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User`s Manual - Grating Solver Development Company

1. tab Vacuum wavelength 1 5 microns Can be changed at will Grating lines mm 125 Or enter grating period of 8 microns Theta 30 Angle of incidence Alpha 45 Unpolarized light Superstrate index 1 Light incident in vacuum Substrate index 1 5 Grating material index 2 Select the editor tab At the upper left you will see the 2D editor button selected 3 Press the custom profile selection button the sawtooth icon The default is a blazed grating Change the angle C3 to 30 degrees and press OK 4 You will now see a grating facet in 2D editor mode The next step is approximate the ideal grating shape by a number of layers You can control the number of layers by selecting the grid button then grid properties and then changing the grid spacing parameters A smaller grid spacing number gives more layers Provided by Daniel Fabricant e mail 23 Nov 2010 16 GSolver V5 2 User guide 5 You now need to set the scale of the grating facet For an 8 micron period you need to enter an 8 in the vertical scale factor box in the editor window so that the layers will have the correct physical scale You can now select the approximation button in the editor window and a layered approximation of the grating is drawn This approximation will be used for the grating calculations Now select the Listing Run tab and press the populate button You can now see the grating layers described numerically on
2. jK y Jk3 k kz A A y K A K Wee k k1 sin cos g iK k kU sin sin p jK where i and j are integer indices ranging from s to s where s is the number of orders retained k1 and k3 are the magnitudes of the k vectors in the superstrate 1 and substrate 3 regions as given previously For one and two dimensional gratings j 0 and the K term is irrelevant A and A are the grating period dimensions in the x and y directions All k vectors and fields have now been defined for the superstrate and substrate regions The fields in each layer are found by application of Maxwell s equations and expansion of the permittivity in a Fourier series This results in a set of coupled wave equations that are solved with algebraic methods resulting in a set of inhomogeneous plane waves 19 2 1 2 Inhomogeneous Plane Wave Intra layer Solutions Considering the n grating layer assuming the standard time harmonic formulation and working in Rationalized MKSA units where B uH D s so E the curl equations are as follows VxE B M 104H Ik LH 0 E V x H D cg E we Ik E Mo Note that extensions to anisotropic media requires that e a 3x3 matrix and for magnetic media o gt HoM The permittivity of free space may be substituted as Zo e0 o P the fact that c 1 E0uo and the vacuum dispersion relation have been used The electric and magnetic fields are rep
3. In general I find that AE remains the algorithm of choice in terms of speed and accuracy There may be situations where the RK algorithm offers superior results for example when AE fails 106 GSolver V5 2 User guide 17 1 6 GAIN The GAIN setting on the Tools menu item is also new to GSolver V5 1 It flags the materials index of refraction evaluation routines to permit negative index of refraction for the Polynomial and Table models In essences a negative imaginary part of the index of refraction implies that energy is gained as the wave propagates through the material This may happen for example in optically pumped laser gain media To use a gain medium you will need to either edit the GSOLVER INI file or use the GSolver materials editor Add a table model for example and enter negative values for the imaginary part of the index of refraction over some wavelength range Then use this new material with the GAIN flag from the Tools menu item set checked 107 GSolver V5 2 User guide 18 Precision Double Double and Quad Double GSolver includes optimized extended precision arithmetic including floating point and complex data types Double Double refers to 128 bit floating point made out of two 64 bit floating point words 106 bit mantissa 21 bit exponent for approximately 32 decimal digit precision And Quad Double refers to 256 bit floating point implementations 212 bit mantissa 43 bit exponent for about 64
4. This is a relative phase calculation in the sense that it computes phases that are relative to each other for the set of reflected and transmitted orders At each stage of the internal calculation fields are normalized by the largest complex eigenvalue in the related algebraic eigenvalue problem as defined by the Fourier components of the layer permitivities and impermitivities and Maxwell s equations This normalization improves numerical stability The absolute magnitudes of the internal fields are calculated only up to some constant complex multiplier The system of equations is configured so that this normalization has no effect on the calculation of the external fields That is the system of equations is ordered so that the external fields come first and back substitution is used to solve only for those fields 9 3 Graphing Select the columns to graph multiple columns can be selected using the shift and control keys along with the mouse Click the chart button The initial chart uses defaults on all settings To alter any feature of the graph click on the various tools Clicking on most items invokes a properties dialog for that item For example to change x axis properties such as display limits and numerical format simply right click on one of the x axis tic mark labels and select properties GSolver s graphing tool is very powerful and includes a large number of display and customization options More information
5. 47 GSolver V5 2 User guide 4 3 3 Spline Curve and Ellipse The spline curve icon is the kidney shaped command icon The operation 1s similar to the poly line tool however the shape is smoothed by a cubic spline estimation through each set of 3 points Double click to complete the figure 4 3 4 Classical Form Generation This tool invokes a dialog whose tools generate Blaze Triangle Sinusoidal General Poly line and General Spline curves The tool icon for the classical form custom profile tool is a black blaze profile triangle orm Profile Construction Blame proe n dened by ore ange The wah i 2caled lo 1 ora Haze angie lo Gnale mesrre aPC Weight icin sume Ur an path Erte the Bo Ea passi mme poed gh dagens o the apak postion ar a fraction of ihe zj Use the list box to select among the several predefined profile types Blaze A blaze profile is defined by a single parameter the blaze angle Enter the blaze angle in grid location C3 in the dialog After the profile is updated clicking the dialog OK button will insert the profile onto the canvas The canvas size is increased automatically if needed 48 GSolver V5 2 User guide General Sawtooth A sawtooth profile is defined by two sub ninety degree angles Enter the angles to define the profile in grid locations C3 and D3 Sinusoid poly This profile is a generalized sinusoid constructed by a piecewise linear approximation T
6. data structure with the Copy Update button on the Listing RUN grid The Populate button copies the Discrete approximation data structure to the Listing RUN grid 10 GSolver V5 2 User guide All horizontal dimensions of the Discrete approximation are relative to the Period which set on the Parameters tab Thus all Widths for each layer must total to 1 0 All vertical dimensions are absolute based on the Units set on the Parameters tab Remembering the above two principles will alleviate many sources of confusion when designing gratings In general it is good practice to populate the Listing RUN grid and do a quick chech the the physical dimentions are as expected The RUN command on the Listing RUN GA and Run tabs operate on the Discrete approximation data structure 1 3 Example Run Quick Start The GSolver V5 1 install directory should include GsolverV50 exe Gsolver ini the materials catalog this users guide and a subdirectory that contains ChartFX ClientServer core dll for graphics and other graphics related dll files If an INI file is not found GSolver will create one with a default for each material class Prior version INI files can be used if a CONSTANT section such as shown below is added CONSTANTS total 3 Ones 1 0 One 25 1 25 0 One 5 1 5 0 This identifies three materials of the following constant refractive indices 1 0 1 25 and 1 5 Besides the C
7. Add OR 01 1R 11 Clear Clear All m Integrated Merit Function Start Stop S amples Checked values extend merit Wavelength Iv 0 4 25 le function over selected range Theta r fo 45 fio M gt parameter os o w fi 5 The merit function is summed over le the parameter range overriding grid E Alpha jo formulas Betal 1 45 Th The Population Weight Cross over and Max Iterations control the operation of the DE run The number of real parameters must be at least 1 is also entered on this dialog Select the solution mode from the drop down list box The Goal settings area of the dialog contains controls for generating input to the DE metric function which will be minimized The DE merit function is as follows 71 GSolver V5 2 User guide merit Sm DE calc Z DE goat i where the sum is over the diffraction efficiency DE by orders given in the Goal settings Each term in the sum includes a weight factor which is also entered in the Goals Setting box Enter the goals in the dialog by entering an order R and T for reflection and transmission together with a weight and goal and add the selection to the list Usually a merit function consists of one or two DE goals The object of the DE algorithm is to find parameters that minimize the merit function a perfect match is at merit 0 With file version 5 1 1 1 the merit function has been expanded to allow for the optimi
8. PI The value of pi POLY X The value of an Nth degree polynomial in X PRODUCT X The product of all numeric values in the argument list RADIANS X Converts the angle expressed in degrees to radians RAND A uniform random number on the interval 0 1 ROUND X n X rounded to n number of decimal places 0 to 15 SIGMOID X The value of the sigmoid function SIN X The sine of X SINH X The hyperbolic sine of X SQRT X The positive square root of X SUMPRODUCT R1 R2 The dot product of the vectors R1 and R2 15 2 Statistical Functions The following lists supported statistical functions TAN X The tangent of X TANH X The hyperbolic tangent of X TRANSPOSE M The transpose of matrix M VECLEN The square root of the sum of squares of the arguments AVG The average arithmetic mean of the arguments CORR R1 R2 Pearson s product moment correlation coefficient for the paired data in ranges R1 and R2 COUNT A count of the non blank arguments F M N F The integral of Snedecor s F distribution with M and N degrees of freedom from to F ERF L U Error function integrated between 0 and L if U is specified then between L and U ERFC L Complementary error function integrated between L and oo FORECAST Predicted Y values for given X FTEST R1 R2 The significance level of the two sided F test on the variances of the data specified by ranges R1 and R2 GMEAN The geometric mea
9. independent Shown is an oval shaped dimple being approximated with 4 index boundaries points where the index may change in the x direction and 4 index boundaries in the y direction The white colored squares are assigned one material index value and the gray colored squares are assigned a second material index value The grid may be made as fine as deemed necessary Using this simple example as a guide arbitrarily complex structures may be built by stacking layers Each layer is independent of the ones above and below so that some layers may have fine structure with numerous index transitions while other layers might be uniform 19 1 5 Relation of Index of Refraction to Permittivity All material indices are entered as n and k values This formulation is used so that the GSolver sign convention for permittivity need not become a source of confusion The index n ik are entered as two real numbers with n gt 0 and k20 GSolver then forms the permittivity as e Inl ilkl InP Ik 2ilnliki The permittivity is used in the grating description vector not the index of refraction The wavelength specified in GSolver is the vacuum wavelength 19 1 6 Solution Routines There are four routines that perform all the calculations for the internal fields and boundary condition matching SolveTE handles the special case of TE mode polarization for one and two dimensional gratings SolveTM handles TM polarization for one and two dimension
10. the Microsoft Office applications 2 3 Importing grating defitinion by text file A grating may be defined in text format external to GSolver and imported using the File Import Text menu item when the Editor tab is active 2 3 1 Grating Text File Format Following is an example grating definition file format for a two layer grating structure The first layer has two regions and the second layer has four regions LAYER 0 213 0 2 DRUDE Lead true 0 8 CONSTANT Ones false LAYER 0 132 0 12 SCHOTT BSC4h true 0 22 HERZBERGER KCL true 0 26 TABLE SIPOLY 10 true 0 4 CONSTANT Ones false END Layers are numberd starting with the substrate and moving up The LAYER description starts with the first layer on top of the substrate which is defined within GSolver on the Parameters tab and is identified with the LAYER 20 GSolver V5 2 User guide keyword The first line following the LAYER keyword is the THICKNESS of the layer in MICRONS be sure to leave the default GSolver units on the parameters tab as Microns Following the thickness line is a line for each block of material within a single period There must be at least one block definition a uniform layer will have width 1 0 A block definition consists of four entries on the same line a relative width based on grating period a material catalog type which must be one of CONSTANT TABLE SCHOTT SELLMEIER DRUDE HERZBERGER and POLYNOMIAL whi
11. the RK or BS method succeeded However I ve only made cursory experimentation I ve also found that BS generally takes the longest time and takes more computer resources to give similar numerical accuracy as AE These comments are only meant as anecdotal and based on the class of grating designs I generally work with 105 GSolver V5 2 User guide 17 1 5 Setting Algorithm Choice The algorithm setting is determined from the Tools menu The two user definable parameters are set by clicking on Tools Control Parameters menu item Control Parameters i x Runge Kutta and Bulirsch Stoer algorithms take two parameters maximum thickness and relative error Cancel Maximum thickness in relative units 2pi z lambda flags automatic layer subdivision which limits integration depth and improves convergence Max Thickness 0 7 Recommended value 0 7 Relative error halts integration when error estimate drops below the relative error on any step iteration Rel Error 10 001 Recommended value 0 001 Max Thickness is the normalized maximum layer thickness allowed If a layer is thicker than Max Thickness it is automatically subdivided Rel Error is the relative error stopping criteria in the Runge Kutta and Bulirsch Stoer integration routines The current algorithm choice AE RK BS is indicated on the status bar and is used for any calculation RUN grating listing tab genetic algorithm tab run tab 3Drun tab
12. 0 0398411 0 0398411 0 001602 0 956955 0 001602 0 960159 1 00000C 0 0400 0 0397173 0 0397173 0 00284298 0 954597 0 00284298 0 960283 4 00000C 0 0500 0 0395579 0 0395579 0 0044321 0 951578 0 0044321 0 960442 4 00000C 0 0600 0 0393628 0 0393628 10 00636462 0 947908 0 00636462 0 960637 1 00000C 0 0700 0 0391317 0 0391317 0 00863483 0 943599 0 00863483 0 960868 1 00000C 0 0388645 0 0388645 0 0112361 0 938663 0 0112361 0 961136 1 00000C 0 0900 0 0385611 0 0385611 0 0141609 0 933117 0 0141609 0 961439 1 00000C 0 1000 0 0382215 0 0382215 0 0174011 0 926976 0 0174011 0 961778 1 00000C 0 1100 0 0378458 0 0378458 10 0209474 0 920259 0 0209474 0 962154 4 00000C 0 1200 0 037434 0 037434 0 0247902 0 912986 0 0247902 0 962566 Fr Foy ery org Gora perg E org ora ery E oy E 4 00000C 0 1300 0 0369864 0 0369864 0 0289191 0 905175 0 0289191 0 963014 4 00000C 0 1400 0 0365035 0 0365035 0 0333232 0 89685 0 0333232 0 963497 1 00000C 0 1500 0 0359858 0 0359858 0 0379909 0 888032 0 0379909 0 964014 1 00000C 0 1600 0 0354342 0 0354342 0 0429104 0 878745 0 0429104 0 964566 4 00000C 0 1700 0 0348495 0 0346495 10 0480691 0 669012 0 0480691 0 96
13. 4 1 GA Design of a Thin Film AR Coating In this example the optimal AR antireflection coating thickness is sought for MgF2 on an Al203 substrate in air at normal incidence at 0 5 um wavelength 1 Start a new grating editor 2 Change the wavelength to 0 5 um on the Parameter tab 3 Change the Substrate material to Table A1203 4 Using the Rectangle tool in the Editor tab create a thin film coating with width of the canvas and of arbitrary thickness 5 Set the material property to Table MgF2 6 Click the Approximation button which creates the piecewise grating data structure 7 On the GA tab click Populate 8 On the GA Options dialog click the Options button enter the following goal ORO 1 click Add 9 This indicates that the specular reflection has O energy the AR condition Click OK 10 In cell B12 enter the following formula if F5 gt 0 amp amp F5 lt 0 2 F5 0 11 Change ES to 0 4 12 Click the GA test button to exercise the parameter selection and verify that the thickness is being updated and constrained non negative and lt 0 2 13 Click RUN GA 14 The merit function Best Energy is updated each time a new minimum is found The final result is updated to the listing The result should be somewhere around 0 09 15 Increasing the maximum constraint may cause the region of multiple minima to be reached If so multiple optimal solutions may be found by executing the GA algorithm multiple times
14. All cell formulas begin with an equals sign and are calculated immediately To toggle between formula view and value view use the menu Formulas gt Formula View The formula engine included in GSolver is very extensive and powerful It includes all common functions and logicals with logical conditional constructs The formula engine is discussed in the Grid Formula chapter Any cell can be used in any formula as long as nested iterations and a few restricted cells are avoided This Listing RUN grid comes equipped with a single free parameter in cell D5 Enter the parameter increment and stop values as indicated in ES and F5 set them to 0 and 80 respectively This will cause the value of theta formula entered in B2 to change from 0 to 40 degrees in steps of 0 5 degrees Now click on the RUN button in cell D9 The first thing that happens is that GSolver cycles through the parameter range For complicated formulas the increment and decrement buttons may be used to single step the grid computation to verify correct behavior After the first run through the parameter loop is reset and then on each parameter increment the current grating list as defined on the grid is sent to the solver routines The solution is written to the Results grid Results tab At the completion of the loop the Results tab is displayed Select any column s to graph by clicking on their headings Multiple columns are selected using the shift and ctrl ke
15. Cale ID Grid View D iv Algebraic Eigenvalue AE Vacuum Wavelength z E ni Runge Kutta RK Bulirsch Stoer BS Grating Period fi Control Parameters UNITS Selection rsion from microns factor or Lines mm 11000 Allow material GAIN 1 0000 0 00000 1 50000 0 00000 Angles of Incidence v 64 Bit Floating Point THETA fo 128 Bit Floating Point 256 Bit Floating Point PHI fo 0 The current Algorithm selection is shown on the status bar in the lower right hand corner am ae y In the piecewise constant grating approximation each layer defines constant regions of complex index of refraction that are independent of depth Using periodic boundary conditions in the transverse directions a 102 GSolver V5 2 User guide Fourier expansions of the index of refraction and Maxwell s equations that relate the transverse electric field E and transverse magnetic field H leads to a general coupled differential equation of the form E d Es e Es dz H H where the z coordinate is normal to the layer M is a constant matrix independent of z in the piecewise constant approximation and is a function of the Fourier coefficients of the permitivity and impermitivity and of the propagation constants defined by the Rayleigh basis for the grating 17 1 1 Algebraic Eigensystem Solution AE Much effort has been given to the stable numerical solution of this differential equation system
16. GSolver open the grating file gs5 containing the desired lookup table and save it to a new filename 92 GSolver V5 2 User guide 14 Material File GSolver ini The materials file GSolver ini format is similar to previous versions New to version 5 1 is a constant material class The various classes are CONSTANT HERZBERGER DRUDE SCHOTT SELLMEIER TABLE The analytic forms for each of these models is given in section 2 6 Following is a description of the ini file The ini file has seven sections the 6 material class sections and the GSOLVER V5 1 section for comments Each material section begins with one of the material class names in brackets Following the class name is a line that gives the number of members of the class which will be read in It has the following format total x where x is the number of materials The ini file may be edited with a text editor However it is important to update the total line if materials are added or deleted Each time GSolver starts it reads the ini file found in the GsolverV50 directory If it does not find an ini file it creates a list of default materials for each class When GSolver exits it re creates the ini file Thus if any material editing is done within GSolver the changes are only written out to disk when GSolver exits In general no checking is done on the material parameter entries other than n gt 0 and k20 The recommend material cla
