UNIGIT RIGOROUS GRATING SOLVER VERSION 2.01.03
Contents
1. versatile rigorous grating so UNIGIT RIGOROUS GRATING SOLVER VERSION 2 01 03 Installation for Version 2 01 02 Copyright by OPTIMOD Jena Germany Developed by Optimod Ricarda Huch Weg 12 D 07745 JENA GERMANY phone 49 03641 825944 cell phone 49 162 9067015 email support unigit com Copyright by Optimod 1 26 October 16 2013 unu S gt oo Sit E M versati e rigorous grating solver eu M NM MU E Nd NEP 2 Fist ot lables eR 2 3 T Basi Examples esc E evans 4 1 1 Plane interface and thin ELE a eo GR RU ri nA 4 LLL Plane di I tUa 4 i 2 A ti Refle tiv Coating ooo sii ir E i 5 1 2 Bus grating example EI ERR QM 6 1 2 1 Preparing the computation e eii toD UND 6 1 2 2 Running a convergence HHTRUEI AN PROS REA COP XV RA YF eto Nave 9 1 2 3 9 112 4 Need to know more a a aE 11 2 One dimensional gratings as osx 13 2T Jungle Layer Ci AU OS oc vasa bere venit o AQ T bU M DON ODE 13 2 2 Simp2. fo mie Insert Replace Delete Cancel Fig 15 Unigit layer for implementing a C method polygon layer The height of the profile is 200 nm and the grating period is 1 micron The grating is illuminated with green light 550 nm under oblique incidence 30 degrees A truncation of 20 seems to be more than sufficient for this example The grating has been implemented in two different ways As Composite Polygon layer to be run by the RCWA solver 1D Classic e As C method CM layer polygon to be run by the C method solver 1C Two files are provided which contain these gratings slope diel 1d ust and slope_diel_1c ust The zeroth order transmission efficiencies are shown in the table 2 below File Solver of Slices Transmission TE 0 Transmission TM 0 slope diel lc ust C method N A 74 97823857 87 31595748 Copyright by Optimod 17 26 October 16 2013 SE t T versatile rigorous grating so slope diel 1d ust RCWA 10 75 00495505 87 52453837 Slicing slope diel ld ust RCWA 25 74 98385363 87 43482951 Slicing slope_diel_1d ust RCWA 150 74 98059365 87 42389105 Slicing Table 2 Diffraction efficiencies 0 order transmission for different 1D gratings Apparently the results are very similar Moreover the RCWA results seem to converge to the C method results with increasing number of slices giving evidence
3. 9 6 degrees the total energy in all propagation orders reflection and transmission is 100 energy criterion 1 2 4 Need to know more Now assume that you are also interested to have a look at the phases you want to see the ellipsometry parameters or you need to know the diffraction angles In the further two cases you could rerun Unigit with changed settings of the output editor in the latter case you even cannot do this Rerunning this simple example is not a big deal and takes only seconds However when it comes to complex 2D gratings a run can easily last several hours In order to avoid time consuming reruns a smarter way was introduced in Unigit in order to retrieve results from an example which has already been run i e the information has already been generated The only thing what you have to do before the original first run of the example or let it better call project is to specify that and where the project information shall be saved This is easily in the output editor by means of checking the box Project File bottom and entering a file name edit control next to it Then the project file can be load just by selecting a project file extension upr instead of a stack file see also section 6 of the Unigit manual After having selected the before generated project file dielgrat 1d upr the way how to generate modified results is quite similar to the regular computation mode One has first to specify the ou
4. TMO 0 5 1 9797E 01 2 8229E 03 6 6188E 01 8 7378E 01 1 9960E 01 2 3919E 03 6 6256E 01 8 7601E 01 0 52 1 9888E 01 1 7398E 03 6 8024E 01 8 8555E 01 2 0047E 01 1 3970E 03 6 8054E 01 8 8743E 01 0 54 1 9419E 01 7 9289E 04 6 9591E 01 9 0026E 01 1 9575E 01 5 6994E 04 6 9605E 01 9 0176E 01 0 56 1 8686E 01 2 3387E 04 7 1300E 01 9 1524E 01 1 8837E 01 1 2258E 04 7 1310E 01 9 1640E 01 0 58 1 8007E 01 1 2138E 05 7 3067E 01 9 2791E 01 1 8152E 01 9 5985E 07 7 3073E 01 9 2882E 01 0 6 1 7573E 01 1 2041E 04 7 3903E 01 9 3333E 01 1 7703E 01 2 0905E 04 7 3916E 01 9 3402E 01 Table 3 Zero order efficiencies in TE and TM polarisation in reflection RE RM and transmission for the parallel multilayer grating RCWA C Method lambda REO RMO TEO TMO REO RMO TEO TMO 0 5 3 6121E 02 2 3741E 02 9 1920E 01 9 5615E 01 3 7467E 02 2 5761E 02 9 2322E 01 9 5483E 01 0 52 3 8851E 02 2 4833E 02 8 8200E 01 9 5919E 01 3 9918 02 2 6852E 02 8 8921E 01 9 5718E 01 0 54 4 6098E 02 2 5940E 02 8 4468E 01 9 6393E 01 4 6867E 02 2 7911E 02 8 5291E 01 9 6181E 01 0 56 5 5352E 02 2 5999E 02 8 0639E 01 9 6991E 01 5 5904E 02 2 7872E 02 8 1445E 01 9 6784E 01 0 58 6 4025E 02 2 5425E 02 7 7808E 01 9 7395E 01 6 4453E 02 2 7188E 02 7 8502E 01 9 7207E 01 0 6 6 9775E 02 2 4813E 02 7 6528E 01 9 7496E 01 7 0114E 02 2 6471E 02 7 7077E 01 9 7335E 01 Table 4 Zero order efficiencies in TE and TM polarization in reflection RE RM and transmission for the non parallel multilayer grating 2 5 Gradient grat
