONICAL makefile HEHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEHHHHHHHHEHHH HEHEHE HHH HEHHHHHHHHAHAAAAH example dat HEHHHHHHHHHHEAHAH a
149. e of stack bridge is fixed by two parameters param_1 posb d ratio of distance of right lower corner from the left boundary line of the period HHHHHHHHHHHHHHHHHHH HHH HH OH 172 HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHH RH HH Oz and period d param_2 posa posb ratio of distance of left lower corner from the left boundary line of the period and distance of right lower corner from the left boundary line of the period Each trapezoid is determined by its hight by the lower side which is the upper side of the adjacent lower trapezoid and by the upper side prescribed by the two parameters param_k_1 b_k d ratio of distance of right upper corner from the left boundary line of the period and period d param_k_2 a_k b_k ratio of distance of left upper corner from the left boundary line of the period and distance of right upper corner from the left boundary line of the period Refractive index of the material of each trapezoid and layer is prescribed as an optimization parameter Number of string parameters ns_geom_param 0 Number of integer parameters ni_geom_param 5 i_geom_param 1 6 indicator of EUV bridge i_geom_param 2 number of different materials number of trapezoids in stack bridge number of non stop layers beneath stack so far no non stop layers are allowed 3 if height of additional layer gt 0 i_geom_param 2 number of different materials number of trapezoids in stack bridge num
150. e right of the j 1 th pin curve No intersections of pin and profile curves are allowed If the defining two lines are lineg 1 pin ccode line Pi pe then we get a SIMPLE SMOOTH PIN CURVE defined by f j t t and f j t ccode The parameter arguments p and p gt of the connection points to the ordinary profile curve are fixed by the second line The remarks on the profile curves apply also here If the defining two lines are 52 lineg _1 pin ccode ccodez line p pe then we get a SIMPLE SMOOTH PARAMETRIC PIN CURVE defined by f j t ccode and f j t ccodez The parameter arguments p and p gt of the connection points to the ordinary profile curve are fixed by the second line The remarks on the profile curves apply also here If the defining two lines are lineg 1 pinz ccode line pi Po dy dy Pere d then we get a SIMPLE NON SMOOTH PIN CURVE defined by f j t t and fy j t ccode The curve has i 1 lt i lt 9 corners with the parameter arguments d d2 d such that 0 lt d lt dy lt lt di The remarks on the profile curves apply also here If the defining two lines are line 1 pini ccode ccodez lines pi pod do d then we get a SIMPLE NON SMOOTH PARAMETRIC PIN CURVE defined by f j t ccode and f j t ccodez The curve has i 1 lt i lt 9 corners with the parameter arguments d dz di such that 0 lt dy lt d2 lt lt di The remarks on
151. e word BOX is a factor for the mesh size of the FEM discretization of the box The box is divided by 2 curves into 3 different material parts The curves are given by the c code in the following 2 times two lines The first code line is the z component of the first curve the second the y component of the first curve the third 45 Figure 18 Coated pin grating determined by two parametric curves the x component of the second curve etc The last three input lines define material points Material with index one is located in that part of the box which is separated by the above curves and contains the point 1 0 8 Material with index two is located in that part of the box which is separated by the above curves and contains the point 1 0 etc Note that the curves must be simple not self intersecting and they must be contained in the box The only allowed intersection points of two different curves are the end points of the curves Finally the material areas separated by the curves in the box must be such that the area with the first index contains a whole strip under the upper boundary side of the box and that the area with the last index contains a whole strip over the lower boundary side If these conditions are violated then the computational result will be complete nonsense Grating data rough_mls name indicates a MULTILAYER SYSTEM WITH ROUGH INTERFACES cf 21 1 nia layers followed by Nmia times nia layers and follo
152. eath and above this rectangular structure there might be additional coated layers of rectangular shape These kind of layers are called coating layers over the grating and coating layers beneath the grating respectively Alternatively a multilayer system input format is possible E g the input MLS ni n2 n3 n4 with ni n2 n3 n4 replaced by non negative integers means N_co_ov nitn2 n3 n4 layers with n1 layers above n2 groups of n3 layers with same widths and materials in the middle and with 7 layers below HHHH HHHH HEHH HHH H HHHH 129 Number 2 HEEHHHAHEHHHHAAAHHHAER RHEE AAEHRA ERR HR RARER RARER RARER Widths of coating layers in micro meter N_co_ov entries Needed only if N_co_ov gt 0 Else no entry and no line For a multilayer system the widths of the n3 layers in the groups must be given only once I e for a multilayer system ni n3 n4 input numbers are needed Widths 0 05 0 03 HEHEHE RRR RR aa RRR aaa Number of coating layers beneath the grating N_co_be 1 HEHEHE AAEN HAARR ARREA EEHEEHE Widths of coating layers in micro meter N_co_be entries Needed only if N_co_be gt 0 H Else no entry and no line Widths 0 05 EERE PEPE ETE EER RNANA PHPP EEEREN RRRA EE Wave length in micro meter lambda Either add a single value e g 63 Either add more values by e g H oy 5 63 64 i 65 69 sfQ 2 The last means that compu
153. econd polygon quadrilateral domain between corresponding segments on the left of first polygon these quadrilaterals must be disjoint last line should be End The z and y coordinates of the points in GEOMETRIES file1 and GEOMETRIES file2 are supposed to be scaled such that the period is one If the program does not find the files GEOMETRIES filel and GEOMETRIES file2 then it takes the files filel and file2 of the current working directory Grating data profile indicates a GRATING DETERMINED BY A SMOOTH PARAMETRIC CURVE i e grating de termined by profile line given as period falt fy t 0 lt t lt 1 where the functions tr f t andt gt f t are defined by the c code of the file GEOMETRIES profile c cf Fig ure 13 where f t t and f t 1 5 0 2 exp sin 6zt 0 3 exp sin 8zt 27 If the program does not find the file GEOMETRIES profile c then it takes the file profile c of the current working directory Grating data profile_par 2 3 15To make it precise suppose P and Q are the common corner points of the two polygonal curves and that Rt R Ri are the consecutive corner points between P and Q on the first polygonal line and R R3 R those on the second polygonal Then the code requires m n and that the coating area between the two polygonal lines is the disjoint union of the triangle PR R 7 the quadrilaterals RIR R R s
154. ements for the solver MIN_ANG_TRI If this is set to a positive number and if no geometry input file name inp is used then the lower bound for the minimal angle of the triangles of the FEM partition is set to this number NMB_OF_DATA Performing an optimization the objective functional is allowed to de pend on 999 values of efficiencies phase shifts energies If this number is not sufficient then the user can enlarge it setting the environment variable to the required number cf Section 10 1 3 STE_DIF_FOR If this is set to a positive number then the standard step size 2 1077 in the difference formula for derivatives w r t the parameters of the multi layer system is changed to this value cf Section 10 2 7 TMPDIR If this is set to the name of an existing directory then the temporary di rectory used for auxiliary files in the programs is changed from the directory prescribed during installation to the new one with the given name WI_GRA_LAY Suppose an optimization of an EUV bridge without the sidewall angle restriction i e with EUV_SWA_90 no is required where the bridge is located strictly in the middle of a period and where no additional layer is generated beside the bridge If the electro magnetic field changes fastly in the vicinity of the interface between cover material and grating then a grading of the FEM grid towards this interface can improve the approximation essentially This grading can be enforced by
155. en the resulting distribution of the materials can be checked entering OPTIMIZE i name dat With this result a picture like the right in the Figure BI appears 95 d h h d h d 0 Figure 32 Bridge composed of trapezoids under light in the EUV range the kth trapezoid measured in um The parameter d_geom_param 5k 3 b d 0 1 is the ratio of the x coordinate of the right upper corner of the kth trapezoid over the period d of the grating The real d_geom_param 5k 2 a b 0 1 is the ratio of the x coordinate of the left upper corner of the kth trapezoid over the x coordinate of the right upper corner The last parameters of the kth trapezoid form the refractive index nz of the material i e the refractive index is n d_geom_param 5k 1 i d_geom_param 5kj The next parameter d_ geom_param 5N 1 is the ratio bo d 0 1 of the x coordinate of the right lower corner of the first trapezoid over the period d of the grating The next real parameter d_geom_param 5N 2 ao bo 0 1 is the ratio of the x coordinate of the left lower corner of the first trapezoid over the x coordinate of the right lower corner If M 0 then there follow three reals for each lower layer included into the optimization part In particular if k is a positive integer less or equal to M then the parameter h p d_geom_param 5 N 3 3k is the height of the Ath layer in um The refractive index of the corresponding material i
156. en we get a profile curve DETERMINED By A NON SMOOTH SIMPLE PARAMETRIC CURVE defined by f j t ccode and f j t ccode2 The curve has i 1 lt i lt 9 corners with the parameter arguments d1 d2 d such that 0 lt d lt d2 lt lt di The profile is vertically shifted by h um The same remarks as in the previous mode apply Beside the above profile curves pin curves are possible Then the meaning of f and fy is changed The curve t f j t fy j t with 0 lt t lt 1 connects the points fali 0 fy 7 0 0 0 and f2 j 1 fy j 1 1 0 The corresponding pin curve is just the affine image of the last curve connecting the two points fe j2 p1 fy J2 p1 and falj2 p2 fy J2 p2 of the profile curve with index ja I e te tp fo J t fe J2 P2 feliz P1 fy J t fyJ2 P2 fy G2 P1 I Fy Ja p fo 9 t fy J2 P2 Fy Ja P1 i CE E T Thus for each of the above profile curves we can define several areas with new material attached to it and bounded by the just mentioned pin curves These areas are listed immediately before the profile curve and in correspondence with their attachment from the right to the left In other words the index j of the profile curve to which the j th pin curve is attached to is the smallest integer l larger than j such that l is an ordinary profile curve and in case of two and more pins the j th pin curve is located to th
157. equirements a larger nurem is useful E g if npor 3 then nprem 31 is a good choice For npor 7 one should take e g nurem 127 and for npor 15 the value Nurem 511 However large nLrem will lead to long computation times at least if nypa is not large In case of large wave numbers long computation times cannot be avoided More hints on how to chose the right npor and nyrrmy will be given in the numerical tests in Section 9 If there is asystem of many layers above or below the grating structure the computation of the non local boundary condition may become slow On the other hand a computation based on piecewise linear approximation like for the FEM matrix might be unnecessary Note that each boundary segment is split into nifem 1 equal parts and the polynomial basis function is approximated by a piecewise linear interpolation over this uniform grid In order to reduce computation time the user can choose a smaller n_rgm for the discretization of the boundary condition To this end he has to set the environment variable BND_n_LFEM to the desired value BND_nyrry This value should be such that BND ntfFem 1 is a multiple of the number NDOF 1 and that NDOF lt BND_np rem lt NLFEM For the computation of the approximate trial functions of the generalized finite element method a finite element system of dimension nifem 1 NLFEm 2 is to be solved If the user sets the environment variable CHOOSE _PMETHOD to yes then the s
158. er bound equal to lower bound and refractive indices included into the set of optimization parameters Number of trapezoids and layers is prescribed Whole stack bridge in one period of the grating sidewall angles can be restricted by penalty terms Lower side of stack bridge is fixed by two parameters param_1 posb d ratio of distance of right lower corner from the left boundary line of the period and period d param_2 posa posb ratio of distance of left lower corner from the left boundary line of the period and distance of right lower corner from the left boundary line of the period Each trapezoid is determined by its hight by the lower side which is the upper side of the adjacent lower trapezoid and by the upper side prescribed by the two parameters param_k_1 b_k b_ k 1 ratio of distance of right upper corner from the left boundary line of the period and distance of right lower corner from the left boundary line of the period HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHH HH HH HH OH OF 177 HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHH RR azz b_0 posb param_k_2 a_k a_ k 1 b_k a_ k 1 ratio of difference of x coordinates of left upper and left lower trapezoid corner over difference of x coordinates of right upper and left lower trapezoid corner a_0 posa Refractive index of the material of each trapezoid and layer is prescribed as an optimization parameter Number of string parameters ns_geom_param
159. er of real parameters nd_geom_param 2 i_geom_param 3 4 m 1 i_geom_param 3 d_geom_param 2 m 1 d_geom_param 2 m corner point of profile curve in other words polygonal grating has i_geom_param 3 2 corners at HHHHHHHHHHHHHHHHH HH HH HHH OH the points 0 0 x_m y_m m 1 i_geom_param 3 Period 0 where 163 x_m d_geom_param 2 m 1 y_m d_geom_param 2 m k 2 i_geom_param 3 1 d_geom_param k real part of refractive index of cover material k 2 i_geom_param 3 2 d_geom_param k imaginary part of refractive index of cover material k 2 i_geom_param 3 3 d_geom_param k real part of refractive index of substrate material k 2 i_geom_param 3 4 d_geom_param k imaginary part of refractive index of substrate material Following parameters must be fixed by setting upper bound lower bound d_geom_param k k 2 i_geom_param 3 1 2 i_geom_param 3 4 Fa CLASS 3 i_geom_param 1 3 polygonal profile curve with proper constraints Polygonal grating defined by profile curve which is piecewise linear between the knots In contrast to CLASS 2 constraints expressed by no selfintersection cf the constraints below is included into the optimization For this class we have implemented only the conjugate gradient algorithm choose ind_opt 1 Number of string parameters ns_geom_param 0 Number of integer parameters ni_geom_param 3 i_geom_param 1 3 i
160. er of the refinement level and 0 05 89 Figure 28 Stack of trapezoids apk bpg 0 1 is the ratio of the x coordinate of the left upper corner of the kth trapezoid over the x coordinate of the right upper corner The last parameters of the kth trapezoid form the refractive index ng of the material i e the refractive index is ny d_geom_param 5k 1 i d_geom_param 5kj The next parameter d_geom_param 5N 1 is the ratio bo d 0 1 of the x coordinate of the right lower corner of the first trapezoid over the period d of the grating The next real parameter d_geom_param 5N 2 ao bo 0 1 is the ratio of the x coordinate of the left lower corner of the first trapezoid over the x coordinate of the right lower corner Finally the refractive indices of the cover and substrate material are given as Neo d_geom_param 5N 3 i d_geom_param 5N 4 Nsu d_geom_param 5N 5 i d_geom_param 5N 6 Setting upper bound equal to lower bound the user must fix these two refractive indices i e the equality du_geom_param j dl_geom_param j must be satisfied for 7 5N 3 5N 6 Clearly the real parts of all refractive indices must be positive and the imaginary parts non negative The imaginary part of the cover material must vanish 90 Note that the presented choice of real parameters has been adapted to the optimiza tion problem in order to avoid additional restrictions Let us call it the internal type of parameters More
161. er set r efficiencies ef phase shiftq p in degrees reflected energy et e7 in per cent transmitted energy e n total energy e et e7 Here n runs through the indices of reflected resp transmitted plane wave modes Note that all these efficiencies phase shifts and energies depend of course on Aj Ok and tm Moreover for the conical diffraction there exist two different efficiencies e and e and phase shifts p and p depending on the polarization splitting cf Section 2 3 The optimization problem f r inf r class 10 1 li lt ri lt ui 19 Tn the case of TE polarization the phase shift is computed by p arg A AF For TM polarization there holds p arg B B Finally in the case of conical diffraction the phase shift pbt is that of the corresponding field component presented in the efficiency e in accordance with the chosen output type cf Section 2 3 In particular for output type 3 Com first type in Section 2 3 and incidence angle 0 the phase shifts of the classical case are computed In the case of output type TE TM second type and incidence angle 0 the phase shifts computed by the conical case are either equal to those of the classical or they differ by an angle of 180 Equality corresponds to a vector s equal to the unit vector in the direction of the z coordinate and deviation by
162. es not stop earlier then the number of iterations is exactly equal to this number A reason to stop earlier is e g that an approximative local solution with a sufficient accuracy has been found before Another reason could be that the gradient computation is inaccurate and the objective functional does not drop in the direction of the negative gradient The next parameter ind_opt is the indicator of the numerical method of optimization and must be a number between 1 and 5 In particular the choice of ind_opt switches to ind_opt 1 Conjugate gradient method with projection ind_opt 2 Interior point method ind_opt 3 Method of augmented Lagrangian ind_opt 4 Simulated annealing ind_opt 5 Newton type method ind_opt 6 Levenberg Marquardt method The first three and the last method are gradient based local optimizers All of these four can deal with the box constraints imposed by the upper and lower bounds of the real parameters 98 However to additional constraints like that for the polygonal profile gratings cf Equation 10 7 in Section 10 2 4 only the implementation of the conjugate gradient method has been adapted Moreover the method of augmented Lagrangian requires the computation of the objective functional f r at parameter sets r outside the class of admissible solutions Since this is meaningful only for the gratings with profiles defined by the graph of a linear function the method of augmented Lagrangian appl
163. ey Rim Rm Rn Re and the triangle R1 QR2 m m m 1 39 Figure 12 Grating determined by polygonal lines 1 0 1 5 0 2 0 3 indicates a GRATING DETERMINED BY A SMOOTH PARAMETRIC CURVE WITH PA RAMETERS i e grating determined by profile line given as period f t f t 0 lt t lt 1 where the functions t gt f t and t gt f t are defined by the c code of the file GEOMETRIES profile_par c The last code uses 2 integer parameters and 3 real parameters named IPARaM1 IPARaM2 RPARaM1 RPARaM2 and RPARaM3 The integer param eters take the values 1 and 0 following the first line of the calling sequence and the real parameters take the values 0 15 1 and 0 following the integer parameter values cf Fig ure 13 where f t t and f t 1 5 0 2 exp sin 6rt 0 3 exp sin 8zt 27 parameter 1 is the index of the curve chosen from GEOMETRIES profile_par c parameter 0 is the number of corners of the curve parameters 1 5 0 2 and 0 3 are scaling param eters in the y coordinate of the curve Note that any number of parameters is possible for a corresponding file GEOMETRIES profile_par c If the program does not find the file GEOMETRIES profile_par c then it takes the file profile_par c of the current working di rectory Grating data profile 0 125 x sin 2 x M_PI x t 40 Figure 13 Grating determined by a smooth parametric curve indicates a GRATING DE
164. f a rectangular area parallel to the axes This inhomogeneous part is determined by a triangular grid and can have already a few layers of coatings involved Beneath and above this rectangular structure there might be additional coated layers of rectangular shape These kind of layers are called coatings over the grating and coatings beneath the grating respectively Alternatively a multilayer system input format is possible E g the input MLS ni n2 n3 n4 with n1 n2 n3 n4 replaced by non negative integers means N_co_ov ni n2 n3 n4 layers with ni layers above n2 groups of n3 layers with same widths and materials in the middle and with 7 layers below Number of coatings 2 HHHHHHHHHHHHHHHHHHHHRHHHHHHHHHHHHHRHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH Widths of coatings in micro m Needed only if N_co_ov gt 0 Else no number no line For a multilayer system the widths of the n3 layers in the groups must be given only once I e for a multilayer system ni n3 n4 input numbers are needed Widths 0 5 0 2 HEHEHE RRRHHAA HAAR AR Aa ARRAS Number of coatings beneath the grating N_co_be 3 HEHEHE EEEE EE E E E RRR HRS Widths of coatings in micro m Needed only if N_co_be gt 0 Else no number no line Widths 0 2 0 3 0 2 FERRETTI RARUA AEN AAAHHH RRRA AR RARAN EHHH TEE PETE TERE TEE HE LATE REEE RHH HH Wave length in micro m lambda Either add a s
