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DIPOG-2.0 User Guide Direct Problems for Optical Gratings over

1. 2 An Q Tn BE y k 2 an 72 Re GF gt 0 Sm BF gt 0 resp Binz YA exp i lant Bi 1472 nel H x y 2 pros exp il ilanr Bry 72 nel Now there are three variants of output data The first computes the third components i e the z components of the Rayleigh coefficients 4 gt gt gt y HO 8 n Milk m Bal Ba ta Pn Anl 4 Br Ta and the efficiencies 12 e nee fat pt Slat 2 9 7 1 1 a E nt Ps SPP 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 n 15 Figure 4 Coordinate system based on x z plane gt X axis and to the direction of propagation of the nth reflected resp transmitted plane wave mode s Qn 4B7 9 x 0 1 0 an 6 7 x 0 1 0 then the output coefficients are the scalar products A si _ apn beq n non 1 2 Va e 3 A _ aq bcp n non eVe a b c ME Qn ze y The efficiencies of the second output are the total efficiencies e of 2 9 and the efficiencies corresponding to the TE and TM parts Br p 2 ki as n n BE nt 9 n i st n i e the efficie
2. DIPOG 2 0 User Guide Direct Problems for Optical Gratings over Triangular Grids A Rathsfeld Weierstra Institut f r Angewandte Analysis und Stochastik Mohrenstr 39 D 10117 Berlin Germany rathsfeld wias berlin de lwl il als Weierstrass Institute for Applied Analysis and Stochastics Abstract This is the description of how to use the programs FEM GFEM FEM CHECK GFEM CHECK FEM PLOT GFEM PLOT FEM FULLINFO and GFEM FULLINFO of the package DIPOG 2 0 The package is a collection of finite element FEM pro grams 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 We note that the DIPOG 2 0 programs require the installation of the pre vious version DIPOG 1 3 or DIPOG 1 4 of the grid generator TRIANGLE 1 4 and of the equations solver PARDISC Additionally some of them need the graphical package openGL or the MESA emulation of openGL together with GLTOOLS 2 4 or alternatively the package GNUPLOT Examples of data and output files are enclosed Don t read the complete user guide Don t you have anything better to do To use DIPOG 2 0 go to the directory
3. n are the so called Rayleigh coefficients The interesting Rayleigh coefficients are those with n U E Indeed these coefficients A7 describe magnitude and phase shift of the propagating plane waves More precisely the modulus LA mitted wave mode and arg A A n n n is the amplitude of the nth reflected resp trans the phase shift The terms with n d U lead to evanescent waves only The optical efficiencies of the grating are defined by Ba AE Bari e il n f n neuthuf n neu 2 4 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 E has been computed approximately then the Rayleigh coefficients can be obtained by a discretized Fourier series expansion applied to the FEM solution restricted to T cf and 2 3 Formula vields 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 11 Analogously to formula 2 1 given in the last subsection for the incident electric field we get HZ y 2 t MPA ey z expl iut 2 5 IR EgV nt Honing y 2 savi exp ikt sind z ik cos y kt w m
4. 730540 002606 034668 268177 066021 162601 numb of nonzero entr rate of nonzero entr memorv for pardiso 92896 0 058254 per cent 12659 kB Reflection efficiencies and coefficients n 0 n 1 n 2 n 3 Reflected energy theta theta theta theta 65 15 21 87 00 0 012070 0 139848 74 0 003512 0 014030 33 0 002098 0 002527 07 0 079551 0 082779 2 179773 Transmission efficiencies and coefficients n 0 n 1 n 1 n 2 n 3 n 4 theta theta theta theta theta theta 26 50 Ya 10 29 54 Transmitted energy 0000 l 1 2 3 1 2 3 4 OOOH 26 52 12 015636 414798 032234 degrees of freedom stepsize of discr numb of nonzero entr rate of nonzero entr memory for pardiso 95 0 498922 0 127814 41 0 613317 0 568916 80 0 134396 0 299329 48 0 040241 0 125325 96 0 038098 0 111135 77 0 101705 0 069144 97 820227 50133 0 09156 369931 0 014719 per cent 59346 kB Reflection efficiencies and coefficients n 0 n i n 3 Reflected energy theta theta theta 65 415 87 00 0 012847 0 143535 74 0 000916 0 015911 07 0 088879 0 060381 2 274312 Transmission efficiencies and coefficients n 0 n 1 n 1 n 2 n 3 n 4 theta theta theta theta theta theta 26 50 T 10 29 54 Transm
5. ae Og PEA Or ana ORs Or add e g 1 68 78 0297 The last means that computation is to be done for the wave lengths 634ix 02 with i 0 1 2 and with wave length 63 i 02 less than 73 Wave length 635 TORRE ETE EAE EERE EEE PEPE BE PE EE Temperature in degrees Celsius from 0 to 400 20 for room temperature Must be set to any fixed number Will be ignored if optical indices are given explicitly Temperature 20 TORO RRE RRA EEE TEE PEE EEE TEE 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 Si TIBr TIGI Cr ZnS Ge Sil 0 12 0 Ti02r Quarz AddOn 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 HHH ho HF HH HH H OF 69 the real part of the corresponding optical index third the imaginary part of the corresponding index Optical index Air HEHEHE A H EEEE E EEEE EEA HHEH Optical indices of the materials of the upper coating
6. fu ja P1 Hfolj fy j f ja p2 fy J2 p ce hi t 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 ja 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 the right of the j 1 th pin curve No intersections of pin and profile curves are allowed If the defining two lines are lines pin ccode lines Pipe then we get a SIMPLE SMOOTH PIN CURVE defined by f j t t and f j t ccode The parameter arguments p and pz of the connection points to the ordinary profile curve are fixed bv the second line The remarks on the profile curves applv also here If the defining two lines are line gt 1 pin ccode ccodez lines P pP then we get a SIMPLE SMOOTH PARAMETRIC PIN CURVE defined by fz j t ccode and f j t ccode The parameter arguments p and p 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 line gt 1 pini ccode lines Pi pa dy dy rer di then we g
7. h This help i Toggle isoline mode decrease control parameter by a factor l Toggle level surface mode m Toggle model display when moving p Dump actual picture to eps file look for eps q Mode control Quit Restore last saved state v Toggle vscale for plane sections w toggle wireframe mode x Show x orthogonal plane section y Show y orthogonal plane section z 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 i To continue the computation of the main program press Bar Space After the installation with the package GNUPLOT the pictures are created interactivelv 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 6 Computation of Efficiencies Using GFEM in CLAS SICAL The same computation from the last section can be performed bv generalized FEM cf the result file enclosed in point 10 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 equation More precisely for integers npor and nirem With 0
8. Add phaseshifts if no additional output is needed but if phase shifts are preferred instead of Rayleigh coefficients yes or no or phaseshifts yes jtik E A T E A A E E TEEPE TE EEE TE ESTEE Hit ett E TE EEE TE EEE PE PE EEE BE HE EERE Att TE HE EE EEE H AE EEE Number of coatings 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 56 coated layers of rectangular shape These kind of layers are called coatings over the grating and coatings beneath the grating respectively Number of coatings 2 tA A APARRA ARER RRE Widths of coatings in micro m Needed only if N_co_ov gt 0 Else no number no line Widths 0 5 0 2 TIED A Number of coatings beneath the grating N co be 3 Ri kisi a HERRERA aS HIEH Widths of coatings in micro m Needed only if N_co_be gt 0 Else no number no line 0 2 R A E A AE RER EEE BE EEE PE EEE HE ADEE Wave length in micro m lambda Either add a single value e g 63 Either add more values by e g NV ii 5 63 ii 64 65 69 ell Pa The last means that computation is to be done for the wave lengths from the Vector of length 5 6377 EGA SR 6900 and
9. The last means that computation is to be done for the angles from the Vector of length 5 3 77 A eS BOF and SF 70 2 hs Or add e g e 1 45 56 27 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 than 56 Note that either the wave length or the angle of incident wave theta must be single valued Angle 30 HEHEHE ANNARRA RRR HERRARNA eta t eta AR RRR aaa Angle of incident wave in degrees phi 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 Either add a single value e g 45 Either add more values by e g 71 ji ze 5 63 64 65 69 0 97 The last means that computation is to be done for the angles from the Vector of length 5 63 77 0764 22 G5 7 86929 and 70 Or add e g 1 45 56 209 i The last means that computation is to be done for the angles 45 i 2 with i 0 1 2 and with angle 45Hix2 less than 567 Note that two of the three the wave length the angle o
10. 1 t f 1 6 0 lt t lt 1 and a second curve f 2 t fy 2 t 0 lt t lt 1 The last connects the point f 1 a1 fy 1 a1 f2 2 0 f 2 0 with fell az faQ a2 fz 2 1 fy 2 1 Moreover 1 f2 2 t f 2 t 0 Sts 1 is a simple open arc above fz 1 8 fy 1 t 0 lt t lt 1 such that 0 lt f 2 t lt 1 0 lt t lt l 12 e over a flat grating with surface x 0 0 lt x lt 1 a material part is attached which is located between the line x 0 0 lt x lt 1 and fe 1 t fy 1 t 0 s t 8 1 Here f 1 t f 1 8 O lt t lt 1 is a simple open arc connecting f 1 0 fy 1 0 Emin 0 with f 1 1 fy 1 1 1 Lmin 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 fy 1 t 0 lt lt 1 Additionally a coating layer is attached located between the first curve AL f 1 8 O lt t lt 1 united with the the two straight line segments 2 0 11 lt lt Lmin and 2 0 1 amin lt lt xo and a second curve fz 2 8 fy 2 t 0 lt t lt 1 The last connects the first point 21 0 f2 2 0 fy 2 0 with the last point 22 0 f2 2 1 fy 2 1 Moreover fe 2 t 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 23 Figure 6 Echelle grating of tvpe A then this can be accomplished bv calling the executable GEN CPIN2 from the subdirecto
11. 5 2 Simple calculation with minimal output 2 22 2 nn nn 42 5 3 Check before computation more infos and plots 44 6 Computation of Efficiencies Using GFEM in CLASSICAL 46 7 Computation of Efficiencies Using FEM GFEM in CONICAL 48 8 Produce a Graph of the Efficiencies 49 9 Parameter Test for GFEM 50 10 Enclosed Files 54 10 1 Geometry input file example inp 222m nun 54 10 2 Data file example dat for CLASSICAL 56 10 3 Data file generalized Dat for CLASSICAL same as conical Dat in CON a A A A a e ee e 66 10 4 Data file example dat for CONICAL 68 10 5 Output file example res of FEM FULLINFO in CLASSICAL 79 10 6 Output file example res of GFEM in CLASSICAL 87 10 7 Output file example res of FEM in CONICAL 89 11 Copyright 94 1 Introductory Remarks and the Structure of the Package 1 1 What is DIPOG 2 0 and Dipog 1 4 DIPOG 2 0 is a finite element FEM program to determine the efficiencies of the diffraction of light bv a periodic grating structure The unbounded domain is treated bv coupling with boundarv element methods DIPOG 2 0 solves the classical TE and TM cases i e the cases of 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 whic