17. New to GSolver V5 1 is the addition of a 5 order Runge Kutta solution method for solving the set of coupled constant coefficient differential grating equations This method includes adaptive step size control accurate to 5 order The implementation of the Runge Kutta follows standard practice There are two user definable parameters associated with the RK method They are the maximum integration depth and a relative error Any grating layer may have an arbitrary thickness Any particular layer may be subdivided into m layers and the RK solution method is then applied to each sub layer In this case the boundary condition solver propagates the fields through the m sub layers At issue is the numerical stability of the RK integrator through a thick layer with potential exponentially growing solutions There is also a trade between layer subdivision and speed of solution The GSolver boundary condition solver S matrix methods are numerically stable Experience shows that a RK numerical integration may become unstable for thick grating layers with complex index of refraction in TM and conical mount illumination By subdividing a thick layer into thinner sublayers the solution can be made numerically stable A heuristic layer subdivision parameter maximum layer thickness of 0 75 221z 4 where z is the thickness and A is the vacuum wavelength controls maximum layer thickness If a layer is thicker than this it is automatically subdivided
18. Profile tool black triangle 6 Select the General Sawtooth form and change the angles to 35 and 90 Click OK 7 Select the newly created object and set its properties to Herzberger ZnSe IRTR 4 This should be the default if the substrate material was set o d 14 GSolver V5 2 User guide 9 10 11 12 13 14 15 16 17 18 19 Select the shape and assure it is moved all the way to the bottom of the canvas Then grab the top center handle and stretch the shape so that it s height is 19 units this equals 1 9 4 7 6 um Click the Approximation button Click the GA tab and Populate the grid Click the Options button and enter the goal as 1T 1 1 1T order DE of 1 weight of 1 Click Add then click OK In cell B7 enter the following formula if f5 gt 2 amp amp 5 lt 5 f5 4 This constrains the DE algorithm to only allow periods between 2 and 5 microns and uses 4 microns as the default Change D5 to 2 and ES to 5 Click the menu item Grid Thickness Formula select All and enter the formula if f6 gt 0 amp amp f6 lt 2 f6 1 This allows for a maximum thickness of 8 and a minimum of 0 Note there are 40 layers in the approximation Set B10 orders retained to 8 Set D6 to 0 and E6 to 2 Click RUN GA While the GA is running the current generation is displayed together with the best merit function value A value of 0 indicates that the goals were met perfectly For this example a
19. The bottom selection in the list is User units This allows configuring the conversion for units not in the drop down box Select this item and then enter the conversion factor in the related text entry box The two quantities that have units in a grating calculation are the wavelength and the period The absolute wavelength is needed for the index of refraction calculation Otherwise all calculations are normalized to the grating period This implied length scaling is permitted since Maxwell s equations are linear NOTE With file version 5 1 1 1 a more consistent use of units has been implemented with a bug fix All forms now expect input in the selected units The parameters with units are Wavelength Period Layer thickness The refractive index transitions within a layer are relative to a Period and so are within the range 0 to 1 3 2 Angles All angles are entered as degrees On the Parameters form are four angle entries that describe the incident plane wave direction and polarization 39 GSolver V5 2 User guide Polar angles O theta and phi are defined as shown in the figure below O is positive for deviation away from the z axis towards the x axis is k z 0 EN y A IZ X positive for counterclockwise rotation around the z axis from the x axis This convention is for the incident plane wave k vector illumination For the reflected components the x axis is used as a reference The two
20. diffraction calculations as GSolver makes no check on index of refraction validity 2 7 2 Constant The Constant material property returns a fixed index of refraction for any wavelength setting A Constant material property is specified with a name and real and imaginary indices of refraction In the INI file these appear as name real value imaginary value The colon after the name serves as a text name delimiter 2 7 3 Drude The Drude model is a well known simple analytic index of refraction model based on a simplified physical model of the material A two parameter model it is not expected to give accurate results at any wavelength particularly above the first model resonance A Drude model material is entered as name pl p2 29 GSolver V5 2 User guide Where pl and p2 are the two real model parameters The relation between the parameters and the model index of refraction is as follows n ik Je 1e e p og p 2 e P2 P ANA pr ql 10000 A where A is the wavelength in microns The square root branch is taken so that n and k are positive quantities All n and k in GSolver are treated as non negative quantities Typically a Drude model might be used to estimate indices of refraction for metals in the infrared region 2 7 4 Sellmeier There are several Sellmeier models in the literature and GSolver s 12 parameter Sellmeier model comprehends several of them A Sellmeier material is entered
21. drawn it is necessary to run the piecewise constant approximation algorithm on the grating objects This is accomplished by clicking on the Approximation radio button in the upper left hand corner of the canvas area The approximation is recalculated each time this button is clicked The algorithm begins operation by scanning through all of the grid points on the canvas filling an array with the material properties of any object it finds always using the top most object for overlapped regions If there is no object found it fills the sampling array with the superstrate property In the case of grouped objects the material property that gets assigned is the first object in the group list Thus it is important to keep track of which objects are grouped and it is good practice to only group objects of like material property 51 GSolver V5 2 User guide Once the sampling array is filled it is then parsed The array is scanned by columns and rows to find the maximum rectangle grouped sampling points that satisfies the piecewise constant approximation For example if every grid point happened to have a material property different from any nearest neighbor then the result would be a maximal set of squares centered on each grid point If adjacent grid cells in a single row have the same material properties then the boundaries between them are removed Columns are scanned before rows Any adjacent rows that are identical are combined i
22. from choice of the definition of the plane wave vector propagation k and i To accommodate the European convention GSolver includes a sign convention check box on the Parameters form just above the orders display This check box only changes the sign of the orders It has no effect on the calculation or the internal representation of the diffraction orders The orders are labeled as shown on the Parameters form 3 5 Substrate Superstrate In line with V5 1 convention all regions of the grating must be assigned a material property The superstrate and substrate material properties are 41 GSolver V5 2 User guide assigned from the Parameters page using the two labeled buttons The button command creates a Material Property Dialog where a material type and entry may be selected 3 6 Saving To Save the current grating use the File gt Save menu command A grating file always has a gs5 type If the Listing Results or GA form data grid contents need to be saved use the Grid gt Save Grid menu command Data from grids are saved as ASCII text txt files 42 GSolver V5 2 User guide 4 Graphical Editor Editor Tab The graphical grating editor as shown below is the primary tool for defining a linear grating structure Note that crossed gratings are handled separately for both editing and calculation and are discussed in the 3DEditor and 3DRun chapters Osea eje Rem pros nocosaA Parameters Editor u
23. in range R TTEST2EV R1 R2 The significance level of the two sided dual population T test for ranges R1 and R2 where the population variances are equal 15 3 Conditional Statistical Functions The following lists supported conditional statistical functions TTEST2UV R1 R2 The significance level of the two sided dual population T test for ranges R1 and R2 where the population variances are not equal 96 GSolver V5 2 User guide VAR The sample variance N weighting of the arguments VARS The sample variance N 1 weighting of the arguments VSUM The visual sum of the arguments using precision and rounding of formatted cell values CAVG C Conditional average CCOUN C Conditional count CMAX C Conditional maximum CMIN Conditional minimum CSTD C Conditional sample standard deviation N weighting CSTDS C Conditional sample standard deviation N 1 weighting CSUM C Conditional sum CVAR C Conditional population variance N weighting CVARS C Conditional population variance N 1 weighting 15 4 String Functions The following lists supported string functions CHAR N The character represented by the code N CLEAN S The string formed by removing all non printing characters from the string S CODE S The ASCII code for the first character in string S EXACT S1 S2 1 if string S1 matches string S2 otherwise 0 FIND S1 S2 N The index of
24. ini is now written to the root directory location of the GsolverV51 exe file More consistent use of units All forms now expect input in the user Units selection made on the Parameters tab The genetic algorithm merit function has been expanded to allow for summing a result over a set of angles or wavelengths This allows for optimization over certain parameter ranges The results of a Grating Listing run or a GA run can now be copied to the internal piecewise grating structure allowing for the results of say a GA run to then be used directly in a Grating Listing run or from Run 1 1 2 New V5 2 Features Version 5 1 release included some 30 interim upgrades with various bug fixes and addition of new features Version 5 2 release includes a new editor patterned on the legacy V4 20 editor A clear understanding of the interrelation between the various grating definition editors and the internal grating array is essential The various user interactions are described in the following section GSolver V5 2 User guide 1 2 GSolver Grating Definition GS4 Editor ar A e e l a ae SAIS Listing Run View Ue 1400 e e i 4 JE Al eth All calculations are performed on the Discrete approximation of the grating structure This is best viewed examined from the Listing RUN view tab Note that changes made on the List RUN view grid are copied to the Discrete approximation
25. look to the column just left of the current cell e Offset values may be constants or expressions Examples e 0 1 refers to the cell above the current cell e 2 refers to the cell two columns to the left of the current cell e 1 refers to the cell to the right of the current cell e 0 1 refers to the cell below the current cell e CSUM C4 C100 1 Joe calculates the sum of all the values in the range C4 C100 for which the cell in the column to the left contains the string Joe e CCOUNT C4 C100 0 1 counts all the cells in the range C4 C100 whose value is greater than the contents of the cell immediately above e verb 1 2 adds 2 to the cell value from the cell to the left 6 4 1 2 Constraint Expressions Constraints are limitations or conditions placed on the variables in the gird They are expressed as algebraic statements appended to formulas A 66 GSolver V5 2 User guide constraint expression can be attached to any formula by typing a semicolon and the constraint conditions after the formula Constraint expressions establish either conditions under which a formula operates or boundaries for valid results of the formula Constraint expressions can be simple equality inequality relationships or arbitrary formulas Any valid grid expression that returns a numeric value is a valid constraint expression However unlike the expression that defines a cell value a constraint expression can
26. matrix Q are the eigenvectors and A is a diagonal eigenvalue matrix The coefficient matrix is reduced to this form through the equivalence of a field normalization that removes the constant diagonal term Multiplying the matrices leads to the following B B Q OQ AWA B B Q QAN hence Qi Q Ay A Q B QA Qy 0 These relations are used to reduce the overall eigensystem computational effort by about a factor of 8 19 2 1 5 Permittivity and Impermitivity For infinite orders the permittivity and impermitivity are strictly inverses of each other For finite orders this relation no longer holds This is the origin of the convergence issues with coupled wave expansions In GSolver both the Fourier coefficients of the impermitivity and 122 GSolver V5 2 User guide permittivity are computed and the inverses are also computed The Fourier coefficients enter into the coefficient matrix calculation as a Toeplitz matrix so the inversion can be done much faster than if it were a general matrix inverse calculation The inverse of a Toeplitz matrix is generally not a Toeplitz matrix The question remains as to where to substitute the Toeplitz impermitivity matrix with the inverse of the permittivity Toeplitz matrix eh and ah are the permittivity and impermitivity Toeplitz matrices stored as vectors to reduce memory ehl and ah are the coefficient matrices computed as the inverses of eh and ah They are not Toeplitz and
27. on grating structures that have complex substrate indices of refraction the imaginary part is ignored on the second run through the parameter set 20 1 TracePro Run Example 1 Start a new GSolver top level window by opening GSolver or by clicking on File gt New 2 Define a grating structure using the Editor tools and click on the Approximation button in the usual manner 3 Click on the RUN tab Click on the menu item Tools gt TracePro gt Setup Run The following Dialog should now be showing cana 129 GSolver V5 2 User guide Trace Pro Property Data xj Use this dialog to set the RUN tab parameters for a TracePro OK properties calculation on the current grating defintion ok After Results have been tabulated return to RUN and click the menu Cancel item Tools gt TracePro gt Write File Two sided calculation fo Theta Start m Information fields les Theta Stop 295 Temperature K 2 Theta Increment Ao Description 0 PhiStar Enter description so Phi Stop 2 Phi Increment 3 Orders fi Wavelength Start microns 2 Wavelength Stop microns 0 1 Wavelength Increment microns This dialog lists the parameter settings that will be executed in the TracePro run l 2 3 Set the various parameters to the desired values Click OK to continue GSolver estimates the number of calls needed to complete the loop and reports the result in a dialog Click OK When the firs
28. reference to the range defined by 99 GSolver V5 2 User guide coordinates N1 through N4 ROW C The row address of the cell referenced by C ROWS R The number of rows in the specified range R S R The string value of the top left cell in range R VLOOKUP X S C The cell in range S that is C columns right of X DFT R The Discrete Fourier Transform of the range R EIGEN M The eigenvalues of the matrix M FFT R The Discrete Fourier Transform of the range R using a fast Fourier Transform algorithm FREQUENCY R B F frequency distribution for R with intervals B INVDFT R The inverse of the Discrete Fourier Transform of the range R INVERT M The inverse of matrix M INVFFT R The inverse of the Discrete Fourier Transform of the range R using a fast Fourier Transform algorithm LINFIT X Y The straight line least squares fit 100 GSolver V5 2 User guide 16 Graphing Options The graphing engine uses ChartFX ClientServer Core dll the ChartFX core graphing engine which provides general chart object options Charts are created from the Results tab and from the Materials Editor The general chart has eight regions that can be customized Customization is accomplished by right clicking on a region to create a menu list and then selecting the options to modify The eight regions are identified in the figure shown below The Axis properties menu is activated by clicking on one of the tick mark labels for the desir
29. the spreadsheet Now you can select the run tab If you check the wavelength box and enter appropriate parameters you can calculate the grating performance in various orders as a function of wavelength Now press the run button When the calculation is complete the results screen pops up Highlight a column and press the chart button to get a plot of efficiency versus wavelength in the chosen order 1 3 4 Yet Another Blaze Procedure 1 2 Set the superstrate substrate Period wavelength and so forth on the Parameters tab Click over to the GS4 Editor and click on the N ties option This brings up a GS4 dialog more on this in the GS4 section below Enter the desired blaze angle in the Balze angle calculator press the enter key to display the result Not the pct value This is the position of the apex of the triangle profile relative to the current Period Click on the handle in the graphic and drag it to so the x position is either the pct value of 1 pct value depending on left right orientation Enter the h value in the Total thickness box which translates to the maximum y dimension for the profile 17 GSolver V5 2 User guide 6 Decide on the number of level you want for the discrete approximation 7 Set the base and top index values and click OK 18 GSolver V5 2 User guide 2 General Principles 2 1 Overview User Interface GUI Data Structures Global Parameters Material ca
30. the RUN button is clicked the parameter in D5 is incremented by the amount in ES until it equals or exceeds the stop value set in F5 60 GSolver V5 2 User guide The order of operation of GSolver when the RUN button is clicked is to first increment the D5 parameter from its current value update all formulas and continue through to the Stop condition All formula dependencies are calculated but error checking is minimal D5 is then reset to the starting value the parameter loops through its range again this time taking the Listing values in column B and running the related diffraction calculation for each step The following example assumes that a grating structure has been defined and populated on the listing grid such as the coated blazed grating example given in 4 3 5 6 2 1 Using Listing RUN to find the diffraction efficiency as a function of incident angle theta example 1 Populate the grid by clicking the Populate button 2 Enter the following formula into cell B2 D5 100 A 3 Enter 0 into cell D5 1 into cell E5 and 80 int cell F5 4 This formula and set up will change the value of the theta parameter from 0 to 80 in steps of 0 01 degrees 5 Click RUN The D5 parameter runs through its range updating cell B2 and then is reset to the starting value The loop is repeated this time running the diffraction calculation The progress bar estimates the time to completion At the end of the calculation the res
31. the first occurrence of S1 in S2 HEXTONUM S The numeric value for the hexadecimal value S LEFT S N The string composed of the leftmost N characters of S LENGTH S The number of characters in S LOWER S S converted to lower case MID S N1 N2 The string of length N2 that starts at position N1 in S NUMTOHEX X Hexadecimal representation of the integer portion of X PROPER S The string S with the first letter of each word capitalized REGEX S1 S2 1 if string S1 exactly matches string S2 otherwise 0 Allows wildcard comparisons treating S1 as regular expression REPEAT S N The string S repeated N times REPLACE S1 N1 N2 S2 The string formed by replacing the N2 characters starting at position N1 in S1 with string S2 97 GSolver V5 2 User guide RIGHT S N The string composed of the rightmost N characters of S STRCAT The concatenation of all the arguments 15 5 Logic Functions The following lists the supported logic functions STRING X N Numeric value of X as a string to N decimal places STRLEN The total length of all strings in the arguments TRIM S The string formed by removing spaces from the string S UPPER S The string S converted to upper case VALUE S Numeric value of S as a string 0 for non numeric S FALSE The logical value 0 FILEEXISTS S 1 if file S can be opened for reading otherwise 0 IF X T F The value of T if X evaluates to 1 or F if X evaluates to 0 ISERROR X Re
32. to the end of the list When grating structures are stored the material property is stored as an index into the material list When a saved grating is loaded materials are loaded by index not by name Therefore if materials are rearranged by editing the INI file this might effect the properties of a saved grating If needed multiple copies of the material INI file can be used Copy the needed file into the root directory where the exe is located before starting GSolver GSolver reads the INI from the 28 GSolver V5 2 User guide root directory If one is not found there a new one is created with minimal entries The material model parameters are stored in the GSolver ini file detailed in the GSolver ini section This ASCII file can be edited with a text editor such as Notepad Described below are the various material models their parameterizations and representation within GSolver 2 7 1 Models GSolver currently has six index of refraction models Constant Drude Sellmeier Herzberger Schott Polynomial and Table Of these models the Constant Drude Polynomial and Table give complex indices of refraction the others are real valued The Table model offers the most flexibility as the entries may be made with a wavelength resolution as fine as desired Each model has approximate validity over a continuous finite range of wavelengths The user must assure that the wavelength values remain within the valid range throughout the
33. wavelength fi f2 for Theta fo fe f2 pif fo 0 2 Alpha fo fo 2 Bta fo fs TotalDepth fo fi fon xPeiod ff f2 for Bo Diffraction Efficiencies 3 Write Fields to File RUN STOP 0 Physical Memory 532 14 Mt 7 44PM 2 Selected parameters generate a loop structure within which the solver routine is repeatedly called As the solutions are found they are written to the Results tab in the normal way Diffraction efficiencies and phases are just as in the case of linear gratings Please refer to the RUN tab chapter for more information 87 GSolver V5 2 User guide 12 Angles Calc Angles Calc is a tool to calculate the diffraction order angles The propagating directions of the orders are determined by the wavelength period incident k vector and the super and substrate indices of refraction and do not depend on the details of the grating structure The grating structure determines the magnitude of energy in each order Urititied GSalver S50 Ble Ese Fomet Formo Yew Tods Grid Component about Ser ee a OROA Syren rome inte ai mn EST Diffraction Order Angle Caleslation Free Parameter horenect 0o 0 0 0 0 0 90 0 0 0 i 0572 lao 50 EJ 190 E E 150 lso lso 000000000 Phepsical Memory 476 46 Nt Tarn The basic layout of the Angles Calculation tool follows the general format of the Listing RUN and GA forms There is a free parameter cell D5