165. f course the refractive index chosen for this material index in the data file name dat must be the same as for the material from which the layer is artificially separated The mesh generator resolves the tiny layer structure by small regular triangles and the size of the triangles close to the layer grows slowly with the distance to the layer Unfortunately the overall number of mesh points notably increases if the width of the additional layer decreases Nevertheless the approximation using a graded mesh is often better than that of a comparable refined uniform mesh Finally we should mention the different level refinement strategies for regular and graded meshes The standard way of mesh refinement is to increase the level in the input file name dat cf Subsection 5 2 If the level l is large such that the maximal meshsize ho 2 is less than the minimal distance of the additional points to the singular point and less than the width of the additional layer then the generated mesh is regular again In other words the degree of grading of the meshes is reduced by increasing the level of discretization Alternatively to increasing the level by one the parameter nypa of the control files generalised Dat and conical Dat cf Sections 6 and 7 can be doubled In this case the degree of grading of the meshes is maintained 3 4 Input file namel inp by TGUI The easiest way to create a geometry input file namel inp
166. f incident light in the plane perpendicular to the grooves of the periodic grating and the case of conical diffraction i e of oblique incidence of light The code is based on the package PDELIB which is a collection of software components to create simulators based on partial differential equations http www wias berlin de software pdelib DIPOG 2 1 provides a conventional FEM approach as well as a generalized FEM version called GFEM The latter is nothing else than the variational approach of the conventional FEM combined with a new trial space for the approximation of the unknown solution To compute follow the subsequent instructions The earlier version Dipog 1 5 does the same for the special case of binary lamellar gratings i e if the different grating material pieces are of rectangular shape with sides parallel to the axes Whenever the user is confronted with binary grating geometries he can use Dipog 1 5 or DIPOG 2 1 However he should prefer the more efficient Dipog 1 5 The fast generalized FEM used in Dipog 1 5 cannot be applied to general polygonal geometries Beside the simulation of the diffraction by gratings DIPOG 2 1 can optimize a few classes of grating structures Local gradient based optimization methods and the global simulated annealing method is used to determine optimal geometry parameters and optimal refrac tive indices respectively The algorithms work for classical and conical diffraction The geometry classes
167. f the actual iteration step and the value of the objective function at the actual iterative solution are printed on the screen If the last iteration step is finished there appears the minimal value of the objective functional and the corresponding set of optimal parameters on the screen Finally in the directory RESULTS a file name2 res cf the enclosed result file in Section is created which contains the input data of the optimization problem some data of the iteration process and the optimal solution The name name2 is to be fixed by the user in the input file name dat Clearly the optimization methods for are based on the same finite element dis cretization which is implemented for the simulation of diffraction cf Sections 2 and p7 Even the gradients of the objective functions can be computed using the same finite el ement matrices If the efficiencies and phase shifts are determined with a discretization error Eqis then the error of the objective functional evaluation is about g and the error of the numerical approximation to the minimal value f ropt is expected to be about eqs Unfortunately the numerical approximation of the optimal solution fopt is expected to have an error about Eqis only Alternatively to the simple call of OPTIMIZE the user may enter one of the following commands If OPTIMIZE is called with the flag f then the optimization is performed like in the case without flag but further data
168. f the grid by 2T h A d npor 1 with A the absolute mesh size of the triangulation and with d the length of the period By the symbol k in the tables we denote the maximal relative wave number length of period d times refractive index divided by wave length The numerical methods are either FEM or GFEM npor MLFEM i e the GFEM with the parameters npor nLFemM and nypa 1 Stars indicate that the accuracy is not reached due to the restricted main memory of the computer The number of grid points ngap is 67 for the first level 75 for the second 169 for the third 600 for the fourth 2 430 for the fifth 8 858 for the sixth 39 698 for the seventh 159 140 for the eighth and 637 914 for the ninth cf the computation time in 6 1 It is impossible to derive a general recommendation from the numbers in the Tables We have indicated the necessary relative mesh size for the fastest method with parameter nypa 1 by bold letters However the methods with doubled npor 1 and nLrem 1 and doubled mesh size one lower refinement level require almost the same computation time and lead to the same accuracy If nurem 1 is large and the grid is of a higher refinement level then the computing time can be reduced by first generating a preliminary grid with the doubled maximal mesh size and second applying an additional uniform refinement step of each triangle into four equal subtriangles Recall that the trial functions for congruent tri
169. from external to internal parameters the input of the upper and lower bounds for the angles and upper side lengths of the trapezoids and for the z coordinates of the lower corner points of the stack are ignored Instead the upper and lower bounds for the internal parameters d_geom_param l with 5k 3 5i geom_param 2 9 are set to 0 9 and 0 1 respectively Similarly the upper and lower bounds for the internal parameters d_geom_param with l 5k 2 and with 5i geom_param 2 8 are set to 0 9 and 0 respectively 10 2 6 Optimization of a polygonal interface inside a general grating The fifth class i_geom_param 1 5 is designed to optimize a small detail inside a fixed com plex grating geometry The basis is a general grating defined by an input file nameG inp This is to be extended by introducing a new polygonal interface curve dividing one of the material areas of the given grating into two cf Figure 31 where the blue rectangle on the left is split into a blue and yellow part on the right by a polygonal consisting of three seg ments The task of optimization is to find the optimal interface among all the polygonal interface curves with fixed end points dividing the fixed area The divided area A is supposed to be convex Moreover A must not reach to the upper and lower boundary lines of the grating cross section The new interface connects two prescribed boundary points P and P of the convex area A which m
170. g gt e g gt 0 15 0v3y 0 45 0 5 0 6 0 75 and 0 9 lt lAmellar 3 4 0 2 0 6 0 2 1 0 5 0 7 50 0 90 0 0 500 0 900 LAMELLAR GRATING rectangular grating consisting of several materials placed in rectangular sub domains with 3 columns each divided into 4 rectangular layers first column with x coordinate in 0 lt x lt 0 2 given in micro meter second column with 0 2 lt x lt 0 6 third columnn with 0 6 lt x lt period period given above whole grating with y coordinate s t 0 2 lt y lt 1 0 first column first layer with 0 2 lt y lt 0 second with 0 lt y lt 0 5 third with 0 5 lt y lt 0 7 and fourth with 0 7 lt y lt 1 second column first layer with 0 2 lt y lt 0 0 second with 0 0 lt y lt 0 50 third with 0 50 lt y lt 0 90 and fourth with 0 90 lt y lt 1 third column first layer with 0 2 lt y lt 0 00 second with 0 00 lt y lt 0 500 third with 0 500 lt y lt 0 900 and fourth with 0 900 lt y lt 1 lAmellar 1 1 0 2 0 8 d SIMPLE LAYER special case of lamellar grating with layer material s t y coordinate satisfies 0 2 lt y lt 0 8 given in micro meter polygon filel GRATING DETERMINED BY A POLYGONAL LINE defined by the data in the file with name GEOMETRIES file1 in GEOMETRIES file1 in each line beginning without there should be the x and y coordinate of one of the consecutive corner points first point with x coordinate 0 last point with x coor
171. g first argument name without tag inp of file to be created second argument angle third argument length of basis 7 fourth argument number of material layers in trapezoid next arguments heights of material layers last argument height of coating layer executable GEN MTRAPEZOID to generate an input file for a grating with several trapezoids one beside the other first argument name without tag inp of file to be created second argument name of input file mtrapezoid INP containing data of grating executable GEN_LAMELLAR to generate an input file for a lamellar grating first argument name without tag inp of file to be created second argument name of input file lamellar INP containing location and widths of layers file lamellar INP to define location and widths of layers in grating generated by GEN_LAMELLAR executable GEN_POLYGON to generate an input file for a grating with polygonal profile curve first argument name without tag inp of file to be created second argument name filel of file with nodes of polygon executable GEN_POLYGON2 to generate an input file for a grating with polygonal profile curve and with coating first argument name without tag inp of file to be created second argument name filel of file with nodes of polygon third argument name file2 of file with nodes of boundary of coated layer file filel to define profile
172. ge these parameters according to your diffraction problem computer memory capacity and computing time requirements How should they be chosen Suppose that ngap is the number of grid points Roughly speaking For the FEM the computation time as well as the necessary storage capacity is proportional to ngp The time for GFEM is proportional to ncp x nuren 1 nupa ncp x Inpor 1 6 1 and the storage to nap x npor 1 7 Hence a doubling of nurem 1 leads to about the same computing time as halving the mesh size of the grid Taking into account 6 1 we recommend to choose nypa as high as possible since the accuracy is almost independent of nypa but the computing time reduces significantly The only exceptional case when a larger nypa is not efficient occurs if the geometry forces the triangulation to have a few 66 large triangles and a huge number of small triangles e g geometries with thin layers In this case the next level uniform partition increases the number of grid points and the computation time significantly whereas a standard triangulation with halved mesh size leads to a small increase of grid points and to about the same numerical error If npor 0 and nirem 1 then the conventional FEM is computed With npor 0 a higher nifem is not recommended For npor gt 1 and not so restrictive accuracy requirements NnLFEM 2Npor 1 is a good choice For npor gt 1 and challenging accuracy r
173. ged over Nnorm iterations then the iteration is stopped cf Step 2 The number of real parameters is nd_ opt 5 These parameters are d opt 1 Maximal stepsize factor maz gt 0 in line search cf Step 2 1 and choose e g Qmar 1 d_opt 2 Constant c1 0 lt c lt 1 in Armijo stopping criterion for line search e g c 0 001 d_opt 3 Threshold acc gt 0 iterate is shifted to the boundary if the distance to the boundary is less than acc cf the projection P in Step 1 and choose Eacc about the expected accuracy of the final solution d_opt 4 Threshold q gt 0 stop if gradient is less than Egra should be about approximation error of gradient or less d_opt 5 Threshold Enorm gt 0 stop if relative change of the norm of the gradient is less than Enorm for the last Nnorm steps cf Step 2 and choose e g Enorm 0 01 The parameters i opt 1 and d_opt i i 1 2 3 5 should be chosen as recommended For the error d_opt 4 Egra of the gradient computation a first estimate can be obtained using the command OPTIMIZE g name dat On the other hand the user can choose niter_max larger than necessary and the positive parameter d_opt 4 smaller than recommended In the worst case a large number of unnecessary iteration steps with very small changes in the iterative solutions are performed at the end of the optimization procedure Even these redundant iteration steps can be interrupted Indeed the
174. geom_param nd_geom_param 5 refractive index of cover material Re n_co gt 0 Im n_co gt 0 n_k d_geom_param 5 k 1 i d_geom_param 5 k refractive index of k th trapezoid in stack Re n_k gt O Im n_k gt 0 k 1 i_geom_param 3 nel d_geom_param nd_geom_param 8 i d_geom_param nd_geom_param 7 refractive index of extra layer material Re nel gt O Im nel gt 0 nl_k d_geom_param 5 i_geom_param 3 1 3 k i d_geom_param 5 i_geom_param 3 2 3 k refractive index of k th layer beneath stack Re nl_k gt 0O Im nl_k gt 0 k 1 i_geom_param 4 n_su d_geom_param nd_geom_param 4 ixd_geom_param nd_geom_param 3 refractive index of substrate material resp material of adjacent lower coatingstrip Re n_su gt 0 Im n_su 0 b_k E eiia reload do a tise Gan tantra sorter page ak terete decals gig agent gt a_k n gt T a if k th trapez ri h_k k 2 v HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHH za OO poset ute e sees Ge eee Se 0 0 posa 0 posb 0 d 0 oa ee hl_k k th layer k 2 EE E E E re a a AEE a S The height hel of the extra layer beside the bridge may be any non negative number h_1 lt hel lt h_2 is not supposed h_k d_geom_param 5 k 4 height of k th trapezoid in stack in micro meter h_k gt 0 b_k b_ k 1 d_geom_param 5 k 3 ra
175. gth 1 HEHEHE AAAA HH EEEE EEEE E E EEEE EEEE EAEE AEAEE H HEHEHE Length of additional shift of grating geometry in micro m This is shift into the y direction i e the direction perpendicular to the grating surface pointing into the cover material Must be a real number Length Q TEED ESE PE PTET TERETE TEI EE EEE TE EH Period of grating in micro meter 0 25 HHEHHHHHHHHHHHHHHHHHHHHHEHHHHHEHHHEH HHH RHEE RHEE HHR AHHH E RHEE R HS PARAMETERS OF GRATING PHRRRNPPHHHHGHPRAHHY NHANH NRR R HARRAPA PEPER HE HRAARAAR RAHNER Parameters nd_geom_param dl_geom_param du_geom_param and ni_geom_param and i_geom_param to describe grating geometry integer parameters i_geom_param i i 1 ni_geom_param In particular i_geom_param 1 class of gratings i_geom_param 2 number of materials real parameters d_geom_param il i 1 nd_geom_param 161 HHHHHHHHHHHHHHHHHHHHHHHHHHH HHH HH HH HF OF bounds of real parameters dl_geom_param i du_geom_param i i 1 nd_geom_param dl_geom_param i lt du_geom_param i name parameters character strings s_geom_param il i 1 ns_geom_param Optimization runs over all real parameters d_geom_param i in dl_geom_param i du_geom_param i with i such that dl_geom_param i lt du_geom_param i The parameters d_geom_param i with dl_geom_param i du_geom_param i keep their values The integer and string parameters switch
176. h 63 i 02 less or equal to 73 Wave length 1 2 I 635 636 002 HEHEHE RAR RRR ARE RRR RRR Temperature in degrees Celsius from O to 400 20 for room temperature Must be set to any fixed number Will be ignored if optical indices are given explicitly Temperature 20 HEHEHE RARER ERE RRR aa EH Optical index refractive index of cover material This is c times square root of mu times epsilon This could be complex like e g 4 298 i 0 073 for Si with wave length 500nm This could be also given by the name of a material like Air Ag Al Au CsBr Cu InP MgF2 NaCl PMMA PSKL SF5 i T1Br T1Cl Cr ZnS Ge 611 0 812 0 Ti02r Quarz AddOn cf Userguide This could be a value interpolated from a user defined table determined by the name of the file file is to be located in the current directory name of file must begin with letter u and may consist of no more than five letters like e g user the file consists of lines each with three real numbers first wave length in micro meter second the real part of the corresponding optical index third the imaginary part of the corresponding index Optical index 1 0 1 0 TERI IE EE EPR PEPETE ETRE THEATER EEE Optical indices of the materials of the upper coating layers HHHH HHH HHH HHH OH OF H This is c times square root of mu times epsilon N_co_ov entries Needed only if N
177. hat the additional factor Ve0Vnt fio in the definition of Hirt guarantees that the incident light wave has a fixed intensity length of Poynting vector independent of the cover material If the cover material is air then the wave is normalized such that the amplitude of the electric field vector is of unit length Like in the TE case the values of the efficiencies and phase shifts of the reflected and transmitted plane wave modes are independent of this normalization factor Only the subsequent Rayleigh coefficients depend on this scaling The z component of the complete field H satisfies the Helmholtz equation A k H 0 in any domain of the cross section plane with constant materials However the trans mission conditions on the interfaces are different We can solve the transmission problem of the Helmholtz equation by FEM Again we have a finite number of transmitted and reflected modes and the Rayleigh expansions hold for E replaced by H More precisely the Rayleigh coefficients are the B of the expansions VR Hey gt B exp i y exp ianz Bi exp ibf y exp iax 2 6 0o Vio O OF ape Jao a B exp i n y exp ianx 2 7 BE Vnt The objective is to compute the Rayleigh coefficients They result from the FEM solution of the new transmission problems and from the discretization of the Fourier series expansion and 2 7 The efficiencies cf 2 4 are computed by ee z a oa n
178. he efficiencies efficiency_2 denote the o the efficiency TM part for TE TM output resp the P part for Jones output For 3 Comp output these first and second efficiency terms are not allowed i e the corresponding n_lin qua_m_re tr are 185 to be set to zero HHHHHHHHHHHHHHHHHHHHHHHH HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH E E E E HH OH If the wavelength or and the incidence angles or and the polarization run over a fixed set of values then the last value of the objective function is replaced by the sum over the last values depending on the wavelengths or and the angles or and the polarization In this case the values w_ equal to w_lin_ene_re w_lin_ene_tr w_lin_ene_to w_lin_re wlin tr j j w_lin_k_re w_lin_k_tr k 1 2 j J w_qua_re w_qua_tr j j w_qua_k_re w_qua_k_ tr k 1 2 j j w_psh_k_re wpsh_k tr k 1 2 j i w_qua_ene_re w_qua_ene_tr w_qua_ene_to and similarly the corresponding prescribed parameters c_ equal to c_qua_re c gua tr j j c_qua_k_re c_qua_k tr k 1 2 j j c_psh_k_re e psh_k tr k 1 2 j j c_ene_re c_ene_tr c_ene_to may depend on the wavelengths or and the angles or and the polarization type If the n_ values of c_ depend on m wavelengths resp angles theta resp angles phi then the input consists of m n_ 1 lines The first contains the indicator WAL resp ATH resp APH and is followed by m
179. he indices are all numbers then the temperature is not used The code words for materials can be Air Ag Al Au CsBr Cu InP MgF2 NaCl PMMA PSKL SF5 Si TIBr TIODI Cr ZnS Ge TiO2r Quarz AddOn and Si1 0 Si2 0 Here Sia b with the real number x a b indicates a blending of SiO and SiO with the refractive index n 2 2 ngio x 1 nsgio For example Air corresponds to an index n 1 Additionally if the user has linked the right sources the refractive indices of the program package IMD can be used as well These are 59 a Al203_hagemann a A1203 palik Ac_llnl_cxro a C_palik a C_windt Ag_hagemann IMD Ag Al1203 e Al203_palik AlAs IMD Al Al palik Al shiles a SiC_llnl_cxro a SiH_palik a SiO2_IIn _cxro a Si_palik a Si_windt92 Au_llnl_cxro Au_palik B203_llnl cxro B4C BaTiO3 Be BeO_palik B C10H804_llnl cxro C5H802 CaF2_palik CH2_llnl_cxro Co203 Co_lInl_cxro CoO CoSi2 Cr3C2_llnl_cxro Cr_Ilnl_cxro Cr_palik CsI palik Cu20 palik Cu_llnl_cxro CuO c ZnSe c ZnS_palik d C_palik a A1203_IInl_cxro a C_hagemann a C a C_windt88 Ag20 llnl_cxro Ag_leveque Ag palik A1203 e_palik Al 3Ga 7As AlAs_palik AIN_llnl_cxro AlSb Al_windt a SiC a Si_llnl_cxro a SiO2 a Si windt88 Au canfield Au_nilsson Au_weaver B203 BaTiO3 e
180. he user should start the local methods from several sets if initial parameters and to take the local optimal solution with the minimal value of the objective functional as a global solution Of course there is no warranty that the true global minimum has been found On the other hand frequently the local minimizer corresponds to a value of the objective function quite close to the minimum Such a local minimum may be just as good as the global optimal solution Clearly one can utilize optimization methods designed to find global minimizers Usu ally these global optimizers are much slower than local methods Other global optimizers use more information on the optimization problem which are not natural in our appli cations The only global algorithm in the package DIPOG 2 1 is the simulated annealing cf Section which is a stochastic method In other words if the parameters are chosen adequately then the solution of simulating annealing is a global solution with prob ability one Choosing the right parameters however is not easy Either the number of necessary iterations might be large and the computing time might be not acceptable or the algorithm might stuck in a local minimizer So there is no warranty even with the global simulated annealing Finally let us mention that the user is free to combine global and local methods First he should apply a certain number of simulated annealing steps Using the final solution of simulated annealing a