5. Version 2 01 01 provides some example stacks which come with the regular installation After installation there are the following stack files available in the subfolder Stacks interface_1d ust arc_ld ust dielgrat_1d ust algrat 1d ust PR BARC 14 15 plain interface between air n 1 and dielectric n 1 5 a plain interface with an anti reflective coating a dielectric grating n 1 5 a metallic grating Aluminum a photo resist grating on a Silicon substrate with a bottom anti reflective coating BARC two layer 1d ust slope diel 1d ust slope diel 1c ust gradient 1d ust rcwa dsin 1d ust ccm dsin 1d ust rcwa double 1d ust ccm double 1d ust stack2d elli ust stack2d arbi ust stack2d rect ust composite 2d ust a double layer Chromium grating on a dielectric interface a symmetric triangular grating in a dielectric material same as previous one but for C method solver a gradient index grating a coated sinusoidal grating with parallel interfaces same as previous one but for C method solver a triangular grating with an overcoat that exhibits a sinus profile same as previous one but for C method solver a 2D grating assembled by a patterned dielectric on Si substrate same as previous but with arbitrary instead of ellipse filling same as previous but with patch filling a 2D composite grating In the following it will be briefly explained how to run these examples in o
6. coming with the installation are rewa_dsin_Id ust and ccm_dsin_1d ust The second example covers the more general case of non parallel interfaces It is shown in figure 17 The grating is triangular and the coating exhibits a trapezoidal profile resulting in non parallel interfaces In this way so called snowing effects occurring after the coating can be modeled Of course a lateral offset can also be invoked see Unigit v2 01 02 manual The associated stack files are rcwa double 1d ust and ccm double 1d ust Copyright by Optimod 18 26 October 16 2013 d 100 nm Fig 16 p 500 nm Ep l 250 nm gt Parallel Multilayer Grating Fig 17 Nonparallel Multilayer grating Simulations have been run for both grating types with both solvers The results for the parallel grating are summarized in 0 5 0 52 0 54 0 56 0 58 0 6 REO 1 9797E 01 1 9888E 01 1 9419E 01 1 8686E 01 1 8007E 01 1 7573E 01 RCWA RMO 2 8229E 03 1 7398E 03 7 9289E 04 2 3387E 04 1 2138E 05 1 2041E 04 TEO 6 6188E 01 6 8024E 01 6 9591E 01 7 1300E 01 7 3067E 01 7 3903E 01 TMO 8 7378E 01 8 8555E 01 9 0026E 01 9 1524E 01 9 2791E 01 9 3333E 01 REO 1 9960E 01 2 0047E 01 1 9575E 01 1 8837E 01 1 8152E 01 1 7703E 01 CM RMO TEO 2 3919E 03 1 3970E 03 5 6994E 04 1 2258E 04 9 5985E 07 2 0905E 04 6 6256E 01 6 8054E 01 6 9605E 01 7 1310E 01 7 3073E
7. 01 7 3916E 01 TMO 8 7601E 01 8 8743E 01 9 0176E 01 9 1640E 01 9 2882E 01 9 3402E 01 Table 3 and those for the non parallel grating in Table 4 An order truncation of 20 and a wavelength loop 0 5 0 6 um was selected for the two examples Moreover AOI s of 60 Copyright by Optimod 19 26 October 16 2013 COO NENEES a EG E Gs L versatile rigorous grating so S degrees for the parallel and normal incidence for the non parallel multilayer grating have been chosen The results are similar although there are visible differences particularly in TM polarization This may mainly be debited to approximation issues in the RCWA Comparisons with a very accurate integral method implementation always showed very good agreement for the C method simulations Furthermore the C method clearly outperformed the RCWA for the given examples in terms of computation speed Due to the large number of slices the RCWA was about 100 times slower compared to the C method solver In addition there are cases were the RCWA slicing becomes quite complicated when the interfaces between the layers overlap in lateral direction This may be the case when the distance between the interfaces 1s smaller than the profile height In spite of these clear advantages there are many cases were one has to rely on the RCWA In conclusion a careful decision whether to use RCWA or CM solver is stronglrecommended RCWA CM REO RMO TEO TMO REO RMO TEO
8. 2 979084 115 230362 80 496628 0 434009 45 350861 109 161903 67 716873 4 163826 47 703072 105 030640 55 967613 7 773154 50 034698 102 078102 44 343903 10 766905 52 346706 99 841942 31 828804 13 439517 54 632942 98 040024 16 879087 15 988197 56 875706 96 485107 3 813079 18 566753 59 038647 95 015381 40 348938 21 365423 61 039059 93 025330 141 000473 25 509546 62 697006 91 612961 166 742508 29 203585 65 270630 90 401749 145 548920 32 943726 67 796883 89 323631 134 979858 36 840511 70 328758 88 369095 128 444458 40 887329 72 871872 87 534004 123 853554 45 062603 75 426155 86 814766 120 373543 49 340286 77 989899 Fig 9 Notepad output of the phases for 0 and 1 order in transmission Analogously tan psi and cos delta or the diffraction angles can be retrieved Copyright by Optimod 12 26 October 16 2013 2 One dimensional gratings 2 1 Single Layer Gratings One single layer binary grating dielgrat 1D ust was already treated in the previous section Now we shall consider a metallic grating Select the input file al grat 1d txt It is made from Aluminum has a period of 0 5 microns and equal lines spaces being 200 nm in height Diffraction Efficiency 78 000 70 000 62 000 54 000 46 000 38 000 1 000 10 750 20 500 30 250 40 000 esults mist000 rm eSults mist000 re Fig 10 Diffraction e