34. x y z Ngase n mod YA sin 2 jz sztan p B cos 2 jz sz tan j 0 where Mbase the base index of refraction and Amoa the modulation index The period is normalized one grating period and s 27 thickness is the so called slant angle and causes modulation in the z direction When g 0 only one layer is needed The general index is then descretized into a number of layers and number of steps within each layer This a piecewise approximation to the index variation It is convenient to choose a step size number of blocks that is commensurate with the grid The Sy tool button Sy is used to generate a fringe pattern that is parallel to the substrate The index of refraction is given by AZ Piso Umod YA sin 257 z B cos 3 j 0 where n z varies in the normal direction only and T is the layer thickness Since GSolver is material oriented the various constant regions within a holographic layer are assigned the constant material property independent of wavelength The material catalog is augmented with as many constant material regions as called for When invoked GSolver first looks through the Constant material list to see if a material already exists with the required index of refraction If one is not found a new material is created and appended to the list When GSolver is exited the GSolver ini material catalog file is rewritten with the new materials 55 GSolver V5 2 User guide 5 GS4 Edito
35. zero value as FALSE Integer operators amp convert their argument s to integer first The maximum number of grid columns is 4096 The maximum number of grid rows is one million Entries that begin with the apostrophe character are treated as text Covered cells cannot be altered This feature is used on any cell that contains labeling information Do not use commas or spaces to delimit pure numbers Scientific notation E convention is supported A range of cells can be referenced absolutely such as A2 A10 Cells may also have relative references 6 4 1 Syntax In general a formula will have the following syntax expression constraint expression comment where expression defines the calculations constraint expression places limits on acceptable values or circumstances under which the calculation should take place and comment may be any text It is recommended that constraints be entered using the if function otherwise the grid will generate an information dialog each time a constraint is violated 6 4 1 1 Expressions Expressions are algebraic formulas comprised of values and operators that define the relationships between values They adhere to the following e A formula must begin with an equal sign 64 GSolver V5 2 User guide e The grid automatically assumes you are typing a formula if you begin with any of the following characters 0123456789 e Spaces are ignored but count again
36. 1 560 2 orders 1 8224 For crossed gratings it is bytes 1760 2 orders 1 560 2 orders 1 8224 Notice that for crossed gratings the size of the memory requirements grows as orders retained to the fourth power In addition both the eigensystem solver and the boundary system solver which operate in a manner similar to matrix inversion require computer operations that grow roughly as the cube of the size of the arrays With just a few orders retained on a crossed grating a significantly large eigensystem problem must be solved for each grating layer In computer terms there are a huge number of multiply accumulate operations needed for finding each eigenvector and eigenvalue and for matching boundary conditions Considering that the basic floating point formation on a typical Intel processor is limited to 64 bits 80 bit internal difficulties with round off error are to be expected Since GSolver is a full vector implementation and a 3D structure is a natural subset of the problem space the 3D structure solver is included However due to symmetry issues the 3D structures generally do not lend themselves to the several convergence and numerical acceleration tricks employed in the normal linear grating solver routines There are in fact 83 GSolver V5 2 User guide four solver routines in GSolver a TE solver a TM solver a general linear grating solver and a full crossed grating solver The first three tak
37. 515 1 00000C 0 1800 0 034233 0 034233 10 0534543 0 858858 0 0534543 0 965767 1 00000C 0 1900 0 033586 0 033586 0 0590525 0 848309 0 0590525 0 966414 1 00000C 0 2000 0 0329101 0 0329101 0 06485 0 83739 0 06485 0 96709 1 00000C 0 2100 0 032207 0 032207 10 0708325 0 826128 0 0708325 0 967793 4 00000C 0 2200 0 0314786 0 0314786 0 0769854 0 614551 0 0769854 0 968521 Saf f 3 3 3 1 00000C 0 2300 0 030727 0 030727 0 0832934 0 802686 0 0832934 0 969273 1 00000C 0 2400 0 0299544 0 0299544 0 0897408 0 790564 0 0897408 0 970046 1 00000C 0 2500 0 0291632 0 0291632 0 0963111 0 778215 k 0 0963111 Ube 0 970837 Brg peg pare 1000000 y gt Physical Memory 535 29 Mt 7 41PM Y The first line of the results grid contain headers that identify each column The general format is R and T for diffraction efficiencies where is the 79 GSolver V5 2 User guide order number On the far right are columns labeled r and t These are the phase data in degrees Between the diffraction efficiencies and the phase data are several columns that list the values of the global parameters If the RUN tab is used to generate the results the first column is a list of the current value of the first parameter checked Thus if multiple para
38. 73 GSolver V5 2 User guide While the GA is running the current generation and best merit function are displayed A merit function value of O indicates that an optimal solution has been found based on the goals given Upon completion the GA loads the values of the best parameters to the grid and creates a table of all the diffraction orders Orders to Orders for T amp R This example is easily extendable to include multiple thin film layers Simply add the materials in the Editor tab and use a separate parameter for each layer thickness 7 4 2 GA Design Example 2 This example considers a sawtooth profile such as might be cut by a DPT diamond point turning machine in ZnSe The problem is to find the optimal sawtooth profile depth and period to maximize transmission in the 1T order i order convention box on Parameters tab unchecked for 3 um wavelength TE polarization and for 30 incident light 1 Start a new grating 2 Change the wavelength to 3 and the substrate material to Herzberger ZnSe IRTR 4 and check the Lambda change update box This tells GSolver to update the index of refraction if the wavelength changes Change Theta to 30 4 From the Editor tab change the canvas properties Edit gt Canvas Properties so that Canvas height is 2 This allows for grating structures that are 2x the grating period creating an approximation with a large number of layers for finer resolution On the Editor tab select the Custom
39. Blaze Triangle Sinusoidal Piecewise linear and Piecewise spline 2 In the Blaze grid profile select the desired blaze angle change the default 35 in cell C3 or leave it as 35 Click OK 3 A blazed profile is created A blaze grating profile is a right triangle Select a material property for the triangle by right clicking 1t 4 At this point it is easy to create a conformal layer for this profile Select the triangle shape just created with a mouse click then hold down the control key and click and drag the triangle A copy of the triangle is created Change the properties of the new triangle Then send it behind the original triangle by right clicking the new triangle and using Order Send to Back Move the second triangle so that a thin conformal layer is created around the original triangle The small gaps left in the lower right and left sides can be filled in with rectangles of the appropriate material settings 15 GSolver V5 2 User guide 5 Click the Approximation button to create the piecewise constant approximation used by GSolver 6 Perform a grating calculation using the RUN or Listing RUN tab 1 3 3 Alternative Blaze Procedure Here we describe how to set up a blazed surface relief transmission grating with the facets towards incident light This example is for a 125 line mm grating with a blaze angle of 30 with light incident at 30 1 Begin by filling in the appropriate information on the parameters
40. E DOUBLE AND QUAD DOUBLE 108 18 1 1 EXAMPLE CALCULATION cnoooccccccconcnnnninonoconcnnnnonananncconccnonenananinss 109 19 DIFFRACTION SOLUTION IMPLEMENTATION 0020 112 19 1 THE GRATING iccssscssccccscscesssscscccccssdecdsescsdcccecsescdsocsssccscasssvsdesssssese 112 19 1 1 STRATIFIED GRATING APPROXIMATION occcononnnnnnoncccncnnononaninisos 112 19 1 2 1 DIMENSIONAL GRATINGS ccccccccsseeccccescecccceseccecceseeceeeeners 112 19 1 3 2 DIMENSIONAL GRATINGS ccsccccccssecccccsccccccescececceseeseceeners 113 19 1 4 3 DIMENSIONAL GRATINGS oocccccononcnnnnoncccncnnononananacicnconononananinos 114 19 1 5 RELATION OF INDEX OF REFRACTION TO PERMITTIVITY 115 GSolver V5 2 User guide 19 1 6 SOLUTION ROUTINES uciiaciciita dial 115 D E A O CL 116 19 21 MAXWELLS EQUATIONS used aie is 117 19 2 1 1 Superstrate and Substrate Solutions ooonoconnnococoncccconacnnnananos 117 19 2 1 2 Inhomogeneous Plane Wave Intra layer Solutions 119 19 2 1 3 Formulation of Eigensystem Solution 121 19 2 1 4 Eigensystem Order ReductiOM coooocnnncccnoncccnonccononcnonnncncnnncnos 122 19 2 1 5 Permittivity and ImpermitivitY ocoocoonnncccnoncccnoncncnnnnnnnnnnninnnanos 122 19 2 2 INTRA LAYER SOLUTIONS BOUNDARY CONDITIONS ccoocconccnno 123 19 2 2 1 Gaussian Elimination i502 uci ssa Ge assess seen 124 19 2 2 2 Stack Matrix Methods add i 125 20 TRACE PRO MATERIAL RUNS eococccccccnccnccccnncccnnccnn
41. EFRACTIVE INDEX COLOR MAP DIALOG cccccsssssssssssssseseees 91 14 MATERIAL FILE GSOLVERLINI cccsscssssssssssesecssscesssoseees 93 15 GRID FORMULA ENGINE ccsccssscscssccsscscscsssscscsscosscssssonsees 94 15 1 MATHEMATICAL FUNCTIONG ccccccccsssssssccccccccccccesescccccccecescssens 94 15 2 STATISTICAL FUNCTIONS tesscssdescsdtestacsesscsedscssatedecssdcedacstecacsossecsocs 95 15 3 CONDITIONAL STATISTICAL FUNCTIONS cocos 96 15 4 STRING FUNCTIONS icicscccssccscccscsssccsecsescesscsssccsesscscsdscasecssecssscesscaeses 97 15 5 LOGIC FUNCTIONS sidesiscscscccccacdoseccdcddccosssddssccssdescssssdesscescs ccosssdcssse 98 15 6 DATE AND TIME FUNCTIONS ccccccccscsssssssccccccccccccssescccccccsccscesens 98 15 7 MISCELLANEOUS FUNCTIONS cccccsssssssssssssssssssssssssssssssssssseseees 99 16 GRAPHING OPTIONS sesscicissssccsiscnssossscdeisacnssetiosnssccnsdenpscuasvasssons 101 17 ALGORITHM SELECTION cssccssssccsscccscssscessscsessccseoess 102 17 1 1 ALGEBRAIC EIGENSYSTEM SOLUTION AB c 0 sccceceeeeeeeees 103 17 1 2 5 ORDER RUNGE KUTTA RK oooconicccccncononincnnnncnnoncncnncncnnonos 104 17 1 3 BULIRSCH STOER METHOD BS cc sccccccececeesessnsececeseeseeenens 105 17 1 4 GENERAL METHOD COMMENTS csseecccceseececcescececceseececeeners 105 17 1 5 SETTING ALGORITHM CHOICE cssseececceseececcecececceseececeenees 106 LAZO GAIN usadas 107 18 PRECISION DOUBL
42. G B fo o fo fo Replace Insert Delete COLOR The Color map has two components lists of piecewise linear breakpoints through RGB space for both the real and imaginary components of the index of refraction Break points may be added and deleted from the list For example given n as the real part of the index of refraction a search is made through the real breakpoint list to find the two break points that bracket n These two points define two RGB coordinates The color of the given n lies along the line between these two points in RGB space as a linear interpolant The same goes for the k value with the list of imaginary component break points This scheme allows for fairly general color assignments to the real and imaginary parts of the index of refraction The representation of the colors is done with a two color fill pattern made up of a cross hatch against a solid background Changes to the default color map are stored with the grating in the gs5 file Therefore when starting GSolver each new grating has the default values 34 GSolver V5 2 User guide To use modified values simply save a simple grating as a gs5 file Loading that file will change the color map to the saved settings 2 8 Materials Editor The Materials Editor Dialog as shown in the figure provides a graphical editing tool for the material catalog items Any changes made using the Materials Editor do not get written to the GSolver ini disk file unti
43. GSolver V5 2 User guide where E is a vector with complex coefficients to account for polarization R are reflected diffracted complex valued E field amplitudes and T are the transmitted diffracted complex valued E field amplitudes kj are the wave vectors for the superstrate 1 and substrate 3 regions is the square root of 1 The kj vectors are known in regions 1 and 3 as well as Einc Orienting the coordinate system so that z points from the superstrate to the substrate k k1 sin 9 cos g x sin 9 sin g Y cos Z 27m kl A E uexp Jk r u sin T cos cos cos T sin g xX sin 7 cos sin cos 7 cos p Y sin 7 sin 0 Z where cos t and sin t are complex numbers such that llull lu l luyl lu l 1 Explicitly sin 7 Complex sin a cos P cos sin f cos T Complex cos a cos f sin a sin P where Complex is a complex number a ib Obvious changes in sign need to be made for coordinates systems where the z axis points from the substrate to the superstrate In each layer the kx and ky components of the k vector are identical because of continuity of transverse field components Knowing the values of the magnitude of the k vector in regions 1 and 3 leads to the following equations 118 GSolver V5 2 User guide k1 k1 sin 9 cos g iK sin sin y jK 9 k1 k k 2 5 7 A x A 2 2A k3 k3 sin cos p iK x sin sin g
44. Grating Solver Development Co www gsolver com User s Manual GSolver Diffraction Grating Analysis for Windows Version 5 2 Grating Solver Development Company www gsolver com GSolver V5 2 User guide GSolver Version 5 2 User s Guide Product Design Author David Fluckiger PhD This Manual Copyright 2006 2007 2008 2009 2010 2012 David Fluckiger All Rights Reserved Trademarks used in the manual are the property of their respective owners 7 Edition July 2012 updated to reflect version 5 2 1 4 of code Grating Solver Development Company Contact information http www gsolver com GSolver V5 2 User guide 1 INTRODUCTION iii iii A RA A E AAA 8 1 1 COVER ria 8 Leda NEW Vaud FEATURES aria 9 19 32 NEW V 5 2 FEATURES lt 4 2 5 028 238 ies en Ai 9 1 2 GSOLVER GRATING DEFINITION sssccccsssssscscsssssesssssssccsssssssccsesses 10 1 3 EXAMPLE RUN QUICK START ccssssscsssssscccsssscssssssccssssssccssscsses 11 1 3 1 BINARY GRATING EXAMPLE o cccccccccccocononononononononononinononanoninininineness 12 1 3 2 BLAZE GRATING EXAMPLE cccccccccccccccscccccsesescsescsesesesesssesceeeseeees 15 1 3 3 ALTERNATIVE BLAZE PROCEDURE ccccccccecccccescsesceesescseseseseseess 16 1 3 4 YET ANOTHER BLAZE PROCEDURE cocccccccccccccconononononononananenenanenes 17 2 GENERAL PRINCIPLES sccccsscasssssisesnstonssensnsonnssentisevnstbssnnsacssivon ste 19 DM OA suduiencnsSuessecusevass booskbacbnssuusa
45. ING CONSTRAINTS sseseesseseessssecessscoceessocecssccecsscseceesoeceesseceessee 72 7 4 1 GA DESIGN OF A THIN FILM AR COATING cccocococonononononononononinononos 73 7 4 2 GA DESIGN EXAMPLE 2 oocccccccocccononononononononononononononononononononononononos 74 8 EXECUTION RUN TAB 8 1 1 RUN CONSTRAINTS occcccnconnnononcnonoccnncnnnnonononnnnononnnnnnonononconononnnanennnos 77 8 1 2 1 ORDER LITTROW ia 78 9 13 WRITE FIELDS TO FE didas 78 Sd CRUNIS TOP ta A ti AAN 78 9 RESULTS TAB ssiotononitocoo conidios ona dina IRA ca Ainas rosana 79 9 1 DIFFRACTION EFFICIENCY cccsssssssscscscsssssscssccccssssssssssccccsscssssssssees 80 D2 AA OTTO 80 93 A etase soetas eeneioe iio tsas asako iaoe ioeie s seisa nie 81 1D 3D EDITOR TAB rides ito aid docena 82 10 1 LIMITS ON SOLVING 3D STRUCTURES sscccssssssssssssscccsssssccsceses 83 10 2 DEFINING A 3D GRATING sccssssscssssnscssscsescosscescccossnssessssoscssoosses 84 10 2 1 HOLOGRAPHIC TOOL isidro til 85 ARA ts soss te sbeses ceset sizesis 86 GSolver V5 2 User guide 1 IDRUN TAB cial did a isaac Es 87 12 ANGLES CALC ii 88 12 1 EXAMPLE DE ANGLE CALC 2 ccccsssssssssssssssssssssssssssssssssssssssseees 89 12 2 DEFINITION OF THE ANGLE CALC ccccssssssssssssssssssssssssssssssseees 89 13 DIALOGS wisscciaceosiedsstues dacwssevsideatsecasscrsvncdsssnndduccosnssdsssscussuvesvcndsdassvdents 91 13 1 REFRACTIVE INDEX SELECTION DIALOG cccccsssssssssssssssseseees 91 13 2 R
46. N OF OLE COORDINATES o ooccccccccococonononononononenininininenenes 37 2 10 2 V4 20C DATA IMPORT CRASH oococcccccccoconoconononononononononononinininacaness 37 3 PARAMETERS TAB ssi ccccsecsccssscsvecassnsssssseccensepenedastecesovasvicssstsereuesc DO 3L UNITS SELECTION vices csoncediscecennscconeudacsosenavcdondeeuseshavacdestusnxdossccandesocd O0 3 2 ANGELES cia OF 3 3 STORES DERINITION a aia L 3 4 ORDER CONVENTION gisssnissscossassossencanscsenstocassecabscvossasnavossusbowosescnaceve FL 3 5 SUBSTRATE SUPERSTRATE eesseseessececsscsccessscceesscoceesseceesscsecsssseeeessee FL A O G2 4 GRAPHICAL EDITOR EDITOR TAB csscessssssssssscsssceees 43 4 1 COORDINATE SYSTEM ccsssssscssssccccsssscscccssssscccsssssccccsssssccscssssscess FD 4 2 CANVAS GRID A A F 4 21 ACCELERATOR KEYS sins A A A n 47 AS E KOLO P id a S 4 3 L RECTANGLE andanada aid 47 4 3 2 PIECEWISE LINEAR POLY LINB cccccccnconnnonnnnnonccnononnanonononocnononnnnns 47 4 3 3 SPLINE CURVE AND ELLIPSE cccccccccccessessscececececeeeessseceeeceeseeenens 48 4 3 4 CLASSICAL FORM GENERATION cccocococononononononnnnnononononononononononononos 48 4 3 5 CONFORMAL COATS wu cccccccssssssssccececececeesenseaecesececeenessaeeeeeseeseeeeas 50 4 3 6 UNIFORM FILL COATS ccccsesscccceccceeeessssececesececeesessaecececeesenenees 50 AIT TEXICAND N O se sehek ea cod vets v ates oss 51 4 4 AUTOMATIC PIECEWISE APPROXIMATION cccccccccscccscscscscccec
47. ONSTANTS section V5 1 INI files must also contain the following sections DRUDE SELLMEIER HERZBERGER SCHOTT POLYNOMIAL TABLE with at least one entry in for each type 11 GSolver V5 2 User guide 1 3 1 Binary Grating Example This section gives a step by step example for creating a single binary layer grating one layer with one index transition 1 Open GsolverV5 1 2 The Parameters form is the global settings home The substrate and superstrate materials may be selected here More details are found in the Dialogs chapter Select a substrate and superstrate material by clicking on the appropriate select buttons 3 Enter the grating period or lines mm wavelength and other parameters A discussion of the angles is given in the Parameters Tab chapter 4 Click on the Editor tab Shown on this tab is the graphical working area called the canvas see chapter 4 The substrate is located at 0 and below referenced to the ruler on the left and is not shown on the canvas 5 This example employs the square rectangle shape button to draw a rectangular structure If not already present use the menu item Tools Customize to add the drawing tools to the toolbar See the section on toolbars if needed Drawing Tools 6 Click on the square tool button Place the mouse cursor anywhere on the active area of the canvas and while holding down the left mouse button drag the mouse to create a rectangle on
48. ace 125 GSolver V5 2 User guide appropriate exponential function By reordering the way the fields are solved in the Gaussian case 16 2 3 1 which is solved top to bottom the stack matrix method solves the fields input to output This reordering of the equations has the added effect of guaranteeing that there are no exponentially growing fields To make this idea explicit consider the fields in one layer top bottom They are written in terms of the eigenvectors and eigenvalues as E E exp Ad C _ S E exp Ad Ep C S B E exp Ad lt E P exp Ad E a E E where the C s are to be solved eliminated by application of the boundary conditions the E s are the eigenvector components and A is the diagonal eigenvalue matrix For the stack matrix method these equations become E E exp Ad C g E exp Ad Er G la E exp Ad E C IS E E exp Ad C U and the matrix multiply results in the stack matrix for this layer E E exp Ad E exp Ad E S S ls exp Ad E E E ae fe J le The next step is to find the algorithm which combines two stack matrices or equivalently combines the fields of two layers This is illustrated below 126 GSolver V5 2 User guide topl bottom1 top2 bottom2 The two stack matrices defined by layers 1 and 2 are combined to form a new stack tha
49. al thickness The linear profile represents a partition between two regions defined by the top and bottom index of refractions GSolver V5 2 User guide 6 Listing Run Tab The Listing RUN tab form is the main interface to the solver routines The internal piecewise constant grating approximation is listed on the grid and the various entries can be modified and constrained with user defined algebraic constraints F untitled GSolver 51 _ ol x Ele Edit Format Formulas wiew Tools Grid Component About Oe are ew pz jsinovmeco Parameters Editor Listing RUN ca Run Resuts 30 5er 30 Run angles cate a el a O a A i i 1 Grating Definition Listing _ nl EJO Free Parameter ha 0 Current Increment Stop eta 0 0 1 100 hii iod 1 Update Decrement increment oes UN Ppt 3 a Io o o o o M d nn O Write Field ith 0 025 Cell List 975 o a an N gt gt En PEE ajajolajajo ololN o oly J a a in on 59 GSolver V5 2 User guide Click on the Listing RUN tab to activate the listing view The grating structure is found in column B together with the grating parameters Column A is used to label the column B entries On the right of the grid are a group of controls used for setting the free parameter Column C displays the cumulative thickness at each laye
50. al gratings and Solve handles general polarization for one and two dimensional gratings Solve3D handles all polarizations for three dimensional gratings The reason that TE and TM polarization modes have their own routines is that in each certain vector components are known to be identically 0 and therefore can be deleted from the general vector formulation SolveTE SolveTM and Solve use a so called stack matrix method to solve for the boundary conditions as well as using certain Toeplitz matrix inversion methods to stabilize the Fourier coefficient representation of the 115 GSolver V5 2 User guide permittivity and impermitivity for each grating layer Solve3D uses Gaussian elimination with partial pivoting to solve for the boundary conditions It is less numerically stable and slower than the stack methods All routines solve for the fields internal to each layer of the modulation region by solving an algebraic eigensystem Symmetry is used to reduce the order of the problem by a factor of 2 for an acceleration by a factor of 8 Some authors have shown that additional symmetry considerations exist for the special case of TE and TM at the added expense of greater problem complexity and memory requirements These alternate methods are not used in GSolver After solving for fields internal to each layer and matching all internal boundary conditions a final boundary condition matching is performed between the input and output