181. hould be chosen as large as possible In other words the computing time the user is willing to spent determines the choice of niter_max The parameters d_opt i i 1 3 4 5 should be chosen as recommended The cooling factor d_opt 2 should be chosen sufficiently large such that a lot of random choices transitions are accepted at the beginning of the iterative process However d_opt 2 should not be too large such that at the end of the iterative process the random choices for the new iterative solutions are almost always rejected Of course this so called freezing of the state should 107 happen at the end only If the iterative solution remains unchanged for the last half of the cooling steps then the cooling factor should be increased Finally the number of restarts i opt 1 should be as large as possible Indeed normally the computation of the objective function is time consuming Consequently the values of niter_max and d opt 2 are usually smaller than what the theoretical analysis requires Using these fast cooling schemes simulated annealing computes rather a locally optimal solution than a global Therefore the restarts will increase the probability that the final solution is the global optimum Again i opt 1 is restricted by the admissible computing time The program of simulated annealing can be used for a pure random search In this case the parameters should be e g i_opt 1 0 d_opt 1 107 d_opt 2 0 9999
182. iC_llnl cxro SiC_palik Si_llnl cxro SiO2 e_palik SiO2_palik SiO_palik Snllnl cxro SnO2 SnTe SrTiO3_palik TaC_lln _cxro Ta Ta_palik Te e Te_palik TiC_Ilnl_cxro Ti_kihara 58 OsO2 PbSe PbS_palik Pd_llnl_cxro PdO Pd_windt PtO2_IInl_cxro PtO Pt_windt Re_lln l_cxro ReOQ2 Re te_lynch Rh203_llnl_cxro Rh Rh_windt Ru RuO4_lln _cxro RuSi Ru_weaver Sc203_llnl cxro Sc Se e Se_palik Si 25Ge 75_palik2 Si3N4_palik Si 5Ge 5_palik1 Si 8Ge 2_palik1 SiC SiC _windt IMD Si SiO2_llnl_cxro SiO _llnl_cxro Si_palik Sn SnO_llnl_cxro SnTe_palik Ta205_llnl_cxro TaC TaN _IInl_cxro Ta_weaver Te e_palik ThF4 TiC Ti_llnl_cxro Ti TiN _llnl_cxro TiN TiN_palik TiO2 e TiO2 e_palik TiO2_IInl_cxro TiO2 TiO2_palik Ti_weaver Ti_windt V203_Ilnl_cxro V203 V205_Ilnl_cxro V205 VC_llnl cxro VC VC_palik V_kihara V llnl cxro V VN_IInl_cxro VN VN_palik VO2_IInl_cxro VO2 VO llnl_cxro VO V palik V weaver W_llnl_cxro W WO2_IInl_cxro WO2 WO3_Ilnl_cxro WO3 W_palik WSi2_IInl_cxro WSi2 W weaver W windt88 W_windt91 Y203_llnl cxro Y203 Y203_palik Y _Ilnl_cxro Y Y_windt Zn_Ilnl_cxro Zn ZnO _Ilnl_cxro ZnO ZnTe ZnTe_palik Zr _lln _cxro Zr lynch Zr Zr N_IInl_cxro ZrN ZrO2_IInl_cxro ZrO2 Zr te_lynch Zr_windt On the other hand the value of the refractive index can be interpolated from a user defined table indicated by the name of the file This file is to be located in the current directory of the computation CLASSICAL CO
183. ical case programs and input files for the optimization with OPTIMIZATION in the directory OPTIM programs object files and input files for the graphical user interface program DIPOG 2 1 GUI in the directory GUI data files for the refractive indices plot programs Possibly there exist subdirectories to install necessary packaged dipog 1 3 tgui gltools tar triangld necessary source files from previous version of DIPOG 1 3 in dipog 1 3 gsl src files to install libhur a necessary sources to create TGUI in the directory GEOMETRIES necessary source files for Fuhrmann s package GLTOOLS 2 4 this produces subdirectory gltools 2 4 during installation necessary source files for Shewchuk s package TRIANGLE 1 4 In case of a simultaneous use of the package each user has its own home directory contain ing the six subdirectories GEOMETRIES CLASSICAL CONICAL OPTIM GUI and RESULTS The subdirectories contain the same example and data files as described above for the directories of the package However the executables are replaced by symbolic links to the executables of the package 5If needed change to the subdirectories and follow the instructions of the corresponding files README txt If not needed delete the subdirectories This subdirectory exists only during internal installation 12 1 5 Environment variables The following environment variables are mandatory LD_LIBRARY_PATH Th
184. ical data of the gratings executable SHOW to visualize the input data name inp exists only with openGL argument name inp executable TGUI graphical user interface to create input files name inp executable GEN_INPUT to generate a general input file no argument executable GEN_ECHELLEA to generate an input file for an echelle grating of type A first argument name without tag inp of file to be created second argument letter A L R third argument depth angle fourth argument width of first part of layer fifth argument width of second part of layer executable GEN_ECHELLEB to generate an input file for an echelle grating of type B first argument name without tag inp of file to be created second argument angle third argument width of layer executable GEN_ECHELLE to generate an input file for a general echelle grating first argument name without tag inp of file to be created second argument letter A for apex angle R for right blaze angle L for left blaze angle or D for depth third argument angle depth fourth argument letter A for apex angle R for right blaze angle L for left blaze angle or D for depth fifth argument angle depth sixth argument width of layer over left blaze side seventh argument width of layer over right blaze side executable GEN_TRAPEZOID to generate an input file for a trapezoidal gratin
185. iciencies n O theta 26 n 1 theta 50 n 1 theta ie n 2 theta 10 n 3 theta 29 n 4 theta 54 Transmitted energy 00 0 012980 0 144468 74 0 000471 0 016141 33 0 000145 0 000968 O7 0 090729 0 055189 2 299969 and coefficients 95 0 502603 0 087406 41 0 642427 0 535243 80 0 174148 0 293832 48 0 069855 0 111978 96 0 062905 0 082480 77 0 036440 0 097038 97 700031 2 3 1 2 3 4 oo Oo NM 2 52 13 026452 102941 733271 1 VALUE ORDER 0 147 139668 533702 752053 462258 103941 059386 000211 136430 447303 715032 675032 h l value l extrap l error 0 818 3 4623 1 3312 0 347 1 7305 0 4006 0 195 1 9703 1 9411 0 1608 0 092 2 0767 2 1617 0 0544 0 045 2 1039 21133 0 0272 2 VALUE ORDER 1 h l value l extrap l error 0 818 0 1178 0 0569 0 347 0 0026 0 0583 0 195 0 0476 0 0350 0 0133 0 092 0 0578 0 0608 0 0031 0 045 0 0594 0 0597 0 0015 NO ERROR ANALYSIS FOR ORD
186. ies only to this grating class The fourth method is a stochastic global optimizer Depending on the value of ind_opt the general optimization procedure requires further ni_opt integer parameters and nd_opt real valued parameters The input is as indicated in Section Finally the performance of the optimization depends on the last input numbers the scaling factors Indeed without proper scaling the iteration might stop at a parameter set far from a local minimum For instance suppose the partial derivative Of Or is much larger than the other partial derivatives Of Or with j 2 N Then the iterative solution moves in the negative gradient direction which is more or less the direction of the first component r1 and the value Of Or reduces until it is in the order of the discretization error of the gradient computation This however means that the gradient at the actual iterative solution has a first component Of Or dominated by the discretization error Moreover this first component dominates the other components Of Or due to the bad scaling Moving into such a gradient direction will sooner or later end up in a direction of no descent and the iteration stops Though the components Of Or with j 2 N are quite accurate they are not used for a correction of the iterative solution towards the local minimum To improve the scaling the parameters r from l u are internally replaced by the scaled parameters r
187. in micro meter e g polygon filel gt GRATING DETERMINED BY A POLYGONAL LINE defined by the data in the file with name GEOMETRIES file1 in GEOMETRIES file1 in each line beginning without there should be the x and y coordinate of one of the consecutive corner points first point with x coordinate 0 last point with x coordinate 1 same y coordinate for first and last point all x coordinates between O and 1 at least two different y coordinates last line should be End e g polygon2 filel file2 gt COATED GRATING DETERMINED BY POLYGONAL LINES i e grating profile line is defined by the data in the file with name GEOMETRIES file1 in GEOMETRIES filel in each line beginning without there should be the x and y coordinate of one of the consecutive corner points first point with x coordinate 0 last point with x coordinate 1 same y coordinate for first and last point all x coordinates between O and 1 at least two different y coordinates last line should be Fnd and the coated layer is enclosed between the polygonal line of GEOMETRIES file1 and the polygonal line of the file with name GEOMETRIES file2 in GEOMETRIES file2 in each line beginning without there should be the x and y coordinate of one of the consecutive corner points first and last point must be corner
188. ine comment lines begin with dummy file name width of additional layer above and below the structure must be positive or could be no for no additional layer BRE AAAAN 139 gt HHHHHHHHHHHHHHHHHHHHHH HHH HHH HH period of grating per number nla of layers above multilayer system number nlb of layers below multilayer system number nmld of multilayers inbetween number nld of layers in each multilayer number ncorn of corner points in each polygonal approximation interfaces randomly generated polygon number nrand of standard Gaussian distributed random numbers needed to generate interfaces automatically created if nrand 0 otherwise nrand has to equal ncorn nla nld nmld tn1b optional standard Gaussian distributed random numbers nla widths of layers above multilayer system nld widths of layers in each multilayer nlb widths of layers below multilayer system standard deviation sigma i of each layer interface for i 0 nla nld nmld nlb 1 correlation length corl i of each layer interface for i 0 nla nld nmld nlb 1 Note that the first of the subsequent refractive indices of the grating must be that of the superstrate resp that of the adjacent upper layer If the width of the additional layers is positive then the last of the
189. ingle value e g 63 Either add more values by e g 115 wave length 63 i 02 less or equal to 73 Sy 5 63 64 i 65 269 sfQ 77 The last means that computation is to be done for the wave lengths from the Vector of length 5 63g SP e642 FS 65 8 697 and 6 702s Or add egs T 63 73 027 The last means that computation is to be done for the wave lengths 63 i 02 with i 0 1 2 and with Wave length 635 TEE PEPPER EE BE EE EE AAEE HH Temperature in degrees Celsius From O to 400 For room temperature set to 20 Will be ignored for explicitly given refractive indices Temperature 20 HEHEHE HARA R aH RRR RAH Optical index refractive index of cover material This is c times square root of mu times epsilon This could be complex like 4 298 i 0 073 for Si with wave length 500nm This could be also given by the name of a material like Air Ag Al Au CsBr Cu InP MgF2 NaCl PMMA PSKL SF5 Si T1Br T1Cl Cr ZnS Ge i1 0 12 0 Ti02r Quarz AddOn cf Userguide This could be a value interpolated from a user defined table determined by the name of the file file is to be located in the current directory name of file must begin with letter u and may consist of no more than five letters like e g user the file consists of lines each with three real numbers first wave length in micro meter se
190. ion Consequently Nmat gt 2 and Nmat 2 holds if the grating structure is manufactured from the same two materials filling the regions adjacent to the grating structure In the case of the grating in Figure I we have two materials with the refractive indices n and n inside the grating structure and two adjacent materials with indices n and n i e Nmat 4 Generally the materials inside the grating structure are to be indicated in namel inp by an index between 1 and Nnmat In particular index 1 24 e eo e o o gt o o o o oe o gt o Figure 5 Pictures of grid produced by SHOW stands for the material immediately over the grating and Nmat for that immediately under it Of course the corresponding Nmat refractive indices are listed in the input file name dat and not in namel inp Finally we remark that the file namel inp contains its name namel without the tag inp This name must include the complete path if the file is not located in the directory GEOMETRIES To check the geometry described by namel inp enter the command SHOW namel inp You will see a first picture cf left picture in Figure 5 with the chosen points of a polygonal structure After pressing Escape or Bar Space you see a second picture cf right picture in Figure 5 with a coarse triangulation and with the different regions later disti
191. ion More precisely for integers npor and nyrem With 0 lt npor with 1 lt nurem and with nurem 1 a multiple of npor 1 the degrees of freedom of the trial space are the function values at the corner points of the triangulations and at the npor points of a uniform partition of each triangle side The restrictions of the trial functions to the triangle sides are polynomial interpolants of the degrees of freedom The restrictions of the trial functions to the triangles are the finite element solutions of the Dirichlet problem for the Helmholtz equation over a uniform triangulation of the partition triangle into nirem 1 x nirem 1 equal subtriangles cf the 65 trial functions over the grid triangle indicated in Figure 25 with npor 3 and nifem 7 Hence GFEM treats the same problems as FEM but is more convenient to treat higher wave numbers and faster for simple geometries In order to accelerate the computation one can use a FEM grid which is a uniform refinement of a coarse grid with each coarse triangle split into nupa X nupa congruent subtriangles In this case the trial space over the congruent grid triangles approximate Dirichlet solutions have to be computed only ones and can be reused several times To use generalized FEM for FEM the executables FEM FEM CHECK FEM FULLINFO and FEM PLOT are to be replaced by the codes GFEM GFEM_CHECK GFEM_FULLINFO and GFEM_PLOT respectively These executable use the same input fi
192. ional graph a Matlab file is produced This file with the tag m can be called from Matlab and shows a graph or an isoline picture of the efficiency phase shift resp energy depending on the wave length and the angles of incidence The name of the Matlab file will be printed on the screen Similarly in the case of classical diffraction and for result files with varying wave lengths or and varying angles of incidence one can enter the command PLOT_GNUPLOT name3 1 Everything is like in the previous case However instead of creating the Matlab file a gnuplot ps file is produced The name of the ps file will be printed on the screen 9 Parameter Test for GFEM Here we present the results of test computations to give an orientation for the choice of the parameters npor and nifem Recall that ngp is the number of grid points For simplicity we present results with nypa 1 only We consider a trapezoidal grating with basis angle of 60 and with one material which covers 60 of the period The height of the trapezoid is 0 3 times length of period and the refractive index is 2 0 Moreover we assume an additional layer which covers the whole period and which has a refractive index of 1 3 and a height of 0 05 times period The substrate has a refractive index of 1 5 and the superstrate is air The grating is illuminated in a classical TE scenario by light of wave length 635nm under an incidence angle of 65 The length of one period is lum 2
193. is data input file if linear terms appear in objective functional if energy terms appear in objective functional if phase shift terms appear without the analoguous structure of efficiency term input HHHHHHHHHHHHHHHHHHHHHHHHHHHH HHH HH HH OH OF Fe SD sqrt 1e 2 E E SD dev SD orig TERRE TERETE PTT EEE EEE EE HHHH EH END HHEHHHHHHHHHHHHHHHHHAHHHEHHHHHHHHEEH HHH HEHE HHRHRHRRE RHEE HEHEHE 202 12 9 Output file example res of OPTIMIZE in OPTIM K gt KK K K K FK FK FK FK 2 K K K 2 2K K FK FK FK K K K K K K K 2 2K 2K 2K FK FK FK K K K 2K KK KK K K K FK FK FK FK FK K K K K 2K K FK FK FK 2K K K K K K K K 2K 2K 2K 2K FK FK FK 2K K x OPTIMIZE GRATING CONICAL CASE x KKK KK 2 2K 2g FK FK FK FK FK K K K K 2 FK FK FK FK 2K K FK K K K K K FK 2K 2K 2K FK FK K K K 2K gt KK K 2K K FK FK FK FK FK K K K K K K FK FK FK 2K 2 FK FK K K K K 2K 2K 2K 2K FK FK 2K 2K K date 9 Aug 2005 10 27 23 refr ind of cov mater 1 0000000 i 0 0000000 n of diff grating mat i5 refr ind 0 5421322 i 0 1500000 refr ind 0 6495191 i 0 0000000 refr ind 0 0000000 i 0 0000000 refr ind 0 0000000 i 0 0000000 refr ind 0 0000000 i 0 6495191 refr ind of substr mat 1 5000000 i 0 0000000 temperature 20 0000000 discret level 3 additional horizontal sh 0 0000000 stretching factor 1 0000000 additional vertical sh 0 0000000 period of grating 1 0000000 i
194. is search path for loading program libraries must contain the address of the solver PARDISO cf Section 1 3 OMP_NUM_THREADS This is to be set to the number of CPUs which should be used for solving linear systems of equations by PARDISO cf Section 1 3 Additionally the following environment variables can be set ADD_STRIPS If this is set to yes and if an optimization in the class of bridges composed of trapezoids under light in the EUV range is performed then auto matically a strip beneath the bridge is included into the FEM domain cf Sec tion 10 2 7 BND_MESH_SIZE If this is set to a positive number and if an optimization is per formed then the bound for the mesh size of the FEM partition at level one is set to this number cf Section 10 2 1p BND_n_LFEM If this is set to a positive integer then for the non local boundary value condition the number of discretization points in each interval of the uni form partition of the upper and lower boundary line is set to this value cf Sec tion 6 CHOOSE_PMETHOD If this is set to yes then like in the p method of the FEM algo rithm with p npor 1 and with elimination of interior degrees of freedom the local trial functions are the solution of a 3p x 3p system of equations cf Section 6 COND_NMB_IT If this is set to a positive integer if the environment variable GET_COND_NUMB is set and if one of the executables GFEM GFEM_FULLINFO or
195. ither add a single value e g 45 Either add more values by e g LT V 5 63 64 65 69 e ets The last means that computation is to be done for the angles from the Vector of length 5 PB Big Pg pA ep gg and T Or add e g E I 45 56 2 The last means that computation is to be done for 160 the angles 45 i 2 with i 0 1 2 and with angle 45 ix2 less or equal to 56 Note that two of the three the wave length the angle of incident wave theta and the angle of incident wave phi must be single valued Angle I 47 48 2 FERRE PETE TET RAAUA TERETE PE AHAAA HRR eee EEE PEE TE PETE TERE TEE HE ETE ERE EE EERE REHA H Length factor of additional shift of grating geometry This is shift into the x direction i e the direction of the period to the right This is length of shift relative to period i e the grating structure given by subsequent input will be shifted by factor times the period given in subsequent input However only the Rayleigh numbers and efficiencies will be computed according to the shift The field vectors in the plots are drawn without shift and the graphics of the executable with tag _CHECK is drawn without shift Must be a real number between O and 1 Length 0 HEHEHE H EEEE EE E E EEEE EEEE EEEE H EEHEHE Stretching factor for grating in y direction Must be a positive real number Len
196. itive ste stop and jump to Step 3 Else compute f r 1 Fy ry41 Slj4i SUjqi dlj 1 duj41 and Vf rj41 If the Armijo criterion Frin f r3 lt aa V Elri sdr 10 16 is fulfilled then stop and leave the line search Else compute a smaller by halv ing its value or by a clever approximate minimization of the univariate function ar f r asdr Repeat Step 2 2 Finally set j j 1 and repeat the Step 2 3 Final output Accept and print r as the local solution Print the corresponding values of f r and P Vf r l The interior point method ind_opt 2 requires one integer input value ni opt 1 This i opt 1 is the threshold integer Nnporm gt 1 choose e g i opt 1 3 Whenever the gradient of the iterative solution remains unchanged over Norm iterations then the iteration is stopped cf Equation 10 15 The number of real parameters is nd_ opt 8 These parameters are d_opt d_opt d_opt d_opt d_opt d_opt d_opt d_opt 1 Initial value g9 gt 0 for parameter of operator F cf Step 2 and choose e g 0o 0 1 2 Reduction factor q 0 lt q lt 1 to reduce parameter 0 of operator F cf Step 2 and choose e g q 0 5 3 Constant c1 0 lt c lt 1 in Armijo stopping criterion for line search choose e g c 0 001 4 Maximal stepsize factor Amar 0 lt amas lt 1 in line search cf Step 2 1 and choose e g max 0 9
197. j 1 y d_geom_param 2j j 1 i geom_param 3 The last four parameters describe the materials More pre cisely the refractive indices of the cover and substrate materials are d_geom_param k 1 i d_geom_param k 2 d_geom_param k 3 i d_geom_param k 4 where k 2i_ geom_param 3 Clearly the real parts of these indices must be positive and the imaginary parts non negative The imaginary part of the cover material must vanish Moreover the refractive indices must be fixed i e du_geom_param k dl_geom_param k k 2i_geom_param 3 1 2i_geom_param 3 4 Given an arbitrary parameter set the self intersection of the polygonal curve is checked in the program For simplicity however the corresponding restriction functionals are not included into the choice of search directions for the optimization algorithms cf Section 10 3 Hence using this class and local optimization methods only local minima away from the boundary of the domain of admissibility can be detected The next class provides a proper treatment of the restriction functionals corresponding to self intersection 10 2 4 Profile grating determined by general polygonal profile curve II The third class i_geom_param 1 3 is a profile grating without coatings defined by a pro file curve which is the cross section of the interface between cover and substrate material Thus the number of materials is i_geom_param 2 2 The profile is supposed to be a polyg ona