12. CHECK GFEM CHECK for check of input exists onlv with openGL argument name dat executables FEM_PLOT and GFEM_PLOT for calculation with plots of resulting fields case of classical diffraction exists only with openGL or GNUPLOT argument name dat executables FEM_FULLINFO and GFEM_FULLINFO for calculation with additional information case of classical diffraction 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 result files name res and name erg produced by executables in CLASSICAL and CONICAL executable PLOT DISPLAV produces two dimensional graph of data on the screen argument name res and indices of modes the efficiencies of which are to be plotted executable PLOT PS produces ps file of two dimensional graph of data ar
13. 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 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 0
14. if Grating data is stack k gt N_mat k 1 Number of materials 4 HEHEHE RRR aa RARER RAH 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 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 FAO 5 1 50 7 ji if Grating data is profiles gt N_mat nti with n nmb curves from ji the file GEOMETRIES profiles c if Grating data is profiles gt N_mat n 1 with n nmb_curves from ii the file GEOMETRIES profiles par c if Grating data is pin gt N_mat 3 if Grat
15. 0 013 9 kkx kK kkxk GFEM 1 3 0 023 7 0 023 7 kK kk xxx GFEM 1 7 0 044 6 0 044 6 ke kek kkxk GFEM 1 15 0 088 5 0 044 6 0 012 8 xxx kik GFEM 3 7 0 022 6 0 022 6 kK kk kkxk GFEM 3 15 0 044 5 0 044 5 kK kk kkxk GFEM 3 31 0 100 4 0 100 4 0 012 7 xxx de 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 ek kkxk 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 kkx GFEM 15 127 0 081 2 0 011 5 xxx ede 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 01 accuracy 53 10 Enclosed Files 10 1 Geometry input file example inp A HHH HHH example inp FEET IEE EP all lines beginning with are comments HH HH H FH HHH HH H FH ST EET EET CREE TE HE E TE PE HE TE HEEE TE H AE E PEELE ETE EEE TD AEE E E E TE HE HE EE PE EEE geometry input file for periodic grating located in directory GEOMETRIES input file for gen polvx HHH HH tA A SPP EE BE EEE A EEEE EEEE E HHE Name of the files without extensions inp Output files will have the same name but with tags polvx and sg Name exampl
16. 0 50 0 90 0 00 0 500 0 900 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 60 i 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 lAmellar 1 1 0 2 0 8 gt a gt 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 e g polygon filef 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 tt 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
17. 1 at least two different v coordinates last line should be End e g polygon2 filel file2 gt COATED GRATING DETERMINED BV 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 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 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 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
18. 10 7 Output file example res of FEM in CONICAL date 10 Feb 2003 10 06 22 ARO FK FK FK FK tu K K 2 2 K K FK FK FK FK tatu FK K K K ad 2K FK FK FK kk K ARO K FK FK FK FK FK K K K K FK FK FK FK FK K K FK FK K K K K FK 2K 2K FK FK 2 K ok kk kk kk CONICAL kk kk kk AR 2g K K K FK FK FK FK tu K K K 2 2K K FK FK FK K FK FK FK K K K ad dd ale KK ARO FK FK FK FK 3K K K K K K K FK FK FK FK FK K FK K K K K FK 2K 2K 2K FK KK Reflection efficiencies and coefficients Order Phi Theta E Z H_z Efficiency O 47 00 30 00 0 19657 0 16355 0 05050 0 15697 9 25813 1 128 80 27 98 0 01482 0 03567 C 0 06213 0 13130 2 30360 2 158 51 86 74 0 07634 0 09663 0 12963 0 05033 0 22670 Reflected energy 11 78843 Transmission efficiencies and coefficients Order Phi Theta EZ 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 89 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 Hz Efficiencv O 47 00 30 00 0 00901 0 00844 0 13596 0 02289 1 91613 1 128 80 27 98 0 00200 0 08454 0 04245 0 01373 0 93218 2 158 51 86 74 0 06824 0 05857 0 06300 0 12368 0 17972 Reflected energy 3 02802 Transmission efficien
19. 4 4 8 12 1 86 conv ord 0 39 0 79 1 99 l conv ord 2 47 6 29 l 0 11 l conv ord 1 78 1 88 JI 1 91 conv ord 0 89 1 57 2 15 l H 4 t iio value extrap l error 93 4264 4 2479 97 9640 0 2896 97 8202 97 8246 0 1459 97 7257 97 5442 0 0513 97 7000 Il 97 6905 0 0257 date 10 Feb 2003 09 56 18 Thank you for choosing dpogtr Bye bye 10 6 date 10 Feb 2003 10 13 19 ARO FK FK FK FK FK dd FK FK FK FK K K FK K K K K 2g 2K 2K FK FK FK FK K K ARO FK FK FK FK FK K K K K FK Ral FK K K K ld 2K 2K FK KK K kk kk kk GDPOGTR kk kk kok KKK KK K K RR K K K K 2 K FK FK FK K K K K K K K 2 2g 2K 2K FK FK ak ak ok K 2K gt gg 2g K 2 K FK FK FK FK tu K K K K FK K FK FK FK 2K K K FK K K K K FK 2K 2K 2K FK FK K 2K ok Reflection efficiencies and coefficients O theta 1 87 65 00 0 013025 0 144735 theta 15 74 0 000363 0 016237 e_ e_ Output file example res of GFEM in CLASS
20. 70 Or add e g 1 63 73 02 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 than 737 Wave length 635 HEHEHE RRR AAA aE RAHA Temperature in degrees Celsius From 0 to 400 For room temperature set to 20 Will be ignored for explicitly given refractive indices Temperature 20 TERA PETRIE A A Optical index refractive index of cover material H HH 57 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 ii like Air Ag Al Au CsBr Cu InP MgF2 NaCl PMMA PSKL SF5 Si TIBr TIC1 Cr ZnS Ge i1 0 12 0 Ti02r Quarz AddOn 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 L 0 A Optical indices of the materials of the upper coatings Needed only if N_co_ov gt 0 Else no number no line Opti
21. AA Ebeh d rL LI IE EC LEEFH y 0 LENE LE PARE Xp iia hse Pu aa IPE O kI i 0 5 PAGT BELLA Pete ERATE i PENES EFPL IIEP K LHF a dil la d 1 TAH A He im Fl A ri ki H Mu fri 15 EM Hi N LLEI l fan 2 um FEK l E Li 1 5 1 0 0 5 0 1 ve ga y 0 4 0 5 Y 0 6 0 5 x 01 08 0 9 1 Figure 21 Imaginary part of transverse component of magnetic field Output of executable GFEM_PLOT in CLASSICAL via GNUPLOT 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 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 45 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
22. DIPOG 2 0 CLASSICAL or DIPOG 2 0 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 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 4 1 1 What is DIPOG 2 0 and Dipog 1 4 o o 4 1 2 Programming language and used packages 22 22 2200 4 13 Get executables comment lines in input files 5 14 Structure of the package re ehr Ela ar ran 5 2 Diffraction Problems for Gratings 9 2 1 The classical TE problem 5 2 u a uw a Sa ar ren 9 2 2 The classical TM problem 2 2 22 222 2a sam 2 ka 11 2 3 Conical probleme 24 x 2 2 a bie SW eG si ya ae ward 12 2 4 References did si da ki 40 ma eb bt a L 17 3 seometry Input 18 3 1 Geometrical data in input file name dat 18 3 2 How to get an input file namel inp 2 2 En nn nn 19 3 3 Code words to indicate special geometries 24 3 4 Stack grating by code words 0 36 4 Input of Refractive Indices 41 5 Computation of Efficiencies Using FEM in CLASSICAL 42 5 1 How to get an input file name2 dat o nn 42
23. K FK FK FK FK ak K K ARO FK FK FK FK FK K K K K tatu FK FK FK 2K FK K FK K K K K FK FK 2K 2K 2K ak K K kk kk kk DPOGTR kk kk kk ARO FK FK FK FK FK K K K K tatu FK FK FK K K K K K K tu tubu 2K FK FK FK ak ok K KKK RR FK FK FK FK FK K K K K tu FK FK FK FK K FK FK K K K K K FK FK 2K 2K 2K ak 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 1 x V v v min method FEM partitioning based on Shewchuk s Triangle matrix assemblv pdelib solver pardiso code generated with DCONV example Comments This is a fantasv grid for the test of gen polvxl Number of materials 4 Minimal angle of subdivision triangles 20 000000 Upper bound for mesh size 0 500000 width of additional strip above and below 0 200000 Grid points 1 0 000000 0 800000 25 0 500000 0 800000 oe 0 000000 0 400000 4 0 250000 0 400000 b 1 000000 0 400000 6 0 750000 0 200000 T 1 000000 0 200000 8 0 000000 0 000000 Di 0 250000 0 200000 10 1 000000 0 200000 iii 0 000000 0 600000 12 1 000000 0 800000 Triangles 1 3 4 mat 2 fac 1 000000 4 6 2 mat 2 fac 1 000000 6 7 5 mat 2 fac 1 000000 3 8 4 mat 2 fac 1 000000 8 9 4 mat 2 fac 0 300000 4 9 6 mat 2 fac 1 000000 6 10 7 mat 3 fac 1 000000 8 11 9 mat 3 fac 1 00000
24. 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 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 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 CONICAL RESULTS MAKES data for the GFEM executables FEM and GFEM for simple calculation case of classical diffraction argument name dat executables FEM
25. angled triangle with hvpotenuse 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 a 60 i e the depth is equal to the period multiplied by sin a cos a and other parameters like above Grating data echelleb 60 0 05 indicates an ECHELLE GRATING TYPE B right angled triangle with one of the legs par allel to the direction of the periodicity cf Figure with angle 60 angle enclosed by hypotenuse and by the leg parallel to the period and with a coated layer of height 0 05 um measured in direction perpendicular to echelle profile height greater or equal to zero Grating data trapezoid 60 0 6 3 0 2 0 1 0 1 0 05 indicates a TRAPEZOIDal Grating isosceles trapezoid with the basis parallel to the direc tion of the periodicity cf Figurej8 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 25 Figure 8 Trapezoidal grating lAmellar 3 4 0 2 0 6 0 2 1 0 0 0 5 0 7 0 0 0 50 0 90 0 00 0 500 0 900 indicates a LAMELLAR GRATING rectangular grating consisting
26. 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 O 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 te 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 which is located between 1 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 fctv 1 0 xmin O with fetx 1 1 fetv 1 1 l xmin O such that 0 lt xmin lt 0 5 is a fixed number such that O lt fctx 1 t lt 1 0 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 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 More
27. di 0 and dz d is allowed If the defining two lines are 36 lines trapezoid di da d3 a lines hi then we get asymmetric TRAPEZOIDAL GRATING profile cf Section 3 3 with the trapezoid starting at x d ending at x da and with the angle a and the height d3 The profile is vertically shifted by h um Note that we require d gt 0 0 lt d lt d3 lt 1 and 0 lt a lt 90 The height must be sufficiently small such that the trapezoid does not degenerate If the defining two lines are line gt 1 profile ccode lines hi then we get a profile curve DETERMINED Bv A SMOOTH SIMPLE CURVE defined bv F 5 t t and f j t ccode The profile is vertically shifted by hi um 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 fet 0 into fet ccode leads to the meaningful code fct 0 if t lt 0 3 feta 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 line gt 1 profile ccode ccodez lines hi then we get a profile curve DETERMINED Bv A SMOOTH SIMPLE PARAMETRIC CURVE defined by f j t ccode and f j t ccodez The profile is v