9. 74 454025 82 930946 32 265106 109 521019 173 037811 100 451668 30 464874 108 259125 171 581833 118 233429 29 547071 107 274605 170 156067 136 577850 29 171785 106 489105 168 781235 153 293030 29 121851 105 859200 167 472977 166 949524 29 258512 105 354744 166 245636 177 501785 19 529768 19 506319 19 436213 19 320227 19 159763 18 957127 18 716064 18 443111 18 151720 17 883398 122 958977 54 291679 17 764856 10 780745 115 540283 48 350090 17 211016 11 504990 111 028351 44 082241 16 593105 12 301006 107 653130 40 359001 15 922061 13 165315 104 943428 36 859035 15 201998 14 096570 102 690407 33 415634 14 435931 15 093785 100 780426 29 909115 13 626781 16 156012 99 144577 26 230577 12 777754 17 282217 97 739044 22 262007 11 892653 18 471186 96 536758 17 858622 10 976230 19 721418 95 524437 12 825907 10 034664 21 030985 94 703598 6 882824 9 076352 22 397306 94 096344 0 404252 8 113173 23 816828 93 759232 9 748634 7 162667 25 284485 93 814857 22 392445 6 251913 26 792969 94 524841 40 524990 5 424640 28 331749 96 468933 67 294182 4 754617 29 886486 100 992378 103 109154 4 370897 31 441431 111 070335 140 123306 4 501498 32 992416 129 761597 172 280975 5 517305 34 581692 146 996185 159 533508 7 774505 36 329300 147 636383 134 479538 10 501047 38 352737 136 535446 113 027092 10 389842 40 615414 124 276207 95 267616 5 970280 4
10. TPUT group Select Single Files New File Efficiency Output Orders from 0 to 0 Check Transmission Define the location where to write the result file under SAVE RESULTS IN and start the computation After the computation is finished you can view the results by clicking the ADD button in the RESULT FILE S group and selecting the corresponding transmission files with the extension te and tm for TE s and TM p polarization correspondingly The result should look similar to that shown in figure 2 Obviously the transmission is 100 96 at 500 nm The AR layer was optimized for 500 nm and normal incidence according to n sqrt n n and h A 4 n Copyright by Optimod 5 26 October 16 2013 Diffraction Effici 0 400 0 475 0 550 0 625 0 700 esults mist000 tm eS ults mist000 te Fig 2 Transmission vs wavelength through a plain AR coated interface 1 2 Basic grating example The first grating example dielgrat_1d ust is a binary grating made of dielectric material It is made from non dispersive material with n 1 5 It has a period of 1 micron and equal lines spaces 400 nm in height For a grating the convergence strongly depends on the truncation number i e where the modal expansion of the electro magnetic fields 1s truncated for the sake of numerical calculation Therefore it is recommended to run firstly a convergence test In the following the basic work flow is explained 1 2 1 Pr
11. eparing the computation After starting Unigit the Central Control Board CCB occurs First it is recommended to load the stack file to work with This is done by clicking the SELECT button in the Stack group upper right In doing so a Windows file requester occurs as it is shown in figure 3 and one has to select the file type txt or ust for searching Unigit stack files In this example the dielgrat 1d ust was selected As indicated in figure 4 the active stack file appears in the CCB The solver is detected and activated automatically due to the type of the stack file Usually it has only to be changed in case one wants to run a 1D conical rather than 1D classical mount The selected solver is shown in radio button group Computation middle left of the CCB Copyright by Optimod 6 26 October 16 2013 Name Anderungsdatum J algrat 1d txt Orte Lj algrat 1d0 txt Lj algrat 1da txt algrat 1dal txt Desktop L arc 14 04 Lj casini c txt D casini cs txt Joerg _ dielgrat 14 54 eckhardt txt A F eli 2d new b Computer L jest lc bt interface 1d txt _ interface 1dSiNx txt 1d b m As a next step it is recommended to choose the loop mode and to enter the so called input data it is saved to the Unigit input file usually UNIGIT_directory input input txt such as wavelength AOI azimuth applies for 1D conical or 2D sim