198. kes the direction sin theta cos phi cos theta sin theta sin phi with the restriction 0 lt theta lt 90 Either add a single value e g 45 Either add more values by e g c V To 7 The last means that computation is to be done for the angles from the Vector of length 5 MBS es CGAI OB 2 OO 4 aiid 70 7 2 s Or add e g 1 45 56 2 The last means that computation is to be done for the angles 45 i 2 with i 0 1 2 and with angle 45 i 2 less or equal to 56 Note that two of the three the wave length the angle of incident wave theta and the angle of incident wave phi must be single valued Angle 47 TERED ESE PETE TTR EE ETE EEEE HH H Length factor of additional shift of grating geometry HHHHHHHHHHHHHHHHHHH HHH HH HH OH op o This is shift into the x direction i e the direction of the period to the right H This is length of shift relative to period i e the grating structure given by subsequent input will be shifted by factor times the period given in subsequent input However only the Rayleigh numbers and efficiencies will be computed according to the shift The field vectors in the plots are drawn without shift and the graphics of the executable with tag _CHECK is drawn without shift Must be a real number between O and 1 Length 0 TERR TTT TTT ETE TEEPE RRHH ARERR ERE TEER BEE EHHH Stretching factor for
199. l curve The class has no name parameters ns_geom_param 0 and ni_geom_param 3 integer parameters The value i geom_param 3 is the number N of knots of the profile curve in the interior of the period i e with x coordinate strictly between zero and period d In other words the grating profile is the polygonal curve connecting the points 0 0 v1 y1 r2 y2 ere N YN d 0 with arbitrary real values x and y such that 0 lt x lt d and such that some additional re strictions are satisfied These additional restrictions guarantee that non adjacent polygonal sides do not intersect each other The number of real parameters nd geom_param is equal to 2i_geom_param 3 5 and the first 2i_geom_param 3 parameters define the profile curve by x d_geom_param 2j 1 y d_geom_param 2j j 1 i geom_param 3 The next four parameters describe the materials More precisely the refractive indices of the cover and substrate materials are d_geom_param k 1 i d_geom_param k 2 d_geom_param k 3 i d_geom_param k 4 where k 2i_ geom_param 3 Clearly the real parts of these indices must be positive and the imaginary parts non negative The imaginary part of the cover material must vanish The last parameter d_geom_param k with the index k 2i_geom_param 3 5 is a threshold parameter for the additional restrictions This threshold and the refractive indices must be fixed i e 88 du_geom_param k dl_geom_param k k 2i_geom_p
200. l data and example files and with the correct links to the executables can be created automatically in the chosen new private home directory by calling the executable MAKEHOME Note that MAKEHOME has just been created by make in the subdirectory MAKES of the installed package Before a user runs the executables he has to set the environment variable LD_LIBRARY_PATH such that the directory containing PARDISO is included Setting the environment variable OMP_NUM_THREADS to a non negative integer he limits the number of used CPUs The solver routine PARDISO runs parallel 4The temporary directory is defined in MHEAD before the installation Its name will be stored in the first line of the file MAKES make_info and can be changed in this file at any time Alternatively the temporary working directory can be chosen by setting the environment variable TMPDIR Most of the subsequent executables can be called without argument Then one gets information on the necessary arguments Usually all input files contain a lot of explanations and informations Indeed each line beginning with the sign is a comment Such lines can be added or deleted without any problem 1 4 Structure of the package The directory containing the README txt file is the home directory of DIPOG 2 1 We sup pose in this user guide that it is named DIPOG 2 1 There exist the following important subdirectories GEOMETRIES input files name inp geometr
201. l solution of optimal design problems for binary gratings J Comput Physics 146 pp 603 626 1998 J Elschner R Hinder and G Schmidt Finite element solution of conical diffraction problems Adv Comput Math 16 pp 139 156 2002 J Elschner R Hinder and G Schmidt Direct and inverse problems for Diffractive Structures Optimization of binary gratings In W Jager H J Krebs eds Mathematics key technology for the future joint projects between universities and in dustry Springer Verlag Berlin Heidelberg 2003 pp 293 304 R Petit ed Electromagnetic Theory of Gratings Springer Berlin 1980 H P Urbach Convergence of the Galerkin method for two dimensional electro magnetic problems SIAM J Numer Anal 28 pp 697 710 1991 For generalized FEM methods applied to the Helmholtz equation modified and described in the subsequent Sections 6 and 7 we refer to I Babu ka F Ihlenburg E Paik and S Sauter A generalized finite element method for solving the Helmholtz equation in two dimensions with minimal pollution Comp Methods Appl Mech Eng 128 pp 325 359 1995 O Cessenat and B Depres Application of an ultra weak variational formulation of elliptic PDEs to the two dimensional Helmholtz problem SIAM J Numer Anal 35 255 299 1998 F Ihlenburg Finite element analysis of acoustic scattering Springer Verlag New York Berlin Heidelberg Applied Mathematical Sciences 132 1998
202. le name2 dat but require the additional input file generalized Dat cf the enclosed file in 12 3 Normally the lat ter is to be located in the current working directory If there is no such file in the current working directory and if the output is written into a directory indicated by a certain path then the code looks for the generalized Dat file also in the directory determined by this path The file generalized Dat fixes the parameters NDOF Additional degrees of freedom on each triangle side Indeed the trial functions on each subdivision triangle are approximate solutions of the Dirichlet problem for the Helmholtz equation s t their restriction to the triangle sides coincides with the La grange interpolation polynomials on the triangle side Here interpolation is taken over a uniform grid with npor 2 interpolation knots including the two end points NLFEM Approximate solution determined by FEM over subdivision triangle where an addi tional uniform FEM partition on each grid triangle is chosen such that the stepsize is side length divided by nitrem 1 NUPA This is for additional uniform partition of all primary grid triangles into nypa X nupa equal subdomains i e the original side of the grid triangle is split into nypa sides of uniform partition subtriangles After this uniform refinement the degrees of freedom and the trial space of approximate Helmholtz solutions are defined using npor and NLFEM Chan
203. line for a polygonal grating generated by GEN_POLYGON or GEN_POLYGON2 file file2 to define polygonal boundary line for the coated layer of polygonal grating generated by GEN_POLYGON2 executable GEN_PROFILE to generate an input file for a profile grating given by c code first argument name without tag inp of file to be created second argument stepsize of polygonal approximation c code file profile c to define profile line for a profile grating of GEN_PROFILE executable GEN_PROFILES to generate an input file for a grating given by many profile lines defined by c code first argument name without tag inp of file to be created second argument stepsize of CLASSICAL polygonal approximation c code file profiles c to define profile lines for a profile grating of GEN_PROFILES executable GEN_PIN to generate an input file for a pin grating given by a profile line defined by c code first argument name of input file to be created second argument stepsize of polygonal approximation c code file pin c to define the profile line for a pin grating of GEN_PIN executable GEN_CPIN to generate an input file for a coated pin grating given by profile lines defined by c code first argument name of input file to be created second argument stepsize of polygonal approximation c code file cpin c to define profile lines for a coated pin grating of GEN_CPIN executable POINT_CLOUD_INP to generate an in
204. ll lines beginning with are comments HHH HH OH HHHH HHHH AA HE TE AE A A A A E E E E A E E A TEEPE EEE PE PEPER TE E AE E AE A TE E PE EE E E AE DE E E BE PP HEE HE HEHE EE E HE HE HEHE TETEE HEHEHE input file for FEM GFEM located in directory CONICAL FERRE TTT HAHAHAH RRRA RRR ETE TEE PETE TERE ETL EAE ATE EET PEPE PEE eT Name of the output file The tag res will be added The file will be written in the RESULTS directory Alternatively a path for the location of the file can be added before the name This must contain at ce least one slash E g for a file name res in the current working directory write name Name example TEE EE EEE EE E EEEE EE EEEE EEEE E H HEH H Should there be an additional output file in the old style of DIPOG 1 3 resp an eps file for FEM_CHECK Add no if not needed Add yes if needed The name will be the same as the standard output file given above but with the tag erg instead of res yes or no yes TERRA TTT ETE ETT TTT eee PeTE TE PETER TETE TEHE A TEEPE PEE BEEBE ee Number of coating layers over the grating N_co_ov The grating cross section consists of a rectangular area parallel to the axes This inhomogeneous part is determined by a triangular grid and can have already a few layers of coatings involved Ben
205. lower boundary line of the grating Now the index of the domain adjacent to the lower boundary line is changed to i_geom_param 2 The number nd_geom_param of real input parameters is equal to 2i_geom_param 2 i geom_param 3 1 The first 2i _geom_param 2 of these real valued parameters define the refractive indices n of the grating materials occupying the domains with the indices k 1 i geom_param 2 by n d_geom_param 2k 1 i d_geom_param 2k Clearly the real parts of all refractive indices must be positive and the imaginary parts non negative The imaginary part of the cover material must vanish The parameter d_geom_param k with k 2i_geom_param 2 j j 1 i geom_param 3 is just the height hj gt 0 cf Equation 10 8 in um of the interior corner point Q over the line through P and P Finally the last of the real input parameters d_geom_param k with the index k 2i_geom_param 2 i geom_param 3 1 is the threshold for the distance of the interior interface corner points to the boundary of the convex domain A cf Figure and the subsequent Equation 10 9 The distance is measured in um in the direction of the normal v to P1 P2 Of course if igeom_param 3 0 then the last real parameter is a dummy Setting upper bound equal to lower bound the user must fix the refractive indices of the substrate and cover material and the last threshold parameter i e the equality du_geom_param j dl_geom_param j j 1 2
206. ls are d_geom_param k 1 i d_geom_param k 2 d_geom_param k 3 i d_geom_param k 4 where k i_geom_param 3 Clearly the real parts of these indices must be positive and the imaginary parts non negative The imaginary part of the cover material must vanish Moreover the refractive indices must be fixed i e du_geom_param k dl_geom_param k k i_geom_param 3 1 i geom_param 3 4 10 2 3 Profile grating determined by general polygonal profile curve I The second class i_geom_param 1 2 is a profile grating without coatings defined by a pro file curve which is the cross section of the interface between cover and substrate material Thus the number of materials is i_geom_param 2 2 The profile is supposed to be a polyg onal curve The class has no name parameters ns_geom_param 0 and ni_geom_param 3 integer parameters The value i geom_param 3 is the number N of knots of the profile curve in the interior of the period i e with x coordinate strictly between zero and period d In other words the grating profile is the polygonal curve connecting the points 0 0 t1 y1 x2 Y2 seen N YN d 0 with arbitrary real values x and y such that 0 lt a2 lt d and such that the non adjacent polygonal sides do not intersect each other no self intersection The number of real parameters is nd_geom_param 2i_geom_param 3 4 and the first 2i_geom_param 3 87 parameters define the profile curve by 2 d_geom_param 2
207. lt 1 0 first column first layer with 0 2 lt y lt 0 second with 0 lt y lt 0 5 third with 0 5 lt y lt 0 7 and fourth with 0 7 lt y lt 1 second column first layer with 0 2 lt y lt 0 0 second with 0 0 lt y lt 0 50 third with 0 50 lt y lt 0 90 and fourth with 0 90 lt y lt 1 third column first layer with 0 2 lt y lt 0 00 second with 0 00 lt y lt 0 500 third with 0 500 lt y lt 0 900 and fourth with 0 900 lt y lt 1 If an input file for a simple layer grating is needed then this can be accomplished by calling the executable GEN_LAMELLAR from the subdirectory GEOMETRIES More precisely if the file GEOMETRIES lamellar INP contains the numbers each number in a separate line 1 1 0 2 and 0 8 then the command GEN_LAMELLAR name lamellar INP creates the file name inp of the desired layer grating with layer material s t the y coordinate satisfies 0 2 lt y lt 0 8 all coordinates are normalized with respect to the period period corresponds to x 1 If an input file for a grating with a polygonal profile line is needed cf Figure 11 then this can be accomplished by calling the executable GEN_POLYGON from the subdirectory GEOMETRIES More precisely if the file GEOMETRIES filel contains the corner points of a polygonal profile line in GEOMETRIES filel in each line beginning without there should be the x and y coordinate of one of the consecutive
208. lt posb d lt 1 posa posb d_geom_param 5 i_geom_param 3 2 ratio of distance of left lower corner from the left boundary line of the period and distance of right lower corner from the left boundary line of the period 0 lt posa posb lt 1 If not both of the parameters posb d and posa posb are fixed then we require 175 HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHH RR zzz posa posb gt 0 for posb d 1 hel h_m d_geom_param nd_geom_param 9 m i_geom_param 5 height of extra layer beside stack bridge over the lower line of the m th trapezoid relative to height of m th trapezoid in micro meter O0 lt hel lt 1 hl_k d_geom_param 5 i_geom_param 3 3 k height of k th layer beneath stack in micro meter hl_k gt 0 k 1 i_geom_param 4 Penalty term parameters phi_min d_geom_param nd_geom_param 2 minimal angle for sidewall angles of trapezoids except the uppermost trapezoid phi_max d_geom_param nd_geom_param 1 maximal angle for sidewall angles of trapezoids except the uppermost trapezoid phi_fac d_geom_param nd_geom_param factor of penalty term In other words to exclude solutions with too large or too small sidewall angles phi a penalty term of the following form is added to the objective functional for each sidewall angle phi l 2 phi_fac lt max 0 phi phi_max l 2 max O phi_min phi gt Following parameters must be fixed by setting upper bound lower boun
209. lution d_opt 2 Threshold 9 _ gt 0 stop if gradient is less than gra should be about approximation error of gradient or less d_opt 3 Threshold norm gt 0 stop if relative change of the norm of the gradient is less than Enorm for the last Nyorm steps choose e g Enorm 0 01 The parameters i_opt 1 and d_opt i i 1 3 should be chosen as recommended For the error d_opt 2 Egra of the gradient computation a first estimate can be obtained using the command OPTIMIZE g name dat On the other hand the user can choose niter_max larger than necessary and the positive parameter d_opt 2 smaller than recommended In the worst case a large number of unnecessary iteration steps with very small changes in the iterative solutions are performed at the end of the optimization procedure Even these redundant iteration steps can be interrupted Indeed the user can invoke the optimization by the command OPTIMIZE f name dat In this case the actual iterative solutions are printed on the screen and pushing the Ctrl C keys the user can stop the iteration whenever he observes that the iterative solutions do not improve In some cases this method is the fasted local algorithm of DIPOG 2 1 For about the same number of iteration steps the number of function and gradient evaluations is less in the Newton type iteration since multiple function evaluations in the line search are avoided The price is that even for accurate gradient
210. m prescribed integer iterations then stop the iteration and go to the next Step 3 Almost unchanged means PW Fr PV FG IPCV Flr with a prescribed small number Enorm Now suppose none of the stopping conditions is satisfied For prescribed 0 gt 0 and q with 0 lt q lt 1 introduce the parameter 0 00 Compute the search direction of the step sdrj sdsl sdsu sddl sddu by solving F sdr sdsl sdsu sddl sddu 0 approximately by the Newton step lt Enorm l 1 norm 10 15 sdr sdsl sdsu sddl sddu VF rj slj suz dlj duj Fo r3 sly suj dlj duj 27Take heed that a symbol with lower index i denotes the ith component of a vector whereas the same symbol with lower index j denotes the jth iterate of the vector in an iterative process 102 Note This that the computation of V F requires the computation of the Hessian V f r is approximated using the Broyden Fletcher Goldfarb Shanno update which is based on first order derivatives only Now knowing the search direction perform the line search 2 1 2 2 2 1 2 2 Initialize amp Omar Eventually reduce a such that sl asdsl gt 0 su asdsu gt 0 dl asddl gt 0 and du a sddu gt 0 hold Set Te rj a sdr Slag sl a sdslj Suj 1 suj a sdsuj dlj41 dl a sddl and duj du asddu If the difference r 41 r is less than the prescribed small pos
211. main program press Bar Space After the installation with the package GNUPLOT the pictures are created interactively with the program on the main terminal window To continue the computation of the main program click the main terminal window and press Enter Return Further if you use the command GFEM_MATLAB name2 then you have the same computational results as in point Additionally you will obtain data files prepared for a Matlab call More precisely using GFEM_MATLAB requires a data file name2 dat for a calculation over a single wavelength a single angle of incidence and a single level of discretization Note that if the last should be larger than one then a single level computation can be enforced by defining the angle of incidence with an incremental input e g the input I 45 46 20 for the angle results in a single computation with an angle of 45 Depending on the input polarization state the program produces the files fet RP_TE m real part of z component of electric field vector for TE polarization fet IP_TE m imaginary part of z component of electric field vector for TE polarization fet IN TE m intensity of electric field vector for TE polarization fet RP_TM m real part of z component of magnetic field vector for TM polarization fet IP_TM m imaginary part of z component of magnetic field vector for TM polarization fet IN TM m intensity of electric field vector
212. micro meter at which the lateral width of the bridge is given is 0 01 number of trapezoidal bridges per period is 7 lateral widths in micro meter of these bridges are 0 05 0 075 0 05 0 05 0 075 0 075 and 0 05 x coordinates in micro meter of the mid points of the bottom lines of these bridges are 0 15 0 3 0 4 0 5 0 6 0 75 and 0 9 e g lAmellar 3 4 0 2 06 0 2 1 0 135 HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHH aE aaa OO a OiT 0 50 0 90 00 0 500 0 900 23 gt LAMELLAR GRATING rectangular grating consisting of several materials placed in rectangular sub domains with 3 columns each divided into 4 rectangular layers first column with x coordinate in 0 lt x lt 0 2 given in micro meter second column with 0 2 lt x lt 0 6 third columnn with 0 6 lt x lt period period given above whole grating with y coordinate s t 0 2 lt y lt 1 0 first column first layer with 0 2 lt y lt 0 second with 0 lt y lt 0 5 third with 0 5 lt y lt 0 7 and fourth with 0 7 lt y lt 1 second column first layer with 0 2 lt y lt 0 0 second with 0 0 lt y lt 0 50 third with 0 50 lt y lt 0 90 and fourth with 0 90 lt y lt 1 third column first layer with 0 2 lt y lt 0 00 second with 0 00 lt y lt 0 500 third with 0 500 lt y lt 0 900 and fourth with 0 900 lt y lt 1 e g JAmellar 1 1 0 2 0 8 gt SIMPLE LAYER special case of lamellar grating with layer material s t y coordinate satisfies 0 2 lt y lt 0 8 given
213. n ne ur U n ne ur 2 8 n Finally we note that the case of an incident wave propagating in a direction of the x y plane together with an arbitrary polarization is the superposition of TE and TM polarization 2 3 Conical problems The essential difference between the classical diffraction of the last two subsections and the conical one is that the direction of the incident light wave is oblique i e it is not restricted to the x y plane Whereas in the classical case the directions of the finitely many reflected and transmitted plane wave modes remain located in the x y plane now they are located on a cone in the x y z space The FEM approach is analogous to the 17 classical case However instead of a transmission problem for a scalar Helmholtz equation Maxwell s system reduces to a coupled system of two scalar Helmholtz equations for the two transverse components E and H of the electric and magnetic field Consequently we have two Rayleigh expansions and two sequences of Rayleigh coefficients More precisely skipping the time harmonic factor we have an incident wave of the form Fiincident aU 2 E exp ilox By 2 H exp iloz py 2 Horeng y z ee with constant vectors E and H and a wave vector k a 8 y such that the wave number k is the modulus of K and the direction of the incoming plane wave is k kt Here k is the same
214. n cf and choose Cea such that Cea times the objective functional is less than one d_opt 3 Threshold mu gt 10 iteration stops if deviation of iterative multipliers is less than mu cf Step 2 and choose its value about the maximum of i the desired accuracy of the constraint conditions and ii the desired accuracy of the minimum value 105 of the objective functional multiplied by Ceat d_opt 4 Threshold 9 4 gt 107 inner iteration stops if norm of gradient of Lagrangian is less than Egra X Cea Should be about discretization error of gradient or less d_opt 5 Threshold ace gt 10713 inner iteration stops if stepsize in line search is less than Eacc iterative solution is considered to be at the boundary if its distance to the boundary is less than ace cf Step 3 and choose e g Eace 10714 d opt 6 Constant c1 0 lt c lt 1 in Armijo stopping criterion for line search in inner conjugate gradient iteration cf Step 2 and choose e g C 0 001 d_opt 7 Maximal stepsize factor Ama gt 0 in line search cf Step 2 and choose e g mar 1 d opt 8 Threshold Enorm O lt Enorm lt 1 stop if relative change in norm of gradient is less than Enorm for the last Nnorm steps cf Equation 10 18 and choose e g Enorm 0 01 The parameters i_opt 2 and d opt i i 1 2 6 7 8 should be chosen as recommended For the error d_opt 4 Egra of the gradient c