28. 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 fctx 2 0 fcty 2 0 with x2 0 fctx 2 1 fcty 2 1 with 0 lt x1 lt xmin lt 1 xmin lt x2 Moreover f ctx 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 0 lt t lt 1 The functions fctx 1 fetx 2 5 fcty 1 and fcty 2 and the parameter xmin are defined by the code of the file GEOMETRIES cpin2 c e g stack 3 profile t 0 2 sin 2 M_PI t 2 profile 0 2 sin 2 M_PI t U profile t 0 O s 2 gt 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 Grating data lamellar A E AE N EIERN Af RER AE E HE A DE E E BE E HEE TE HEHE EE HE HE BE HE E TETEH EREHE Number of different grating materials N_mat This includes the material of substrate and cover material For example if Grating data is namei gt Number of materials given in file if Grating data is echellea gt N_mat 3 with coating height 20 N mat 2 with coating height 0 if Grating data is echelleb gt N_mat 3 with coating height 20 N_mat 2 with coating height 0 if Grating data is trapezoid gt N_mat number of material layers 3 for coating height 20 N mat number of materia
29. five letters like e g user The file consists of at Al most 112 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 imaginarv part of the index At the end of each line a comment beginning with the sign ff can be added Optical indices with negative real or imaginarv parts are not admitted As seen in the example presented above first the index of the cover material is given Then the indices of the materials of upper coated lavers follow These are rectangular lavers 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 lavers is zero then no lines with optical indices are needed Next the indices of the materials of lower coated lavers 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 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
30. layers This is c times square root of mu times epsilon H N_co_ov entries Needed only if N_co_ov gt 0 Else no entry and no line Optical indices 1 2 1 3 HA i 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 H HHRH ERRAR RAPAREN TEE AHRR TEL EAE EEE ERE EE PEE Optical index of substrate material This is c times square root of mu times epsilon Optical index 1 5 1 0 A Ai Tvpe 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 3 Com HEHEHE HRA RAR aa RRR aaa Type of polarization and coordinate system for incoming wave vector i 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 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
31. li 0 502603 0 087406 0 642427 0 535243 0 174148 0 293832 0 069855 0 111978 0 062905 0 082480 0 036440 0 097038 97 700031 Rm extrap error 1 3312 l 0 4006 1 9411 0 1608 2 1617 0 0544 2 1133 0 0272 cc 103941 059386 000211 136430 e 447303 e 715032 e 1 13 e e e 675032 026452 102941 733271 extrap NO ERROR ANALYSIS FOR ORDER 3 4 VALUE ORDER 3 4 4 l value extrap 4 4 20802 0 2682 0 1595 0 1545 0 1397 0 1353 0 1364 0 1358 4 REFLECTED ENERGY 4 4 l value l extrap 6 5736 2 0360 2 1798 2 1754 2 2743 2 4558 2 3000 2 3095 4 4 ORDER 0 4 4 l value l extrap 22 1644 27 0660 27 9758 28 1831 l 27 5337 27 6783 27 4473 27 4263 4 ORDER 1 4 4 l value extrap 4 4 56 7616 55 5900 H 4 H 4 4 Hh 4 H 4 4 H 4 h 4 4 en l 1 36 2 14 2 73 l TE en conv ord on 4 52 2 46 l 2 61 nn n
32. lt 1 where the functions tl gt fcetx t and t gt fcty t are defined by the c code of the file GEOMETRIES profile c e g profile par 2 3 1 61 i oO PO WN 0 38 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 t 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_PIx 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 de
33. of binary gratings I Direct problems and gradient formulas Math Meth Appl Sci 21 pp 1297 1342 1998 J Elschner and G Schmidt The numerical 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 J ger 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 Babuska 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 35 255 299 199
34. of several materials placed in rectangular subdomains cf Figurejp with 3 columns each divided into 4 rectangular lav ers first column with x coordinate in 0 um x 0 2 um second column with 0 2 um x 0 6 pm third column with 0 6 um lt x lt period period given above whole grating with y coordi nate s t 0 2 ym lt y lt 1 0 um first column first layer with 0 2 ym lt y lt 0 um second with 0 um lt y lt 0 5 um third with 0 5 um lt y lt 0 7 um and fourth with 0 7 wm lt y lt 1 um second column first layer with 0 2 ym lt y lt 0 0 um second with 0 0 ym lt y lt 0 50 um third with 0 50 um lt y lt 0 90 um and fourth with 0 90 um lt y lt 1 um third column first layer with 0 2 um lt y lt 0 00 um second with 0 00 um lt y lt 0 500 um third with 0 500 ym lt y lt 0 900 um and fourth with 0 900 ym lt y lt l um Grating data lAmellar 11 0 2 0 8 indicates a SIMPLE LAYER special case of lamellar grating with layer material s t y coordinate satisfies 0 2 ym lt y lt 0 8 um 26 Figure 9 Lamellar grating Grating data polygon filel indicates a GRATING DETERMINED BY A POLYGONAL LINE cf Figure 10 defined by the data in the file with name GEOMETRIES filel in GEOMETRIES filel in each line begin ning 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 coo
35. 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 Dr pols 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 70 TM RI ib iS ida RRR REE RRR aaa 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 Ri RRR RRRA R AAPP REE RRR Raa Angle of incident wave in degrees theta 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 ec V 102 27
36. subtriangles After this uniform refinement the degrees of freedom and the trial space of approximate Helmholtz solutions are defined using npor and NLFEM Change 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 nep x Mirem IP gt nar x npor 1 6 1 nupa and the storage to nap x npor 1 2 Hence a doubling of nirem 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 nupa 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 large triangles and a huge number of small triangles e g geometries with thin lavers In this case the next level uniform partition increases the number of grid points and the computation time significantlv 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 n
37. that 0 lt f 1 t 0 lt t lt 1 Additionally a coating layer is attached located between the first curve fs 0 411 01 0 lt t lt 1p and a second curve 1 EB hd 0 lt t lt 1 The last connects the first point f 1 a1 f 1 a1 fz 2 0 f 2 0 with the last point ee 1721 fy 2 1 Moreover fa 2 t 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 The functions f 1 f 2 fy 1 and f 2 and the parameters aj as and Zmin are defined by the code of the file GEOMETRIES cpin c Grating data cpin2 indicates a COATED PIN GRATING DETERMINED By Two PARAMETRIC CURVES TYPE 2 ci Figure 18 where Emin 0 2 fall t Emin 1 2 min t fy 1 t 0 5 sin 7t and f 2 1 0 5 2min 1 Lmin t fy 2 t 0 8 sin at i e over a flat grating with surface x 0 0 lt a lt 1 a material part is attached which is located between the line z 0 0 lt x lt 1 and 404114 LO US te 1 Here MALO LO Os t 1 is a simple open arc connecting f 1 0 fy 1 0 Emin 0 with f 1 1 f 1 1 1 Tmin 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 at tached located between the first curve f 1 fy 1 t 0 lt t lt 1 united with the the 32 Figure 15 Grating
38. the following important subdirectories The 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 GEOMETRIES input files name inp geometrical data of the gratings executable SHOW to visualize the input data name inp exists only with openGL argument 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 tvpe 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 laver fifth argument width of second part of laver executable GEN ECHELLEB to generate an input file for an echelle grating of tvpe B first argument name without tag inp of file to be created second argument angle third argument width of laver executable GEN TRAPEZOID to generate an input file for a trapezoidal grating first argument name without tag inp of file to be created second argument angle third argument length of basis fourth argument number of material lavers in trapezoid next arguments heights of material lavers last argument height of coating laver exe
39. 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 0 and 1 Length 0 A Stretching factor for grating in v direction Must be a positive real number Length il Rik RRR RRR RAE HH HH HIER 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 O FERRE A Period of grating in micro m 1 IHR A 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 HHH HHH HH H HH 59 HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHH RR azz 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
40. with 0 lt fe j t lt 1 for0 lt t lt 1 If the defining two lines are linea echellea R d lines hi then we get an ECHELLE TYPE A grating profile with right blaze angle greater than 45 with depth di and without coated laver cf Section 8 3 The profile is verticallv shifted by hi um 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 di by a then the left interior angle is a degrees If the defining two lines are lineg 1 echelleb a line j hi then we get an ECHELLE TYPE B grating profile with angle a and without coated layer cf Section 13 3 The profile is vertically shifted by hi um Note that 0 lt a lt 90 If the defining two lines are lines binary do di da da lines fy then we get a BINARY GRATING PROFILE with 2 teeth with height do with transition points di da da and without coating layer In other words before the shift the grating function is zero between 0 and d between d and dz between ds _ gt and da _1 and between ds and the period d The grating function is dy between d and da between d and d4 between da _3 and da _ gt and between da _ and da _ The profile is vertically shifted by hi um Note that i lt 6 0 c d O lt di lt d2 lt lt da lt d For i equal to 1 or 2
41. 0 9 12 6 mat 3 fac 1 000000 6 12 10 mat 3 fac 1 000000 INPUT DATA 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 80 file width in grating geometry 0 40 micro m 0 30 micro m 0 20 micro m REFRACT INDICES cover material 1 00 10 00 layers above grating 1 20 i 0 00 layers below grating 2 30 i 0 00 2 20 i 0 00 2 10 10 00 substrate material 2 00 1 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 period 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 i 0 00 2 30 i 0 00 nmb of levels f comp 5 INFO OF SOLUTION LEVEL 1 degrees of freedom 813 stepsize of discr 0 81789 Il a 00 an o numb of nonzero entr rate of nonzero entr memory for pardiso 0 886426 per cent 536 kB Reflection efficiencies and coefficients n O theta 65 00 0 039594 0 181811 e_ 0 3 462309 n 1 theta 15 74 0 008538 0 021079 e_ 1 0 117800 n 2 theta 21 33 0 031693 0 008219 e 2 0 236279 n 3 theta 87 07 0 102928 0 466177 e_ 3 2 757185 81 Refl