12. ernal processing of the different layer types It is left to the user to increase the sampling density of the gradient grating to check whether the diffraction spectrum is converging to that of the binary grating Diffraction Efficiency 40 000 34 000 28 000 22 000 16 000 2 bus sai civ cutters pe XX 30 000 37 500 45 000 52 500 60 000 radient 32 000 tt radient 32 000 11 ts dielgrat000 te ts dielgrat000 tm Fig 18 Comparison of zeroth order transmission for a binary RCWA grating vs a gradient grating Copyright by Optimod 21 26 October 16 2013 uu Bia mms m T ort 44 BB i versatile rigorous grating soly 3 Two dimensional gratings 3 1 Grating types Again there are basically three different types of 2D gratings in Unigit see manual layer editor 2D namely the patch fill the ellipse fill and the arbitrary fill The two latter options are more general but in a sense less accurate Here a sort of equidistant discretization is done which is quite similar to the so called 1D gradient grating On the other hand the patch filling can be considered as a sort of RCWA hard transition for 2D However it is limited to rectangles only while the other two option enable a variety of 2D cross sections such as ellipses diamonds and corner rounded rectangles In this example of the tutorial a simple 2D grating shal
13. fficiencies in SSH Output Value Efficiency Efficiency amp Phase C Amplitude amp Phase C Phase C Efficiency total C TanPsi CosDelta C Complex Value C Diffracti Output Orders x y Minimum f0 o Maximum 0 o Show Info in DOS Window 1 2 2 Running a convergence test Then run the computation just by clicking the start button or Ctr S hotkey The results can be retrieved by means of the NOTEPAD button in the Result files group bottom right It Fig 5 Unigit Output Control Editor 21 Cancel should be similar to those presented in table 1 below Truncation 1 ooOo oocneom Table 1 Depending on the required accuracy a truncation between 4 and 9 seems to be appropriate TE 15 798 20 421 20 257 20 015 20 031 20 018 20 024 20 022 20 025 20 024 Zero order transmission of a dielectric grating TM Truncation TE TM 24 333 11 20 025 21 411 20 129 12 20 025 21 407 21 623 13 20 026 21 412 21 246 14 20 026 21 41 21 428 15 20 026 21 413 21 361 16 20 026 21 411 21 411 17 20 026 21 414 21 39 18 20 026 21 413 21 41 19 20 027 21 414 21 401 20 20 027 21 413 1 2 3 Running an AOI loop Now you are probably interested in the diffractive behavior of this grating To this end check the radio button Theta i in the LOOP group and insert 0 60 1 and change the grating period to 0 3 microns use the stack editor Due to the changed pi
14. fficiencies vs truncation order of a metallic grating 0 order reflection Again you should first run a convergence test Check the TRUNCATION button enter 1 40 1 to specify the truncation loop and 0 7 for the wavelength Your convergence curves reflection should look like those of figure 10 In spite of the improved models implemented in UNIGIT the TM convergence pink curve for metallic gratings is sometimes worse compared to TE The next recommendation is again a loop over the AOI 0 50 1 with a truncation setting of 25 The resulting reflection curves for the 0 and 1 order are shown in figure 11 Clearly a sharp change in the zeroth orders can be observed when the first orders start to propagate at AOI 23 57 degrees Copyright by Optimod 13 26 October 16 2013 FH Diffraction Efficiency 0 000 12 500 25 000 37 500 50 000 esults mist000 re esults mist000 rm esultsmistm01 rm esults mistm01 re Fig 11 Reflection efficiencies 0 amp 1 order vs AOI for a metallic grating 2 2 Simple Multilayer gratings There are two multi layer stack files available which come with the installation of version 2 01 01 of Unigit e PR BARC ID ust A patterned photo resist above a BARC layer on a Silicon substrate The grating period is 0 5 microns The PR lines are 300 nm high and have a line width of 250 nm The 100 nm thick BARC layer is not patterned Af
15. flection files with the extension re and rm for TE s and TM p polarization correspondingly The result should look similar to that shown in figure 1 You can easily see the 4 reflection for both polarizations at normal incidence and the Brewster angle at about 56 degrees Of course you can choose a finer angular grid to find the zero in TM at 56 31 degrees Copyright by Optimod 4 26 October 16 2013 j comme HHEHHEEHHEH i vus EE E ver rigorous gr Unigit Diagram by URCOMTEC 1 0 0 8 e a e p Diffraction Efficiency 02 Er ee rp e 00 20 40 60 80 AOI Degrees Axes Font j Curves V Legend AutoScale min max Sze 15 5 fi mE ToCipboard x AOI Degrees afo 9o V ConnectPonts Delete Cancel y Diffraction Efficenc 1 0 59 Pensize Nomarker update Fig 1 Reflection vs AOI from a plain interface n 1 5 for TE and TM polarized light 1 1 2 Anti Reflective Coating Select the input file are 1d txt and check the radio button Lambda in the LOOP group to run a loop over the wavelength and insert a start and a stop value as well as a step width 0 4 0 7 0 02 meaning run the wavelength from 400 nm through 700 nm in step of 20 nm Insert a 0 for the AOI and insert the truncation order from to 0 0 Then verify the output conditions by clicking the EDIT button in the OU