215. n the upper resp lower boundary lines must be rational More precisely for left and right end points A and B the node points must be of the form A B A xr with r p q and 0 lt p lt q lt 1000 3 5 Code words to indicate special geometries One can indicate special grating geometries in the input file name dat by special code words We explain these here Grating data echellea R 0 3 0 03 0 04 indicates an ECHELLE GRATING TYPE A right angled triangle with hypotenuse parallel to the direction of the periodicity right interior angle greater than 45 cf Figure 6 with depth of 0 3 um i e triangle height 0 3 um and with coated layers of height 0 03 um resp 0 04 um over the first resp second part of the grating measured in direction perpen dicular to the echelle profile height greater or equal to zero Grating data echellea L 0 3 0 03 0 04 indicates an ECHELLE GRATING TYPE A right angled triangle with hypotenuse parallel to the direction of the periodicity left interior angle greater than 45 with parameters like above Grating data echellea A 60 0 03 0 04 indicates an ECHELLE GRATING TYPE A right angled triangle with hypotenuse parallel to the direction of the periodicity with left interior angle 60 i e the depth is equal to the period multiplied by sin a cos a and other parameters like above 34 Figure 6 Echelle grating of type A Grating data echelleb 60 0 0
216. n n_qua_2_tr n_qua_tr numbers in n_qua_tr lines n_qua_tr numbers in n_qua_tr lines n_qua_tr numbers in n_qua_tr lines TERMS FIRST TE or S TRANSMITTED EFFICIENCY lines lines lines EFFICIENCY lines lines c_qua_2_tr n_qua_2_tr numbers in n_qua_2_tr lines QUADRATIC TERMS FIRST TE or S REFLECTED PHASE SHIFT n_phs_i_re 1 w_phs_1_re n_phs_i_re numbers in n_phs_1_re lines 30 o_phs_1_re n_phs_1_re numbers in n_phs_1_re lines 0 c_phs_1_re n_phs_1_re numbers in n_phs_1_re lines 91 464440 QUADRATIC TERMS FIRST TE or S TRANSMITTED PHASE SHIFT n_phs_1i_tr 0 w_phs_1_tr n_phs_1_tr o_phs_1_tr n_phs_i_tr c_phs_1_tr n_phs_i_tr QUADRATIC TERMS SECOND n_phs_2_re 0 w_phs_2_re n_phs_2_re o_phs_2_re n_phs_2_re c_phs_2_re n_phs_2_re QUADRATIC TERMS SECOND n_phs_2_tr numbers in n_phs_1_tr numbers in n_phs_1i_tr numbers in n_phs_1_tr lines lines lines TM or P REFLECTED PHASE SHIFT numbers in n_phs_2_re numbers in n_phs_2_re numbers in n_phs_2_re TM or P TRANSMITTED 195 lines lines lines PHASE SHIFT 0 w_phs_2_tr n_phs_2_tr numbers in n_phs_2_tr lines o_phs_2_tr n_phs_2_tr numbers in n_phs_2_tr lines c_phs_2_tr n_phs_2_tr numbers in n_phs_2_tr lines QUADRATIC TERMS REFLECTED ENERGY w_qua_ene_re 0 c_ene_re no line if w_ene_re 0 QUADRATIC TERMS TRANSMITTED ENERGY w_qua_ene_tr
217. n theta cos phi cos theta cos phi sin phi with the restriction 90 lt phi theta lt 90 If type of polarization is TE TM TP then the incident light beam takes the direction sin theta cos phi cos theta sin theta sin phi with the restriction 0 lt theta lt 90 Either add a single value e g 45 Either add more values by e g EE V 5 63 64 65 69 Oe Tra The last means that computation is to be done for the angles from the Vector of length 5 TO Reg pA BB ct Gaa ad LO Pe Or add e g wc 1 45 662 The last means that computation is to be done for the angles 45 i 2 with i 0 1 2 and with angle 45 i 2 less or equal to 56 Note that either the wave length or the angle of incident wave theta must be single valued Angle T 30y Sle 2 A EE E EE AE E E AE TE TEPER ESE ETSI EEE EEE TE H HE AE HE HEEE E E PETE EEE EE E E HE EE E HE AE HEERE HE Angle of incident wave in degrees phi HHHHHHHHHHH HHH HHH HHH OH OF a i If type of polarization is pol then the incident light beam takes the direction sin theta cos phi cos theta cos phi sin phi with the restriction 90 lt phi theta lt 90 If type of polarization is TE TM TP then the incident light beam takes the direction sin theta cos phi cos theta sin theta sin phi with the restriction 0 lt theta lt 90 E
218. n we get an ECHELLE TYPE A grating profile with right blaze angle greater than 45 with depth d and without coated layer cf Section 3 5 The profile is vertically shifted by h wm Note that 0 lt d and d is less than half the period d If the input letter R is replaced by L then the left interior angle is greater than 45 Moreover if the input letter R is replaced by A and the following number d by a then the left interior angle is a degrees If the defining two lines are lineg 1 echelleb a line hy then we get an ECHELLE TYPE B grating profile with angle a and without coated layer cf Section 3 5 The profile is vertically shifted by h um Note that 0 lt a lt 90 If the defining two lines are lineg 1 echelle La R 8 lines hy then we get a GENERAL ECHELLE grating profile with a left blaze angle a in degrees and with a right blaze angle 8 in degrees Instead of the two inputs L a and R 8 one can choose also the inputs A y for an apex angle y in degrees or D d for a depth of the grating equal to dp in wm Moreover any combination of two inputs of the types A y L a R 8 and D d is accepted However the choice A y and D d might be ambiguous By definition it fixes an echelle grating with right blaze angle larger than the left To get the flipped grating with left blaze angle larger than the right the input should be A y D d
219. n_ lines each containing a number c_ 186 HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHH RHR Raz for one value of the wavelength resp angle theta resp angle phi and for one order of mode If the n_ values of c_ depends on m1 wavelengths m2 angles theta resp ml wavelengths m2 angles phi resp m1 angles theta m2 angles phi then the input consists of mi m2 n_ 1 lines The first contains the indicator W T resp W P resp T P and is followed by mi m2 n_ lines each containing a number c_ for one pair of values wavelength angle theta resp wavelength angles phi resp angle theta angle phi and for one order of mode If the n_ values of c_ depend on mi wavelengths m2 angles theta m3 angles phi then the input consists of mil m2 m3 n_ 1 lines The first contains the indicator WIP and is followed by m1 m2 m2 n_ lines each containing a number c_ for one triple of values wavelength angle theta angle phi E g c_ lambda1 mode1 c_ lambda1 mode2 c_ lambdal moden_ c_ lambda2 mode1 c_ lambda2 mode2 c_ lambda2 moden_ c_ lambdam mode1 J C_ lambdam mode2 c_ lambdam moden_ 77 c_ lambda1 phi1 mode1 c_ lambda1 phil mode2 c_ lambdai phiil moden_ c_ lambdal phi2 mode1 c_ lambdal phi2 mode2 c_ lambda1 phi2 moden_ c_ lambda1 phim2 mode1
220. nary part of a refractive index cf the following subsections of Section 10 2 Suppose now that this should be fixed to a constant value i e the real parameter is not included into the set of optimization parameters Simi larly as for the simulation by DIPOG 2 1 cf Section 4 such parameters can be chosen from predefined lists For instance if the refractive index in the simulation would be determined by the name AlAs then the parameter input d_geom_param j of the real part of the refractive index can now be given as the string Re AlAs Clearly the parameter input d_geom_param k of the corresponding imaginary part of the refractive index is to be given as Im AlAs In any case if a real resp imaginary part of a refractive index is determined by such a string input then the corresponding imaginary resp real part must be defined the same way Moreover the corresponding bounds dl_geom_param j du_geom_param j dl_geom_param k and du_geom_param k must be determined with the same input strings Note that for more than one wavelengths involved in the computation of the objective functional DIPOG 2 1 cannot optimize the refractive indices depending on the wavelengths Therefore for multiple wavelengths all real parameters must be fixed to a constant value independent of the optimization process but of course depending on the wavelength In other words for multiple wavelengths the input by strings is mandatory
221. nce to boundary slack variable is less than this should be the expected accuracy accuracy threshold eps_ste stop iteration if improvement step of iterates is less than this should be a tenth of the previous accuracy accuracy threshold eps_gra stop iteration if norm of reduced gradient is less than this should be about discretization error of gradient calculation threshold eps_norm if relative difference of two squared norms is less than this number then the norms are considered to be the same e g te 3 METHOD ind_opt 3 maximal number liter_max of conjugate gradient steps in inner iteration e g 50 number n_norm of same gradient norms after which the algorithm stops e g 3 value rho for parameter in algorithm factor in augmented Lagrangian e g 5 calibration factor c_cal of objective functional in modified Lagrangian sum of objective functional plus perturbation term smaller value enforces better fulfillment of constraints 198 HHHHHHHHHHHHHHHHHHHHHHH HHH HHH HH HH OH OF d_opt 3 d opt 4 d_opt 5 d_opt 6 d _opt 7 d_opt 8 HAEE PET AE HE EE A HE HE EE EEE HHHH HHH HH HHH HH HH HOH OF should be such that value of objective functional multiplied by c_cal is much less than one threshold eps_mul if deviation in iterates of multiplyer is less than this number divided by r then iteration stops should be about maximum of i the accuracy of
222. ncoming light wave length I 0 6350000 0 6360000 0 0020000 type of output res TE TM type of polarization TP polarization angle 20 0000000 I 30 0000000 31 0000000 2 0000000 I 47 0000000 48 0000000 2 0000000 angle of incidence theta angle of incidence phi data of generalized FEM n_DOF 0 n_LFEM 0 n_UPA 0 203 grating parameters class number 5 number of materials 5 number of real param 13 real param 1 1 0000000 real param 2 0 0000000 real param 3 in 1 0000000 1 5000000 real param 4 in 0 0000000 0 5000000 real param 5 in 1 0000000 1 5000000 real param 6 in 0 0000000 0 5000000 real param 7 in 1 0000000 1 5000000 real param 8 in 0 0000000 0 5000000 real param 9 1 5000000 real param 10 0 0000000 real param 11 in 0 1350241 0 0953109 real param 12 in 0 2417941 0 1350241 real param 13 0 1000000 number of integer param 6 integer param 3 2 integer param 4 11 integer param 5 9 integer param 6 3 number of char strg par 1 char strg par 1 initial parameters real parameter 3 1 3500000 real parameter 4 0 1000000 real parameter 5 1 2500000 real parameter 6 0 1500000 real parameter 7 1 1500000 real parameter 8 0 1000000 real parameter 11 0 0500000 real parameter 12 0 0500000 objective functional 0 0550000 eff_1 tra 1 11 0418380 2
223. nerate interfaces automatically created if nrand 0 otherwise nrand has to equal ncorn nla nld nmld tn1b optional standard Gaussian distributed random numbers nla widths of layers above multilayer system nld widths of layers in each multilayer nlb widths of layers below multilayer system standard deviation sigma i of each layer interface for 1 0 nla nld nmld nlb 1 correlation length corl i of each layer interface for i 0 nla nld nmld nlb 1 Note that the first of the subsequent refractive indices of the grating must be that of the superstrate resp that of the adjacent upper layer If the width of the additional layers is positive then the last of the subsequent refractive indices of the grating must be that of the substrate resp that of the adjacent lower layer e g rough_mls k name gt MULTILAYER SYSTEM WITH ROUGH INTERFACES k TIMES just like above However k random realizations are computed and the output values are replaced by the mean values over 124 example the k computations Standard deviations are added too Grating data AEEA E E HE AE E AE EE E EE AE EAE PEEP STREP E HE E E HE DE TE HETE E HE EE E AE HEEE HE HE EEE PEE E EE GE E E E E HE EE EE E E HEETE E Number of different grating materials N_mat This includes the material of substrate and cover material For exam
224. ng optical index third the imaginary part of the corresponding index Optical index Air HEHEHE RRR RAE RRR aaa Optical indices of the materials of the upper coating layers This is c times square root of mu times epsilon N_co_ov entries Needed only if N_co_ov gt 0 Else no entry and no line For a multilayer system the indices of the n3 layers in the groups must be given only once HHH HH HH HHH HH HF OF I e for a multilayer system ni n3 n4 input lines are needed Optical indices 1 2 1 3 HEHEHE A E EEEE E E EEE EEEE EEA EHH EHE Optical indices of the materials of the lower coating layers This is c times square root of mu times epsilon N_co_be entries Needed only if N_co_be gt 0 Else no entry and no line Optical indices 1 6 HEHEHE RRR Raa RRR aaa Optical index of substrate material This is c times square root of mu times epsilon Optical index 1 5 i 0 FERRETTI HARUAR ENR APANA HRR RA ARRERA EE PETE TE PEPER EAE HE ERE EERE PEE BEE TET Type of output results Either TE TM results in terms of TE and TM part of Wave Either Jones results in terms of Jones vector representation br 3 Com 2 results in terms of the component in the z axis that is in the direction of the grooves For more details cf Section 2 3 in USERGUIDE ps Type 3 Com HRR RRRRRH HARUAR PEPPER TET TET EE EE TEE N RR RR ARARA REEERE HHRHH HHHH
225. nguished by different optical indices in different colours Press Escape or Bar Space to end the check If you enter SHOW v namel inp then additionally an eps file of the picture is produced Alternatively to create namel inp one can call the executable GEN_INPUT from the subdirectory GEOMETRIES and work interactively Just enter the command GEN_INPUT This program prompts you for everything needed Nevertheless we recommend the first way of copying and modifying an existing file 25 The programs of DIPOG 2 1 are based on a coupling of finite elements and boundary elements over the upper and lower boundary lines of the FEM domain This coupling requires that the refractive indices of the materials on both sides of the boundary lines coincide Therefore the FEM domain is extended by additional rectangular strips adjacent to the upper and lower boundary lines The positive width relative to the period of these two strips is fixed in the namel inp file e g by the lines Width of additional strip above and below 0 5 The material of the upper strip is the grating material of index one that of the lower has index Nmat In order to have the same material on both sides of the upper and lower boundary lines the refractive index of the lowest upper coated layer resp the cover material must be the same as that of the grating material with index one and the refractive index of the highest lower coating layer resp the su
226. ningful then the refractive indices should be ordered according to the location from above to below If an input file namel inp is used then the optical index of a subdomain with the material index j is just the j th optical index following below If grating is lAmellar then first material is cover material last material is substrate and all other materials are ordered from left to right and inside the columns from below to above For technical reasons the index of the material adjacent to HHHHHHHHHHHHHHHHHHHHHHHHHHHH HH HHH HH HH OH OF HH HH HH HH H OF 141 the upper line of the grating structure must coincide with that of the material in the adjacent upper coated layer resp in the adjacent superstrate Similarly the index of the materials adjacent to the lower line of the grating structure must coincide with that of the material in the adjacent lower coated layer resp in the adjacent substrate N_mat numbers are needed Optical indices 1 3 10 16 FI 0 TERE IEIE EE BPE PEPE PETE EERIE HETIL EEE eee TEE RRRA RAAR RAHNER Number of levels Lev In each refinement step the step size of the mesh is halved Lev refinement steps are performed Alternatively one can prescribe an bound for the maximal error of the efficiencies E g the input e 1 means that the level for the computation is the smallest positive integer such that all efficiencies are
227. nkst 1 2 2 2 P_i P_ i 1 gt EPSilon Period b j 1 2 nks m 1 2 j 1 j 2 nks 1 i nks 1 m 1 nks j m m if m lt j m m 2 if m gt j P_j P_m P_j P_ n 1 P_m P_ m 1 gt EPSilon P_m P_ m 1 c no intersection of P_ i 1 P_i and P_ j 1 P_j for i j 1 2 3 nks 1 if li jl gt 1 Following parameters must be fixed by setting upper bound lower bound HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHH HH HH HH OH OF d_geom_param k k 2 i_geom_param 3 1 2 i_geom_param 3 5 Fe CLASS 4 i_geom_param 1 4 stack of trapezoids ie ea ee a ee eS i Stack grating consisting of several trapezoids with refractive indices included into the set of optimization parameters Number of trapezoids is prescribed 165 HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHH aaa OO Whole stack in one period of the grating Lower side of stack is fixed by two parameters param_1 posb d ratio of distance of right lower corner from the left boundary line of the period and period d param_2 posa posb ratio of distance of left lower corner from the left boundary line of the period and distance of right lower corner from the left boundary line of the period Each trapezoid is determined by its hight by the lower side which is the upper side of the adjacent lower trapezoid and by the upper side prescribed by the two parameters param_k_1 b_k d ratio of distance of right upper corner from the left b
228. of additional strip above and below 0 200000 Grid points 000000 0 800000 500000 0 800000 000000 0 400000 250000 0 400000 000000 0 400000 750000 0 200000 000000 0 200000 000000 0 000000 250000 0 200000 000000 0 200000 000000 0 600000 000000 0 800000 000000 000000 000000 4 mats 1 2 mat 1 5 mat 1 4 mat 1 000000 4 mat 2 fac 0 300000 1 1 1 i 1 2 fac 2 2 2 2 6 mat 2 fac 1 000000 3 3 3 3 fac fac fac v v v OONo Qe or oor OF o Oo Oo Oo ji O 7 mat 3 fac 1 000000 9 mat 3 fac 6 mat 3 fac 12 10 mat 3 fac 000000 000000 000000 ji j FS eS PP es ee DROWDDF AWA A m N Ko INPUT DATA 143 COATED LAYERS nmb of upper layers 2 corresponding widths last width last width of dat file width in grating geometry 0 50 micro m 0 40 micro m nmb of lower layers 3 corresponding widths first width first width of dat file width in grating geometry 0 40 micro m 0 30 micro m 0 20 micro m REFRACT INDICES cover material 100 2 000 layers above grating e N H jo Q O layers below grating 2 30 1 0 00 2 20 i 0 00 210 1 0700 substrate material 2 00 i 0 00 FURTHER DATA temperature 20 00 degrees Celsius wave length 0 635 micro m angle of inc theta 65 00 degrees polarization type TM GRATING grating pe
229. of each triangle in x direction must be less than one half period first add the nodes of all the triangles later give the triangles by the indices of their nodes HE Se H H EEEE EEE EEEE Each point in a separate line Scaled to period 1 Input ended by 1 1 Grid points Le oF OOF oO fF Oo Oo Oo O 000000 0 800000 500000 0 800000 000000 0 400000 250000 0 400000 000000 0 400000 750000 0 200000 000000 0 200000 000000 0 000000 250000 0 200000 000000 0 200000 000000 0 600000 000000 0 800000 1 TERRA EEE EE EEEE A E E E EEEE E EEEE EAEE AEAEE H HE Triangles H H HHH Each given in a separate line by 5 parameters namely by index of first point by index of second point by index of third point by index of material and by additional factor for maximal mesh size of partition inside the triangle 113 Input ended by 1 1 1 1 1 riangles 2 1 000000 000000 000000 000000 300000 000000 0 7 3 1 000000 1 9 3 1 000000 2 6 3 1 000000 6 12 10 3 1 000000 at ab Sh a1 Sl OrPrPRONA POrRrRPRPRE T T3 4622 6752 38 42 8942 4962 6 81 91 BEEBE RE A T E E HERESIES EERE AP ETE EET PERE TE LESTE TEEPE TE EEE TE ETE E HEDE HE H EHP PE EE End A E E E HE A AE E TE EP EEE REE TE HE E TE PE HE TE AEE SPE AEE EH PEE TE AEE E HEE E E HE E E EEE EE 12 2 Data file example dat for CLASSICAL makefile HHEHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH
230. olution of this sparse finite element system is reduced to the solution of an npor 1 Npor 2 dimensional finite element system where the trial functions are the piecewise linear interpolations of the bubble functions for the p version of the finite element method The polynomial degree is p npor 1 Thus the generalized finite element method turns into a p method with elimination of internal degrees of freedom and with finite element matrix approximated by piecewise linear interpolation 7 Computation of Efficiencies Using FEM GFEM in CONICAL The computation in the case of conical diffraction is completely the same as for the classical computation cf the result file enclosed in point 12 7 The same names of executables can be used as in Section 2 The only differences are All computations are to be done in CONICAL instead of CLASSICAL Of course the now used input file name2 dat cf the enclosed data file in 12 4 is longer than that of the classical case To create such a file copy the example file example dat in CONICAL not that in CLASSICAL The input file for the generalized method generalized Dat is the same but has a 67 T T Tanne A Tr nare 2 eff 0 1 1 i 1 1 1 1 1 0 63 0 64 0 65 0 66 0 67 0 68 0 69 0 7 0 71 0 72 wave_length Figure 26 Efficiencies depending on wavelength Output of PLOT_PS new name Now it is called conical Dat 8 Plot a Graph
231. olution r and go back to Step 2 5 Final output Accept and print ropt as the final solution Print the corresponding value f Top For comparison print the last iterative solution r and the correspond ing value f r The method of simulated annealing ind_opt 4 requires one integer input value ni_opt 1 This iopt 1 is the number of restarts nest gt 0 The number of real parameters is nd_opt 5 These parameters are d_opt 1 Initial temperature tini gt 0 chose tin equal to the variation of the objective functional or set tini 0 to invoke an automatic choice of tini d_opt 2 Cooling factor crac 0 5 lt Cfact lt 1 in each iteration step the temperature is multiplied by Cfact a slower logarithmic cooling scheme is applied if crace 1 choose e g Cfact 0 95 d_opt 3 Stopping threshold Estop O lt Estop lt 1 algorithm stops if the difference of the values of the objective function at actual and previous step differ by a value greater than zero but less than Estop choose e g Estop 0 d_opt 4 Initial value oimn gt 0 of radius o jn min fu l of neighbour hood where the random search for a new iterate is performed choose o 1 d_opt 5 Reduction factor fac 0 lt fact lt 1 to reduce the radius of neighbourhood in each step multiplying o by fact choose e g fact to be the square root of the cooling factor C fact The maximal number of iterations niter_max s