51. al index is then descretized into a number of layers and number of steps within each layer the steps are taken equally in the x and y directions This creates a checkerboard layer approximation Since GSolver is material oriented the various constant regions within a holographic layer are assigned the constant material property independent of wavelength The material catalog is augmented with as many constant material regions as called for When invoked GSolver first looks through the Constant material list to see if a material already exists with the required index of refraction If one is not found a new material is created and appended to the list When GSolver is exited the GSolver ini material catalog file is rewritten with the new materials 85 GSolver V5 2 User guide 10 3 Sine Tool The Sine Tool is used to create a crossed sinusoidal height boundary between the superstrate and the substrate The boundary is between the top and bottom regions These top and bottom regions do not have to have the same material properties as the super and substrates 86 GSolver V5 2 User guide 11 3D RUN Tab The 3D RUN tab is the only user interface to the full vector crossed grating solver routine laa Ele Edit Format Formulas Yiew Tools Grid Component About eee 2 woe hk ofa Parameters Editor Listing RUN GA Run Results 30 Editor 3D Run Crossed Grating Editor Start Stop Inc
52. and the fields are propagated through the sublayers using the solution of the Maxwell s equations common to each sublayer 104 GSolver V5 2 User guide The other parameter is a relative error term which is used to halt the RK adaptive step size algorithm at some minimum estimated relative error in the solution The default for this value is 0 001 It is tempting to make the relative error very small in the hope of getting more numerical precision While this is true in principle it is seldom practical Care should be exercised in setting the relative error and generally the default setting results in numerical solutions that are much better than 0 001 remember it is a maximum error estimate the actual error may be much smaller RK may also benefit from greater numerical precision 17 1 3 Bulirsch Stoer Method BS Most of the comments made about the RK method apply to the Bulirsch Stoer method The two user adjustable parameters of the RK method have the same function in the BS method This method is distinguished from the RK method by subdividing the interval into many steps propagating the function through a midpoint method and applying rational function approximations There is a large literature on both the RK and BS differential equation solver methods 17 1 4General Method Comments Limited experimentation have demonstrated that the AE method is the fastest I have not found any situations where the AE method failed while
53. angles a alpha and beta are used to define the polarization state If B 0 the illumination is linearly polarized For transverse electric TE polarization the principal E field is normal to the plane of incidence defined by k and the z axis For transverse magnetic TM polarization the principal E field is in the plane of incidence E2 GSolver V5 2 User guide a is the angular deviation of the principle E field direction away from TE towards TM a 0 for pure TE and a 90 for pure TM B determines the magnitude of the secondary E field which is perpendicular to the principal E field and k and 90 degrees out of phase in time If the principal and secondary E fields have equal magnitude the wave is circularly polarized In general 45 lt B lt 45 Labeling the principle E field as El and the secondary E field as E2 f is the angle shown in upper right in the figure 3 3 Stokes Definition The polarization state is also defined by the Stokes vector Since there is a well defined plane wave the Stokes vector has three relevant components S1 S2 S3 They are calculated as follows El cos a E2 sin a these are the magnitudes of El and E2 S1 E1 E1 E2 E2 S2 2 E1 E1 cos S3 2 E1 E1 sin These components are shown on the Parameters form as read only 3 4 Order Convention Different coordinate conventions lead to a different numbering of the diffracted orders These various schemes arise
54. as a name followed by 12 comma delimited parameters For example BK7 0 5 1 1 03961 0 231792 1 01147 0 0 0 0060007 0 0200179 103 561 0 0 gives one Sellmeier formulation for the common glass BK7 The index of refraction is calculated according to the following formula Ca A i Cas io A Cor where the c s are the various model parameters and A is wavelength in microns The internal GSolver representation of wavelength is in microns as are all quantities with length units 30 GSolver V5 2 User guide This model is purely real Since it does not estimate the imaginary part of the index of refraction k is set to 0 transparent for Sellmeier materials 2 7 5 Herzberger GSolver s Herzberger model is a 20 parameter real index of refraction model A typical INI file entry for a material of Herzberger type is shown MgO RTR 5 1 0 00309946 9 61396e 006 1 72005 0 0 00561194 0 0 0 0 0 028 0 0 0 0 1 09862e 005 where the name MgO IRTR 5 in this example is followed by the 20 comma delimited model parameters The index of refraction is calculated according to qe 4 n cA taroa 24 Cri 7 E ES e a ares A 0 0028 There is no parameter labeled c so the first list entry starts with cz This is a real index of refraction model the imaginary part of Herzberger models are set to 0 2 7 6 Schott GSolver incorporates a six parameter Schott index of refraction model A Sc
55. ation The STOP button puts a message in the queue to terminate the loop The loop will stop at the first convenient break point When the RUN is finished the Results tab is activated with results written to the Results grid as they are computed 11 GSolver V5 2 User guide 8 1 2 1 order Littrow The 1 order Littrow check box invokes a simple Littrow condition As the wavelength changes Theta is altered so that the first order Littrow condition in reflection is satisfied A Littrow constraint is generally run as a function of wavelength only 8 1 3 Write Fields to File The Write Fields to File command directs GSolver to record the computed complex E fields as an ASCII text file Clicking on this control activates a file save dialog enter the name of the file into which to record the complex E fields The data can be saved to a new or existing file If the file already exists the results will be appended to its end The Complex E fields are the field amplitudes of every order retained in the calculation at the top interface of the grating reflected fields and at the bottom interface transmitted fields In general all orders have non zero magnitude However they may not be propagating For a field in some order to propagate the related k vector z component must have a real component Grating structures can support large amplitude evanescent fields which can be visualized as standing waves These fields are essential in the sol
56. ay be set independently If the width of the canvas is resized to increase the grid sampling resolution for example the various components can be selected and stretched to the new width Recall that the canvas width is 1 grating period independent of how the viewport of the canvas is configured If the canvas is resized smaller any grid objects that now are off the canvas must be resized to the new canvas size 4 2 Canvas Grid The grid spacing is set in the Canvas Properties dialog and the grid represents the resolution of GSolver s piecewise constant approximation There is a minimum grid spacing determined by the monitor resolution together with the mapping modes settings The easy way to increase resolution is to simply set the canvas size to some larger value putting more monitor pixels at disposal On the other hand there is a point beyond which increased resolution has no benefit or effect on the outcome of the calculation This will be grating specific and depends on the relative changes in the material properties A rule of thumb is design to 1 10 The snap to grid feature may be turned on off from the Canvas Properties dialog This feature attempts to size all components so that boundaries are on grid points This is often convenient for sizing components but can be inconvenient if components have incommensurate 46 GSolver V5 2 User guide dimensions with respect to the grid spacing In this case the snap to grid
57. bsconvestwavvecssweoess 19 22 LIRA AND DROP lia 19 2 3 IMPORTING GRATING DEFITINION BY TEXT FILE cccccoscccccccccceeee 20 2 3 1 GRATING TEXT FILE FORMAT 0 cccccccccccccccccccccscscecscscecseecseeeseeseeses 20 24 IMPORTING V 4206 ois scccess ussasnacscceausesdessiner deesdseseasenatesconynctcssosadeesses 22 DiS MORSA T T aii 23 2 6 TOOLBARS AND MENUS spsssceseesiceosndscetoesasincaldeesmeadesbaacesnasontecsasoaceales 23 264 MENU BAR soria a a S A 24 PA P VN EEEE E E EEE EEEE NE eles 25 DOS DRAWING naati i E E E Seeds de 26 264 IROTATE ao ae 26 LO AY OUT A AS Se E 26 IA AS EIE a D LENE EE E E E Ea E 27 LD NUDGE A AE E E E e E ORNS 27 2 0 8 STRUCTURE 4 0 a e a EE Boats a a a a S ees 27 2 6 9 ZOOM A AS RS a 28 26 10 CANVAS ua a at N 28 2 1 INDEX OF REFRACTION sc sccscsssisvesscdcasssccsssvesecsbecssosssscebocdescsceesisceses s 28 Dilo MODELS Lt Oe Be eek Pa E o e Betas Pte ARN eS 29 D722 CONSTANT ist raid 29 XRD IDR UDE Sillas 29 ZA IS BLEMPIER tal aia acid 30 GSolver V5 2 User guide 2 THERZBERGER coimas on Seen Gs ads BN Se 31 2270 SCHOT t occas N haste asa ea 31 AT POLYNOMIAL si cioicdeo sci exies coc aid 32 ITE E ee ern Ae Ta 32 ZO COLOR MAP iiss sotana bans a a ast ches delia 33 2 8 MATERIALS EDITOR 6 siiadscdeccerstesbaccecccocensisedendediessniscdsvacdssdiccessisesiscs OD 2 9 TYPES OF SAVEDDATA ssussscccerviccovedscessexstucoagednedasedvaccenoeSaccenexsedesonce OO 2 10 KNOWN BUGS ada T 2 10 1 INVERSIO
58. can be turned off Note that the better place for fine tuning dimensions is on the piecewise constant representation of the grating structure accessible on the Listing RUN tab 4 2 1 Accelerator Keys To delete a region select it and then key shift del To copy a region select it and while holding down the control key drag the object with the mouse Objects can be dragged from one grating canvas to another for multiple concurrent GSolver objects To copy an object for pasting into another canvas or any OLE enabled application right click the object and select copy To paste right click the canvas and select paste 4 3 Tools Several tools available for drawing grating profiles are discussed in the following sections When any region is selected the boundary is augmented with handles small gray squares that can be dragged to resize the object 4 3 1 Rectangle The Rectangle tool icon is used to add a uniform layer thin film A uniform layer is a rectangle that spans the width of the canvas Or it may be used to add a binary transition region 4 3 2 Piecewise Linear poly line The poly line tool icon is a triangle A linear poly line region is defined by a starting point defined by clicking the canvas after the tool selection moving the mouse to a new point and clicking again and repeating Each click generates a boundary line from the prior click location to the current click Double clicking will complete the region
59. ccncccnnncccnccnes 129 20 1 TRACEPRO RUN EXAMPLE sssesesesesccccccecccsesesesesesessssssssnsssssssssss 129 21 REFERENCES nustanscictesceriaca dano isses strna ne esst ossei sotos tocos seda 131 GSolver V5 2 User guide 1 Introduction 1 1 Overview Introduced in 1994 GSolver is a full vector implementation of a class of algorithms known as Rigorous Coupled Wave RCW Analysis These algorithms give a numerical solution of Maxwell s equations for a periodic grating structure that lies at the boundary between two homogeneous linear isotropic infinite half spaces the substrate and the superstrate The solution is rigorous in the sense that the full set of vector Maxwell s equations are solved with only the following two simplifying assumptions 1 a piecewise linear approximation to the grating construction and 2 a truncation parameter for the Fourier series representation of the permittivity and impermitivity within each grating layer GSolver is set up to work with linear isotropic homogeneous materials Within GSolver a grating is specified by a series of thin layers Each layer consists of box shaped regions of constant indices of refraction By allowing the scale of this approximation to decrease a spatially continuous grating structure can be approximated to any desired accuracy Version 5 1 uses the same hardware key system as previous versions of GSolver and is forward compatible with the older keys 32 bit paral
60. ch are the seven index models used in GSolver the catalog entry NAME of the material which must be listed in your GSolver ini and loaded into GSolver The final entry is a flag true false which tells GSolver to update the index value if the wavelength changes or not If a block definition line does not have four entries errors will occur The sum of the block widths must total 1 0 otherwise an error condition is set and the file read aborts A typical file import would proceed as follows Open a new GSolver instance set the Parameters to some nominal values using microns then click on the Editor tab and then click from the menu File Import Text This action will initiate a file read dialog box Navigate to the text file which contains your grating definition as explained above and open it GSolver will read the file and update the Editor window 21 GSolver V5 2 User guide NOTE DO NOT CLICK the APPROXIMATION button if you do not want GSolver to approximate the layer widths and thicknesses to the Editor grid spacing Rather go directly to either Run or Listing Run and populate the grid When the text file is read in the grating is already defined as a piecewise constant structure Therefore the internal grating structure is updated automatically and there is no reason to approximate it using the Editor tool 2 4 Importing V4 20c GSolver V5 1 incorporates an entirely new user interface with new features and ex
61. decimal digit precision These numeric types are implemented in software using optimized calls to emulate a full IEEE complient operation including optimized routines for function calls exponentiation trig functions and so forth Since operations with these data types are implemented in software the corresponding run times will increase dramatically Having run simple test on a series of multiply accumulates assuming that there is no dedicated 64bit floating point multiply accumulate hardware instruction I find that Double Double multiply accumulate takes about 125 times a Double and a Quad Double takes about 1000 times as a Double These numbers are given only as very rough estimates I have not taken into account any memory fetch optimizations nor compiler optimizations The purpose for adding this option to GSolver is to provide a mechanism for increasing the bit depth for the eigensystem solution Often but not always increasing the numerical precision will stabilize the solution If on of the extended precision options is selected it is applied to whichever solution algorithm is selected Selection of the numerical precision is made from the tools menu The three choices are 64 bit Floating Point normal hardware implementation 128 bit Floating Point Double double precision 256 bit Floating Point Quad double precision 108 GSolver V5 2 User guide The selection is valid from the Run tab 3D Run tab and the Listi
62. del entry property to each material region V4 20 objects are generally of constant value so new 22 GSolver V5 2 User guide material entries are created as needed for these types If a V4 20c binary file contains saved catalog materials these may not be converted correctly This situation can also occasionally cause a program crash see the known bugs section which is being addressed 2 5 Forms GSolver is a form driven application The various data fields that define a grating and the intended calculations are arranged by class on different forms The forms are labeled Parameters Editor Listing RUN GA Genetic Algorithm Run Results 3D Editor 3D Run and Angles Calculation Separate chapters are devoted to the descriptions of each Forms are activated by clicking on tabs and the data fields in each form provide interfaces to the internal data structure of which there is one for each top level GSolver window Each data document represents one grating structure with its related global and calculation run parameters 2 6 Toolbars and Menus Several toolbars are provided for access to various GSolver functions Most of the tools relate to the graphical grating design interface The toolbars can be customized by adding and removing buttons and grouping them as desired These toolbar buttons are described below The Tool Customization dialog is activated by clicking on the menu item Tools Customize Use the toolbar Custo
63. depend on the type of monitor used The other screen modes are as follows MM_HIENGLISH one logical unit is 0 001 inch MM_HIMETIRIC one logical unit is 0 01mm MM_ISOTROPIC one logical unit is 1 pixel in both x and y MM_LOENGLISH one logical unit is 0 01 inch MM_TEXT one logical unit is one pixel MM_TWIPS one logical unit is 1 1440 inch While it is possible to change the mapping mode for printing purposes it is generally recommended that the mapping mode not be altered On the Measurements and Size dialog together with the mapping mode are entries to determine the relation between logical and physical extents for viewing and printing The canvas Drawing scale should usually be set to the drawing units default is centimeters 45 GSolver V5 2 User guide The canvas area can also be modified from the Canvas Properties dialog This dialog simply assumes that the canvas width and height are some number times the grating period The grating period is taken as arbitrary for viewing purposes The Default canvas size shown on the Size and Units tab of the Measurements and Size dialog is 10 cm by 10 cm Thus the default viewing scale is 10 cm 1 grating period If the canvas width is set to 2 2x on the Canvas Properties dialog while all other settings remain at default values the canvas will be drawn with a 20 cm width which would then represent one grating period The canvas width and height scales m
64. dragged past an adjoining boundary To edit the material simply double click the region to bring up the material selection dialog There are two layer delete options Del current removes the current layer from the grating and Del All removes all grating layers The layer thickness is also indicated and may be edited directly absolute units There are two tools to insert grating layers To insert a single layer use the N_x button which inserts a layer with N transitions To insert a grating profile such as triangle blaze or some sort of linear profile use the N ties button The ties refers to the number of piecewise linear parts there are to the profile 1 lt ties lt 3 57 GSolver V5 2 User guide 5 1 N ties Profile Tool xi Frandes z Levels 25 Total thickness 1 Ones Top index AL Bottom index Relative handle location X 0 552083 Y 0 240979 Pt 1 r Blaze angle calculator Enter angle deg 15 b2 Period p AN b1 b2 Period b1 b2 The profile tool is shown in the figure above Essentially a linear profile may be defined up to three linear sections by clicking and dragging the tie points The relative location of the tie points are shown in the location region Note that the width is relative to the Period and the vertical dimension is relative to the Total thickness entry That is the the maximum vertical position drawn in the figure is set to Tot
65. e advantage of the various known methods for stabilizing and accelerating the coupled wave problem solution The 3D solution is retained in GSolver primarily for backward compatibility For those aware of the limitations imposed by the 64 bit floating point hardware on the numerical algorithms the algorithm allows investigation of a fairly large crossed grating problem space Problems that need only a few retained orders such as all orders being evanescent are usually readily solved It is generally recommended that there be no more than a few real propagating orders and that materials with large phase shifts such as metals be avoided The convergence behavior of all 3D grating structures should be investigated since the general approach to convergence often has an oscillatory component 10 2 Defining a 3D Grating GSolver handles all gratings as piecewise linear approximations For a 3D grating this approximation consists of a unit cell with periodic boundaries This cell has a width of one grating X period and a length of one grating Y period The thickness is arbitrary and cells may be stacked as desired Each unit cell may be divided into any desired number of checkerboard regions Each region is then assigned a material property which gives it an index of refraction The basic tool for defining a 3D unit cell is the NxM tool This tool creates an NxM checkerboard with two materials For any layer any region of the unit ce
66. e clipboard and pasted into other applications control c control v 36 GSolver V5 2 User guide 2 10 Known Bugs 2 10 1 Inversion of OLE Coordinates The Editor Graphics are created in a normal Cartesian coordinate system with the origin in the lower left hand corner of the canvas viewport The conventional coordinate system for windows objects has the origin in the upper left hand corner of the window with the y axis increasing down Thus when graphical objects are dragged into other OLE objects including print objects the figures are upside down Since this issue touches a lot of code it is difficult to change but is being addressed 2 10 2 V4 20c Data Import Crash There is an unfortunate bug that can creep into V4 20c binary data files gs4 owing to the existence of two 4 20 binary schemas In V4 20 an additional data field was created to attempt saving the V4 20 links list as well as any assigned material properties with the grating data This required adding data fields to the binary schema with a flag to identify which type of file was being loaded The logic that was used is not sufficiently robust to correctly align the binary data in every case This causes an unrecoverable read error when a file containing catalog data is read by a different GSolver 4 20 version This problem is being addressed The current work around is to eliminate material catalog links in 4 20 data before importing it to V5 1 This problem can al