16. for the correct implementation of both methods A similar example could also be run comparing all three methods i e the Rayleigh Fourier approach beside the C method and the RCWA solver However here one should reduce the profile height to make sure that the RF method still converges It is left to the user as an exercise 2 4 Complex Multilayer Stacks In section 2 2 examples for simple binary multilayer stacks have been already presented In this section more sophisticated gratings shall be discussed assembled from several layers of different material that are separated by arbitrary rather than straight boundaries such as for example triangle trapezoidal or sinusoidal interfaces Basically these types of gratings can be run either with the RCWA solver 1D classic or in many cases preferably with the C method solver 1D C method An exception are overhanging and very steep profiles meaning profiles with slopes close to vertical although there are workarounds for the former Here we provide two examples a trapezoidal grating that is coated by a layer of uniform thickness everywhere and a triangular grating over coated by a layer with an inverse sinusoidal profile Grating files are given for RCWA as well as for C method simulation for both examples The first example is shown schematically in figure 16 The thickness of the intermediate layer is constant everywhere rendering the two interfaces parallel The associated stack files
17. i ust 24 19229348 24 68655616 512 x 512 Stack2d arbi ust 24 13369369 24 63391415 Copyright by Optimod 22 26 October 16 2013 teen TM Stack2d fect ust 24 13640498 124 63724637 Table 5 Reflection efficiencies 0 Order for 3 different 2D grating realizations 3 2 Speed up Another exciting new feature of Unigit version 1 02 04 upwards is the smart order management which permits speed up factor of 10 and higher It should be pointed out however that the speed up should be very carefully used The best way to always be on the safe side is to verify the speed up factor by means of an extended convergence analysis i e not only to vary the order but also the speed up factor during convergence runs and compare the results to those obtained with highest order and speed factor 1 Here the speed up factor was set to 7 84 and the previous 2D example was repeated The results are shown in Table 6 below The patch fill example w o speed up is added for reference highlighted in green Grating file Speed up Zero Order TE Zero Order TM Discretization Reflection Reflection Stack2d elli ust 7 84 24 13003663 24 64187327 2048 x 2048 Stack2d arbi ust 7 84 24 11269207 24 62607999 Stack2d rect ust 7 84 24 11603230 24 62961443 Table 6 Reflection efficiencies 0 Order for the example in table 3 with 2D speed up The results are still very close The maximum dev
18. iation within the results is still below 0 1 for TE and even below 0 05 for TM case The computation time could be reduced from about 20 seconds to about 2 3 seconds for the patch and arbitrary fill grating confirming the predicted speed up factor Copyright by Optimod 23 26 October 16 2013 4 Composite 2D Gratings In Unigit the third dimension of 2D gratings can be implemented quite similarly as for 1D gratings in two different ways First the brute force but sometimes cumbersome way is to build the stack up from individual layers by means of the stack editor Both the copy cut and paste features as well as the work with sequences pre manufactured sub stacks may greatly facilitate this approach However Unigit offers also a more elegant way that can be accessed through the CONE 3D option in the layer editor When activating this layer type the input window as shown in figure 19 comes up UNIGIT Layer Editor hh 1 Name name Layer Type Cone 3D v Thickness 1 um Refractive Index Medium 1 Direct Input Real Part 1 000 Imaginary Part 0 000 m Refractive Index Medium 2 Ellipse Direct Input Real Part 1 000 Imaginary Part 0 000 Edit dx dy E sy Power Angle Tp pz o p p P fe Boom fs ps b p PR fp Power vertical 1 HofSlices 5 Shown Slice Boltom Slicing File Resolution 1288 x 256 M Ortho EIC Fig 19 Layer editor for a 2D composite layer He
19. ings Gradient gratings are gratings that rely on a gradual change of the refraction index rather than a stepwise index profile as given across a common RCWA slice One can also conceive a gradient grating as a thin film layer with a periodically but gradually changing refraction index In Unigit gradient gratings are created and described by means of the soft transition RCWA slice for details see Unigit manual section layer editor It might be of interest to know that a gradient layer may converge to a standard RCWA slice when increasing the number of entrances to infinity but using only a limited number of refraction index values A simple example will be discussed in the following Remember the dielectric grating dielgrat l1d ust of section 1 2 Suppose that you run an AOI loop for a range of theta 1 30 through 60 degrees Next setup a gradient grating with the same geometry and refraction Copyright by Optimod 20 26 October 16 2013 index as dielgrat 1d This grating can be found under the name gradient ld ust in the Unigit stack folder It consists of 32 equidistant index values namely 16 in a row with index 1 5 and 16 with index 1 So it resembles closely dielgrat 1d Therefore one would expect a similar diffraction behavior The zeroth order transmission versus AOI is plotted for both cases and both polarizations in figure 18 Apparently the results are almost the same The differences arise from the different int