42. 1 and 0 01 choosing the refinement levels 4 4 and 5 respectively 16Fastest method means the one with the smallest complexity estimate 6 1 51 METHOD k 25 20 k 50 39 d 8 d 16 FEM xk xk KA k GFEM 0 012 8 xk xk GFEM 0 023 7 xk xk GFEM 0 023 7 0 012 8 GFEM 3 7 0 012 7 kk x GFEM 0 022 6 0 012 7 GFEM 0 044 5 0 012 7 GFEM 0 044 5 0 044 5 GFEM 0 011 6 kk GFEM 0 050 4 0 022 5 GFEM 0 050 4 0 022 5 GFEM 0 100 3 0 022 5 GFEM 15 31 0 011 5 kK GFEM 0 050 3 0 011 5 GFEM 15 255 0 050 3 0 025 4 GFEM 0 081 2 0 025 4 Table 1 Relative mesh size h refinement levels necessary to reach 1 accuracy METHOD k 25 20 d 8 k 50 39 d 16 FEM KKK Xx Xk xk GFEM GFEM GFEM kk k kk 0 012 8 Xx Xk xk Xx Xk xk K kK GFEM GFEM GFEM GFEM kk 0 012 7 0 022 6 0 022 6 Ak kk kii kii 0 012 7 GFEM GFEM K kK 0 011 6 0 050 4 0 050 4 kiki KKK 0 011 6 0 022 5 GFEM kk 0 025 4 0 050 3 0 050 3 KKK Xix 0 011 5 0 025 4 Table 2 Relative mesh size h refinement levels necessarv to reach 0 1 accuracv METHOD k 3 15 k 6 30 12 60 k 25 20 k 50 39 d 1 d 2 d 4 d 8 d 16 FEM 0 013 9
43. 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 0 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 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 fetx 1 rg2 cty 1 arg2 fetx 2 1 fety 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 2 t lt 1 0 lt t lt 1 The functions fctx 1 fctx 2 fcty 1 and fcty 2 and the parameters argl arg2 and xmin are defined by the code of the file GEOMETRIES cpin c e g oping gt COATED PIN GRATING DETERMINED BY TWO PARAMETRIC CURVES TYPE 2 76 i e over a flat grating with surface x 0 0 lt x lt 1 a material part is attached which is located between 1 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 fetx 1 1 fetv 1 1 l xmin O such that 0 lt xmin lt 0 5 is a fixed number such that O lt fctx 1 t lt 1 0 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 fctx 1 t
44. 3 7 n LFEM 2xn DOFH1 b challenging accuracv requirements or large wave numbers n DOF 3 with n LFEM 31 or n DOF 7 with n LFEM 127 or n DOF 15 with n LFEM 512 HHH HH HH HHH H HOH 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 vou wish to compute on level 1 041 1 set the Ho H HH H HF 66 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 HHH THREE H EEEE EEEE 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 Here interpolation is taken over uniform grid with n_DOF 2 interpolation knots Value should satisfy 0 lt n_DOF lt 100 value 3 HEHEHE AAAA AAA H H A A EEE E EAEE A E EE EEEE E EE EEEE EEEE n LFEM Approximate solution determined bv 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 conventi
45. 5 micro meter measured in direction perpendicular to echelle profile height greater or equal to zero 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 lAmellar 3 4 0 2 0 6 20 2 1 20 ooo 0 00 73 i 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 lAmellar 1 1 0 2 0 8 e gt SIMPLE LAYER spec
46. 8 F Ihlenburg Finite element analysis of acoustic scattering Springer Verlag New York Berlin Heidelberg Applied Mathematical Sciences 132 1998 J M Melenk and I Babu ka The partition of unity method Basic theory and appli cations Comp Methods Appl Mech Eng 139 pp 289 314 1998 17 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 0 CLASSICAL FEM name dat Or cd DIPOG 2 0 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 additional shift of grating geometiv This is shift into the x direction This is length of shift relative to period 0 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 1 Grating data namel Here namel refers either to a file namel inp w
47. 8 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 fastestif method with parameter nupa 1 by bold letters However the methods with doubled npor 1 and InLFEM 1 and doubled mesh size one lower refinement level require almost the same computation time and lead to the same accuracy If Inirem 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 triangles 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 Inirem 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 nirem 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
48. C CURVE cf Figure 16 where Emin 0 2 falt i ea 7 0 0 5 sin rt i e over a flat grating with surface z 0 0 lt x lt 1 a material part is attached which is located between x 0 Owes and if fylt O lt t lt 1 Here O fyt 0 lt t lt 1 is a simple open arc connecting fz 0 f 0 Emin 0 with fall fy 1 1 Emin 0 such that 0 lt Zmin lt 0 5 is a fixed number such that 0 lt f t lt 1 0 lt t lt 1 and such that 0 lt f t 0 lt t lt 1 The functions f f and the parameter x are defined by the code in GEOMETRIES pin c Grating data cpin indicates a COATED PIN GRATING DETERMINED By Two PARAMETRIC CURVES cf Fig ure 17 where Lmin 0 2 bi 02 4 0 8 fe 1 t tee en fy 1 t 0 5 sin zt and f 2 t fy 2 t 0 lt t lt 1 is the polygonal curve connecting the four points fs 1 a1 fy 1 a1 fe 1 a1 fy 1 0 5 0 1 f2 1 a2 fy 1 0 5 0 1 and fa 1 a2 fy 1 a2 i e over a flat grating with surface r 0 0 lt x lt 1 a material 31 Figure 14 Grating determined by a simple smooth parametric curve part is attached which is located between x 0 0 lt x lt 1 and f2 1 t fy 1 t 0 lt t lt 1 Here f 1 t f 1 t 0 lt t lt 1 is a simple open arc connecting fa 1 0 fy 1 0 Ein 0 with LD AD 1 2 0 such that 0 lt Tin lt 0 5 is a fixed number such that 0 lt f 1 t lt 1 0 lt t lt 1 and such
49. E and the triangle RLQR 22 cf Figure 15 where n 3 204 t 7141 sin 2rt fy 2 t sin 2rt 0 5 and f 3 t sin 27t 1 If an input file for a pin grating determined by a simple non intersecting profile line given as K f t fy t 0 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 Tmin and the profile line f t f 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 step size equal to 0 06 times the length of period cf Figure 16 where Emin 0 2 felt Emin 1 23min t and fy t 0 5 sin rt If an input file for a coated pin grating determined by the simple non intersecting pro file lines given as fr j t f 3 8 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 tin 41 42 and the profile lines f 3 8 fy J 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 step size equal to 0 06 times the length of peri
50. ES profile par c e g profile 0 125 sin 2 M_PIx t gt GRATING DETERMINED BY A SIMPLE SMOOTH FUNCTION i e grating determined by sine profile line given as 1 t fcty t 0 lt t lt 1 where the function t gt fcety 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_PIx 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 t gt fcty j t are defined by the c code of the file GEOMETRIES profiles c Ce e g profiles_par 1 2 75 i 3 0 5 0 50 e gt GRATING DETERMINED BY SMOOTH PARAMETRIC CURVES WITH PARAMETERS i e grating determined by profile lines given as fctx j t cty j t 0 lt t lt 1 j 1 n nmb_curves where t
51. ICAL 2 111777 0 060075 n 2 theta 21 33 0 000035 0 001347 n 3 theta 87 07 0 091483 0 053904 Reflected energy 2 308647 Transmission efficiencies and coefficients n O theta 26 95 0 502786 0 085087 n 1 theta 50 41 0 644183 0 532792 n 1 theta 7 80 0 176698 0 293270 n 2 theta 10 48 0 071550 0 111089 n 3 theta 29 96 0 063904 0 080675 n 4 theta 54 77 0 032720 0 097400 Transmitted energy 97 691353 2 3 i 2 3 4 27 52 13 031272 085740 720512 Reflection efficiencies and coefficients n O theta 65 00 0 013043 0 144752 n 1 theta 15 74 0 000365 0 016240 n 2 theta 21 33 0 000044 0 001384 n 3 theta 87 07 0 091547 0 053854 Reflected energy 2 309310 Transmission efficiencies and coefficients si 2 3 4 1 92 3 E OOON 27 52 13 031559 084372 719528 n O theta 26 95 0 502838 0 084941 n 1 theta 50 41 0 644282 0 532607 n 1 theta 7 80 0 176861 0 293228 n 2 theta 10 48 0 071650 0 111036 n 3 theta 29 96 0 063954 0 080553 n 4 theta 54 77 0 032481 0 097406 Transmitted energy 97 690690 END date 10 Feb 2003 10 30 53 Thank you for choosing Bye bye gdpogtr 88 000400 136395 424602 687989 741238 112318 060098 000423 136472 427416 682703 745111
52. LINFO 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 file name2 dat but require the additional input file generalized Dat cf the enclosed file in 10 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 47 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 step size is side length divided by nifem 1 NUPA This is for additional uniform partition of all primary grid triangles into nupa X nupa equal subdomains i e the original side of the grid triangle is split into nupa sides of uniform partition
53. RAMETRIC CURVE i e grating de termined by profile line given as f t f t 0 lt t lt 1 where the functions t f t and t gt f t are defined by the c code of the file GEOMETRIES profile c cf Figure where f t t and f t 1 5 0 2 exp sin 6zt 0 3 exp sin 87t 27 Grating data profile_par 2 3 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 f t ft 0 lt t lt 13To make it precise suppose P and Q are the common corner points of the two polygonal curves and that Ri R Ri are the consecutive corner points between P and Q on the first polygonal line and Ri 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 Ri the quadrilaterals Ri R R3 Rj sisa A pe ee R and the triangle Ri QR m 1 0 m m 1 28 Figure 11 Grating determined bv polvgonal lines 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 inte
54. The subdirecto ries 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 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 1 We suppose that a plane wave is incident from above If needed change to the subdirectories and follow the instructions of the corresponding files README txt If not needed delete the subdirectories 6This subdirectory exists only during internal installation with a direction located in the x y plane i e in the plane perpendicular to the grooves and under the incident angle 0 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 t
55. amplitude 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 0 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 f enclosed by the x axis and by the projection of the electric field vector E to the 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 0 The Rayleigh expansions above 14 gt Ds Figure 3 Coordinate system based on x z plane resp below the grating take the form E z y 2 ET exp iloz By qz J exp il ijanz Bry yz E 2 An Hiz y 2 Hi exp ifaw By y2 ya exp ilona Bty 12 EZ
56. ant material as well as some transmission conditions on the interfaces between materials of different refractive indices The wave number k is equal to w c times the refrac tive index of the material Thus we can determine E 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 1 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 coatingsf On It and above T resp on I and below I7 the component E admits an expansion into the Rayleigh series of the form E v y gt Atexp 18 y exp ianz Ag exp isfy exp iar 2 2 EAT gt A exp 1879 exp ians 2 3 BE k 2 a y De wn Aine l A ni gt c 0 vnt a k sind an kt sing In TFor technical reasons in the FEM code it is important to have the same material on both sides of the boundary lines T 10 x Incident wave AN Reflected modes Transmitted modes Figure 1 Cross section of grating Here d is the period of the grating and the complex constants A we E fn EZ lan lt kt if Im kt 0 0 if Sm k gt 0