67. eceeeeee OL 4 4 1 GRATING REPRESENTATIONS occococococononononononononnnnnononononononononononononos 52 4 5 HOLOGRAPHIC TOOL Vp SH eocosoosonconccrsonoccorsonacesosrancnncorannacconanacoss OF 5 1 N TIES PROFILE TOOL sccccsseseessseseseesensostssncssosssosonsscossceaessssserseeneore OO GSolver V5 2 User guide 6 LISTING RUN TAB sisssiccsssvsdessscvasncvsvccenssuncscaccssnsdeisecsavecasvessotesesstse 59 6 1 1 MAKE GRID CURREN Enine saneren rE S AT ia 60 6 2 PARAMETER CONTROL sseceesseceesssseceesscceesseceessccecessseceesseceesseceessee 60 6 2 1 USING LISTING RUN TO FIND THE DIFFRACTION EFFICIENCY AS A FUNCTION OF INCIDENT ANGLE THETA EXAMPLE cccceesessecececeeeeeenees 61 6 2 2 ABORT BUTTON iii a vo Recs esd on Bod fs a Biv e anes 61 6 2 3 EXAMPLE OF VARYING THE THICKNESS OF THE GRATING 645 62 6 2 4 EXAMPLE OF LITTROW CONSTRAINT ccccccccsseessssceeescceeeessnseaeees 62 637 CELLS TE a ias 63 6 4 FORMULA ENGINE eisscsssstccsccstccsssascocccecoesesunesescesocostecseessesccsosseseeuess 63 OAT ANTAS AI ET ava ee Res 64 6 4 1 1 EXPRESSIONS viii ddr lin 64 6 4 1 2 Constraint EXpressions it diiaja 66 7 GENETIC ALGORITHM GA TAB 7 1 OVERVIEW OF DIFFERENTIAL EVOLUTION e eesseseeesssccecssceceesseceesseo 69 7 2 GUIDING PRINCIPLES ssa scsscdscessccesvadscecsbsnnedssandsvecoseueabedescoducevesseersncs 69 7 3 SETTING GA OPTIONS scscssscecennscvovsiasccsssnevacondasoncesesvaddonsdsdcdoooessedesses 70 7 4 APPLY
68. ect a small distance in the indicated direction but do not allow moving past canvas boundaries 2 6 8 Structure structure A Ta a Ch Ea The Structure command buttons can be used to alter the z order which object is on top of overlapping objects When creating the piecewise constant approximation of the grating the top most object is used at each grid sample point to define which material property to use 27 GSolver V5 2 User guide 2 6 9 Zoom Zoom QA The zoom and pan commands affect only the canvas view they have no effect on the internal object dimensions which are sized to units of the grating period 2 6 10 Canvas The Canvas command buttons include the undo and redo commands also accessible with control z and control y as well as the grid and canvas property dialogs 2 7 Index of Refraction Each enclosed region in the model is assigned a material property from which an index of refraction is calculated The default is for each material to use the substrate property as assigned on the Parameters form GSolver comes with a number of predefined material properties in each model class This list is not exhaustive and it is expected that materials will be added as needed by the user This can be done directly editing the Gsolver ini file with a text editor such as Notepad Or it can be done from within GSolver using the Material Editor NOTE It is recommended that new materials in any model class be added
69. ed axis Canvas Title 2101 x File Copy To Clipboard Yiew Tools Help 2 Ea amp be SR 2 BE SP ER amp 4 00 aa mm ka 0 50 ls 30 70 5 3 060 oR E 0 50 x SumR Boao iit 5 020 0 20 01 ss ooh semi 00 00 2200 33 00 no 5500 600 77 00 88 00 Axis Axis Labels 2 Data Legend In addition to the region specific menus a number of general properties are accessible for modification from the chart tool bar and menu items The chart object includes a data editor Data is copied from the source when the chart is created Changing any data items by using the chart data editor has no effect on the original data source 101 GSolver V5 2 User guide 17 Algorithm Selection GSolver has three distinct algorithms for solving the inter layer set of coupled differential equations determined by Maxwell s equations and the Fourier expansion of the permitivity and impermitivity of the layer The three solution methods are the Algebraic Eigensystem solution traditional method for GSolver a 5 order Runge Kutta method and a Bulirsch Stoer method with Richardson extrapolation and rational approximation The algorithm of choice is selected from the Tools menu Prunctied csowest toa File Edit Format Formulas View Tools Grid Component About CA Sa ae ese Nees TracePro Parameters eto tistingeRUN Calculator Angles
70. ed into the general grating storage array after which a RUN will use the results of the GA run Previously the only way to update the internal grating array was directly from the 2D Editor Approximation function 7 1 Overview of Differential Evolution DE parameterizes optimization problems A specific set of parameters a vector is associated with a merit function whose extremum is to be found A set of parameter vectors is a population The algorithm will alter various population member entries with other entries as well as creating new entries from random distributions Population members with good merit functions are chosen more often than others to serve as a source of parameters New vectors are made up with these good parameters as well as new random members and the entire set is iterated until some stopping criterion is reached See the web page above for tutorial information 7 2 Guiding Principles Applying DE to the grating problem creates some difficulties For example most parameterizations of the grating problem require careful 69 GSolver V5 2 User guide attention to physical constraints to produce meaningful solutions The piecewise constant approximation suggests a direct parameterization based on the geometric constraints cell thickness and width However the cells that make up the grating are constrained in the following ways all cells in a layer must have the same thickness the sum of the widths o
71. eneral three dimensional case the coefficients are coupled between x and y The Fourier coefficients of the permittivity and impermitivity are directly computed and used The inverses of these two matrices could also be calculated and similar use could be made of them Inter layer Solution Inhomogeneous Plane Wave Expansion 120 GSolver V5 2 User guide 19 2 1 3 Formulation of Eigensystem Solution Applying the curl equations to the assumed form of the fields within the n layer and eliminating the z terms four coupled equations remain dS K iij 2 dz Ik S vij 7 IK v I k A Eliza a Koa soa pq Y 1k S IkU 2 y E A at dz IK yj t at k Sen xpq ypa ypa p pg dU xij xy acy TY ES pq 1 T i Say KS pg dU a A k u Ela ky ES gq HIS ay K 45 5 where p i g and q j h This represents a set of coupled first order differential equations that are solved by standard eigenvector techniques Sx Sy Ux and U are functions of z only and their functional values are needed at the top and bottom of each layer They turn out to be the sums over terms of the eigenvector times exp 4 z where A is the appropriate eigenvalue and z is either 0 top of layer or tn the n layer thickness All solvers SolveTE SolveTM Solve Solve3D use the same eigensystem solver The coefficient matrix is formed the matrix is balanced the Hessian is found and the system is solved with LR decomposit
72. enter as two dimensional arrays By pairing ah with ehl and eh with ahl convergence of the TM and general polarization is greatly accelerated Indeed for one and two dimensional gratings only a few evanescent orders are needed to converge to better than the 1 level for dielectrics For larger phase shift materials such as metals three to eight evanescent orders are generally sufficient Some experimentation with the number of orders retained is recommended to demonstrate convergence The diffraction efficiency may not converge monotonically but could have an exponentially damped sinusoid component 19 2 2 Intra layer Solutions Boundary Conditions The fields in each region are readily solved To complete the problem the tangential electric and magnetic fields must be made continuous across each interface The general problem can be set up as a series of matrix manipulations that include finding at least one inverse per boundary The internal layers required to track only the x and y components of the E and H fields and are therefore 4s x 4s dimensional where s 2 orders 1 The superstrate and substrate require calculation of the x y and z components of E and H and are 6s x 6s dimensional The Stack matrix methods of solving the boundary conditions are significantly different than the Gaussian elimination method Both are described below The first step in either case is to reorder the equations so that the final matrix inversion co
73. entifies a color with an index of refraction entry is identical for both the real and the imaginary pars Following is a description of the algorithm for the real part n 91 GSolver V5 2 User guide Refractive Index Color Map cancel Real part Color Map Imaginary part Color Map 1 50000 0 00000 250 250 250 1 00000 1 00000 50 250 501 50000 250 250 50 2 00000 50 50 250 2 50000 150 250 250 15 00000 250 50 50 4 00000 20 20 50 250000 2 00000 Controls R G B fi 50 250 50 Replace Insert Delete coron Given a value n find the two entries in the appropriate list that bracket it such that n Snn where n and nz are consecutive members of the list Then draw a line in RGB space from the n RGB coordinate to the nz RGB coordinate and find the RGB coordinate for the linear interpolant given by n between n and nz The resulting color is used for the n value The Color Map may be altered as needed break points may be added removed and changed The COLOR button may be used to find a specific COLOR RGB coordinate Use the Replace Insert and Delete buttons together with the Control entries n R G B to modify the list The color lookup list is stored with the grating in the gs5 file When GSolver is started it creates a new color lookup table based on default values To use a custom lookup table simply save a grating file with the desired lookup table Then after starting
74. er However the more complex the grating structure the more Fourier coefficients are needed to describe it sufficiently Experience has shown that generally the number of terms needed is a strong function of the period to wavelength ratio and not so much as the wavelength to grating feature ratio All classical diffraction gratings fall into the 2 dimensional category Most of these gratings may be defined as a surface height relief of the substrate Hes EA AAA AAA AAA gt Period The figure illustrates an example of a lamellar approximation to a sawtooth grating profile Only a single period needs to be defined Simple surface relief gratings such as blaze sinusoidal triangular and binary are constructed by a thin layer stack approximation Each layer consists of some number of index transitions 1 for the present example where the index transition point is placed at the appropriate location so that the stack approximates the surface relief desired Any degree of accuracy may be imposed by increasing the number of layers However once the individual layer thickness becomes significantly smaller than a wavelength less than a few percent little change is noted in the solution More complicated grating structures might require several index transitions per layer 113 GSolver V5 2 User guide Any complex number may be entered as an index of refraction The GSolver code enforces correct sign conventions so that no material ex
75. f the cells in a row must equal 1 since the unit canvas width is one period In addition to the grating geometry parameters the global parameters may also be used These include grating period wavelength and the angles that define the illumination plane wave The actual material properties in any particular cell are problematic Since GSolver ties all indices of refraction to an actual material in a catalog it is difficult to allow arbitrary changes in these indices However a limited solution is to create specialized materials that cover the range of the index of refraction the search space at known specific wavelengths Then using those materials the wavelength can be allowed to vary over the discrete space of wavelengths so that different index of refractions are selected in a controlled manner 7 3 Setting GA Options The DE options are set in a dialog shown below which is invoked by clicking the Options button cell F17 on the default GA grid 70 GSolver V5 2 User guide Genetic Algorithm Settings Differential Evolution ES Population nominally 10x total number of parameters Clncel Weight 07 nominally 0 8 D lt F lt 1 larger values gt faster convergence Cross over 0 3 nominally 0 9 0 lt CR lt 1 Max Iterations 25 Stopping criterial Number of Real Parameters E Select differential solution mode Goal settings for selected orders Order Goal Weight RandtBin Fl Enter DE Goals fo R y fo fi
76. h results from the Approximate function Two other grating representations are also created for the Listing RUN tab and the GA tab In both cases the Populate function copies the results of the Approximate function representation to the respective grids When the RUN Listing RUN tab and RUN GA GA tab buttons are pressed the solver routines are invoked with the grating definition given on their respective grid To copy any modifcations from the Listing RUN grid or the GA grid to the grating definition generated by the Approximate function simple click the Copy Update button on the grid If you make modifications to the grating on say the Listing RUN grid and then want to use those modification to do a GA run you will need to first Copy Update the Listing RUN grid to update the Approximate data structure and then Populate the GA grid 4 5 Holographic Tool Vp Sy New to Version 5 1 22 is an additional holographic tool VH is used to generate volume holograms with fringes close to the normal direction to the grating SH is used to generate a fringe pattern parallel to the grating a Bragg reflector SH creates a series a thin film slabs in the piecewise approximation whose index of refraction varies periodically 54 GSolver V5 2 User guide The Vy tool button Vy assumes that some material base index has some density modulation The modulation is defined by a sinusoidal variation The index of refraction is given by n
77. hapter 2 6 2 Main 25 GSolver V5 2 User guide CEC x Deol 50 89 The default Main command button bar contains the following commands New start a new GSolver window Load load a saved gs5 grating structure Save a grating structure Cut copy and paste apply to graphical as well as to data items Print 2 6 3 Drawing The default Drawing command button bar contains commands to generate various graphical items Included on this command bar are additional buttons to instantiate the Materials Editor and the Color Map dialogs Other drawing commands line polyline text fields bitmaps and ports are included for convenient grating design annotation and markups and are not otherwise used for actual grating structures 2 6 4 Rotate Rotate E Gatas These commands are used to rotate graphical objects The default canvas properties include a snap to grid allows only discrete moves based on grid spacing and angle snap discrete angles based on grid spacing These snap properties may be toggled on or off 2 6 5 Layout Layout H ang 26 GSolver V5 2 User guide The Layout commands are used to size multiple selected drawing objects to each other Use the shift key in concert with mouse button to select multiple objects 2 6 6 Align Tel ess The Align commands are used to align multiple selected drawing objects with each other 2 6 7 Nudge Nudge ES The Nudge commands move the selected obj
78. he number of line segments that define the boundary may be set to any reasonable number The default is 15 Enter the number of line segments in grid location D2 There are two columns of coefficients labeled A and B in the equation below with A entries in grid locations C4 13 and B entries in grid locations D4 13 The general sinusoid profile is defined by the following formula where AO is in grid location C3 Al and B1 are in grid locations C4 and DA and so forth y A YA cos x B sin x i 1 N Any desired number of coefficients may be used The table may be extended simply by entering nonzero values in grid locations Cn where n gt 13 and so forth General poly line This form provides a method for constructing an arbitrary poly line approximation to a general profile Enter the coordinates of the poly line end points as an X Y pair in grid locations An Bn starting with n 2 All x coordinate entries must be in the range O lt x lt 1 where 1 represents one grating period unit y coordinate values should be in the range y gt 0 y is absolute 0 lt x lt 1 x is scaled by Period General cubic spline This tool is very similar in operation to the General poly line The difference is that every 3 consecutive points are taken as a cubic spline definition Thus the number of points entered in the table should be a multiple of 3 These two tools allow for precise point location for both a poly line and spline profile def
79. hibits gain The n and k values are entered as positive real numbers with n strictly greater than 0 The solver makes k k k which indicates absorption with the sign convention used Use of the lamellar approximation permits arbitrarily complex grating structures to be analyzed 19 1 4 3 Dimensional Gratings Three dimensional gratings are crossed gratings periodic diffraction structures etched along both the x and y dimensions GSolver uses the same 3D solver that was used in versions 1 0 through 4 0 This means that the TM convergence is not accelerated The reason is that the fundamental theorem of algebra does not apply to polynomials in more than one variable However there remains an expectation that in time the problem can be significantly mitigated using the ideas that have satisfactorily solved the two dimensional convergence problems Some recent progress in this area has been made if the grating as appropriate symmetries However since these ideas only apply to a certain class of gratings they cannot be used for the general case As an example suppose that a dimpled surface is to be analyzed The following figure illustrates how an elliptical mesa in a unit cell of the crossed periodic structure might be approximated Y Period X Period 114 GSolver V5 2 User guide In the figure note that the x and y periods are independent and that the number and placement of the index transition regions are also
80. hott material entry example is shown below BK7 2 27189 0 0101081 0 0105925 0 00020817 7 64725e 006 4 24099e 007 This is the Schott model for the glass BK7 The index of refraction for the Schott model is calculated according to the following formula 1 2 A E AS yo as n A 31 GSolver V5 2 User guide where A is in microns and the six parameters are labeled c2 through c7 The Schott material model is real so materials of type Schott return k 0 2 7 7 Polynomial The Polynomial model allows for tenth order polynomials to define both the real and imaginary parts of an index of refraction This requires 20 real parameters to define a material model of type Polynomial The basic INI gives a few hypothetical materials For example type2 1 5 0 2 0 0 0 0 0 0 0 O 0 1 0 095 0 1 0 O 0 0 0 0 0 defines a polynomial mode of name type2 The 20 comma delimited parameters are used to calculate real and imaginary indices of refraction according to the following formulas 9 CA 0 9 1 Caer i 0 where the absolute value signs assure that n and k are both non negative 2 7 8 Table The Table model is the most general material model It consists of a material name followed by a list of entries Each line in the list consists of three numbers wavelength n and k The wavelength is in microns n ke If a wavelength evaluation is done at a wavelength that is not in
81. ined on the Parameters tab The vertical dimensions are absolute However it is often convenient to construct a grating with a predetermined number of layers with a certain total thickness The Vertical Scale Facter multiples the vertical scale scale factor gt 0 by the desired quantity In short the canvas vertical scale is relative to the Vertical Scale Factor The Editor tools and operating principles are discussed below 44 GSolver V5 2 User guide 4 1 Coordinate System The canvas width is equal to 1 grating period for all unit settings ie 10 units on the ruler The canvas origin is the lower left hand corner The two rulers that span the canvas on the left and top represent units relative to the canvas view The substrate lies in the region lt 0 on the vertical scale and is not accessible from the canvas The substrate and superstrate material properties are assigned on the Parameters form All drawing must be done on the canvas There are two dialogs that hold the definitions of how the canvas is displayed They are the Canvas properties and Measurements and size dialogs Although the canvas width always maps to 1 grating period this unit length can also be mapped to some number of pixels on the screen This is done in the following manner GSolver sets the default coordinate mapping style to MM_LOMETRIC which is interpreted as one logical unit equals 0 1 mm on the monitor The actual dimensions
82. inition 49 GSolver V5 2 User guide 4 3 5 Conformal Coats Conformal coats are created by selecting the profile group or object copying it and shifting the copy vertically The copy properties are then set and the new object is sent to the bottom of the z stack or the original object is brought to the front This procedure is illustrated with the following example Conformal coat on a blaze profile 1 2 3 4 Start a new canvas by clicking on File gt New Click on the custom tool icon select the blaze profile tool default and click OK This puts a blaze profile on the canvas Right click the blaze profile object and set the material property While holding down the control key click on the blaze profile object and drag it up towards the top of the canvas This makes a copy of the profile object so there are now two on the canvas Right click on the new copy and set the material properties to the coating material Move the copy so that the peak of the copy is a few grid units above the original most of the copy will overlap the original Right click on the new copy again and then click on the Order option and then Send to back You may want to turn off the snap to grid property of the canvas To fill in the small areas at the base of the coating use the rectangle tool to create two small rectangles to fill in the area remaining Use the order property to put the new objects behind the original The gra