20. l be considered which enables to compare all three grating types directly for the same geometry The grating consists of a 100 nm thick dielectric layer on top of a Silicon substrate Rectangular holes with CDx 250 nm and CDy 375 nm are etched into the dielectric This geometry has been implemented differently in three Unigit layer stacks Stack2d elli ust with ellipse fill option Stack2d rect ust with patch fill option Stack2d arbi ust with arbitrary gradient filling The grating was illuminated normally with a plane wave having 800 nm wavelength The truncation order was set to 6 meaning that 2361 orders in both directions resulting in a still quite moderate total number of 169 have been maintained during computation This is justified because of the low contrast material and the pitch over wavelength ratio gt 1 The Table 5 below shows the resulting zeroth order reflection for TE and TM polarization Apparently all values are quite close to each other The maximum deviation is about 0 2 This can be further reduced by increasing the sampling density of the ellipse and arbitrary filling layers One example is also included in the table Here the discretization of the ellipse fill was increased from 512 x 512 to 2048 x 2048 highlighted in green color Evidently the results are now much closer than before Grating file Zero Order TE Reflection o Zero Order TM Reflection o Discretization Stack2d ell
21. le acest app esito bibi Buen osi 14 23s Sloped gratings AA TEREN SETS 16 20 Complex Multilayer Stacks ua orci ulpa ER uia antica 18 2 5 Gradient 20 22 Bele E 22 SEE RENE 23 4 Comp sit 2 INOS Roper UN Na PEU RP REIR RTL ret 24 5 en CROCO GR SR brad ccu du us pete 26 List of Tables Table 1 Zero order transmission of a dielectric grating ssssseeeeer 9 Table 2 Diffraction efficiencies 0 order transmission for different 1D gratings 18 Table 3 Zero order efficiencies in TE and TM polarisation in reflection RE RM and transmission for the parallel multilayer 138 20 Table 4 Zero order efficiencies in TE and TM polarisation in reflection RE RM and transmission for the non parallel multilayer grating sess 20 Table 5 Reflection efficiencies 0 Order for 3 different 2D grating realizations 23 Table 6 Reflection efficiencies 0 Order for the example in table 3 with 2D speed up 23 Copyright by Optimod 2 26 October 16 2013 Introduction Unigit
22. nd sinusoidal layer This type of layer can be mixed with RCWA slices or composite layers to form a multilayer stack When specifying these types of layers a special solver routine based on the Rayleigh Fourier method will be applied during simulation The drawback of this method is that it is not rigorous and provides only accurate results for shallow profiles Another alternative is the C method solver This method is based on a rigorous modal algorithm derived from Maxwell s equations Not only is this algorithm usually much faster than the RCWA but it also provides much faster convergence and accuracy in many cases Copyright by Optimod 16 26 October 16 2013 ae Both alternatives do not require the profile decomposition in slices resulting in the great speed enhancement In the following one example of a triangular grating shall be considered as a case study The grating consists of patterned dielectric material with index 1 5 The profile is shaped as a symmetric triangle as shown in figure 15 UNIGIT Layer Editor Name name Layer Type EB Mat ele Thickness 12 Refractive Index Medium 1 Direct Input Real Part 1 000 Imaginary Part 0 000 Refractive Index Medium 2 Direct Input Real Part 1 500 Imaginary Part 0 000 Polygon Points relative Layer z versus x x z 0 0000 0 0000 0 5000 0 2000 1 0000 0 0000 SS a ae
23. nsional two dimensional three dimensional Copyright by Optimod 26 26 October 16 2013