57. ar equations solver PARDISO together with some LAPACK and BLAS routines For good visualization the package openGL or at least 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 visu alization i e without openGL and GNUPLOT To get informations on the necessary packages we refer to DIPOG schmidt wias berlin de rathsfeld wias berlin de TRIANGLE http www cs cmu edu quake triangle html GLTOOLS http www wias berlin de software gltools GNUPLOT http www gnuplot info 3Contrarv to the original meaning of binarv several lavers are admitted PARDISO Olaf Schenk unibas ch O Schenk K G rtner W Fichtner Efficient Sparse LU Factorization with Left Right Looking Strategy on Shared Memory Multiprocessors BIT Vol 40 158 176 2000 O Schenk K Gartner Solving unsymmetric sparse systems of linear equations with PARDISO to appear in Journal of Future Generation Computer Systems 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 0 home directory To do so produce a header file MHEAD in the subdirectory MAKES of the DIPOG 2 0 home directory Just copy one of the example files MHEAD_SGI MHEAD_LINUX or MHEAD_DEC to MHEAD and change the operating syst
58. at is the same but has a new name Now it is called conical Dat 8 Produce a Graph of 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 13 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 23 To interrupt the presentation of the picture press Enter Return Alternativelv one can enter the command PLOT PS name3 1 2 3 4 49 Evervthing 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 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
59. atings 2 3 i 0 2 2 aes 21 A Optical index of substrate material Al Wave length in micro m lambda 635 Temperature 20 Optical indices of grating materials 152 user Ly AN 2 3 1 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 the indices are all numbers then the temperature is not used The code words for materials can be Air Ag AP Au CsBr Cu InP MgF2 NaCl PMMA PSKL SES Si TIBr TI Cr ZnS Ge TiO2r Quarz AddOn and Sil 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 nsio x 1 ngio For example Air corresponds to an index n 1 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 CONICAL Its name must begin with the letter u and may consist of no more than
60. cal indices ii ee os GO 122 1 50 A Optical indices of the materials of the lower coatings Needed only if N_co_be gt 0 Else no number no line Optical indices 220 ti 00 2 2 i 0 2 1 i 0 TERRA A Af Optical index of substrate material 2 20 Fi 20 TEEPE A Angle of incident wave in degrees theta Either add a single value e g 45 Either add more values by e g ji 5 ii 63 64 65 69 TOs 99 The last means that computation is to be done for the angles from the Vector of length 5 63 77 864 9 657 y 6972 and 70 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 angle 45 i 2 less than 56 98 Note that either the wave length or the angle of incident wave must be single valued Angle of incident wave 65 THREE OO EE HHE Type of polarization Either TE TM or TE TM Type TM A Length factor of additional shift of grating geometry This is shift into the x direction i e the i 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
61. cies and coefficients Order Phi Theta E_Z H Z Efficiency 0 47 00 160 53 0 31129 0 27182 0 49973 0 42367 62 7 7211 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 0 08036 0 05007 0 10289 0 07817 2 25198 2 158 51 138 27 POS EMS FO IO AOS Transmitted energy 96 97198 Reflection efficiencies and coefficients 90 Order Phi Theta E_z H_z Efficiency 0 47 00 30 00 0 06129 0 00846 0 09079 0 04955 1 45263 1 128 80 27 98 0 02946 0 01247 0 02120 0 02318 0 20498 0 05922 0 00385 0 01123 0 00823 0 02441 2 158 51 86 74 Reflected energy 1 68202 Transmission efficiencies and coefficients Order Phi Theta E_Z H_z Efficiency 0 47 00 160 53 0 38108 0 19799 0 58998 0 31418 66 49400 1 20 54 135 99 0 22472 0 00453 0 37512 0 09299 15 48188 1 128 80 161 77 0 07243 0 23512 0 06585 0 25190 15 85666 2 158 51 138 27 0 05169 0 00215 0 03000 0 03203 0 48544 Transmitted energy 98 31798 Reflection efficiencies and coefficients Order Phi Theta E_z H_z Efficiency O 47 00 30 00 0 06068 0 01649 0 07617 0 03097 1 07145 1 128 80 27 98 0 02951 0 00963 0 01652 0 00861 0 13367 2 158 51 86 74 0 04706 0 03015 0 00604 0 01967 0 02330 Reflected energy 1 22842 Transmission efficiencies and coefficients 91 Order Phi Theta E_z H_z E
62. ckness 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 ff 3 2 How to get an input file namel inp Change to subdirectory GEOMETRIES Copy an existing file like e g example inp cf the enclosed file in 10 1 and change its name into e g namel inp cd DIPOG 2 0 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 beginning with ff is a comment To check the result 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 distinguished by different optical indices in different colours Press Escape or Bar Space to end the check 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 As mentioned above special gratings like echelle gratings lamel
63. coating layer resp of the cover if there does not exist anv 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 anv 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 10 2 in the subdirectory CLASSICAL is needed To get this change the directorv to CLASSICAL copv one of the existing files with tag dat e g the file example dat and call it name2 dat cd DIPOG 2 0 CLASSICAL cp example dat name2 dat Change name2 dat in the editor according to your requirements You will find the nec essarv information as comments in the file name2 dat Indeed each line beginning with isa 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 10 6 the name of which is announced on the screen Xou will find all Ravleigh coefficients the efficiencies and energies on both 42 the screen and in thi
64. cty 1 t 0 lt t lt 1 such that O lt fctx 2 t lt 1 0 lt t lt 1 The functions fctx 1 fctx 2 fcty 1 and fcty 2 and the parameter xmin are defined by the code of the file GEOMETRIES cpin2 c e g stack 3 profile t 0 2 sin 2 M_PI t Ze profile 0 2 sin 2 M_PI t H 1 Grating data example HH HH Number of different grating materials N_mat This includes the material of substrate and cover material if gt if gt if gt if i hh For example Grating data is namel Number of materials given in file Grating data is echellea 7 N mat 3 with coating height 20 N mat 2 with coating height 0 Grating data is echelleb N_mat 3 with coating height 20 N mat 2 with coating height 0 Grating data is trapezoid N_mat number of material layers 3 for coating height 20 N_mat number of material layers 2 for coating height 0 Grating data is lAmellar km 23 namel inp N mat k times m plus 2 Grating data is polygon filel N_mat 2 Grating data is polygon2 filet file2 N_mat 3 Grating data is profile N_mat 2 Grating data is profile 2 3 II N_mat 2 Grating data is profile 0 125 sin 2 M_PI t N_mat 2 Grating data is profile 0 5 0 5 cos M_PI 1 t 0 25 sin M_PIx t N_mat 2 64 gt N_mat 4
65. cutable 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 lavers file lamellar inp to define location and widths of lavers in grating generated bv GEN LAMELLAR executable GEN POLVGON to generate an input file for a grating with polvgonal profile curve first argument name without tag inp of file to be created second argument name filel of file with nodes of polvgon executable GEN POLVGON2 to generate an input file for a grating with polvgonal 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 CLASSICAL file filel to define profile line for a polvgonal grating generated bv GEN POLVGON or GEN_POLYGON2 file file2 to define polvgonal boundarv line for the coated laver of polygonal grating generated by GEN POLVGON2 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
66. determined bv smooth parametric curves two straight line segments x 0 xi lt lt Emin and 2 0 l min lt T lt T2 and a second curve fz 2 8 fy 2 t 0 lt t lt 1 The last connects the first point 21 0 fz 2 0 f 2 0 with the last point 2 0 f 2 1 f 2 1 Moreover fa 2 t 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 24 lt 1 0 lt t lt 1 The functions 7 1 2 1 and f 02 and the parameter min are defined by the code of the file GEOMETRIES cpin2 c 33 Figure 16 Pin grating determined bv parametric curve Figure 17 Coated pin grating determined bv two parametric curves Figure 18 Coated pin grating determined bv two parametric curves Tvpe 2 35 3 4 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 line 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 j t fy 3 t 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
67. e Besseren conv ord se 4 98 0 60 l 1 88 l Tr nn conv ord nn l 2 43 1 04 2 36 l nn Sure conv ord nn 0 195 52 7622 0 092 52 7521 0 045 52 7150 8 VALUE ORDER 1 h value 0 818 9 2569 0 347 11 1626 0 195 12 6196 0 092 13 4623 0 045 13 6750 9 VALUE ORDER 2 h value 0 818 3 8619 0 347 1 6091 0 195 2 0156 0 092 2 0208 0 045 2 0265 h value 0 818 0 9660 0 347 0 7818 0 195 1 4148 0 092 1 1687 0 045 1 1029 h value 0 818 0 4156 0 347 1 7545 0 195 1 0322 0 092 0 7882 0 045 0 7333 H 4 H 4 H 4 Hh 4 4 57 5905 52 7520 52 7661 17 3500 14 6183 13 7469 12 VALUE TRANSMITTED ENERGY d H 4 4 AS O w O Ko H 4 4 r 00 N e 00 4 o o N Pp N H 4 4 o N O N N 86 Hh
68. e A HE Comments ii Input must be ended by a 0 in an extra line H These comments will appear in several output and result files Comments This is a fantasy grid for the test of gen_polyx 0 A Number of materials 4 THREE OO Minimal angle of subdivision triangles 20 000000 TORRE Upper bound for mesh size 0 500000 A Width of additional strip above and below Automatic choice of small width if this value is 0 Width 0 200000 FRA A A 54 Grid points HA 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 bv two piecewise linear functions in x first add the nodes of all the triangles later give the triangles bv the indices of their nodes bb A Ab Each point in a separate line Scaled to period 1 Input ended by 1 1 7 Grid points 4040040440000 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 u HL A Triangles H Each given
69. e of DIPOG 1 3 Add no if not needed Add ves if needed The name will be the same as the standard output file given above but with the tag t erg instead of res yes or no yes HEHEHE HRA RRA aE RRR aaa 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 fi HHHH coated layers of rectangular shape These kind of layers are called coating layers over the grating and coating layers beneath the grating respectively Number 2 HEHEHE HHA RAR aa RRR Ra 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 0 05 68 0 03 Ri ii Sisa HERRERA aH HARE aaa Number of coating layers beneath the grating N_co_be al HEHEHE RRR RRA aS RRR Raa 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 Widths 0 05 H A Ai Wave length in micro meter lambda Either add a single value e g 63 Either add more values by e g y 5 ji 63 64 65 ji 69 AQ 79 The last means that computation is to be done for the wave lengths from the Vector of length 5