83. ion The fields are then calculated at the top and bottom of the layer z 0 and z d the thickness of the n layer The E fields are unnormalized and may be multiplied by any quantity without affecting the computation of the resultant diffraction efficiencies This fact is used to normalize the internal fields so that no exponential is ever taken with a positive real exponentially growing argument Without this precaution there would be exponentially growing solutions that cause instabilities for deep gratings The full coefficient matrix system that needs to be solved is of order s where s 2m 1 and m is the number of orders retained GSolver uses a 121 GSolver V5 2 User guide symmetric order calculation so that the calculated orders range from m to m This enhances convergence by producing a balanced Fourier sum Examination of the coefficient matrix reveals certain symmetries that permit a reduced system to be solved and the remaining eigenvectors and eigenvalues are related with simple algebraic expressions to reduce the system solutions This is an important step as the solution of the eigensystem in each layer is a principal component of the solution time budget the other component being the boundary condition solver 19 2 1 4 Eigensystem Order Reduction GSolver solves a reduced eigensystem due to the following structure E le ola al Ba 9 Q Q Q Q 0 A where the B matrix is the coefficient
84. is given in the graphing chapter 81 GSolver V5 2 User guide 10 3D Editor Tab Some crossed grating 3D structures can be solved with GSolver The 3D Editor is the interface for defining these structures and is essentially the same as in prior versions of GSolver F untitled GSolver 50 jo x UO IMINO NO _ Ele dt Format Formulas ew Tools Grid Component About Ree ee Woe hols Parameters Edtor Listng RUN Ga Run Resuts 3D Editor 30 Fun Layer thickness 0 Delete Current layer p Insert 3D structure Physical Memory 536 02 Mt 7 43 PM NOTE with file version 5 1 1 1 the 3D Editor Tab must have units set to Microns And all entries with units are interpreted in Microns 82 GSolver V5 2 User guide 10 1 Limits on Solving 3D Structures The coupled wave solution is based on solving the truncated infinite set of Maxwell s equations by the well known algebraic eigensystem method This implies that for each layer of the grating regions a full complex eigensystem is solved followed by a boundary conditions solver This process is iterated through each layer from the substrate to the superstrate The behavior and solvability of general complex eigensystems has been thoroughly studied for decades The size of the eigensystem including auxiliary memory arrays for holding intermediate results using 64bit floating point structures is roughly estimated as bytes 1760 2 orders
85. it to the model list Replace the current material selection Delete to remove the current selection For the Table Model materials the second set of buttons are used to edit the list and the first set of buttons are used to update the Table material object in the Table catalog The Chart provides a graph of the currently selected material and any changes The graph has a number of interactive properties including the ability to drag data points with the mouse see the chapter on graphing 2 9 Types of Saved Data There are three types of data files that are created saved and loaded within GSolver First there is the grating definition file with all of the related data structures and parameters This is a binary file of type gs5 The second type of data files contains the saved contents of the various grids Listing Run GA and Results The grid contents are saved as ASCII and can be viewed and manipulated by any text editor The contents of the grids can be saved and loaded allowing for archiving various data runs based on a single grating definition They can also be loaded into other programs for further analysis The third type of file supports saving the graphs The graphs are generated with the SoftwareFX client server graphics interface including the Microsoft GDI libraries for optimal display device interaction These graphs can be saved in a variety of formats cfx bmp and emf All graphic images can be copied to th
86. l the current version of GSolver is closed However the internal materials tables that are created by reading the GSolver ini file when GSolver starts are modified and any changes are available for immediate application x int wzd save coy Table Model 7 ENEE 0 211964 1 862 2 776 0 213807 1 83963 2 75288 0 215683 1 793 2 757 0 217591 1 79319 2 75488 0 219534 1 806 2 755 0 221512 1 82019 2 75475 DE BR Ne fF DE ma 1 0 223526 1 838 2 754 Wavelength microns 0 225577 1 85831 2 75175 sl y 0 227666 1 88 2 748 Lambda 1 fi Regraph Lambda 2 0 4 Edit selected table line nontable model entry 24 7944 15 46 37 682 Selected Table listing gt Ma Insert Replace Delete Insert Replace Delete There are two sets of edit buttons on the bottom of the dialog Insert Replace and Delete The first set applies to all materials The second apply to the table entries themselves in a line by line fashion The first step in editing a material is to select a model from the drop down list box the next is to select a specific material The selected material 35 GSolver V5 2 User guide properties are then loaded in the large list box for Table Models or in the Edit box for the other materials Make any modifications to the material parameters in the edit box and select the edit operation desired Insert to add
87. la D5 100 To check the result increment or decrement the parameter in D5 or simply have the Grid show all the cell formula entries using menu item Formulas gt Formula Expressions 6 Enter 0 in D5 10 in D6 and 200 in D7 7 Click RUN The total grating thickness is varied from 0 to 2 6 2 4 Example of Littrow constraint The Littrow condition constrains the incident angle Theta by the wavelength to period ratio One way to implement this constraint is Theta asin lambda 2 Period Additional checking should be done to ensure that the argument of the asin is correct and that IThetal lt 90 The example below shows one way to implement a Littrow condition constraint using the wavelength as the free parameter fixing the Period If this example is worked following the previous example clear the layer thickness formulas by using the dialog tool to enter the formula 0 1 in all the layer cells 1 Populate the grid if needed 2 In cell B6 the wavelength parameter enter D5 100 1 62 GSolver V5 2 User guide 3 In cell B2 Theta cell enter degrees asin if abs B6 2 B7 lt 1 B6 2 B7 0 4 Enter O in DS 1 in D6 and 99 in D7 This will change the wavelength from 1 to 1 99 microns Click RUN 6 Another way to constrain this formula is as follows degrees asin if abs B6 2 B7 lt 1 B6 2 B7 0 gt 0 amp amp lt 90 2 Note there are a variety of ways to enter this constraint see section 5 2 1 2 The appe
88. lel port and USB type keys In general the GSolver executable is static linked This means that it is a stand alone application and does not rely on a host of Microsoft DLLs However the basic graphics charting engine requires the ChartFXO clientserver core dll as well as the GDI library which is a native component for most Microsoft OS These additional libraries are installed in the local GSolver directory install directory support to minimize possible conflicts with the host system and other applications GSolver uses the system registry to store the user tool bar and menu selections basic form layouts and working file names The materials catalog is called GSolver ini The ini file type is a hold over from earlier versions of GSolver GSolver V5 2 User guide 1 1 1 New V5 1 Features Version 5 1 represents a major rework of previous versions of GSolver V4 20c is the prior version Many features have been added many others expanded Following is a list of the principal differences between V5 1 and previous versions Graphical Grating Editor Automatic piecewise approximation construction Greatly expanded genetic algorithm for automatic design General algebraic constraints and equation editor Improved graphing Object linking and embedding for interfacing to other programs with drag and drop capability Modified interface with independent floating GSolver windows The materials file Gsolver
89. ll can be altered by simply clicking on it A new material may then be assigned to that region by clicking on the Material button and selecting a new material The boundary lines between regions of a cell can be moved by dragging with the mouse or altered by entering new values in the X and Y text boxes at the top of the form 84 GSolver V5 2 User guide The 3D layer structure is numbered starting with the first layer which lies on the substrate and then increasing for each layer above The Holographic tool and the Sine tool automate two popular 3D grating profiles The basic 3D grating parameters are entered on the Parameters Tab as usual the additional Y period is entered on the 3DEditor tab 10 2 1 Holographic Tool A holographic grating may require 1 or many layers depending on whether the modulation of the photoresist has density variation in the z direction or simply in the x and y directions The tool handles either situation The tool assumes that some material base index has some density modulation The modulation is defined by a sinusoidal variation of up to 10 order The index of refraction is given by n x y z n ny gt _A sin jK x sz tan where n is the base index of refraction and nz is the modulation index K is the in plane spatial k vector 27 A 27 Ay and s 27 thickness is the so called slant angle and causes modulation in the z direction When y 0 only one layer is needed The gener
90. meters are checked the first column will run through its values and then repeat as often as required by the loop If the Listing RUN tool is used to generate the results then the first column represents the value of the parameter in cell D5 see the Listing RUN chapter 9 1 Diffraction Efficiency The diffraction efficiency is defined as follows DE E conj E E conj E E conj E re k py Keine where subscript k diffraction order kzrx is the z component of the k diffraction order as defined by the grating equation and R is reflection R gets changed to T for the transmitted orders Since all orders are represented as plane waves if the z component of the diffracted k vector is O or pure imaginary then DE is identically O for that order 9 2 Phases Phase for each order is defined as follows given the complex E field for the n order equal E first calculate sz sign re E xim E the sign of the z component of the usual vector cross product Next compute re E im E HSZ X arccos dd E im E 80 GSolver V5 2 User guide on the range n lt p lt n The sign of the operation is used for the transmitted orders and the sign for the reflected orders This defines the complex angle between the real and imaginary components of the complex E field vector The dot in the above formula is the usual vector dot product No attempt has been made to do phase unwrapping
91. mize dialog to turn on and off any toolbar and use the Command page of the toolbar Customize dialog to add or remove buttons from any toolbar Command buttons may be dragged from one toolbar to another To remove a button from a toolbar simply drag it from the toolbar to the Command button palette Toolbars are docking enabled 23 GSolver V5 2 User guide 2 6 1 Menu bar OE File Edit Format Formulas View Tools Grid Component About The File menu item includes the commands for saving and loading saved gs5 GSolver V5 1 grating files It also hosts the printing commands The print commands will be active for any form that supports printing Use the Import GS4 command to attempt to import a Version 4 20 grating file This command is active for the Editor tab form To import a gs4 file start from a new GSolverV50 window Click the Editor tab and then click the import command on the File menu GSolver V5 1 will import the file assigning constant materials for all of the grating regions If a required constant material is not found a new constant material is created There may be problems importing gs4 gratings that have non constant materials see the known bugs section The Edit menu command list includes numerous actions that apply to the grid data structures in the Listing RUN form the GA form and the Results form The bottom section of the Edit menu contains a group of commands that apply to the graphical Editor form Although mo
92. n or bottom up ending up with the sets of equations in R and T This final system in R and T are then solved The internal fields are iteratively solved reusing the same matrix work storage area for each layer in turn Thus the fields are first calculated in the first layer and numerical values are found for all field components at the top and bottom of the layer These fields are solutions to Maxwell s equations within the layer They can be multiplied by any arbitrary constant which allows scaling such that numerical overflow is avoided The fields are then solved for in the next layer at the top and bottom The internal interface boundary condition is then solved using Gaussian elimination to remove the internal interface leaving field solutions at the top of the first layer and the bottom of the second layer This proceeds through the grating stack ending with field solutions at the superstrate interface top and substrate interface bottom These fields are then matched using the remaining boundary conditions and divergence free conditions to solve for the R s and T s in terms of the w s and k s 19 2 2 2 Stack Matrix Methods Boundary condition solving based on the stack matrix method is illustrated in the following figure input fields output fields Each interface has fields approaching it and leaving it the substrate has only outgoing fields The input and output fields are all multiplied by an layer interf
93. n of the arguments HMEAN The harmonic mean of the arguments LARGE R N The N largest value in range R MAX The maximum of the arguments 95 GSolver V5 2 User guide MEDIAN The median middle value of the range R1 MIN The minimum the arguments MODE The mode or most frequently occurring value MSQ The mean of the squares of the arguments PERCENTILE R N The value from the range R that is at the N percentile in R PERCENTRANK R N The percentile rank of the number N among the values in range R PERMUT S T The number of T objects that can be chosen from the set S where order is significant PTTEST R1 R2 The significance level of the two sided T test for the paired samples contained in ranges R1 and R2 QUARTILE R Q The quartile Q of the data in range R RANK E R O The rank of a numeric argument E in the range R SSQ The sum of squares of the arguments RMS The root of the mean of squares of the arguments SMALL R N The N smallest number in range R SSE The sum squared error of the arguments STD The population standard deviation N weighting the arguments STDS The sample standard deviation N 1 weighting SUM The sum of the arguments T N T The integral of Student s T distribution with N degrees of freedom from o to T TTEST R X The significance level of the two sided single population T test for the population samples contained
94. ndix contains a listing of all the functions available in the grid formula engine as well as formula syntax 6 3 Cell List Cell F15 contains the label Cell List Any cell designation such as B2 C12 G23 etc that is entered in the column below F15 will be included on the Results List For example assume that you want to include a listing of the contents of cell B2 theta which may be the results of a formula calculation on the Results listing Simply enter B2 in column F below the Cell List label During a Run cycle GSolver scans down column F starting at row 16 for a list of cell designations Then for each parameter update it records those current cell contents in the Results listing This provides a way to include any grating definition results in the Results listing for additional analysis 6 4 Formula Engine Any cell entry that begins with an equals sign is interpreted as a formula Each formula is evaluated immediately and the results are shown in the same cell To expose the formula scripts of a cell select it and use the menu item Formulas Formula Expression 63 GSolver V5 2 User guide Formulas are limited to 512 characters Formula recursion is not supported That is a formula cannot depend either directly or indirectly on its own output value All calculations are done in double precision Calculations with logical operators amp amp Il consider a non zero value as TRUE and a
95. ng Run and GA tabs when an item in the respective grids are selected The numerical precision selection applies regardless of which solution algorithm is selected see Chapter 16 Generally the Algebraic Eigenvalue method should be used It solves the system of differential equations simultaneously 18 1 1 Example Calculation This example illustrates the value of increased precision The following Blaze grating in conical mount with elliptic polarization is used Start GSolver From the Parameters tab Change the substrate to Table model and AL Change the Lines mm to 1200 Change the wavelength to 0 5 microns Change THETA to 17 5 degrees Change PHI to 10 degrees Change BETA to 10 degrees From the Editor Tab Selecte the Custom Profile Tool menu gt Component Special Profile tool Select Blaze default and set the blaze angle to 17 5 cell C3 Select OK Verify that the material is Table AL right click the figure and check materials property Select the Approximation button to load up the piecewise constant approximation From the Run Tab Select Orders with Start 1 and Stop 25 Click RUN 109 GSolver V5 2 User guide When the calculation is finished plot the 1R through the SumR columns You should see something like the following Untitled chart 6 3 iS 2 o E W e O z a 1 iJ i i 12 00 15 00 18 00 21 00 24 00 Orders Notice that at order 19 and 24 the calculation appea
96. nnects the superstrate with the substrate The Gaussian system used in the Solve3D Fnlmat3D code is conceptually easier and is described first followed by the Stack matrix methods 123 GSolver V5 2 User guide 19 2 2 1 Gaussian Elimination At the first boundary going from the superstrate down the equivalence of the tangential E and H fields require that UO Ry S 0 ij 1 4 990 Ryy S yy 0 Syoolk i k u k Rey key Ryy KU 0 joo lk Uy Ti ku A D k R kU 0 zy xij where R u S and U are defined in Sections 16 2 1 and 16 2 2 For s total orders 2 orders 1 each field component x y z has s entries At the internal interface between the n and n 1 layers the tangential fields require S xij d Si 0 Sy d Sp 0 Ut d U1 0 n U i d U y 0 yij where d is the thickness of the n layer At the last interface with the substrate the boundary conditions dictate that TS d TES Sad kaTa KT UR k Vay k yTy kU d where d is now the thickness of the n layer This set of equations together with the divergence free conditions no charge is accumulating within the grating lead to two more equations Ki Ray k vip Ryy ky Ray 0 Ky La Loar Li ky Daj 0 124 GSolver V5 2 User guide which represent a solvable system The trick is to solve the internal fields iteratively eliminating dependence on the internal fields top dow
97. nto one row of twice the thickness Since the sampling is done at the grid spacing it is convenient to leave the snap to grid property turned on It takes a little discipline not to attempt to create finer detail Some experimentation with a grating structure will indicate what feature size is required for accurately estimating the diffraction efficiency 4 4 1 Grating Representations 52 GSolver V5 2 User guide There are four internal representations of the grating structure The following is an overview of their interrelationship and indications of which representation is sent to the solver routines to calculate the diffraction efficiency See sec 1 2 figure for additional detail Approximate e Listing RUN o Results T Y D A a populate al Copy Update Run Tab Results 53 GSolver V5 2 User guide The Canvas on the Editor Tab contains a list of geometric primitives and represent the grating structure within a single period No diffraction efficiency calculations are done using the list of geometric primitives Whenever the Approximate button is clicked the list of geometric primitives are converted to a piecewise constant approximation This internal representation is the basic grating description which is passed to the solver routines For example working from the Run tab clicking the RUN button will invoke the solver routines with the grating structure whic
98. of refraction The grating region which may also be thought of as a modulation region naturally divides itself into three categories depending on the number of geometric degrees of freedom In GSolver these have been termed 1 2 or 3 dimensional gratings The same code is used for 1 and 2 dimensional gratings Three dimensional or crossed gratings mix or couple the x and y dimension layer Fourier coefficients which complicates the convergence optimizations used elsewhere 19 1 1 Stratified Grating Approximation All gratings are constructed as piecewise constant lamellar stacks This is described in detail in the following paragraphs 19 1 2 1 Dimensional Gratings A one dimensional grating or modulation region has a single degree of freedom A stack of uniform isotropic thin films is an example of a one 112 GSolver V5 2 User guide dimensional modulation region The only geometric degree of freedom is the thickness of each layer along the z direction 19 1 3 2 Dimensional Gratings Two dimensional gratings consist of a stack of layers each consisting of a periodic array or stripes of homogeneous materials The orientation from layer to layer must be the same In GSolver the uniform dimension is taken as the y direction the periodic dimension is the x direction and the thickness of the layer is the z direction There are no restrictions on the number of layers or the number of materials within each period in each lay