24. rder to give some idea about the prospects of Unigit Additional assistance can be obtained if necessary by means of the context sensitive help by pressing F1 or clicking the button Before plunging into the various more advanced examples the basic functionality of how to run an application will be shown in more detail with some basic example Copyright by Optimod 3 26 October 16 2013 1 Basic Examples 1 1 Plane interface and thin film 1 1 1 Plane Interface After starting UNIGIT select the input file interface_1d txt by means of the SELECT button of the STACK group The name of the stack file is then shown under ACTIVE STACK FILE Check the radio button Theta i in the LOOP group to run a loop over the angle of incidence AOI and insert a start and a stop value as well as a step width 0 89 1 meaning run from 0 through 89 degrees in step of one Insert a wavelength value in the corresponding field don t care much the refraction index is fixed and insert the truncation order from to 0 0 Then verify the output conditions by clicking the EDIT button in the OUTPUT group Select Single Files New File Efficiency Output Orders from 0 to 0 Check Reflection Define the location where to write the result file under SAVE RESULTS IN and start the computation After the computation is finished you can view the results by clicking the ADD button in the RESULT FILE S group and selecting the corresponding re
25. re the user can define the top and the bottom cross section shape of his grating pattern just like the same way as he is used to with the ellipse filling feature But as opposed to it now he defines a real 3D body The vertical shape of this body is defined by the vertical power Here the 1 results in a straight line while a value 71 gives a convex and a value 1 results in a concave shape respectively Furthermore the user can define the number of slices Actually a equidistant slicing routine takes care to automatically translate the compact 3D description in something digestible for the RCWA solver Finally the user can sweep up and down through the 3D body by means of the Show Slice option Here the cross section shape of the activated slice number 1s plotted The figure 20 below shows such a sequence of cross sections for the body defined above Copyright by Optimod 24 26 October 16 2013 wm Slice Top wn Slice 4 wn Slice 2 m Slice Bottom Fig 20 Sequence of cross section when sweeping through a 2D composite layer Copyright by Optimod 25 26 October 16 2013 AOI AR BARC CCB CD CM PR RCWA TE TM ID 2D 3D 5 Acronyms Angle of Incidence Anti Reflective Bottom Anti Reflective Coating Central Control Board Critical Dimension Coordinate Transformation Method a k a C method Photo resist Rigorous Coupled Wave Approach Rayleigh Fourier Method Transverse Electric Transverse Magnetic one dime
26. sec Exit Fig 4 Unigit Central Control Board with selected stack file Eventually one has to specify the flavor of output by means of the output editor tab It is activated just by clicking the EDIT button in the Output group middle right of the CCB Regarding the detailed functionality of the output editor it is referred to the corresponding section of the user manual Related to this example the following actions have to be done Choose single files in radio button group Output files upper right Check the Transmission box in the same group below Check the radio button New in the same group Check the output formats Fix and 8 digits Choose the option Efficiency in the Output value group upper right Choose the output orders 0 through 0 lower right Check the box Efficiency in 96 After having done this steps correctly the output editor should look like depicted in figure 5 Copyright by Optimod 8 26 October 16 2013 Outp ut Editor gt s r Output Files C Output File Allin One C Single Files New File C Append Reflection Transmissic Filename T Matrix Filename T Matrix C Program Files Filename R Matrix TE Filename R Matrix TM E TE TM O ptimod Unigit Results dielgrat_1d upr Output Format C Float C 4 Digits Fix 8 Digits C Sci IV E
27. tch the convergence is supposed to be better So a truncation order of 5 should be sufficient in what follows Copyright by Optimod 9 26 October 16 2013 output file and choose output orders minimum 1 and maximum 0 In addition it is recommended to check the Project file box and enter a project file name in the output editor Then run Unigit again The reflection and transmission efficiencies are depicted in figures 6 and 7 below These plots can be simply generated by selecting the corresponding result files and hitting the diagram button as described in the manual see section 1 Diffraction Efficiency 0 000 i i 0 000 15 000 30 000 45 000 60 000 esultsXtest000 re esultsXest000 rr esults testm01 re esultsXtestm01 rr Fig 6 Reflection efficiencies vs AOI for the dielgrat 1d example Diffraction Efficiency 0 800 a 0 600 E M RM H 0 200 0 000 0 000 15 000 30 000 45 000 60 000 esults test000 te esults test000 tr esultsXtestm01 te esultsXtestm01 tr Fig 7 Transmission efficiencies vs AOI for the dielgrat 1d example Copyright by Optimod 10 26 October 16 2013 T E versat e rigorous grating sop From the results one can see that the minus first order in reflection occurs at AOI gt 41 8 degrees e the minus first order in transmission appears at AOI gt