70. e to below as fe fy t O lt t lt 1 7 1 n where n and the functions t gt f j t and t f j t are defined by the c code of the file GEOMETRIES profiles c cf Figure 15 where n 3 ht t Jo sin 27t fy 2 t sin 27t 0 5 and f 3 t sin 2rt 1 Grating data profiles par 1 2 3 0 5 0 50 indicates a GRATING DETERMINED BV SMOOTH PARAMETRIC CURVES WITH PARAM ETERS i e grating determined bv n non intersecting and periodic profile lines given from above to below as K f j t f 3 t O st lt 1 j 1 n where n and the functions tr f 3 t and t gt f j t are defined by the c code of 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 15 where n 3 H t Alt sin 2rt 112 4 Saar 0 5 30 Figure 13 Grating determined bv a simple smooth function and f 3 t sin 2rt 0 5 0 50 parameter 3 is the number n of boundary and in terface 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 Grating data pin indicates a PIN GRATING DETERMINED By PARAMETRI
71. ected energy 6 573574 Transmission efficiencies and coefficients n O theta n 1 theta n 1 theta n 2 theta n 3 theta n 4 theta 26 50 E 10 7291 54 Transmitted energy 438747 866995 163526 062429 007596 078023 22 56 256892 861878 966041 415630 LOW degrees of freedom stepsize of discr numb of nonzero entr rate of nonzero entr memory for pardiso 95 0 132885 0 41 0 034493 0 80 0 228543 0 48 C 0 171167 0 96 0 096782 0 TT 0 001540 0 93 426426 3197 0 34705 23379 0 228739 per cent 2606 kB 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 004059 0 0 003041 0 0 009015 O0 0 007285 0 2 035991 Transmission efficiencies n O theta n 1 theta n 1 theta n 2 theta n 3 theta n 4 theta 26 50 Ts 10 29 54 Transmitted energy 131487 001482 008718 148711 and coefficients 251543 681800 302391 115618 050452 160324 2 3 1 2 3 4 O O OH 2l 589973 11 609071 781807 754536 degrees of freedom stepsize of discr 95 0 439727 0 41 0 521995 0 80 0 061566 0 48 0 021529 0 96 0 071286 0 77 0 002095 0 97 964009 12628 0 19522 82 164372 761614
72. em the paths and the flags according to your computer system Then go to the home directory cd DIPOG 2 0 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 0 home directory containing the first four subdirectories These subdirectories together with all 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 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 0 We sup pose in this user guide that it is named DIPOG 2 0 There exist
73. en 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 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 43 Real Part Figure 20 Real part of transverse component of magnetic field Output of FEM PLOT via openGL in CLASSICAL All other calculations are performed with this level Clearlv there is no warranty that the efficiencies really deviate by a number less than e from the true values 5 3 Check before computation more infos and plots Instead i
74. equal to zero If an input file for a lamellar grating is needed rectangular grating consisting of several materials placed in rectangular subdomains cf Figure 9 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 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 acc
75. ertically shifted by hi um The same remarks as in the previous mode apply If the defining two lines are linea _ profilei ccode lines hi di da aap d then we get a profile curve DETERMINED By A NON SMOOTH SIMPLE CURVE defined by F 5 t t and f 5 t ccode The curve has i 1 lt i lt 9 corners with the parameter arguments di ds d such that 0 lt di lt d c lt d i The profile is vertically shifted by hi um The same remarks as in the previous mode apply If the defining two lines are lines profilei ccode ccode lines hi di da Es di then we get a profile curve DETERMINED By A NON SMOOTH SIMPLE PARAMETRIC CURVE defined by f j t ccode and fy j t ccodez The curve has i 1 lt i lt 9 corners with the parameter arguments di da d such that 0 lt dj lt d lt lt di The profile is vertically shifted by h um The same remarks as in the previous mode apply 37 Beside the above profile curves pin curves are possible Then the meaning of f and f is changed The curve t gt felj t f 5 t with 0 lt t lt 1 connects the points felj 0 f 3 0 0 0 and f 3 1 f 3 1 1 0 The corresponding pin curve is just the affine image of the last curve connecting the two points fz J2 p1 fy Ja p1 and fa j2 P2 fy J2 P2 of the profile curve with index ja I e im dato falj t folja pa feliz P1 fyl t fy J2 P2 fu a pr
76. et a SIMPLE NON SMOOTH PIN CURVE defined by f j t t and fy j t ccode The curve has 1 lt i lt 9 corners with the parameter arguments di da d such that 0 lt d lt d2 lt lt d i The remarks on the profile curves apply also here If the defining two lines are 38 line gt 1 pini ccode ccodez lines Pi P2 d d ia di then we get a SIMPLE NON SMOOTH PARAMETRIC PIN CURVE defined by f j t ccode and f j t ccode The curve has i 1 lt i lt 9 corners with the parameter arguments di da d such that 0 lt dy lt dy lt lt d i The remarks on the profile curves apply also here For example Figure 19 presents the stack grating generated by Grating data stack 5 echellea R 0 3 1 2 profile 0 3 x sin 2 x M_PI xt 0 8 pin 0 5 0 5 xcos M PI x t sin M PI xt 0 6 0 9 pin1 0 if t lt 0 5 fet t else fct 1 t 0 1 0 4 0 5 profile t 0 0 39 Figure 19 Stack grating 40 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 co
77. f incident wave theta and the angle of i incident wave phi must be single valued Angle 47 HEHEHE HRA RR Aa HRA 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 IHR A Stretching factor for grating in v direction Must be a positive real number Length ale TIED A Length of additional shift of grating geometry in micro m i This is shift into the v direction i e the direction perpendicular to the grating surface pointing into the cover material Must be a real number Length O HEHEHE RA HE HH HH HO HORROR RRR ARRAPA PAHE RERNA AHHH HAHAHAHA RRA RORRRRRRRRRARAAR Period of grating in micro meter 1 HA A Af Grating data 72 i Either this should be e g namel if namel inp is the input file with the geometry data in sub directory GEOMETRIES Alternativelv a path for the location of the file
78. f 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 Instead if you use the command FEM_FULLINFO name2 then the same is done as in point Additionally there appears more information in cluding the full input data and the convergence history cf the enclosed file in point on the screen and in the result file Instead if openGL or GNUPLOT 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 B ta 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 44 Ma Field 22279 2 5 Imag P ATLL ESS ia 2 5 skye HELL Le AR PERLE 2 bee Le KELLI 2 Core LEE FR q 1 FEBE ELL LEER EL F FRE 2 a 15 ba LONA NA EE ea ARUE TER RITTER A OA RE PR FELL A 1 itiha PENG PAUP PRAT Ii II PRE KLE E AS AE bi MG ga A a zz ET FE ELSE ER eee 0 5 i AI Ke NI Ni pl Ne ne ta Kc ALEEEFHFA EY i FREE A ELLY FR AA PRUTE N
79. fficiency 0 38965 0 12856 0 61617 0 20649 61 81559 0 25045 0 03050 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 il 20 54 135 99 1 128 80 161 77 2 158 51 138 27 CESA LON ASS OOO AE ES Transmitted energy 98 77158 Reflection efficiencies and coefficients Order Phi Theta E_z H_z Efficiency O 47 00 30 00 0 06175 0 01688 0 07173 0 03108 1 02091 1 128 80 27 98 0 02812 0 00775 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 Hz Efficiency 0 39161 0 10975 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 O 47 00 160 53 1 20 54 135 99 1 126 80 161 77 2 158 51 138 27 TES AE ES LOS OOOO AE AOS Transmitted energy 98 83852 date 10 Feb 2003 10 06 25 Thank you for choosing CONICAL Bye bye 93 11 Copyright Responsible programmer 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 Nachtiga
80. ger parameter values cf Fig ure 12 where f t t and f t 1 5 0 2exp 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 parameters in the y coordinate of the curve Note that any number of parameters is possible for a corresponding file GEOMETRIES profile_par c Grating data profile 0 125 x sin 2 x M PI x indicates a GRATING DETERMINED BY A SIMPLE SMOOTH FUNCTION cf Figure 13 i e grating determined by a profile line given as t f t 0 lt t lt 1 where the function t gt f t is defined by the c code f t 0 125 sin 2rt do not use any blank space in the c code Grating data profile 0 5 0 5 x cos M PI x 1 t 0 25 x sin M PI x t indicates a GRATING DETERMINED BY A SIMPLE SMOOTH PARAMETRIC CURVE cf Fig ure 14 i e grating determined by ellipsoidal profile line given as f t fy t 0 lt t lt 1 where the functions t gt f t and t gt f t are defined by the c codes f t 29 Figure 12 Grating determined bv a smooth parametric curve 0 5 0 5 cos r l t and f t 0 25 sin rt respectively no blank space in c codel 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 abov
81. gument name res and indices of modes the efficiencies of which are to be plotted header MHEAD note that MHEAD is to be adapted to vour computer svstem before installation executable MAKEHOME for another user produces new version of four 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 programs and input files for grating and grid data dpogtr programs and input file for the FEM computation in the classical case gdpogtr programs and input file for the GFEM computation in the classical case conical programs and input file for the FEM computation in the conical case conical2 programs and input files for the GFEM computation in the conical case results plot programs Possibly there exist subdirectories to install necessary packages dipog 1 3 necessary source files from previous version of DIPOG 1 3 in dipog 1 3 gsl src files to install libhur a gltools tar necessary source files for Fuhrmann s package GLTOOLS 2 4 this produces subdirectory gltools 2 4 during installation triangld 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 containing the four subdirectories GEOMETRIES CLASSICAL CONICAL and RESULTS
82. h is a collection of software components to create simulators based on partial differential equations http www wias berlin de software pdelib DIPOG 2 0 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 4 does the same for the special case of binary lamellar grating 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 4 or DIPOG 2 0 However he should prefer the more efficient Dipog 1 4 The fast generalized FEM used in Dipog 1 4 cannot be applied to general polygonal geometries Additionally to the computation of efficiencies Dipog 1 4 computes optimal binary grat ings for given efficiency sequences or for prescribed energy restrictions The implementation of optimal design for DIPOG 2 0 is still in progress For Dipog 1 4 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 or c language and based on the UNIX system The pro grams require the previous version DIPOG 1 3 or DIPOG 1 4 the grid generator TRIANGLE 1 4 and the line