232. olygor quadrilateral domain between corresponding segments on the left of first polygon these quadrilaterals must be disjoint last line should be End then the command GEN_POLYGON2 name filel file2 creates the file name inp of the desired polygonal grating If an input file for a grating determined by profile line given as f t fy t O lt t lt 1 is needed then this can be accomplished by calling the executable GEN_PROFILE from the subdirectory GEOMETRIES More precisely suppose the profile line f t f t 0 lt t lt 1 is given by the functions t gt f t and t f t defined by the c code in the file GEOMETRIES profile c Then GEN_PROFILE name 0 06 creates the file name inp of the desired profile grating where the profile curve is approxi mated by a polygonal line with a stepsize equal to 0 06 times the length of period cf Figure 13 where f t t and f t 1 5 0 2 exp sin 6rt 0 3 exp sin 87t 27 If an input file for a grating determined by more than one non intersecting and peri odic profile lines given as fs j t fuj t O lt t lt 1 j 1 n is needed then this can be accomplished by calling the executable GEN_PROFILES from the subdirectory GEOMETRIES More precisely suppose n and the profile lines f j t fy g t O lt t lt 1 are given by the c code in the file GEOMETRIES profiles c Then GEN_PROFILES name 0 06 11To make it precise
233. om_param 2 i_geom_param 2 i_geom_param 3 1 lt 1000 Following parameters must be fixed by setting upper bound lower bound d_geom_param 2 i_geom_param 2 i_geom_param 3 1 d_geom_param k k 1 2 i e the refractive indices of the cover material d_geom_param k k 2 i_geom_param 2 1 2 i_geom_param 2 i e the refractive indices of the substrate material HHHHHHHHHHHHHH HHH HHH HHH HH HHTHHHHHHHHHHHHHHH CLASS 6 i_geom_param 1 6 EUV bridge Stack of several trapezoids bridge in grating with refractive indices included into the set of optimization parameters Non stop layers of different heights beneath the stack with refractive indices included into the set of optimization parameters In other words Some of the lower additional layers can be added through the set of optimization parameters Fixed further layers can be added in the GRATING ILLUMINATION part of this input file Extra layer beside stack height gt zero If upper line of this layers contains a corner of the trapezoids in the stack then the height of the extra layer and all trapezoid heights of trapezoid beneath this line must be fixed by setting upper bound equal to lower bound and refractive indices included into the set of optimization parameters Number of trapezoids and layers is prescribed Whole stack bridge in one period of the grating sidewall angles can be restricted by penalty terms Lower sid
234. om_param i lt 1 Finally note that the automatic generation of an additional strip for the FEM domain is switched off compare Section if the classical case of TE polarization is considered i e type of polarization is TE angle of incidence is zero and output data is presented in the TE TM form of Section 2 3 and if the x coordinates of the lower corner points of the stack of trapezoids are fixed to period times rational number du_geom_param 5 N 1 dlgeom_param 5 N 1 du geom param 5 N 2 dlgeom_param 5 N 2 du geom_param 5 N 1 a k l E N h lt 1000 du geom param 5 N 2 du_geom_param 5 N 1 k gt A k lo N l lt 1000 In the case that the generation of an additional strip is switched off the material of this strip i e the material of the substrate resp of the adjacent lower layer must not be counted in the number of materials given in parameter igeom_param 2 In order to avoid any restriction to rational values the automatic generation of an additional strip can be enabled by setting the environmental variable ADD_STRIPS to yes 10 3 Numerical methods of optimization 10 3 1 General parameters of optimization algorithm All the optimization algorithms implemented for DIPOG 2 1 are iterative So the first control parameter is in common Its the number niter max gt 0 of maximal iterations The level dependent input is described at the end of Section If the iteration do
235. omponent to d_opt 2 d_opt 3 upper lower bound is less than this then point is considered to be at boundary should be about desired accuracy threshold eps_gra for gradient to stop iteration should be about discretization error of gradient calculation threshold eps_norm if relative difference of two squared norms is less than this number then the norms are considered to be the same e g te 2 200 FHR RRHH HAA LEVENBERG MARQUARDT METHOD ind_opt 6 ni_opt 0 nd_opt 4 d_opt 1 scaling factor mu for initial regularization of symmetrized Jacobian d_opt 2 stopping threshold epsilon1 stop if gradient norm J T el _inf lt epsilon1 d_opt 3 stopping threshold epsilon2 stop if increment norm Dp l 2_2 lt epsilon2 Dp 2_2 with Dp 2_2 the term of previous step d_opt 4 stopping threshold epsilon3 stop if least square functional e 2_2 lt epsilon3 ERRRRREEEHUMH HHHH Maximal number of iterations 10000 Indicator for optimization method al conj grad meth ni_opt ak number of integer param i_opt 3 n times the same grad norms gt stop nd_opt 5 number of real param d_opt ii maximal stepsize factor in line search 0 001 constant c_1 in Armijo criterion 1e 3 expected accuracy threshold 1e 2 gradient accuracy threshold 1e 2 grad norm deviation threshold Scaling parameters d_geom_scal i i 1 1 1 1
236. omputation a first estimate can be obtained using the command OPTIMIZE g name dat On the other hand the user can choose niter_max and i opt 1 larger than necessary and the positive parameters d_opt i i 3 4 5 smaller than recommended In the worst case a large number of unnecessary iteration steps with very small changes in the iterative solutions are performed The redundant iteration steps of the outer iteration however can be interrupted Indeed the user can invoke the optimiza tion by the command OPTIMIZE g name dat In this case the actual iterative solutions are printed on the screen and pushing the Ctrl C keys the user can stop the iteration whenever he observes that the iterative solutions do not improve 10 3 5 Simulated annealing Suppose the optimization problem is to find a minimum i e to find an admissible vector Topt in R such that f Top lt f r holds for any admissible r RY Here a vector r RY is called admissible if the coordinates r of r satisfy l lt ri lt u and if the constraint conditions gm r lt 0 are fulfilled for any m 1 M The functionals f and gm are continuously differentiable and possibly non linear Simulated annealing consists of the following Steps 1 5 1 Initialization of restarts Set the actual number J of restarts to zero Read the user defined value of the initial temperature tini that of the neighbourhood radius 0jn and the user supplied
237. on 4 To this end all the files from the IMD directory imd imd nk dir must be copied to the DIPOG 2 1 directory refr_ind_data Of course the user must observe the copyrights of the package IMD To get informations on the above mentioned necessary packages we refer to DIPOG http www wias berlin de software DIPOG TRIANGLE http www cs cmu edu quake triangle html GLTOOLS http www wias berlin de software gltools GNUPLOT FLTK IMD http cletus phys columbia edu windt idl levmar http www imm dtu dk pubdb views edoc_download php 3215 pdf imm3215 pd PARDISO http www computational unibas ch cs scicomp software pardiso 1 3 Get executables comment lines in input files If the executable programs do not exist then generate them by using the file makefile located in the DIPOG 2 1 home directory To do so produce a header file MHEAD in the subdirectory MAKES of the DIPOG 2 1 home directory Just copy one of the example files MHEAD_SGI MHEAD_LINUX or MHEAD_DEC to MHEAD and change the operating system the paths and the flags according to your computer system 4 Then go to the home directory cd DIPOG 2 1 and add the commands make clean make If the package is installed then the programs can be used by different users simultaneously To this end each user should have a private DIPOG 2 1 home directory containing the first six subdirectories These subdirectories together with al
238. on the several iteration steps will be printed on the screen With the flag s the optimization is performed like in the case without flag and afterwards a plot of the resulting optimal grating is shown on the screen The call of OPTIMIZE with one of the flags i g and p does not invoke any optimization method Such a call is designed to prepare the optimization The flag i results in an input check which includes a plot of the initial grating like those in Figures and and the computation of the corresponding value of the objective functional A call with g checks the local error of the gradient computation An approximation for the absolute and relative error of the gradient at the initial solution is determined Since the true value of the gradient is unknown the true gradient is replaced by the approximate gradient computed on a refined FEM mesh If OPTIMIZE is called with the flag f then the value of the objective function and the parameters of the actual iterate are shown on the screen After a few iteration steps the differences in the objective function and the parameters of consecutive iterates might be very small such that the user wants to stop the iteration and to switch to the next higher level determined by the incremental input for the level This can be accomplished with the kill command The user opens another window and enters ps al grep optim in order to find out the process number PID of the process optimize optimizem
239. onal curve running from interface point P1 to interface point P2 inherits the material index i_geom_param 6 and the index of the second is appointed to i_geom_param 2 1 The domain adjacent to the lower boundary with material index i_geom_param 2 1 before the subdivision now gets the material index i_geom_param 2 Number of real parameters nd_geom_param 2 i_geom_param 2 i_geom_param 3 1 k 1 i_geom_param 2 d_geom_param 2 k 1 real part of refractive index of grating material with index k d_geom_param 2 k imaginary part of refractive index of grating material with index k k 2 i_geom_param 2 j j 1 i_geom_param 3 d_geom_param k height of j th corner of new polygonal interface curve over line through the given interface points Pi and P2 which are located at boundary of the convex domain with material index i_geom_param 4 and which are the left and right end points of the new polygonal interface curve in other words the j th corner point is P P1 P2 P1 j m 1 t n where n is the vector normal at the segment 171 P1 P2 and where t d_geom_param k is a positive or negative real in micro meter 1000 lt d_geom_param k lt 1000 d_geom_param 2 i_geom_param 2 i_geom_param 3 1 threshold for distance of interface corner point to boundary of convex domain U which is to be split distance is measured in direction of normal to P1 P2 this is a dummy if i_geom_param 3 0 in micro meter 0 lt d_ge
240. org a a a a a O EEE E E 14 14 16 17 22 23 23 24 32 33 34 50 55 60 60 60 61 65 10 2 2 Profile grating determined by a polygonal profile function 87 10 2 3 Profile grating determined by general polygonal profile curve I 87 0 2 4 Profile grating determined by general polygonal profile curve T1 88 fete ee ene Rene ee eee eee eRe 89 10 2 6 Optimization of a polygonal interface inside a general grating 91 aa 95 eee e O ee e aa 98 ee ee 98 Hk GbE Ga GS amp aod 99 Joke Eee ok eee ee a Meee ee eee a ee 101 ode Her de ek Geto eo ee we ee 104 e MR Yok alec a tee ee a ees ce eee 106 esha the De ii aoe BA 108 oe hh Bohs Dos ae ies ae ace 110 bopa Gries oes ee Oe PSs oa G amp Bes he SB oe ee Bd 110 111 12 Enclosed Files 112 arg he OE Oe Bee ee es pe ae 112 iii es eo ke 114 12 3 Data file generalized Dat for CLASSICAL resp conical Dat in CONI oe Pee ees ee ee ee ee reece Se 127 yo ales He aed Tee ao ee ee aoe 129 ere 142 OON 150 SEET A 152 Sere cer eee 156 oA At sonal sone 203 206 1 Introductory Remarks and the Structure of the Package 1 1 What is DIPOG 2 1 and Dipog 1 5 DIPOG 2 1 is a finite element FEM program to determine the efficiencies of the diffraction of light by a periodic grating structure The unbounded domain is treated by coupling with boundary element methods DIPOG 2 1 solves the classical TE and TM cases i e the cases o
241. oundary line of the period and period d param_k_2 a_k b_k ratio of distance of left upper corner from the left boundary line of the period and distance of right upper corner from the left boundary line of the period Refractive index of the material of each trapezoid is prescribed as an optimization parameter Number of string parameters ns_geom_param 0 Number of integer parameters ni_geom_param 2 i_geom_param 1 4 i_geom_param 2 number of different materials 2 number of trapezoids Number of real parameters nd_geom_param 5 i_geom_param 2 4 k 1 i_geom_param 2 2 h_k d_geom_param 5 k 4 height of k th trapezoid in stack in micro meter h_k gt 0 b_k d d_geom_param 5 k 3 ratio of distance of right upper corner from the left boundary line of the period and period d 0 lt b_k d lt 1 a_k b_k d_geom_param 5 k 2 ratio of distance of left upper corner from the left boundary line of the period and distance of right upper corner from the left boundary line of the period 0 lt a_k b_k lt 1 If not both of the parameters b_k d and a_k b_k are fixed then we require a_k b_k gt 0 for b_k d 1 n_k d_geom_param 5 k 1 i d_geom_param 5 k refractive index of k th trapezoid in stack Re n_k gt O Im n_k gt 0 k 5 i_geom_param 2 9 posb d d_geom_param k ratio of distance of right lower corner from the left boundary line of the period and period d 0 lt posb d lt 1 166 HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HH
242. p and go to Step 3 Also if the gradient is almost unchanged in the last Nnorm prescribed integer iterations then stop the iteration and go to the next Step 3 Almost unchanged means that VirL rj41 Mj 41 MUj 1 VrL rji41 1 My 41 1 MUj 1 1 VL 7541 mlj muza Ste 0 18 holds for 1 1 norm With a prescribed small number Enorm If all the stopping criteria fail then increase j by one and repeat Step 2 Final output Accept and print r as the local solution Determine the reduced gradient P Vf r by setting to zero all those components of Vf r for which Irjli lt li ace and Vf r gt 0 and for which rj gt ui Eacc and Vf r lt 0 Print the corresponding values of f r and P Vf r The method of augmented Lagrangian ind_opt 3 requires nicopt 2 integer input val ues The first integer parameter i opt 1 is the maximal number liter max gt 0 of conjugate gradient steps in the inner iteration of Step 2 The second i_opt 2 is the threshold in teger Nnorm gt 0 choose e g i opt 1 3 Whenever the gradient of the iterative solution remains unchanged over Nnorm iterations then the iteration is stopped cf Step 2 The number of real parameters is nd_opt 8 These parameters are d_opt 1 Parameter value gt 0 in augmented Lagrangian cf and choose e g 0 5 d_opt 2 Calibration factor Cea gt 0 of objective functional in modified Lagrangia
243. p of the upper coating material above the natural starting point of the strip to be added automatically Similarly suppose that the width of the additional strips is chosen automatically and that there exist lower coated layers below the grating geometry The natural starting point of the additional lower strip is the point of the grating geometry with the lowest y coordinate which belongs to an area occupied by a material different from that of the adjacent lower coated layer Again for technical reasons the grating geometry without the additionally added layers must not contain a strip of the lower coating material below the natural starting point If the coated layer adjacent to the boundary line is very thin then the automatically added additional layer in the FEM domain is thin and a huge number of small triangles appear in the triangulation The resulting large number of degrees of freedom can be avoided in the case of classical TE polarization where the coupling of finite elements and boundary elements does not require the same material on both sides of the boundary lines 26 Avoiding an additional strip below resp above the FEM domain requires two assumptions The upper resp lower polygonal boundary line of the FEM domain should form a horizontal straight line segment f All corners with maximal resp minimal y coordinate must have an x coordinate which is the product of period times a rational number k l with l lt 1000 To s
244. ple if Grating data is namel gt Number of materials given in file namel inp if Grating data is echellea gt N_mat 3 with coating height gt 0 N_mat 2 with coating height 0 if Grating data is echelleb gt N_mat 3 with coating height gt 0 N_mat 2 with coating height 0 if Grating data is echelle ae gt N_mat 3 with coating heights gt 0 N_mat 2 with coating heights 0 if Grating data is trapezoid gt N_mat number of material layers 3 for coating height gt 0 N_mat number of material layers 2 for coating height 0 if Grating data is mtrapezoid gt N_mat number of material layers in each bridge 2 if Grating data is lAmellar k m F 2 gt N_mat k times m plus 2 if Grating data is polygon file1 gt N_mat 2 if Grating data is polygon2 file1 file2 gt N_mat 3 if Grating data is profile gt N_mat 2 if Grating data is profile 2 3 3 2 gt N_mat 2 H if Grating data is profile 0 125 sin 2 M_PI t gt N_mat 2 if Grating data is profile 0 5 0 5 cos M_PI 1 t 0 25 sin M_PI t gt N_mat 2 if Grating data is profiles gt N_mat n 1 with n nmb_curves from the file GEOMETRIES profiles c
245. ponds to a period of 1 um and to the code words echelle A 90 D 0 3 0 03 0 04 Grating data trapezoid 60 0 6 3 0 2 0 1 0 1 0 05 35 Figure 7 Echelle grating of type B indicates a TRAPEZOIDAL GRATING isosceles trapezoid with the basis parallel to the di rection of the periodicity cf Figure 8 with angle of 60 angle enclosed by basis and the sides with a basis of length 0 6 um consisting of 3 material layers of heights 0 2 um 0 1 um and 0 1 um respectively and with a coated layer of height 0 05 um greater or equal to zero Grating data mtrapezoid 2 0 02 0 06 90 80 0 04 3 0 16 0 21 0 18 0 25 0 50 0 75 indicates a MULTI TRAPEZOIDAL GRATING trapezoids with bases parallel to the direction of the periodicity cf Figure 9 each trapezoid consists of 2 layers with height 0 02 um and 0 06 um respectively side wall angle of these trapezoidal layers are 90 and 80 the number of trapezoids is 3 and the lateral width of the trapezoids measured at a height of 0 04 um is 0 16 wm 0 21 wm and 0 18 wm respectively finally the midpoints of the trapezoids have the lateral distances 0 25 wm 0 50 um and 0 75 um to the starting point of a period Grating data 1Amellar 3 4 0 2 0 6 0 2 1 0 36 Figure 8 Trapezoidal grating 0 Oo 0 7 0 0 0 50 0 90 0 00 0 500 0 900 indicates a LAMELLAR GRATING rectangular grating consisting of several materials placed in rectangular subdomains cf Figure 10
246. pt 3 threshold eps_acc if difference of component to upper lower bound is less than this then point is considered to be at boundary should be about desired accuracy d_opt 4 threshold eps_gra for gradient to stop iteration should be about discretization error of gradient calculation d_opt 5 threshold eps_norm if relative difference of two squared norms is less than this number then the norms are considered to be the same e g 1e 2 HERE AE RE HE TEHE HE AE HEH EEE INTERIOR POINT METHOD ind_opt 2 197 fi_opt 1 nd_opt 8 HHHHHHHHHHHHHHHHHHHHHH HHH HHH HH HH HF OF HHEHHHHHHHHHHHHHHH AUGMENTED LAGRANGIAN ni opt 2 nd_opt 8 HHHHHH HHH HH HHH HHH i_opt 1 d opt 1 d_opt 2 d_opt 3 d_opt 4 d_opt 5 d_opt 6 d opt T7 d_opt 8 adopt ills i_opt 2 d_opt 1 d_opt 2 number n_norm of same gradient norms after which the algorithm stops e g 3 initial value rho_0 for parameter in algorithm parameter of operator F e g 0 1 reduction factor q to reduce parameter in algorithm for each new iteration parameter of operator F e g 0 5 Constant c1 in Armijo stopping criterion for line search in conjugate gradient e g 0 001 factor alpha_max the initial bound for step size factor in line search e g 0 9 accuracy threshold eps_acc considers the parameter values to be on the boundary if dista
247. put MLS ni n2 n3 n4 with ni n2 n3 n4 replaced by non negative integers means N_co_ov ni n2 n3 n4 layers with ni layers above n2 groups of n3 layers with same widths and materials in the middle and with 7 layers below Number 0 TERRIER TET TeTETETE TESTER PEPER TTT TET eee Ee Pe TE TEETER ETE HEA TEEPE PETE TE PEE BEE eT Widths of coating layers in micro meter N_co_ov entries Needed only if N_co_ov gt 0 Else no entry and no line Widths HEATHER AHHH AAHHHA ER AHHRR ARERR RARER ARERR RARER RARER ARR Number of coating layers beneath the grating N_co_be 0 HHEEHHHRAEHHHHAAAHHHAER RHEE AA AHHH A RARER REAR ERA AERA REHASH Widths of coating layers in micro meter N_co_be entries Needed only if N_co_be gt 0 Else no entry and no line For a multilayer system the widths of the n3 layers in the groups must be given only once I e for a multilayer system ni n3 n4 input numbers are needed Widths TERETE IE EE EPP PETTERS EEE EE IHHHEH Wave length in micro meter lambda Either add a single value e g 63 Either add more values by e g ay 5 157 63 64 65 69 O T The last means that computation is to be done for the wave lengths from the Vector of length 5 SAGI E GAT Ae EBI Ee 60 and 70 Or add e g ee TE 63 273 3027 5 The last means that computation is to be done for the wave lengths 63 i 02 with i 0 1 2 and with wave lengt