99. or Tab for greater detail It can be made finer in two ways 1 by changing the grid spacing by selecting Grid Properties from the Grid menu which can also be activated by right clicking in the canvas area or 2 by changing the canvas resolution a number of view units equals one grating period which can be accessed under the menu entry Edit gt Canvas Properties Also the actual layer and inter layer geometric dimensions of any piecewise constant feature are accessible and modifiable on the Listing RUN tab Click the Run tab This brings up the standard global parameter list similar to that of prior versions of GSolver Using the check boxes select one or several parameters enter limits and then click the RUN button The calculated results are shown on the Results Tab Alternatively click on the Listing RUN tab to bring up the single parameter editable list option On the Listing RUN tab click the Populate button to load the list from the current internal piecewise constant construct If the Approximation button on the Editor form has not been clicked this construct is empty and so nothing will change The piecewise constant listing is discussed in the Listing RUN chapter For this example the Listing RUN will be used for a couple of simple calculations To create a run with the angle of incidence changing Enter the following formula into grid B2 D5 2 13 GSolver V5 2 User guide 18 19 20 21 22 23 24
100. panded capabilities In particular all materials are now tied to material properties that assign indices of refraction for each region including the superstrate and the substrate This requires new data structures that do not exist in previous versions of GSolver An import function is provided to attempt conversion of V4 20c binary grating files gs4 into the V5 1 format In many cases the material index of refraction properties are assigned constant values in V4 20c On import the constant index of refraction properties are translated into material type CONSTANT see Gsolver ini file format and this property is added to the current material list automatically if it is not found Holographic gratings are approximated as a set of constant index of refraction regions Depending on the granularity of the index modulation this may lead to a very large number of materials of type CONSTANT To import a V4 20 GSolver object open a new GSolver window and click on the Editor tab Then click on the menu item File gt Import GS4 20c A file open dialog is created in which you should select the existing gs4 file After selection and clicking the OK button GSolver creates a dummy 4 0 data structure and loads the gs4 binary object into it It then reads through the data structure and creates a V5 1 data structure from it using default values for any information that are not assigned In particular V5 1 must assign a material type INI model and mo
101. period of around 2 0 a total thickness of about 7 6 and an 1T order with approximately 96 9 efficiency are typical Change the mode settings under GA Options to examine the behavior of different DE evaluation schedules Following is a modification to the above example to find the best grating for simultaneously maximizing transmission in the 1T and IT for thickness period and angle of incidence The best such configuration will have 1T IT 0 5 1 2 Enter the following in B2 if F7 gt 0 amp amp F7 lt 80 F7 40 Enter 0 in D7 and 80 in E7 75 GSolver V5 2 User guide 3 Change the goals on the Options dialog to 1T 0 5 1 and 1T 0 5 1 4 Click RUN GA A typical run may result in Theta 2 Period 2 7 and thickness 2 7 with IT 42 and 1T 47 76 GSolver V5 2 User guide 8 Execution RUN Tab Just as in previous versions of GSolver the RUN tab provides immediate access to several key grating parameters Ce as Ee Edt Foma Pomiuss Yew Ioas Gwi Comoonert hat gS athe g aa rper ato En ja Ren re orar one Run Parameters Stat Stop Ine A Tat cider Lilio conaran Wavekegh A joi Ci thala a CIN Pewakal Memory 517 28 Me 7 40 PM 8 1 1 Run Constraints The RUN tab parameters that are checked define a loop over which the selected parameters vary Select a parameter and enter the start increment and stop values for it Click on RUN to start the calcul
102. product If there is no grating then S is initialized to the identity and the final calculation simply turns out to be a solution of the interface Fresnel equations The implementation of the stack method is somewhat altered if the problem is purely a TE or a TM mode In that case certain vector components are identically zero 128 GSolver V5 2 User guide 20 Trace Pro Material Runs GSolver has an integrated tool for generating a material property file essentially a BRDF for a grating suitable for reading into TracePro The idea is to set up a grating and loop through all the angles and wavelengths for radiation incident from the superstrate to the substrate normal mode as well as from the substrate to the superstrate reverse normal For normal parameter selections a large number of GSolver runs may be generated On completion of the normal mode the results of the calculation are stored to temporary arrays and the grating structure is then temporarily inverted the superstrate and substrate materials are swapped and the order of the grating layer structure is inverted and the series of loops are run through again Finally the run data from both sets of runs normal and inverted are written to a text file in a format suitable for import into TracePro and the grating structure is returned to its normal orientation GSolver expects the superstrate to have only real valued indices of refraction If the TracePro tool is run
103. r The GS4 Editor is new to GSolver V5 2 it includes editing syntax of V4 20 and is a layer by layer graphical editor A representation of the editor is shown in the following figure I bs Ele Edit Format Formulas View Tools Grid Component About osa Be 9 5 ems Free y o noOLoOCABm e Parameters Graphical Ector 684 Editor ssngRUN ca run Resuts sedi 3DRun Angescac Define grating profile with GS4 syntax X fk 0 20318725099601 Layer thickness 0 0249999942779 Current Layer 6 Add Insert Layers Add on top Insert at current layer N_x Single layer N transitions N ties Linear profile N tie point 3D View dbl dick to set index Click drag transitions 9 29AM INUM AE This editor operates directly on the internal discrete approximation and each layer in the grating can be displayed in the graphic window on the left A layer consists of various regions which each region identified by a 56 GSolver V5 2 User guide width relative to a period and an index of refraction Each block in the layer can be directly edited by clicking on and dragging the boundary lines or after selecting a boundary line by clicking and dragging it simply entering the relative location in the text box Boundary lines are numbered from left to right Each boundary is limited by the position of the boundary on the immediate left and immediate right Thus a boundary may not be
104. r fields are obtained in the case of no diffraction these are the Fresnel fields When klz or k3z become purely imaginary the related DE is identically 0 This happens for evanescent waves which can be thought of as standing waves at the boundary surface since the grating is being illuminated by an infinite extent plane wave The fields internal to each grating layer are found by solving Maxwell s equations with the permittivity expanded as a Fourier series This leads to the solution of the field equations with a number of unknown amplitude coefficients These are found by application of the boundary conditions for the tangential components of the E and H fields By ordering the equations properly all internal fields may be eliminated from the calculation greatly simplifying the solution and allowing it to proceed in an iterated manner reusing the same work arrays The final boundary condition call results in the incident reflected and transmitted fields 19 2 1 Maxwell s Equations The solution of the field equations within each layer and the region of the problem is found by solving the first order Maxwell 19 2 1 1 Superstrate and Substrate Solutions The superstrate and substrate are semi infinite homogeneous media It is common knowledge that a traveling wave solution to Maxwell s equations in terms of the E field may be written as E E R exp kl r ij E gt T exp k3 r d2 ij 117
105. r for information purposes Layer 1 is the first layer of the grating structure and interfaces to the substrate Each subsequent layer lies on top of the previous layer The first thing to do is to populate the grid with the grating definition structure created on the Editor using the Approximation function Any empty grid cell can be used to hold intermediate results The grid supports split views for both vertical and horizontal splitting Grid specific menu items are grouped under Format Formulas View and Grid Many of the controls operate on specific elements of the grid In those cases the element must be selected before the controls are activated 6 1 1 Make Grid Current Make grid current option copies the current piecewise grating structure including parameters as represented on the grid to the internal grating storage array It has no effect on the Canvas graphic representation of the grating This allows for any updates changes to the grating structure to be used between the Grating Listing tab the GA tab and the RUN tab For example after a GA run the resulting grating structure can be copied into the general grating storage array after which a RUN will use the results of the GA run 6 2 Parameter Control The Listing RUN grid is equipped with a single parameter which is located in cell DS This cell can be incremented and decremented by the increment value in cell E5 using the button controls in F7 and E7 When
106. r Phoenix Editions 2003 W Magnus S Winkler Hill s Equation Dover Phoenix Editions 2004 131
107. r components of the k vector in the superstrate region 1 and the substrate region 3 for order m my The x and y kvector components are the same in the super and substrates due to boundary 89 GSolver V5 2 User guide conditions The diffraction order angles are then calculated in the usual way from these components 90 GSolver V5 2 User guide 13 Dialogs 13 1 Refractive Index Selection Dialog The superstrate and substrate materials properties are set with the following dialog which is similar to the material selection property page used to set material properties for the Editor canvas objects Refractive Index Selection E OK Apply Cancel IV Lambda change update Index 1 47977 0 00000 Table Model X KCL he Color Map The superstrate and substrate are not represented on the Editor canvas so the color assignments are not relevant 13 2 Refractive Index Color Map Dialog Every material property has two colors that represent the real and imaginary parts of the index of refraction The mapping between a color and the value of the index of refraction is set using the Color Map dialog The dialog contain two lists that represent the range for the real n and imaginary k parts of that index of refraction Each list is comprised of a number of entries in increasing order together with a color defined in RGB space 8 bit coordinates Numbers in the list are called break points The algorithm that id
108. raphical object is created Note the material property does not take a default value but is set according to the substrate material property Edit gt Measurements and Size This dialog controls that canvas viewport units and display scheme It is recommended that the default settings be used See the Editor form chapter for more details Edit gt Canvas Properties This dialog allows control of the relative canvas size number of canvas units to 1 period See the Editor form chapter for more details Edit gt Color Map This Dialog controls the color look up table for the real and imaginary parts of the index of refraction See the Refractive Index Color Map Dialog for more details Edit gt Material This command brings up the Materials editor dialog See the Materials Editor dialog for more details The Format and Grid menu items all apply to the data grids on the various forms As will be noted it is possible separately from the grating to save as a text file the data on any grid Thus several different parameter runs may be saved and reloaded for a single grating configuration The grid data only is saved and loaded with this feature not the internal piecewise constant data structure Several commands apply only if a grid item is selected If a command does not apply it is grayed out disabled The Formulas menu item group applies to the Listing RUN and GA grid formula engine which is detailed in the Grid Formula Engine c
109. reference the cell in which it resides by using the symbol For example the formula A1 A2 gt 2 amp amp lt B5 Il C7 means the value of the current cell is the sum of cells Al and A2 and that value must be either greater than 2 and less than or equal to the value of cell B5 or equal to the value of cell C7 The benefit of constraint expressions is maximized when combined with current cell reference support as indicated in the above example Constraint violation generates an error dialog box and so can be inconvenient in loops An alternative is to us the if 1 2 function This function returns 1 if the condition in the first slot is true otherwise it returns the result in 2 For example if d5 100 gt 90 28 d5 100 lt 90 d5 0 will return the value d5 100 if it is between 90 and 90 otherwise the function returns 0 67 GSolver V5 2 User guide 7 Genetic Algorithm GA Tab The GSolverV5 1 genetic algorithm has been significantly enhanced It is based on Differential Evolution DE a class of genetic algorithms for which there is a large literature The Differential Evolution home page is http www icsi berkeley edu storn code html F untitled GSolver 50 e Edit Format Formulas View Tools Grid Component About Reese Wook Oe Parameters Editor Listng RUn GA Run Resuts 3D E tor 30 Run HA A RARA Se E ees GA Grating Definition Listing Real Parameters Min
110. resented as 119 GSolver V5 2 User guide E S JexpKk r ij H Uy 2 exp Jk r ij where S and U are unknown complex valued vector field functions The key to the coupled wave expansion within each layer is to replace with a truncated Fourier series representation Both e the permittivity and the inverse of e impermitivity are required The permittivity within each layer is a function of x and y only for the general three dimensional grating case but not of z These are periodic functions in A and Ay elx y gt amp expUK gx IK hy gh elx y Dee expUK gx IK hy gh where the Fourier coefficients are constants e and are only strictly inverses when gh ranges from to 00 When the series are truncated they may no longer be inverses This is the origin of some of the convergence problem generally and manifested principally in the TM polarization case By properly adjusting the coefficients of the truncated series so that the permittivity and impermitivity are inverses for any truncation order convergence is greatly facilitated For one and two dimensional gratings the permittivity and impermitivity are functions of a single variable and it is fairly straightforward to enforce the proper relationships The one and two dimensional code calculates the Fourier coefficients of the permittivity eh the impermitivity ah and the inverses of each of these eh1 ah1 For the g
111. rs to become unstable the SumR result for order 24 is NAN indicating a problem Return to the Run tab and select the menu item Tools 128 Bit Floating Point and then click on RUN again the calculation will take several times longer to run When complete the plot now looks like 110 GSolver V5 2 User guide Examination of the Results table values indicate that the calculations at 19 and 24 orders appears better behaved 111 GSolver V5 2 User guide 19 Diffraction Solution Implementation This chapter contains a general description of the theory used by GSolver to solve the grating equations After a review of the general ideas behind the grating structure the formulation for Maxwell equations and various solution techniques are discussed 19 1 The Grating A general periodic grating structure is defined to exist at the interface of two semi infinite homogeneous isotropic media called the superstrate and the substrate In GSolver the superstrate is characterized by a single real index of refraction making it a lossless dielectric This permits the definition of the normalized incident plane wave anywhere in the medium The substrate is characterized by a single complex number The transmitted energy or rather complex E field is calculated at the top or entrance into the substrate If the substrate is lossy then the fields will decay away exponentially depending on the size of the imaginary component of the index
112. so manifest itself between different installs of V4 20c 37 GSolver V5 2 User guide 3 Parameters Tab This chapter describes the various data fields and controls on the Parameters tab form which is shown in the following figure a o File Edit Format Formulas View Tools Grid Component About S Bea et WOeKkKo Parameters eto Listing RUN GA Run Results 3D Editor 3D Run Default Parameter Values Mic v a Vacuum Wavelength fi microns Microns 2 UNITS Selection Eno Pais e eae fi conversion from microns factor or Lines mm 17000 a Superstrate Index select 1 00000 0 00000 Substrate Index select 1 50000 0 00000 Angles of Incidence selec THETA fo IV order convention PHI 0 m POLARIZATION ALPHA fo BETA fo or Stokes Parameters Physical Memory 500 24 ME 7 44 PM Z The figure in the lower right depicts the grating showing the illumination k vector plane wave along with the various diffracted orders 38 GSolver V5 2 User guide The various Parameter entries are discussed in subsequent sections 3 1 Units Selection The internal representations of wavelength and grating period are in units of microns For display different units may be chosen for these quantities Several common length units are available for selection from the Units drop down list box Selecting one of these defines the units conversion factor for the display
113. ss is the Table model Table models are the most general and easily extensible Also most index of refraction data is recorded in tabular form 93 GSolver V5 2 User guide 15 Grid Formula Engine Quick Reference Guide to Built in Functions 15 1 Mathematical Functions The following lists the supported mathematical functions Function Description ABS X The absolute value of X ACOS X The arc cosine of X ASIN X The arc sine of X ATAN X The 2 quadrant arc tangent of X ATAN2 X Y The 4 quadrant arc tangent of Y X CEIL X The smallest integer greater than or equal to X COS X The cosine of X COSH X The hyperbolic cosine of X DEGREES X Converts the angle expressed in radians to degrees DET M The determinant of the square matrix range M DOT R1 R2 The dot product of the vectors R1 and R2 EXP X e raised to the X power FACT N The value of N FLOOR X The largest integer less than or equal to X FRAC X The fractional portion of X GAMMA X The value of the gamma function evaluated at X GRAND A 12th degree binomial approximation to a Gaussian random number with zero mean and unit variance INT X The integer portion of X LN X The log base e natural log of X LNGAMMA X The log base e of the gamma function evaluated at X LOG X The log base 10 of X LOG10 X The log base 10 of X LOG2 X The log base 2 of X MOD X Y The remainder of X Y MODULUS X Y The modulus of X Y 94 GSolver V5 2 User guide
114. st of these commands are self explanatory notes for particular commands are presented below Edit gt Components This command activates the Components dialog Each object on the Editor canvas is identified with a default name This dialog helps navigate a grating construct with numerous objects It also provides an alternative way to call up the properties dialog for any particular object Edit Properties The Properties dialog can be used to define the properties of the selected Editor drawing object The essential property is the Material which defines the index of refraction for the object Other properties exist for convenience and include the object name edit flags indicating whether a property can be altered the thickness of the boundary line and the color By default the color is tied to the index of refraction see the Editor Form chapter However the color can also be set independently of the assigned index of refraction for display purposes 24 GSolver V5 2 User guide only since the display color has no effect on the assigned material property Note that the boundary line is generally not used in creating the piecewise constant approximation of the canvas objects However if the boundary line is made thick enough and intersects a grid point it may be included and assigned default superstrate properties by the piecewise constant algorithm Edit Default Properties The Default Properties are used when a new g
115. st the 512 byte limit e Ranges look like B12 G29 A1 D5 Operators order of precedence 14 12 12 12 112 12 11 11 10 10 lt lt 9 gt gt 9 lt 8 823 112 1 Unary percent Exponentiation Unary plus Unary minus Bitwise complement integer Logical not Multiplication Division Remainder integer mod Addition Subtraction Shift left bit shift on integer Shift right bit shift on integer Less than Greater than Less than or equal Greater than or equal Equal Not equal Bitwise And or string concatenation Bitwise XOR integer Bitwise Or Logical And Logical Or Conditional like the C language conditional 65 GSolver V5 2 User guide In formulas with more than one operator the grid formula evaluates operators in the order of precedence presented above with highest precedence first That is AND OR NOT operators are evaluated after inequality operators in a logical expression and multiplication division operations are performed before subtraction addition operations in an arithmetic expression Operators at the same precedence level are evaluated from left to right Indirect Reference An indirect cell reference specifies a row column offset from the current cell location with the follow syntax t column offset row offset e If one value is included the grid formula assumes that it is a column offset For example the offset reference 1 tells the grid formula to
116. stna RUN GA Run Resuts 3DEdtor 30 Run anges cate E Grating Editor Cetea scaler rp if Approximation ua 12 36PM BE Z The main area is called the canvas where all drawing takes place The canvas represents an edge on view of the grating The substrate exists in the region below the canvas The area above the substrate is the superstrate The canvas is the modulation region where the grating exists This is illustrated in the following figure 43 GSolver V5 2 User guide Once a figure is drawn using the primitive shapes and automatic profile tools a click on the Approximation radio button in the upper left corner invokes the piecewise constant approximation routine This routine examines the canvas at the each of the canvas grid points and creates a piecewise constant approximation the internal representation GSolver uses for diffraction calculations If drawing objects overlap the object on top determines the index of refraction for that region If several objects are grouped the grouped object is treated as a single object with the first object in the group setting the material property for the group New to version 5 1 18 is the Vertical Scale Factor in the upper righthand comer of Editor form The scale factor multiples the vertical dimension when the Approximation function is called GSolver assumes that all horizontal dimensions are relative to a grating period def