83. he 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 esp 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 1 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 fetx 1 fetv 1 l xmin O 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 te 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 which is located between 1 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
84. his inclined plane i e the angle between D and the orthogonal projection of D to the x z plane To change between the xy system and the xz system the following formulae are useful sind COSfxz Day arcsin sin sind Ory arcsin V1 sin fyz SIN dyz Ong arccos 0080 Cosa A sin en arcsin Lay l if sind COS mr gt 0 iss tas cat sin Zn Pay z vm else 1 cos20 COS bry 8 In other words 7 6 and are the spherical coordinates of k k and 0 is not the angle enclosed by k kt and the positive y axis but the angle enclosed by k k and the negative y axis 13 gt lt gt X Figure 2 Coordinate system based on x y plane 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 gine 2 Bin 2 Bir Hire n z rl z A 2 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
85. ial case of lamellar grating with layer material s t y coordinate satisfies 0 2 lt y lt 0 8 given in micro meter e g polygon file1 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 f 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 BV POLVGONAL LINES i e grating profile line is defined bv the data in the file with name GEOMETRIES file1 in GEOMETRIES file1 in each line beginning without there should be the x and v coordinate of one of the consecutive corner points first point with x coordinate O last point with x coordinate 1 same v coordinate for first and last point all x coordinates between O and 1 at least two different v 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 in each line beginning without there should be the x and y coordinate of one of the consecu
86. iciencies cf 2 4 are computed bv E si e an n n ne ur U n ne ur 4 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 12 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 Ravleigh expansions and two sequences of Ravleigh coefficients More preciselv skipping the time harmonic factor we have an incident wave of the form Fineident T Y Z E exp ilox By ya Hie t y z H exp ilaz py ya with constant vectors E and H and a wave vector k a B 79 such that the wave number k is the modulus of k and the directio
87. 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 Input ended by 1 1 1 1 1 1 000000 1 000000 1 000000 1 000000 0 300000 1 000000 3 1 000000 3 1 000000 3 1 000000 12 10 3 1 000000 O ON 1 1 1 1 1 o 99 End A 78720 2027272020272 787 2020278727702 72 RARA RARA AAA 10 2 Data file example dat for CLASSICAL H A AHHHHHH HHH example dat RHEIN all lines beginning with are comments HHH HH HF HHH HH HF ES EPEAT TE EERE TE EE E EE HE PEPE EE EPR AE HETE E PE EE PE HE AE EE E E HEE EE HE LEE HE PE HEEE HE input file for FEM GFEM located in directory CLASSICAL it A Name of the output file 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 EE least one slash E g for a file name res in the current working directory write name Name example HA A Should there be an additional output file in the old style of DIPOG 1 3 ii Add no if not needed Add ves if needed The name will be the same as the standard output file given above but with the tag erg instead of res
88. ing data is cpin 7 gt N_mat 4 if Grating data is cpin2 HHHHH HH HHH HH HH HH HOF HL 50 2 3 i lt 0 HH HH HH Number of levels Lev it In each refinement step the step size of the mesh is halved dE 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 H estimated error less than 1 per cent Number 3 pett Ab bb TE E TEPER E E E ETE E EE E E HE EE EHE DE TE HETE LEP AE HE HETE A b TE E EE EE TP PEE HE 65 End FERRERO RARA RO PERERA ROPERO 7202027878 7002 0078720200787 202 027278720202 72 7202007872000 0 027872020872 72020 272787202 02787202 HHH 10 3 Data file generalized Dat for CLASSICAL same as con ical Dat in CONICAL Rik Sika RRR RRA HH HIHI HHH HHH A AHAAH generalized Dat FAB HHHHRRHHE EHI EHH all lines beginning with are comments HHH HH OH HHH HHH HF jtik tti TE RESET TPE EE HERREN E E E DE EE E E E D E E E E A TE E EE H HEHEHE input file for GFEM located in directory CLASSICAL contains constants for numerical method in program GFEM Hood HH HF o Recommendation for n_DOF and n_LFEM a mild accuracy requirements and wave numbers not too large n DOF 1
89. ith 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 3 of this section 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 pathl namel 18 In particular for a geometrv input file in the current working directorv 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 length 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 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 thi
90. itted energy 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 77 0 049270 0 095506 97 725688 83 3 1 2 3 4 27 52 13 020839 168650 788186 970304 047639 002377 159453 975753 762242 619565 076732 057844 139668 533702 752053 462258 degrees of freedom stepsize of discr numb of nonzero entr rate of nonzero entr memory for pardiso 200431 0 04493 1481809 0 003689 per cent 278626 kB Reflection efficiencies and coefficients n 0 n 1 n 2 n 3 theta 65 theta 15 theta 21 theta Si Reflected energy 00 74 33 07 0 012980 0 144468 0 000471 0 016141 0 000145 0 000968 0 090729 0 055189 2 299969 Transmission efficiencies and coefficients n 0 n 1 n Sl n 2 n 3 n 4 theta 26 theta 50 theta its theta 10 theta 29 theta 54 Transmitted energy 95 41 80 48 96 TT 1 2 3 a a Oi UN 2 2 Sac Le 4 0 1 VALUE te h a a a oe G 0 818 0 347 0 195 0 092 0 045 G in i i a 2 VALUE in is u a a i is ORDER 0 ee value GA GA GAS GHA CA E Sl Si A GG l 3 4623 1 7305 1 9703 2 0767 2 1039 i a GHA dB a ih a a IR si a Gun Jan cini ORDER 1 a Sn ns i iy St tu Sl li a
91. l qmr solver The programs are based on codes written by K Gartner direct solver cgs solver R Schlundt gmres solver J Fuhrmann T Koprucki H Langmach PDELIB adaption of GLTOOLS B Kleemann G Schmidt A Rathsfeld adaption to the grating diffraction problem generalized FEM Owner of program Weierstra Institut f r Angewandte Analysis and Stochastik D 10117 Berlin Mohrenstr 39 Germany im Forschungsverbund Berlin e V Wissenschaftsgemeinschaft Gottfried Wilhelm Leibniz e V References see Section 2 4 Acknowledgements The author gratefully acknowledges the support of the German Ministry of Education Research and Technology under Grant No 03 ELM3B5 RO O O K K K k k x Copyright 2003 x KK k k K k k k K k k k k k 94
92. l layers 42 for coating height 0 if Grating data is lAmellar km 23 i cc namel inp gt N mat k times m plus 2 if Grating data is polygon filef gt N_mat 2 if Grating data is HHHHH Hr HF HF HF HH HHH HHH HF FH OF EE polygon2 file1 file2 77 i hh gt N mat 3 if Grating data is profile gt N_mat 2 if Grating data is profile 2 3 29 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_PIx 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 nti 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 7 gt N_mat 4 if Grating data is cpin2 gt N_mat 4 if Grating data is stack k gt N_mat k 1 Number of materials HERHHHAHAHHHHAAHHHHARR EERE ARERR RARER A 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 inde
93. lar 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 type A is needed right angled triangle with 19 e eo oe o o Am o o o o o gt o Figure 5 Pictures of grid produced by SHOW 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 45 degrees M
94. lt npor with 1 lt nirem and with Inirem 1 a multiple 46 O Point Degree of freedom Function Interpolation polvn 0 1 of grid values Function FEM s of bou Function value 1 0 T7 values Figure 22 Trial basis function over a single grid triangle 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 nurem 1 x nurem 1 equal subtriangles cf the trial functions over the grid triangle indicated in Figure 22 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 FUL
95. n Jio for the z component of the incident magnetic field H ident Note that the additional factor VeoVnt fio in the definition of Hireident 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 y By exp 18 y exp ianr Bi exp 16 y exp iaz 2 6 E 00 Vene gt gt B exp 1871 exp ians 2 7 Bee VAT 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 eff
96. n of the incoming plane wave is k k Here kt is the same wave number as in the classical TE and TM case The direction k k must be prescribed bv the user of DIPOG 2 0 This can be characterized bv two parameters namelv by the angles 0 and which are the spherical coordinates of a B y We emphasize that 9 and are the spherical coordinates of a f 7 and not those of the normalized wave vector k kt a B JB Contrary to this the angles 0 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 xy system we define the direction as cf Figure 2 Dry sind COShzy COSO yy COSPzy sind Here sy is the angle of inclination of the plane containing the direction D and the z axis from the y z plane Angle is the angle of direction D inside this inclined plane i e the angle between D and the orthogonal projection of Dz to the x y plane For the xz system the direction is given by cf Figure f3 D sind COSO yz COS sin sing Here is the angle of inclination of the plane containing the direction D and the y axis from the x y plane Angle 0 is the angle of direction D inside t
97. ncies of the projection of the nth reflected resp transmitted wave mode to the component with electric resp magnetic field polarized in s n direction Finally the third variant computes the S and P parts of the electric field i e the components of the Jones vector If s n is defined as above and if p n is the direction orthogonal to s n and the direction of propagation of the nth reflected resp transmitted plane wave mode Qn EBF 7 x S Qn E67 y x s then the S and P parts of the Rayleigh p coefficients are ap be qx n 1 c Va e 1 Bis n be p aq n 1 2 va amp 16 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 si resp p direction a E n Se As pt p The efficiencies of the third output are the total efficiencies e of 2 9 and the efficiencies 2 l n T 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
98. 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 2um 4um Sum and 16um In other words we have chosen the geometry generated by the code words Grating data trapezoid 60 albc 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 imaginarv part of the Ravleigh coefficient of
99. o the grooves i e it shows in the z direction Hence if uo is the magnetic permeability of vacuum and c the speed of light then the transverse z coordinate of the electric field is given as 216 Esta z t EN pl w XI 2 1 tat 1 incident Oi A opt o E x y z Jo exp ik sind x ik cos y kT w po o Nn 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 v n 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 n 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 const