248. put file for an advanced grating with point clouds to enforce a mesh grading at the corner points first argument name without tag inp of file to be improved second argument number of cloud points at circular layer third argument number of layers fourth argument threshold angle for corners with point clouds output geometry input file with a added to the name before the tag inp executable GEN_CPIN2 to generate an input file for a coated pin grating of type 2 given by profile lines defined by c code first argument name of input file to be created second argument stepsize of polygonal approximation c code file cpin2 c to define profile lines for a coated pin grating of GEN_CPIN2 input files name dat non geometrical data of the gratings data file generalized Dat data for the GFEM executables FEM and GFEM for simple calculation case of classical diffraction argument name dat executables FEM_CHECK GFEM_CHECK for check of input exists only with openGL argument name dat executables FEM_PLOT and GFEM_PLOT CONICAL OPTIM for calculation with plots of resulting fields case of classical diffraction exists only with openGL or GNUPLOT argument name dat executable GFEM MATLAB for calculation with plots of resulting fields case of classical diffraction plots in form of Matlab file argument name dat executables FEM_FUL
249. qua_i_tr ass transmitted 2 gt w_qua_1_tr efficiency_1 c_qua_i_tr S55 j o_qua_1i_tr j jet j n_lin_2_re aan reflected gt w_lin_2_re efficiency_2 j o_lin_2_re jet j n lin 2 tr transmitted gt w_lin_2_tr efficiency_2 j ONIN 2 tr j 1 j n_qua_2_re gt reflected 2 gt w_qua_2_re efficiency_2 c_qua_2_re j o_qua_2_re J j 1 j n_qua_2_tr transmitted 2 gt w_qua_2_tr efficiency_2 c g a_2 tr J N j o_qua_2_tr j j 1 j n_psh_i_re reflected 2 gt w_psh_t_re phaseshift_1 c_psh_i_re 184 HHHHHHHHHHHHHHHHHHHHHHHHH HHH HH H HH OF j o_psh_i_re j I j n_psh_i_tr transmitted 2 gt w_psh_1_tr phaseshift_1 6 psh ltr gt j o_psh_1_tr j jal j n_psh_2_re S55 reflected 2 gt w_psh_2_re phaseshift_2 c_psh_2_re ass j o_psh_2_re j j j n_psh_2_tr gt transmitted 2 gt w_psh_2_tr phaseshift_2 c_psh_2_tr j o_psh_2_tr j jet j reflected 2 w_qua_ene_re energy c_ene_re transmitted 2 w_qua_ene_tr energy c_ene_tr total 2 w_qua_ene_to energy c_ene_to 2 2 Here p c sin Pix p c 360 HHEHHHHHHHHHHHHHHHHEE HHH HHHHHH HHH HH HHH HHH H OH refl transm Note that the angles of phase shift phaseshift_m are between 180 and 180 o refl transm Note that the efficiencies efficiency_1 denote the o the efficiency TE part for TE TM output resp the S part for Jones output refl transm Note that t
250. ractive indices The wave number k is equal to w c times the refrac tive index of the material Thus we can determine by the standard method for elliptic differential equations by the FEM Using the periodicity of the problem and standard cou pling techniques with the boundary element method the domain of numerical computation can be reduced to a rectangle 2 cf Figure i This covers one period of the grating and is bounded by the horizontal lines located inside the last upper and first lower coating layer counted from above to below resp in the cover material and substrate material for gratings without coatings Above resp beneath the grating structure including all the layers the component EF admits an expansion into the Rayleigh series of the form gt A exp i87y exp ians Ag exp ibdy exp iaz 2 2 E x y E a y gt A exp i8 y exp iant 2 3 BE 4 k a P kt wn Aine 1 n n a c 0 Vnt 2 a k sin An kt sind n Here d is the period of the grating and the complex constants A are the so called Rayleigh n TFor technical reasons in the FEM code it is important to have the same material on both sides of the boundary lines 15 Incident wave Reflected modes Transmitted modes Figure 1 Cross section of grating coefficients The interesting Rayleigh coefficients are those with n UF pe Aan if Sm k 0
251. rall lt Enorm l 1 s 5 Nnorm with a prescribed small number Enorm If the stopping conditions fail then perform a line search 2 1 2 2 2 1 Initialize amp amaz 2 2 Set rj41 r a sd Ifthe difference between rj and r is less than the machine accuracy stop and jump to Step 3 Else compute f rj41 and Vf rj41 If rj 1 is admissible and if the Armijo criterion Fj ri lt aa VS re 84 10 11 is fulfilled then stop and leave the line search Else compute a smaller by halv ing the actual value or by a clever approximate minimization of the univariate function amp gt f r a sd Repeat Step 2 2 Determine a new search direction by the following non linear conjugate gradient for mula sdj41 P Vf 541 B sd 10 12 aaa Via Vila V ICH a fo Vi r3 VEC Set j j 1 and repeat Step 2 26 This change of rp and the corresponding change of the subsequent iterates r before the application of the projection of P V f ro avoids a lot of unnecessary time consuming tiny iteration steps toward the boundary 100 3 Final output Accept and print r as the local solution Print the corresponding values of f r and P Vf r The conjugate gradient method ind_opt 1 requires one integer input value ni_opt 1 This i opt 1 is the threshold integer Nnporm gt 1 choose e g i opt 1 3 Whenever the gradient of the iterative solution remains unchan
252. riod 1 00 micro m grating height 1 60 micro m fem grid height 2 00 micro m nmb of materials 4 corr refract indices 1 20 i 0 00 1 50 i 0 00 1 70 2 0 00 2 30 i 0 00 nmb of levels f comp 5 INFO OF SOLUTION LEVEL 1 144 degrees of freedom stepsize of discr numb of nonzero entr rate of nonzero entr memory for pardiso 813 5859 0 81789 0 886426 per cent 536 kB Reflection efficiencies and coefficients n O theta 65 00 n 1 theta 15 74 n 2 theta 21 33 n 3 theta 87 07 Reflected energy Transmission efficiencies 0 039594 0 181811 0 008538 0 021079 0 031693 0 008219 0 102928 0 466177 6 573574 and coefficients 2 3 1 2 a 4 NOOW 22 56 256892 861878 966041 415630 oOW Oo n O theta 26 95 0 132885 0 438747 n 1 theta 50 41 0 034493 0 866995 n 1 theta 7 80 0 228543 0 163526 n 2 theta 10 48 0 171167 0 062429 n 3 theta 29 96 0 096782 0 007596 n 4 theta 54 77 0 001540 0 078023 Transmitted energy 93 426426 INFO OF SOLUTION LEVEL 2 degrees of freedom 3197 stepsize of discr 0 34705 numb of nonzero entr 23379 rate of nonzero entr memory for pardiso 0 228739 per cent 2606 kB Reflection efficiencies and coefficients n O theta 65 00 n 1 theta 15 74 n 2 theta 21 33 n 3 theta 87 07 Reflected energy
253. ro e g echellea L 0 3 0 03 0 04 gt ECHELLE GRATING TYPE A right angled triangle with hypotenuse parallel to the direction of the periodicity left interior angle gt 45 degrees with parameters like above e g echellea A 60 0 03 0 04 gt ECHELLE GRATING TYPE A right angled triangle with hypotenuse parallel to the direction of the periodicity with left interior angle Alpha 60 degrees i e depth period times sin Alpha times cos Alpha and other parameters like above e g echelleb 60 0 05 gt ECHELLE GRATING TYPE B right angled triangle with one of the legs parallel to the direction of the periodicity with angle 60 angle enclosed by hypotenuse and by the leg parallel to the period and with a coated layer of height 0 05 micro meter measured in direction perpendicular to echelle profile height greater or equal to HHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHH HHH FH 134 HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHH RR za Oz zero e g echelle L 60 R 30 0 05 0 1 gt GENERAL ECHELLE GRATING with left blaze angle 60 degrees with right blaze angle 30 degrees with coated layer over left blaze side of height 0 05 micro meter measured perpendicular to profile height gt 0 and with coated layer over right blaze side of height 0 1 micro meter measured perpendicular to profile height gt 0 must be O if previous height is 0 Instead of the two inputs T 6
254. ry then the file is to be specified by adding its 60 path as Name of output file path name3 Again the tag res resp erg will be added In particular for an output file in the current working directory use Name of output file name3 The computation proceeds over several levels where the mesh size is halved at each new level The maximal number of levels is indicated at the end of name2 dat e g by Number of levels 3 However if efficiencies are computed for more than one angle of incidence or for several wave lengths or if a single value of angle wave length is given by incremental or vector mode i e beginning with the letter I or V then the computation is performed for the highest level only The ideal choice for the level would be the minimal positive integer such that the solution falls under a certain error bound We have implemented the following choice If name2 dat contains Number of levels eE with the lower case letter e and with a number greater than zero then the code computes the efficiency for the levels 1 2 but no more than 15 until the maximum of the differences of efficiencies corresponding to two consecutive levels is less than e Thus assuming a monotonic convergence the smallest level for the given error bound e is the smaller one of the last two consecutive levels The efficiencies will be presented on the screen and in the output files for
255. s Nik d_geom_param 5k 1 i d_geom_param 5k The real parameter of index nd_geom_param 9 is the relative height of the upper boundary line of the extra layer beside the bridge inside the trapezoid of index i geom_param 5 i e 96 it defines the height ho of the extra layer beside the bridge by the formula i_geom_param 5 1 ho D hk F hrel k 1 he hi geom paramjs d geom param nd geom param 9 The refractive index of the corresponding layer material is no d_geom_param nd_geom_param 8 i d geom param nd geom param 7 The refractive indices of the cover and substrate material are given as Nco d_geom_param nd_geom_param 6 i d_geom_param nd_geom_param 5 Ns d_geom_param nd_geom_param 4 i d_geom_param nd_geom_param 3 The last three real parameters restrict the sidewall angles a and p fork 1 N 1 cf Figure 32 For these angles the real numbers Ymin d_geom_param nd_geom_param 2 Pmax d_geom_param nd_geom_param 1 are the lower and upper bound respectively However instead of requiring the strict fulfillment of Ymin lt Ak lt Pmar ANd Ymin lt bk lt Ymar k 1 N 1 we shall enforce the bounds only weakly by adding the penalty term N 1 YP fac max 0 Qk Ymax max 0 Pmin az k 1 max 0 8 Pmaz max 0 Qmin Bey 10 10 to the objective functional 10 2 The calibration factor Yfa is the last real parame ter d_geom_par
256. s suggests to determine approximate solutions r 7 1 2 3 by the following itera tion scheme of the Gau Newton method Choose an initial solution r and for any given 108 rj define rj41 by 1 ra srt Ar Ary VEE VE VE le ra In the Newton type algorithm of DIPOG 2 1 the correction term Ar is slightly modified in order to guarantee that the new iterate satisfies the constraints l lt r lt u More precisely if the components of the new iterate rj do not fall into the the interval l u i 1 N the r 1 is determined as the optimal solution of the following quadratic problem with box constraints 2 ve e Fran e e min rj l lt rjj lt u i 1 N Obviously this is a modification of the Gau Newton method in the spirit of the SQP methods The Newton type method ind_opt 5 requires one integer input value ni opt 2 The i opt 1 is the threshold integer Nnorm gt 1 choose e g iopt 1 3 Whenever the gradient of the iterative solution remains unchanged over Nnorm iterations then the iteration is stopped The number i opt 2 is maximal number of iteration for which an increase of the objective functional is accepted choose e g iopt 2 5 The number of real parameters is nd_opt 3 These parameters are d_opt 1 Threshold gt 0 iterate is shifted to the boundary if the distance to the boundary is less than Eace choose Eace about the expected accuracy of the final so
257. s the initial solution he should apply a local method to end up with an improved final solution 83 10 1 5 Maximum likelihood estimator Suppose only efficiency data is measured and suppose that the measured values are sample values of random variables More precisely suppose that the measured c is normally distributed with expectation et rop and variance oi P alei rope 0 where b is the back ground noise level and where a is a yet unknown constant factor If the measurements are independent then the joint distribution of the measurement data takes the form ch eft rp iY o a r o e Matron H o 1 o 3 Ll i ane ao gt Hence given a fixed measurement data ctt the maximum likelihood estimator deter mines those model parameters r and a for which the density o a r ctt is maximal In other words the maximum likelihood approach determines r and a minimizing the func tional 2 t ei r TEOSA f r a gt log aeit r 0 VEN over the domain defined by 0 lt a and l lt ri lt ui The optimization can be switched to this maximum likelihood method by changing the input data file name dat as follows Firstly all weights for energies and all numbers for the involved linear efficiency resp quadratic phase shift terms must be set to zero Secondly the weight inputs must be like
258. sin theta sin phi Either TM means that incident magnetic field is perpendicular to wave vector and to normal of grating plane and incoming wave vector is presented in xz system as sin theta cos phi cos theta sin theta sin phi Either TE TM means TE and TM two calculations Either TP means polarized electro magnetic field and incoming wave vector is presented in xz system as sin theta cos phi cos theta sin theta sin phi Or pol means polarized electro magnetic field and incoming wave vector is presented in xy system as sin theta cos phi cos theta cos phi sin phi Type 4 gr TEED EE PEPE TET TEEPE PETTITTE TE Parameter of polarization If type of polarization is pol or TP then this is the angle in degrees between x axis axis in plane of grating grooves which is perpendicular to grooves and projection of electric field vector onto x z plane of grating grooves Needed only if polarization is of type pol or TP Else no entry and no line Parameter 20 AA E HE TE E AE A E A E E E E EEE T T TE E E AE TE PE AE TE E EE TE LETTE TE E TE E TEEPE TE E E BE PP HEE HE HEHE EE E HE BE HEE TETEE HE Angle of incident wave in degrees theta 159 HHHHHHHHHHHHHHHHHHH HHH HHH HH FH If type of polarization is pol then the incident light beam takes the direction si
259. straints i e M 0 In case of the class of Section 10 2 4 the M ajoa th Gm r lt 0 are the inequalities in Equation 10 7 99 1 Initialization Take the initial solution rp from the input file provided by the user Compute f ro and the gradient V f ro of f at ro In case that ro is an interior point of the set of admissible vectors then choose the search direction sdo V f ro If either a component ro of ro is equal to the bound u resp l or if gm ro 0 holds for some m then set sd equal to the projection of V f ro to the tangent cone of directions pointing from rg toward admissible points Note that if the difference of the component ro to the bounds u resp l is less than the small prescribed threshold Eace then we set ro equal to u resp l and the search direction is chosen as for the boundary point In any case we write sdo P Vf ro Set the iteration index j to 0 2 Iteration step If the number of iterations j is larger than the prescribed number niter_max then stop the iteration and go to the next Step 3 Else compute the norm of the reduced gradient P V f r If this is less than the prescribed gradient threshold then stop the iteration and go to the next Step 3 Also if the reduced gradient is almost unchanged in the last Nnnorm prescribed integer iterations stop the iteration and go to the next Step 3 Almost unchanged means PLEV Fra PIEVE ri PI VF
260. subsequent refractive indices of the grating must be that of the substrate resp that of the adjacent lower layer c e g rough_mls k name MULTILAYER SYSTEM WITH ROUGH INTERFACES k TIMES just like above However k random realizations are computed and the output values are replaced by the mean values over the k computations Standard deviations are added too Grating data lamellar HHEHHHHEHHHHHHHHHHEHAH HEH HEHHEHEH HEH HAH HEHEH HEHEHE HEH RHE RHE HEH RH Number of different grating materials N_mat This includes the material of substrate and cover material For example if Grating data is namel gt Number of materials given in file namel inp if Grating data is echellea gt N_mat 3 with coating height gt 0 N_mat 2 with coating height 0 if Grating data is echelleb gt N_mat 3 with coating height gt 0 N_mat 2 with coating height 0 if Grating data is echelle H gt N_mat 3 with coating heights gt 0 N_mat 2 with coating heights 0 H if Grating data is trapezoid gt N_mat number of material layers 3 for coating height gt 0 N_mat number of material layers 2 for coating height 0 if Grating data is mtrapezoid gt N_mat number of material layers in each bridge 2 140 if Grating data is lAmellar k m 2 2 gt
261. t line should be End e g profile gt GRATING DETERMINED BY A SMOOTH PARAMETRIC CURVE i e grating determined by profile line given as fctx t fcty t 0 lt t lt 1 where the functions gt fctx t and t gt fcty t are defined by the c code of the file GEOMETRIES profile c e g profile_par 2 3 a 0 1 5 0 2 0 3 22 gt GRATING DETERMINED BY A SMOOTH PARAMETRIC CURVE WITH PARAMETERS i e grating determined by profile line given as fctx t fcty t 0 lt t lt 1 where the functions t gt fctx t and tl gt fcty t are defined by the c code of GEOMETRIES profile_par c the last code uses 2 integer parameters and 3 real parameters named IPARaM1 IPARaM2 RPARaM1 RPARaM2 RPARaM3 the integer parameters take the values 1 and 0 following the first line of the calling sequence and the real parameters take the values 1 5 0 2 and 0 3 following the integer parameter values Any number of parameters is possible for a corresponding file GEOMETRIES profile_par c e g profile 0 125 sin 2 M_PI t gt GRATING DETERMINED BY A SIMPLE SMOOTH FUNCTION i e grating determined by sine profile line given as t fcty t 0 lt t lt 1 where the function t gt fcty t is defined by the c code fcty t 0 125 sin 2 M_PI t do not use an
262. t without these layers must not contain more than nine different materials Finally the input of the geometry by code words cf Sect is confined to one line which excludes lamellar stack and box gratings as well as profile gratings with parameters Recall that the interface program TGUI allows to construct complex geometries and to include them as namel inp files cf Sect 3 2 111 12 Enclosed Files 12 1 Geometry input file example inp makefile HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHH HHHHHH HEHEHE HEHEHE HERE HHHH EER example inp HHPHHHHHHHHHHH HEH all lines beginning with are comments HHH H HH HF HHH H HH FH AE E EE RESET TEPER ESE EEE TE EEE EET TE HE AE AE REE EEE E SIRE AE E E E E HE EEE BE HEE geometry input file for periodic grating located in directory GEOMETRIES input file for gen_polyx HHHH TRAE E EEEE EEA A E E EEEE E EEEE EEE EEE E HHE Name of the files without extensions inp Output files will have the same name but with tags polyx and sg Name example HERE E EEE PEER THEE EEE EE SEE ERE Comments Input must be ended by a 0 in an extra line These comments will appear in several output and result files Comments This is a fantasy grid for the test of gen_polyx 0 TERE TET TET TETETETEPE EET PTETE TE TET Tee ee aE TEEPE PETE TERE TEAL HE ALTE ERE EE EEE PEE
263. tation is to be done for the wave lengths from the Vector of length 5 8 6377 50 640 A65 E07 and aT Or add e g eS T 263 273 10277 The last means that computation is to be done for the wave lengths 63 i 02 with i 0 1 2 and with wave length 63 i 02 less or equal to 73 Wave length 635 PHRNRNPPHHHHGHHRAHHN PHAR HN RRHHR RRAN AYE PERE EHEN R NARH Temperature in degrees Celsius from O to 400 20 for room temperature Must be set to any fixed number Will be ignored if optical indices are given explicitly Temperature 20 TERRE TET TeTETETETETERE PEPER TET TET eee EP TE TEEPE REEL EE EEE PE BE BT Optical index refractive index of cover material This is c times square root of mu times epsilon This could be complex like e g 4 298 i 0 073 for 130 Si with wave length 500nm This could be also given by the name of a material like Air Ag Al Au CsBr Cu InP MgF2 NaCl PMMA PSKL SF5 Si TIBr T1Cl Cr ZnS Ge i1 0 12 0 Ti02r Quarz AddOn cf Userguide This could be a value interpolated from a user defined table determined by the name of the file file is to be located in the current directory name of file must begin with letter u and may consist of no more than five letters like e g user the file consists of lines each with three real numbers first wave length in micro meter second the real part of the correspondi
264. te solution of the equations F r slj suj dlj duj 0 where o o gq with 0 lt q lt 1 tends exponentially to zero and where the approximate solution rj sl suj dlj du is obtained by one step of Newton s method choosing rj_1 slj 1 Suj 1 dlj _1 du _1 as initial solution More precisely the interior point method consists of the following Steps 1 3 1 Initialization Set the index j of the iteration to zero choose rp from the initial values given by the user in the input file and suppose the strict fulfillment of the restrictions l lt ro lt u i 1 N Define slo and sup using the Equations and 10 14 Read the initial value 09 from the input of the user and introduce the dual slack variables dlp and duo by dlo 0 slo and duo 0 suoli Increase j by setting 7 1 2 Iteration step If the number of iterations j is larger than the prescribed number niter_max then stop the iteration and go to the next Step 3 Else compute f r and the gradient Vf r Determine the reduced gradient P V f r by setting to zero all those components of V f r for which rj lt li ace and Vf r gt 0 and for which rj gt Wi Eace and Vf r i lt 0 Here acc is a small prescribed threshold If P V f r is less than the prescribed small constant 9 4 then stop the iteration and go to the next Step 3 Also if the reduced gradient is almost unchanged in the last Nnor