28. ter selecting this stack file and opening the stack editor the window should look like shown in figure 12 Copyright by Optimod 14 26 October 16 2013 UNIGIT Stack Editor Superstrat Thickness o um Direct Input Real Part 1 000 Imaginary Part 0 000 r Layers Grating Period x 0 5 um Number of Layers 2 i vp Thickness nm 1 RCWwA Slice Hard Transition 300 000 BARC 2 Thin Film 100 000 r Substrat Thickness D um File Location C Program FilesNoptimodNunigitsNKD ata SILICON NK Edit m Comment Cancel OK et Fig 12 Unigit stack editor showing the PR BARC 1D ust grating stack Obviously the stack is assembled by two layers a RCWA hard transition on top of a thin film exactly as been described above e Two layer 1D ust A patterned Chromium double layer on a dielectric substrate The grating pitch is 0 5 microns The geometry is depicted in figure 13 This example gives a hint how to assemble non binary gratings from rectangular slices An automated slicing tool is available in the full version of UNIGIT 100 nm 250 nm Fig 13 Schematic cross section of the grating implemented in the file Two layer 1D ust The associated stack editor is shown in figure 14 below Copyright by Optimod 15 26 October 16 2013 UNIGIT Stack Editor m Superstrat Thickness 0 um Direct Input Real Part 1 000 Imaginary Part 0 000 m La
29. tput conditions by means of the output editor and then run the extraction by means of the same button which is used for starting a regular run A successful extraction 15 indicated between the CANCEL and EXIT button The phases for the minus first and zeroth order for reflection and transmission are shown in figures 8 and 9 respectively Copyright by Optimod 11 26 October 16 2013 TE Phase TM Phase 157 313339 17 930563 157 311188 17 967262 157 304077 18 077332 157 289963 18 260683 157 265152 18 517145 157 223541 18 846415 157 155319 19 247938 157 043640 19 720623 156 855087 20 261843 156 493500 20 862036 157 477646 22 000513 158 878052 23 249418 159 777939 24 420305 160 480728 25 599087 161 064087 26 806774 161 558853 28 050274 161 977936 29 330267 162 324936 30 643234 162 597122 31 981678 162 786484 33 333485 162 879578 34 680584 162 856537 35 996780 162 688980 37 244614 162 336655 38 371040 161 741867 39 302422 160 820343 39 941456 159 447433 40 177021 157 440735 39 943497 154 560486 39 434765 150 664520 39 676540 146 733826 43 343769 149 171021 53 505356 168 944473 67 672752 175 325195 79 258354 172 469910 86 694618 173 533859 91 453552 175 335587 94 692612 177 162766 96 905693 178 898911 98 179131 179 436630 98 328629 177 796448 96 844849 176 121872 92 301071 35 359726 111 354233 1
30. ulations truncation order and polarization applies for 1D classical For this basic example the following values and settings have to be entered Enter 0 5 microns for the wavelength Enter 0 for the AOI Theta 1 Select the truncation loop by clicking the Truncation radio button Enter truncation orders 1 20 and 1 for start end and step Moreover one has to choose a location where to save the results file to by means of the edit control field and the selection button at the bottom of the CCB After having set all these things correctly the CCB should look like shown below Copyright by Optimod 7 26 October 16 2013 r Stack m Loop Lambda jas nm Select NotePad r Incidence Angle polar Select NotePad C Theta i o sees has V TE Computation TM Computation Set Input r Wavelength in microns Rayleigh Orders Edit New ri Start fi End 20 Step f E dit Project Control Get Stack r Stack File C Program Files Optimod Unigit Stacks dielgrat_1d txt r Computation 0 File s 1D Classic s C 1D C Method 1D Conical Ben C 2D gt Result File s x Diagram m Save Result in D Program Files optimod unigit Results test a Notepad The computation time of the last run 3 was B
31. yers Grating Period x 0 5 pm Number of Layers 2 j No Typ Thickness nm W layer 2 1 RCWwA Slice Hard Transition 100 000 EK layer 1 2 RCWwA Slice Hard Transition 100 000 Add Layer Edit Move up dowr Copy Cut Paste Delete Seq Save Ctrl C Copy the highlighted Layers r Substrat Thickness um Direct Input Real Part 1 500 Imaginary Part 0 000 Edit Comment Fig 14 Unigit stack editor showing the PR BARC 1D ust grating stack 2 4 Sloped gratings Unigit version 2 01 01 offers a few different ways for the simulation of sloped gratings The term sloped is understood as anything different from binary e g sinusoidal triangular or trapezoid Unigit offers two basic methods to input sloped gratings the composite Fourier and the composite polygon layer The details of how this layers can be implemented are given in the Unigit manual see the corresponding sections in the layer editor During the input of this type of gratings a slicing routine is run and a stack of RCWA slices is generated automatically There is of course also another cumbersome way of building up any interface or layer from scratch which shall be only mentioned for reason of completeness namely piling up RCWA slices by means of the stack editor However there are alternatives for describing and simulating sloped gratings The first alternative are the Rayleigh Fourier polygon a
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