117. superstrate and substrate 19 2 Theory This section contains a presentation of the theory of the solution of the diffraction problem The grating is defined as a piecewise constant periodic lamellar structure at the boundary between the semi infinite superstrate and substrate regions The grating is illuminated by an incident plane wave determined by five parameters a wavelength and four angles Two angles determine direction of travel from the superstrate to the substrate and two angles determine the state of polarization The incident E field has unit magnitude The only dimensioned measures in the problem are the wavelength and the grating period s Since these lengths enter as ratios their units are not important as long as they are the same such as microns GSolver computes the fields internal to each layer and explicitly at the boundaries These fields are complex inhomogeneous plane waves The reflected diffracted fields and transmitted diffracted fields are computed by eliminating all internal fields and solving for the fields at the top and bottom interface The diffraction efficiency is defined to be 116 GSolver V5 2 User guide DE R Re k1 kz DE y T Re k3 kz where Il ll is the norm sum of the magnitude squared of the vector components and the subscripts ij refer to the ij diffracted field and Rj T are the complex vector E field magnitudes For i j 0 the specula
118. t connects top and bottom2 and so forth down through the layers If S is the stack for layer 1 and S is the stack for layer 2 the combined stack is found to be 1 7 S7 1 S 83 Sa Si I S 83 SiS S H ie 1 1 2 bl S 1 S3S 3 1 8 1 38 S LU LU Note that one 4s x 4s matrix inversion is replaced by two 2s x 2s inverses Thus the stack method is not only faster it is more stable than the Gaussian method The stack methods are used in all solvers other than the 3D grating case The final stack matrix is complicated by the fact that the external fields require all vector components and are therefore of a different dimension When the grating layers have been solved there is found a single stack matrix S This is placed in the following 6s x 6s final boundary condition solver as follows UO R 1 S k S R 1 S k x 0 u id 12 1x 11 y UO S k x S 7 k x R S k x 0 0 k 0 T 0 0 0 0 k JT 0 T where the sub blocks are appropriately dimensioned GSolver calls the matrix that pre multiplies the RT vector the F matrix This matrix is 127 GSolver V5 2 User guide formed in Fnlmat TE TM and a final partial inverse is performed Since the bottom rows of the terms on the left are all 0 only those rows that are needed in the inverse are computed The notation kx indicates a matrix that when multiplied by a vector results in a vector cross
119. t set of loop conditions are completed a second dialog pops up asking if the user wishes to invert the grating and complete the second set of loop conditions Click OK On completion a file dialog pops up Enter the name of the text file in which to record the results If that file already exists the contents are over written 130 GSolver V5 2 User guide 21 References E G Loewen et al Grating Efficiency theory as it applies to blazed and holographic gratings Apl Opt 16 77 2711 E Popov et al Lamellar metallic grating anomalies Apl Opt 33 94 5214 M G Moharam et al Diffraction analysis of dielectric surface relief gratings JOSA A 72 82 1386 M G Moharam et al Rigorous coupled wave analysis of metallic surface relief gratings JOSA a 3 86 1780 L Li Multilayer modal method for diffraction gratings of arbitrary profile depth and permittivity JOSA A 10 93 2581 L Li Convergence of the coupled wave method for metallic lamellar diffraction gratings JOSA A 10 93 1184 T K Gaylord et al Zero reflectivity high spatial frequency rectangular groove dielectric surface relief gratings Apl Opt 25 86 4562 T K Gaylord et al Analysis and applications of optical diffraction by gratings Proc IEEE 73 85 894 M Neviere E Popov Light Propagation in Periodic Media Marcel Dekker Inc 2003 L Brillouin Wave Propagation in Periodic Structures Dove
120. talog Graphical object description Grid data Grid Formula Engine Canvas drawing editor Graphing utility Maxwell equn solver Piecewise constant grating structure Each top level GSolver window as depicted in the figure above is designed to operate on a single grating structure The objects in blue are shared components The objects on the left are unique to each top level GSolver window Therefore if multiple GSolver windows are open each with some grating structure entered then all parameter settings are unique to that window s particular grating When GSolver is started the first order of operation is to look for the INI file in the local directory where GSolver was launched If GSolver does not find one it creates a new one with default materials of each type When GSolver shuts down its final operation is to rewrite the INI file with the current material structure Thus if there are multiple GSolver windows open from the same directory the last one closed will overwrite the INI file This should be kept in mind when using the GSolver material editor to add or otherwise change the material catalog 2 2 Drag and Drop GSolver V5 1 is built as an OLE Object Linking and Embedding object This permits the various graphical fields to be dragged between different 19 GSolver V5 2 User guide GSolver windows as well as any other OLE enabled application such as
121. that varies from the Current entry to the Stop value in increments of the value in cell E5 Formulas are used to set up a particular order calculation Refer to Chapter 5 for a description of the use of formulas in a grid 88 GSolver V5 2 User guide 12 1 Example DE Angle Calc This example creates a table of the 1 order angles as a function of the incident angle assuming the substrate is type Constant One 5 and the superstrate is type Constant Ones 1 Click on the Angles Calc tab 2 Enter a 1 in cell B2 3 In cell B3 enter the following D5 4 Click the RUN button cell D9 The table displays values in degrees for the diffracted angle for the 1R order labeled deTheta the 1T order labeled deTheta3 and the relevant Phi angle Formulas can be used to alter any of the parameters in cells B2 through B6 For crossed 3D gratings click on the 3D button The table is now augmented with an additional order since orders are then defined in the x and y directions 12 2 Definition of the Angle Calc The Angle Calculation for the diffraction order is defined as follows where A wavelength and A are the X and Y periods define dx Alh dy A A For normal gratings Y oo dy 0 Set Nsup Superstrate index of refraction and Nsw N3 K3 complex substrate index of refraction Now calculate k N1sin 0 cos m dx sal kl E k k k N1sin 0 sin m dy k3 _ SPN k k These are the vecto
122. the canvas 7 Move the mouse cursor into the interior of the rectangle and right click This brings up an item property menu Select Properties 8 Select a material for the rectangular region just created In principle any shape may be made and assigned a property For overlapping shapes the region on top is used when making the grating definition 9 Drag the rectangle to the bottom of the canvas so it rests on the substrate region 10 The units of the canvas are normalized to 1 grating period The view region can be sized to any reasonable size however the width 12 GSolver V5 2 User guide 11 12 13 14 15 16 17 of the canvas is 1 period no matter how the canvas is sized for display purposes This is explained in detail in the Editor chapter Recalling that periodic boundary conditions are assumed the single rectangle drawn in the canvas represents a binary grating looking edge on Once the grating is defined with the graphical editor an internal piecewise constant approximation can be created This gives the representation used in the RCW analysis Click on the Approximation radio button in the upper left corner of the canvas area to create the piecewise constant approximation Each time this button is clicked and only then the internal representation of the piecewise constant construct is recalculated The spatial resolution of the piecewise constant construct is determined by the canvas grid see Edit
123. the table GSolver linearly interpolates the table For example the following is a partial entry for silver AG AG 0 186412 0 995 1 13 0 187836 1 00425 1 14938 0 189282 1 012 1 16 32 GSolver V5 2 User guide 0 19075 1 0195 1 16813 0 192242 1 028 1 18 0 193757 1 0375 1 19438 0 195296 1 048 1 21 0 196859 1 05963 1 22563 0 198448 1 072 1 24 0 200063 1 08481 1 25125 0 201704 1 098 1 26 0 203373 1 11163 1 26563 0 205069 1 125 1 27 0 206793 1 13719 1 275 0 208547 1 149 1 28 0 210331 1 16144 1 28531 0 212146 1 173 1 29 0 213992 1 18188 1 29281 0 215871 1 19 1 295 For a wavelength selection of 0 20 for example the table value entries at 0 198448 and 0 200063 would be linearly interpolated This is a common method of tabulating indices of refraction and users are encouraged to use the Table model 2 7 9 Color Map Indices of refraction both real and imaginary parts are represented by colors on the Editor canvas The two colors are given as a background and a cross hatching pattern Assigning of the colors is through a user definable color map interpolation scheme 33 GSolver V5 2 User guide Refractive Index Color Map mm coed Real part Color Map Imaginary part Color Map 0 00000 0 O 0 0 00000 250 250 250 1 00000 150 150 50 1 00000 50 250 50 1 50000 250 250 50 2 00000 50 50 250 2 50000 150 250 250 15 00000 250 50 50 4 00000 20 20 50 Controls R
124. ting listing generated by the piecewise constant approximation which is the internal representation of the grating for purposes of calculation can be edited for precise thicknesses This is discussed in Chapter 5 4 3 6 Uniform fill Coats Uniform fill coats are made in a similar N manner to the conformal coats discussed N 50 GSolver V5 2 User guide above Simply select the rectangle tool set its properties and size width as wide as the canvas and then put it at the bottom off the z order All other objects will be on top and any canvas space is now covered by the fill material defined by the rectangle object 4 3 7 Text and Lines For convenience GSolver contains text and line tools that can be used to annotate grating structures on the canvas Text and line tools do not have a material property and should not be used as part of the grating approximation They do have a color property but this is not tied to any particular material However if it happens that a line or text object intersects a grid point it is possible that the piecewise constant approximation algorithm will assign some undefined property to that location Therefore the text and grid lines should be used for annotation only if they affect the piecewise constant approximation algorithm they should be either temporarily covered sent behind something moved slightly or not used 4 4 Automatic Piecewise Approximation Once the grating profile is
125. turns 1 if X contains an error otherwise 0 ISNUMBER X 1 if X is a numeric value otherwise 0 ISSTRING X 1 if X is a string value otherwise 0 TRUE The logical value 1 AND 0 if any argument is 0 1 if all arguments are 1 otherwise 1 NAND O if all arguments are 1 1 if any argument is 0 otherwise 1 NOR 0 if any argument is 1 1 if all arguments are 0 otherwise 1 NOT X 0 if X 1 1 if X 0 otherwise 1 OR 0 if all arguments are 0 1 if any argument is 1 otherwise 1 XOR 1 if any argument is not 0 or 1 otherwise 0 if the number of arguments valued 1 is even 1 if the number of arguments valued 1 is odd 15 6 Date and Time Functions The following lists the supported date and time functions DATE Y M D The date value for year Y month M and day D DATEVALUE S The corresponding date value for a given string S DAYS360 S E The number of days between two dates based on a 30 360 day count system DAY DT The day number in the date time value DT EDATE S M The date time value representing number of months M 98 GSolver V5 2 User guide before or after start date S EOMONTH S M The date time value of the last day of the month M months after S for M or M months before if M is HOUR DT The hour value 0 23 of date time value DT MINUTE DT The minute value 0 59 of date time value DT MONTH DT The number of the month in date time value DT NETWORKDAYS S E H Number of
126. ults tab is activated automatically The results grid is configured to hold the results during the first run through of the parameter loop The user can therefore click over to the Results tab and watch the table being filled At the end of the calculation GSolver eliminates zero value columns 6 2 2 Abort Button It is possible to set up grating structures that take a very long time to both allocate memory and run the defined calculation The Abort button attempts to stop the loop However the calculation only responds to the 61 GSolver V5 2 User guide abort message at certain convenient places For example if the operating system is off trying to allocate sufficient RAM the abort will not happen until the OS returns from that activity 6 A 3 Example of varying the thickness of the grating Populate the grid by clicking the Populate button assumes you have defined a grating with the Editor 2 Select the Thickness Formula tool using menu item Grid gt Thickness Formula 3 On the Thickness formula dialog click on the All button This tells GSolver that the formula should be entered in every layer thickness cell on the grid If you only want to vary the thickness of some of the layers then click to deselect other layers from the list A selected layer is indicated with a sign 4 Enter the following into the entry text box on the dialog D5 100 5 Click OK The grid layer cells should now be populated with the formu
127. using the algebraic eigensystem method of solution In particular all growing exponential solutions may be avoided with proper renormalization of the eigenvalues or ordering of the solution space GSolver also takes advantage of the intra layer field normalizations which are never actually evaluated absolutely The internal fields are scaled to eliminate any growing exponential The remaining numerical stability issue is in the calculation of the eigensystem of M and limitations due to finite round off precision arithmetic see Chapter 17 GSolver incorporates an exact truncated Fourier series inversion the impermitivity is the inverse of the permitivity independent of truncation order matrix balancing reduction to Hessian form and QR decomposition to solve the eigensystem However for large eigensystems lots of orders retained in the Fourier representation of the permitivity and for slow convergence of the Fourier system numerical round off error can propagate through the eigensystem solution leading to meaningless 103 GSolver V5 2 User guide results In that case additional numerical precision can often stabilize the solution There are no user modifiable parameters associated with the Algebraic Eigensystem AE method of solution The numerical formulation of the solution is generally stable for arbitrary grating layer thickness and for arbitrary polarization This method is recommended 17 1 2 5 Order Runge Kutta RK
128. ution of the boundary conditions which explains why some gratings require a large number of orders beyond the number of real propagating orders be retained to get convergence or physically meaningful results This is particularly true for gratings that contain materials where relative differences in complex index of refraction are large and the field is in TM mode 8 1 4 Run Stop The RUN button executes a loop defined by the selected parameters There is always at least one calculation in a RUN which represents the current version of all the parameters even if no parameters are selected The STOP button simply posts a message to the GSolver message queue to halt the current calculation This is a non deterministic action and GSolver will respond to the STOP command at a convenient point 78 GSolver V5 2 User guide 9 Results Tab The Results tab holds the grid where the calculated diffraction efficiencies are posted Reintroduced in GSolver 5 1 is a simple phase calculation which is also listed untitled A File Edit Format Formulas Yiew Tools Grid Component About kee e 2 Woe hk Of Parameters Eator Listngrun ca Rin Results 30 Eator 30 Run E 9 0000 0 04 0 04 0 96 0 96 4 00000C oor 00 0 0399824 0 0399824 sone 78361 0 959661 a 78361 0 960018 4 00000C 0 0200 0 0399294 0 0399294 10 000712901 0 958645 0 000712901 0 960071 1 00000C
129. value Max Value lo 1 0 369141 lo 11 0 113068 lo 11 0 0106201 Weight 0 70000000 CrossOver 0 30000000 Population 25 Maxiterations 25 Method Rand Bin GA tes Options N e y Physical Memory 537 96 Mt 7 39PM a 68 GSolver V5 2 User guide The GSolver DE implementation allows for an arbitrary number of real parameters and all of the popular evolution strategies However some care and skill are required to find any particular grating efficiency solution If the solution constraints are too far afield of any physical solution the algorithm is likely to simply wander around in parameter space and never produce a satisfactory result However the application of DE to grating design is a very powerful tool in optimization This chapter explains DE and how it applies to grating design NOTE I file version 5 1 1 1 an additional menu item as been added to the Menu Grid menu called Make grid current This option will copy the current piecewise grating structure including parameters to the internal grating storage array It has no effect on the Canvas graphic representation of the grating This allows for any updates changes to the grating structure to be used between the Grating Listing tab the GA tab and the RUN tab For example after a GA run the resulting grating structure can be copi
130. working days between S and E NOW The date time value of the current system date and time SECOND DT The seconds value 0 59 of the date time value DT TIME H M S The time value for hour H minute M and second S TIMEVALUE S The corresponding time value for a given string value S TODAY The date value of the current system date WEEKDAY D Integer representing the day of the week on which day D falls 1 is Sunday 7 is Saturday YEAR DT The year value of date time value DT YEARFRAC S E B The portion of the year represented by the number of days between start date S and end date E 15 7 Miscellaneous Functions The following lists miscellaneous supported functions CELLREF N1 N2 A reference to the cell in column N1 and row N2 CHOOSE N The N argument from the list COL C The column address of the cell referenced by C COLS R The number of columns in the specified range R HLOOKUP X S R The cell in range S that is R rows beneath X INIT X1 X2 The first argument on the first recalculation pass and the second argument on all subsequent recalculation passes when Grid is performing iterative calculations INTERP2D R1 R2 N The interpolation value for a 2 dimensional vector INTERP3D R X Y The interpolation value for a 3 dimensional vector MATCH V R T The relative position in range R of value V based on positioning criteria T N R The numeric value of the top left cell in range R RANGEREF N1 N2 N3 N4 A
131. ys along with the mouse in the usual manner The many options available for graphical display are discussed in the Graphing Options chapter Return to the Listing RUN tab Reset the parameter D5 to 0 and change cell B2 to 10 for a fixed 10 degree incidence angle Enter the following formula in cell B6 wavelength if D5 100 gt 5 D5 100 5 This is a from of a conditional entry The wavelength remains constant 0 5 microns if D5 100 lt 0 5 Otherwise it changes linearly with D5 as given in the formula In cell B7 enter the following formula 1 4 D5 100 14 GSolver V5 2 User guide This formula changes the grating period from 1 to 2 linearly as D5 changes from 0 to 100 25 Change the orders field to 5 26 Click the Run button and examine the results Note that the thickness of any layer can be entered as a constant or through a formula on the grid listing With this capability all film thicknesses layers can be accurately set the finite grid resolution of the canvas does not limit layer thicknesses 1 3 2 Blaze Grating Example For the Blaze grating use the tool button that shows a Blaze profile in black This is a general tool that includes common grating design tasks Note that the dimensions need to be considered carefully as pointed out in section 1 2 x Blaze Tool Aa Y Henno Button 1 From within the Editor tab click on the Blaze tool button This brings up the Custom Profile Construction dialog which includes
132. zation over a range of angles and or wavelengths If a parameter in the Integrated Merit Function area is selected the diffraction efficiency is solved for the range of parameters selected and the above merit function is accumulated The parameter range is uniformly sampled from the Start to Stop range with Sample samples The resulting merit function is used by the GA Note By using this feature the calculation of the merit function can be significantly increased It is recommended that the number of samples be set to a small number initially to gauge the CPU time The GA grid contains the same functionality formula engine as the Listing RUN grid The formula engine is described Section 5 2 7 4 Applying Constraints The GA parameters are selected from a uniform random population given by Min Value and Max Value for each parameter in the list It is very important to remember that DE makes algebraic combinations of current parameters to use as new parameters Therefore actual parameters are not constrained to the Min and Max bounds Each formula should also have a constraint on the chosen parameter so that the result gives a physically realizable value For example if a cell width is to be altered you must enter a constraint on the other cells in that 72 GSolver V5 2 User guide grating layer so that that total width sum of all the cells in a layer sums to 1 The order of operation is illustrated in the following examples 7

User`s Manual - Grating Solver Development Company

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