265. th of period e g by 1 um Additional geometrical information fixed in file name dat concerns the coated layers In principle the grating part is a rectangular domain cf Q in figure 1 Above and below this part we can add a few number of coated layers in form of strips parallel to the upper and lower side of the rectangle The numbers of these layers together with the corresponding thickness is given in special lines of name dat Explanations of the lines in name dat can be found directly in the neighbouring comment lines starting with symbol 3 2 How to get an input file namel inp The elementary way to create namel inp is the following Change to subdirectory GEOMETRIES Copy an existing file like e g example inp cf the enclosed file in 12 1 and change its name into e g namel inp cd DIPOG 2 1 GEOMETRIES cp example inp namel inp Change namel inp in your editor emacs vi according to your requirements You will find the necessary information as comments in the file namel inp Indeed each line be ginning with is a comment For example the number Nmat of different materials is fixed in namel inp by the input lines Number of materials Nmat We emphasize that this number must include the two materials located immediately over and under the grating area since two rectangular layers from these adjacent regions are added to the area of FEM computat
266. the computation of the objective function and its gradient is created with a maximal stepsize of hg2 where ho is the coarsest stepsize For example the level is set to 3 by Number of levels 3 If the computation of the objective functional is time consuming then it might be better to perform a few iteration on a coarser FEM level first and to utilize the coarse level optimal solution as the initial solution on the fine level Therefore an incremental input is possible For instance Number of levels Mh 2 indicates that the level takes all values 1 j x 2 with j 0 such that 1 j7 2 lt 5 Hence using the initial values the optimization starts with an iteration on level 1 The optimal solution of level 1 is passed to the initial solution of the level 3 iteration After the level 3 iteration is finished the level 3 solution will be the initial solution of level 5 Hopefully this is a better initial solution than that given by the user in name dat and after a smaller number of iterations the optimal solution of level 5 is reached After the input of the level the parameters w w w wbt w w w w wt w n n n nm n n l GG eo Gc and of the objective functional f in 10 2 are given For instance the lines n QUADRATIC TERMS TRANSMITTED EFFICIENCY n_qua_tr 2 w_qua_tr 2 3 79 o_qua_tr 1 0 canat 10
267. the constraint conditions and ii the accuracy of the minimum value of the objective functional multiplied by c_cal threshold eps_gra if norm of gradient of Lagrangian is less than this number times c_cal then inner iteration stops should be about discretization error of gradient calculation threshold eps_acc if step size in line search of inner cg algorithm is less than this then line search is stopped moreover for computation of reduced gradient the point is considered to be at boundary if its distance to the boundary is less than eps_acc should be less than error of the parameter solution set e g 1 e 15 constant c1 in Armijo stopping criterion for line search in conjugate gradient e g 0 001 bound alpha_max for initial bound for step size factor in line search e g 5 threshold eps_norm if relative difference of two squared norms is less than this number then the norms are considered to be the same e g te 3 SIMULATED ANNEALING ind_opt 4 ni_opt 1 a ope ltl nd_opt 5 d_opt 1 d opt 2 number n_rest of restarts algorithm starts from initial solution and from n_rest randomly chosen other solutions e g 0 initial temperature t_ini should be about the oscillation of the objective functional if this is 0 then initial temperature will be determined automatically e g 0 cooling factor c_fact starting from an initial temperature 199 H
268. the phaseshifts for different polarization are included into one term only If the type of polarization and coordinate system for the incoming wave vector is TE TM if the type of output is TE TM and if the angle phi of illumination is zero classical case then the phase shift are computed first for TE and then for TM and terms like psh_i_re reflected TE TM 2 gt w_psh_1_re PS_1 c_psh_i_re 55 j o_psh_i_re j jet j psh_1_tr transmitted TE T M 2 gt w_psh_i_tr PS_1 gph itr 192 j o_psh_1_tr j j 1 j with reflected TE TM reflected TE reflected TM Pood phaseshift_1 phaseshift_2 o_psh_i_re o_psh_i_re o_psh_i_re j j j transmitted TE TM transmitted TE transmitted TM Pout phaseshift_1 phaseshift_2 o_psh_1_tr o pshul tr o_psh_1i_tr J j j are of interest To indicate that terms of this type are to be included instead of the terms depending on one phaseshift only add a CL TE TM before the input of the numbers n_psh_1_re and n_psh_1_tr respectively E g n_phs_1_tr CL TE TM 2 In such a case an input of weights w_ and prescribed values c_ beginning with POL is wrong since the phaseshifts for different polarization are included into one term only HHHHHHHHHHHHHHHHHHH HHH HHH HHH HH AEA PETE TERETE AE E E AE TE EAE HEHE HE If the weights for the efficiency terms should be the reciprocal squared uncertainty and if this uncertainty shoul
269. the profile curves apply also here For example Figure 22 presents the stack grating generated by Grating data stack 5 echellea R 0 3 1 2 profile 0 3 sin 2 x M_PI x t 0 8 pin 0 5 0 5 x cos M_PI x t sin M_PI x t 0 6 0 9 pinl 0 if t lt 0 5 fa t else fae 1 t 0 1 0 4 0 5 profile t 0 0 53 Figure 22 Stack grating 54 4 Input of Refractive Indices The optical properties of the materials involved in the grating are characterized by the refractive indices Hence for each material piece the corresponding index must be added through the input file name dat This is done in lines like the following Optical index refractive index of cover material Air Optical indices of the materials of the upper coatings 1 1 1 2 Optical indices of the materials of the lower coatings 2 3 i 0 ae Pl ae 21 A Optical index of substrate material Al Wave length in micro m lambda 635 Temperature 20 Optical indices of grating materials 1 2 user Ly a 0 2 3 i 0 As seen in this example the indices can be added as real or complex numbers e g 1 1 resp 2 2 i 5 or as code words of known materials e g Al for aluminum In the last case there is a program which interpolates the refractive index from a table in dependence on the temperature and on the wave length The temperature enters only through such materials given by code words If t
270. then the phase shifts are computed first for TE and then for TM and terms like n_psh_i_re reflected TE TM 2 gt w_psh_t_re ps_1 c_psh_it_re j o_psh_i_re j jet j npsh ltr transmitted TE T M 2 gt w_psh_1_tr ps_1 c psh i trl j o_psh_1_tr j iA j n_psh_2_re reflected TE TM 2 gt w_psh_2_re ps_2 c_psh_2_re j o_psh_2_re j jet j n_psh_2_tr transmitted TE TM 2 gt w_psh_2_tr ps_2 psh_2 tr J j o_psh_2_tr j ja j 191 HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HR HERE zzz E E ps ps ps ps n_ m with reflected TE TM reflected TE reflected TM 1 phaseshift_1 phaseshift_1 o_psh_1_re o_psh_1_re o_psh_1_re j j j transmitted TE TM transmitted TE transmitted TM 1 phaseshift_1 phaseshift_1 o_psh_1_tr o_psh_1_tr o_psh_i_tr j j j reflected TE TM reflected TE reflected TM 2 phaseshift_2 phaseshift_2 o_psh_2_re o_psh_2_re o_psh_2_re j j j transmitted TE TM transmitted TE transmitted TM 2 phaseshift_2 phaseshift_2 o_psh_2_tr o_psh_2_tr o_psh_2_tr j j j are of interest To indicate that terms of this type are to be included instead of the terms depending on one phaseshift only add a TE TM before the input of the numbers n_psh_1_re n_psh_2_re n_psh_1_tr and n_psh_2_tr respectively E g n_phs_2_tr TE TM 1 In such a case an input of weights w_ and prescribed values c_ beginning with POL is wrong since
271. this level In other words a computation for a level higher by one than that of the output is necessary for this variant If a computation over several angles wave lengths or polarizations is required then the optimal level will be determined for the first angle for the first wave length resp for the TM polarization only All other calculations are performed with this level Clearly there is no warranty that the efficiencies really deviate by a number less than from the true values 5 3 Check before computation more infos and plots Instead if openGL is available and if you wish to check your input data then use the command FEM_CHECK name2 All the input information without output data will appear on the screen and in the result file Moreover there will appear a picture of the grating geometry with indicated refractive indices on the screen The picture looks like that on the right hand side in Figure If 61 Real_Part Figure 23 Real part of transverse component of magnetic field Output of FEM_PLOT via openGL in CLASSICAL indicated in the data file name dat then an eps file of the cross section is produced Instead if you use the command FEM_FULLINFO name2 then the same is done as in point 5 2 Additionally there appears more information in cluding the full input data and the convergence history cf the enclosed file in point 12 5 on the screen and in the result file Instead if openGL or G
272. tio of distance of right upper corner from the left boundary line of the period and distance of right lower corner from the left boundary line of the period O0 lt b_k b_ k 1 lt 1 b_0 posb a_k a_ k 1 b_k a_ k 1 d_geom_param 5 k 2 ratio of difference of x coordinates of left upper and left lower trapezoid corner over difference of x coordinates of right upper and left lower trapezoid corner O lt a_k a_ e 1 bk a_ Ce 1 lt 1 a_0 posa k 1 i_geom_param 3 posb d d_geom_param 5 i_geom_param 3 1 ratio of distance of right lower corner from the left boundary line of the period and period d 0 lt posb d lt 1 posa posb d_geom_param 5 i_geom_param 3 2 ratio of distance of left lower corner from the left boundary line of the period and distance of right lower corner from the left boundary line of the period 0 lt posa posb lt 1 If not both of the parameters posb d and posa posb are fixed then we require posa posb gt 0 for posb d 1 hel h_m d_geom_param nd_geom_param 9 m i_geom_param 5 height of extra layer beside stack bridge over the lower line of the m th trapezoid 180 HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHH RHR HR E E relative to height of m th trapezoid in micro meter O0 lt hel lt 1 hl_k d_geom_param 5 i_geom_param 3 3 k height of k th layer beneath stack in micro meter hl_k gt 0 k 1 i_geom_param 4 Penalty term parameters phi_min d_geom_param n
273. ude of the electric field vector is of unit length The values of the efficiencies and phase shifts of the reflected and transmitted plane wave modes are not effected by the normalization factor Only the subsequent Rayleigh coefficients depend on this scaling The polarization type must be prescribed by the user of DIPOG 2 1 There are three possibilities The first is to choose TE polarization with the electric field vector E pointing in the direction perpendicular to the wave vector incidence direction and to the y axis The second is TM polarization with the magnetic field vector H pointing in the direction perpendicular to the wave vector incidence direction and to the y axis Note that the direction perpendicular to the wave vector and to the y axis is by definition the z axis if wave vector and y axis should be collinear The third choice is to prescribe the angle Y cf Figure 4 enclosed by the x axis and by the projection of the electric field vector E to the x z plane 19 Dy Figure 3 Coordinate system based on x z plane Fixing the incident field the resulting total field is determined and can be computed by FEM It remains to describe the output data of DIPOG 2 1 The Rayleigh expansions above resp below the grating take the form E z y z E exp iloz By yz gt A exp il ilang Bry t yz IF neZ H x y z H exp ilox By y2 Do exp i lans
274. um 4m 8m and 16m In other words we have chosen the geometry generated by the code words Grating data trapezoid 60 a1 bc where a is 0 6 times the length of the period b is 0 3 times period and c is 0 05 times period Now the accuracy and the best choice of parameters depend on the maximal relative wave number which is period times refractive index over wave length We have checked the accuracy in percent For instance one percent accuracy means that The absolute error of the percentage of energy reflected by the grating is less than 1 the value itself is less than 100 The absolute error of the percentage of light transmitted into the minus first order is less than 1 the efficiency itself is less than 100 The absolute error of real or imaginary part of the Rayleigh coefficient of the minus first reflected mode is less than 0 01 the modulus of the coefficient itself is less than one The corresponding relative mesh sizd h and the corresponding number of refinement levels starting from 1 for the coarsest necessary to achieve an accuracy up to 1 0 1 and 0 01 respectively are presented in the Tables Here we define the relative mesh 16 Of course the values presented in the Tables are taken from a discrete set of test values only Indeed we have computed only those relative mesh sizes which result from the halving the mesh size strategy realized in the code by switching to higher refinement levels 69 size h o
275. ust be in the list of grid points in nameG inp Moreover the new polygonal interface is sought in form of a graph of a piecewise linear function defined over a uniform partition of the straight line segment P P2 In other words there is a positive integer N such that the new interface is the polygonal curve connecting the end points P and P through the corner points Q1 Qo Qu located in the interior of the convex area A The orthogonal projections of these 91 000 i 0 000 300 i 0 200 200 i 0 300 500 i 0 000 eere Figure 29 Example grid for stack of trapezoids Qk onto the the straight line segment P P2 are supposed to form a uniform partition of P P gt Hence if v is the unit normal perpendicular to P P and pointing to the left of P Po then the interior corners are given as cf Figure 30 Qe Pit IP Pil thay B L N 10 8 where hy denotes the hight of Qk over P Po The refractive indices of the grating materials except those of the substrate and cover material are included into the set of optimization operators A special case of this class switched on by choosing i geom_param 3 0 is the optimization of only the refractive indices without any geometry parameter namely the optimization of the refractive indices in the fixed grating geometry nameG inp The number ns_geom_param of name parameters is one and s_geom_param 1 contains the name nameG of the file
276. wed by nj lowest layers This system is described by the file name which is contained in the directory GEOMETRIES alternatively name must contain the path of the file This input file name contains the following ordered data each in a separate line comment lines begin with dummy file name width of additional layer above and below the structure must be positive or could be no for no additional layer period of grating 46 Figure 19 Coated pin grating determined by two parametric curves Type 2 number Nia of layers above multilayer system number ny of layers below multilayer system number Nma of multilayers inbetween number nq of layers in each multilayer number neorn of corner points in each polygonal approximation interfaces randomly generated polygon number Nyana of standard Gaussian distributed random numbers needed to generate interfaces automatically created if n ang 0 otherwise Nrand Neorn NMiatNidNmla N optional standard Gaussian distributed random numbers Nia Widths of layers above multilayer system nia widths of layers in each multilayer np widths of layers below multilayer system standard deviation o i of each layer interface for i 0 Mia NiaMmia Ny 1 correlation length corl i of each layer interface for i 0 Mia NidNmia Nw 1 Note that the first of the subsequent refractive in
277. which is located between x 0 0 lt x lt 1 and fctx 1 t fcty 1 t 0 lt t lt 1 Here fctx 1 t fcty 1 t 0 lt t lt 1 is a simple open arc connecting fctx 1 0 fcty 1 0 xmin 0 with fctx 1 1 fcty 1 1 1 xmin 0 such that 0 lt xmin lt 0 5 is a fixed number such that 0 lt fctx 1 t lt 1 O lt t lt 1 and such that 0 lt fcty 1 t O lt t lt 1 Additionaly a coating layer is attached located between the first curve f ctx 1 t fcty 1 t 0 lt t lt 1 and a second curve fctx 2 t fcty 2 t 0 lt t lt 1 The last connects the point fctx 1 arg1 fcty 1 arg1 fctx 2 0 fcty 2 0 with fctx 1 arg2 fcty 1 arg2 fctx 2 1 fcty 2 1 Moreover fctx 2 t fcty 2 t 0 lt t lt 1 is a simple open arc above fctx 1 t fcty 1 t 0 lt t lt 1 such that O lt fctx t lt 1 O lt t lt 1 The functions fictx ctx 2 fcty 1 and fcty 2 and the parameters argi arg2 and xmin are defined by the code of the file GEOMETRIES cpin c 138 HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHH a aaa OO stack 3 profile t 0 2 sin 2 M_PI t 2 profile 0 2 sin 2 M_PI t 1 profile t 0 0 STACK GRATING i e a stack of 3 profile curves shifted by 2 1 0 micro meter in vertical direction For more details see the description in the USERGUIDE e g pOX 0 1 gt e a i 1 3 2e 0 6 3 1 0 4 cos 2 M_PI t 0 4 sin 2 M_PI t a 028 2 0
278. witch off the automatic generation of additional strips in the FEM domain the user must work with a geometry input file namel inp containing the lines Width of additional strip above and below no Sometimes the additional strip is required below the grating structure but it should be switched off above it This can be indicated by Width of additional strip above and below no_up 0 Here the number following the no_up is the thickness of the additional layer beneath the grating Similarly an additional strip above and no strip below can be indicated by Width of additional strip above and below no_lo 0 If the automatic generation of additional strips is switched off then the input counter of the materials Nmq as well as the list of refractive indices of the grating must not include the materials of the omitted additional strips As mentioned above special gratings like echelle gratings lamellar trapezoidal and simple profile gratings need not to be generated by an input file namel inp Special code words will generate automatically hidden files of this type However in some situations the user might wish to change the automatically generated namel inp files He might wish to add small modifications to the geometry or he wants to change the meshsize To do this the user can create the otherwise hidden namel inp files explicitly by the following executables If an input file for an echelle grating of
279. y blank space in the c code e g profile 0 5 0 5 cos M_PI 1 t 0 25 sin M_PI t gt GRATING DETERMINED BY A SIMPLE SMOOTH PARAMETRIC CURVE i e grating determined by ellipsoidal profile line given as fctx t fcty t 0 lt t lt 1 where the functions t gt fcty t and t gt fcty t are defined by the c codes fctx t 0 5 0 5 cos M_PI 1 t and fcty t 0 25 sin M_PI t respectively do not use any blank space in the c codes e g profiles 121 HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HH HERE zzz gt GRATING DETERMINED BY SMOOTH PARAMETRIC CURVES i e grating determined by profile lines given as fctx j t fcty j t 0 lt t lt 1 j 1 n nmb_curves where the functions t gt fctx j t and t gt fcty j t are defined by the c code of the file GEOMETRIES profiles c e g profiles_par 1 2 3 0 5 0 50 we gt GRATING DETERMINED BY SMOOTH PARAMETRIC CURVES WITH PARAMETERS i e grating determined by profile lines given as fctx j t fcty j t 0 lt t lt 1 j 1 n nmb_curves where the functions t gt fctx j t and t gt fcty j t are defined by the c code of the file GEOMETRIES profiles_par c the last code uses 1 integer parameter and 2 real parameters named IPARaM1 RPARaM1 RPARaM2
280. y is zero then the phase shift is not defined Even for small efficiencies a very fine FEM grid is needed to obtain acceptable approximate values for the phase shifts For example if the user wants to design a beam splitter into four transmitted directions then he would choose e g the objective functional wp nm n n f r enra a eaaa ey 25 10 3 Choosing the functional f r e would result in some kind of anti reflection grating If the user is concerned with the synthesis problem inverse problem then the efficiencies are determined by measurements and the grating realizing these values is sought Knowing that the sought grating is in a certain class of gratings described by the parameter sets r the inverse problem is equivalent to the minimization of f r So le c with c chosen as the corresponding measured efficiencies If the type of polarization and the coordinate system for the incoming wave vector is TE TM then the phase shifts are computed first for the TE polarization and then for n n nix 74 the TM case We denote this dependence by adding a TE and TM respectively Instead of the terms B abt pit at Sow pet 2t 10 4 Dar pho a Date pte in 10 2 one might wish to include B 1 j put TE 1 TM F gt D p TE p2t TM ety T 10 5 Dt L pb TE
281. yer is attached located between the first curve fctx 1 t fcty 1 t 0 lt t lt 1 and a second curve fctx 2 t fcty 2 t 0 lt t lt 1 The last connects the point fctx 1 arg1 fcty 1 arg1 fctx 2 0 fcty 2 0 with fctx 1 arg2 fcty 1 arg2 fctx 2 1 fcty 2 1 Moreover fctx 2 t fcty 2 t 0 lt t lt 1 is a simple open arc above fctx 1 t fcty 1 t 0 lt t lt 1 such that 122 HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHH RR azz O lt fctx t lt 1 O lt t lt 1 The functions ftctxC 5 6te 2 s fcty 1 and fcty 2 and the parameters arg1 arg2 xmin are defined by the code of the file GEOMETRIES cpin c ecg pinz gt gt e g gt COATED PIN GRATING DETERMINED BY TWO PARAMETRIC CURVES TYPE 2 i e over a flat grating with surface x 0 0 lt x lt 1 a material part is attached which is located between x 0 0 lt x lt 1 and fctx 1 t fcty 1 t 0 lt t lt 1 Here fctx 1 t fcty 1 t 0 lt t lt 1 is a simple open arc connecting fctx 1 0 fcty 1 0 xmin 0 with fctx 1 1 fcty 1 1 1 xmin 0 such that 0 lt xmin lt 0 5 is a fixed number such that O lt fctx 1 t lt 1 O lt t lt 1 and such that 0 lt fcty 1 t 0 lt t lt 1 Additionaly a coating layer is attached located between the first curve fctx 1 t fcty 1 t 0 lt t lt 1 and a second curve fctx 2 t fcty 2 t 0 lt t lt 1 The last connects the point x1 0

DIPOG-2.1 User Guide Direct Problems for Optical Gratings

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