100. od cf Figure 17 where a 0 2 ay 0 2 ay 0 8 TAL Emin 1 nd fy 1 t 0 5 sin rt and K f 2 t fy 2 t 0 lt t lt 1 is the polygonal curve connect ing the four points fa 1 a1 fy 1 a1 fe 1 a1 fy 1 0 5 0 1 fz 1 a2 fy 1 0 5 0 1 and fe 1 a2 faQ a2 If an input file for a coated pin grating of tvpe 2 determined bv the simple non intersecting profile lines given as K f j t fy 3 t 0 lt t lt 1 j 1 2 is needed 19T 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 fe t fy t 0 lt t lt 1 is a simple open arc connecting fz 0 f 0 min 0 with fe 1 fy 1 1 tmin 0 such that 0 lt Emin lt 0 5 is a fixed number such that 0 lt f t lt 1 0 lt t lt 1 and such that 0 lt f y t 0 lt t lt 1 UI 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 f 1 8 0 lt t lt 1 Here f 1 t 1 8 0 lt t lt 1 isa simple open arc connecting f 1 0 fy 1 0 amp min 0 with f2 1 1 f 1 1 1 min 0 such that 0 lt min lt 0 5 isa fixed number such that 0 lt f 1 t lt 1 0 lt t lt 1 and such that 0 lt f 1 1 0 lt t lt 1 Additionally a coating layer is attached located between the first curve K f
101. omplished 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 Figurej10 then this can be accomplished by calling the executable GEN POLVGON from the subdirectorv 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 v coordinate of one of the consecutive corner points first point with x coordinate 0 last point with x coordinate 1 same v coordinate for first and last point all x coordinates between 0 and 1 at least two different v coordinates last line should be 21 End then the command GEN POLVGON name filel creates the file name inp of the desired polvgonal grating If an input file for a grating determined bv two polvgonal profile lines is needed cf Fig ure 11 then this can be accomplished by calling the executable GEN_POLYGON2 from the subdirectory GEOMETRIES More precisely if
102. onal FEM method with elimination of interior nodes of grid triangle i e real mesh size is mesh size shown in result file divided bv n_DOF 1 If n_DOF lt n_LFEM method resembles p method or PUM Value should satisfy 1 lt n_LFEM lt 2043 and n_LFEM 1 must be a multiple of n_DOF 1 value 63 A BE TEEPE PE PETE TEE EEE 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 FH Value should satisfy 1 lt n_UPA lt 128 value 1 THREAD EEEE that s it HEHEHE RRR RA aH HH HIHI 67 10 4 Data file example dat for CONICAL H HEE RI example dat PORRA HHH all lines beginning with are comments HHH HHH HHH HH H HA input file for FEM GFEM located in directory CONICAL i Rik ia HRA AEE RRR aaa 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 ER least one slash E g for a file name res in the current working directory write name Name example A Should there be an additional output file in the old styl
103. ons t gt f t and t gt 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 bv a polvgonal line with a step size equal to 0 06 times the length of period cf Figure 12 where f t t and f t 1 5 0 2 exp sin 6zt 0 3 exp sin 8zt 27 If an input file for a grating determined by more than one non intersecting and peri odic profile lines given as f j t fy y t 0 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 3 8 fy J t O lt t lt 1 are given by the c code in the file GEOMETRIES profiles c Then GEN_PROFILES name 0 06 creates the file name inp of the desired profile grating where the profile curves are ap proximated bv a polvgonal line with a step size equal to 0 06 times the length of period 9To make it precise suppose P and Q are the common corner points of the two polygonal curves and that Ri Ri R are the consecutive corner points between P and Q on the first polygonal line and Ri R3 R2 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 Ri the quadrilaterals Ri RiRZRi f Hepler T
104. oreover 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 accomplished 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 20 profile height greater or equal to zero If an input file for a trapezoidal grating is needed isosceles trapezoid with the basis parallel to the direction of the periodicitv cf Figure 8 then this can be accomplished by calling the executable GEN TRAPEZOID from the subdirectorv 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
105. ot 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 requirements a larger nirem is useful E g if npor 3 then nifem 31 is a good choice For npor 7 one should take e g nirem 127 and for npor 15 the value nurem 511 However large nirem 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 nirem will be given in the numerical tests in Section Bl 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 10 7 The same names of executables can 48 T T Tanne ji Tr nare 2 gt eff 0 l l ji l l l l l 0 63 0 64 0 65 0 66 0 67 0 68 0 69 0 7 0 71 0 72 wave length Figure 23 Efficiencies depending on wavelength Output of PLOT PS 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 10 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 D
106. over 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 2 t lt 1 0 lt t lt 1 The functions fetx 1 fetx fcty 1 and fcty 2 and the parameters argi arg2 and xmin are defined by the code of the file GEOMETRIES cpin c ep pind 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 1 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 fetx 1 1 fetv 1 1 l xmin O such that 0 lt xmin lt 0 5 is a fixed number such that O lt fctx 1 t lt 1 0 lt t lt 1 and such that 0 lt fcty 1 t O lt t lt 1 Additionalv 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 63 gt 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 point x1 0 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 f
107. rdinates between 0 and 1 at least two different y coordinates last line should be End Grating data polygon2 filel file2 indicates a COATED GRATING DETERMINED BY POLYGONAL LINES cf Figure 11 i e the grating profile line is defined by the data in the file with name GEOMETRIES filel 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 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 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 27 Figure 10 Grating determined by a polygonal line last point of second polygon quadrilateral domain between corresponding segments on the left of first polygon these quadrilaterals must be disjoint last line should be End Grating data profile indicates a GRATING DETERMINED BY A SMOOTH PA
108. 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 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 zero 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 lAmellar 3 4 0 2 0 6 0 2 1 0 O 045 79 7 0 0
109. rv GEOMETRIES More precisely suppose in and the profile lines fe 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 step size equal to 0 05 times the length of period cf Figure 18 where Emin 0 2 De Esa 1 22min ad Ban and fl 0 5 Emin 1 Emin t fy 2 t 0 8 u 3 3 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 lavers 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 0 lt fr 2 t lt 1 0 lt t lt 1 The functions f 1 f 2 fy 1 and f 2 and the parameter min are defined by the code of the file GEOMETRIES cpin2 c 24 Figure 7 Echelle grating of tvpe B echellea L 0 3 0 03 0 04 indicates an ECHELLE GRATING TYPE A right
110. s 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 easv access to the data To enhance readabilitv bv 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 normallv 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 directorv different from CLASSICAL or if the output file should be written into a different directorv then the file is to be specified bv adding its 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 bv incremental or vector mode i e beginning with the letter I or V th
111. termined 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 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 te gt GRATING DETERMINED BV SMOOTH PARAMETRIC CURVES WITH PARAMETERS i e grating determined bv 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 62 i 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
112. the file GEOMETRIES filel contains the corner points of a polygonal profile line in GEOMETRIES filel in each line beginning without 44 there should be the x and v 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 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 polvgorf quadrilateral domain between corresponding segments on the left of first polvgon these quadrilaterals must be disjoint last line should be End then the command GEN POLVGON2 name filel file2 creates the file name inp of the desired polvgonal grating If an input file for a grating determined by profile line given as fz t f 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 functi
113. 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 7 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 size h of the grid by h on h an d npor 1 with h 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 RLFEM i e the GFEM with the parameters npor NLFEM and nypa l 14 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 15Note 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 50 Stars indicate that the accuracv is not reached due to the restricted main memorv of the computer The number of grid points ngp 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 69
114. tive 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 74 i corresponding segments on the left of first polvgon 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 tl gt fcetx t and t 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 2 2 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 tl gt fotx t and t 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 O 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 Anv number of parameters is possible for a corresponding file GEOMETRI
115. x 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 the upper line of the grating structure must coincide with that of the material in the adjacent upper coated laver resp in the adjacent superstrate Similarlv 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 laver resp in the adjacent substrate N mat numbers are needed Optical indices 1 3 10 1 6 1 0 HH HH HE Number of levels Lev In each refinement step the step size of the mesh is halved Lev refinement steps are performed 78 Alternatively one can prescribe an bound for the maximal error of the efficiencies E g the input e 1 means H 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 TORRE AE EERE EEE EERE BEBE PEPE PEE End IS TEPER TERETE EEE TE EERE TE EE E TE HE ETERS PEE ET PEE EEE E EE EE E EE PTT Ae 10 5 Output file example res of FEM FULLINFO in CLASSI CAL ARO FK FK FK FK FK Rd FK FK FK K FK FK K K K K K

DIPOG-2.0 User Guide Direct Problems for Optical Gratings over

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