Synthesizing Geometries for 21st Century Electromagnetics

Transcription

1 ECE st Century Electromgnetics Instructor: Office: Phone: E Mil: Dr. Rymond C. Rumpf A 337 (915) rcrumpf@utep.edu Lecture #18 Synthesizing Geometries for 21st Century Electromgnetics Synthesis of Sptilly-Vrint Plnr Grtings 1 Lecture Outline Preliminry Concepts The grting vector Grting phse Generting sptilly-vrint plnr grtings Etrs More efficient grid strtegy Deformtion control Sptilly-vrint plnr grtings on curved surfces Slide 2 1

2 The Grting Vector K. The Grting Vector in Two Dimensions y The grting vector K is very similr to wve vector k. The direction of K is perpendiculr to the grting plnes. The mgnitude of K is 2 divided by the period of the grting. 2 K Given the slnt ngle, it is clculted s 2 K ˆ cos ˆ ysin It llows convenient clcultion of the nlog grting. cos K r Slide 4 2

3 The Grting Vector in Three Dimensions r r K r r ˆ yyˆzzˆ vg cos position vector 2 K K K ˆK yˆkzˆ y z K Slide 5 The Anlog Grting y For simplicity, clcultion of the nlog grting will be written s r cos K r or r Re ep jk r If insted you wish to generte physicl nlog grting, it will need to be scled to convey permittivity vlues. r cos K r vg Slide 6 3

4 The Binry Grting Anlog Grting Binry Grting Controlling the Fill Frction r Kr cos r cos Kr 0 r cos Kr 0.8 Genertion of the grting using cosine function gives us smooth nlog profile. This requires functionlly grding the mteril properties to relize this. Tht is hrd. We cn use threshold to convert the nlog profile to binry grting. This is much esier to physiclly relize. We cn djust the threshold vlue to control the duty cycle, or fill frction, of the grting. 7 Grting Phse r 4

5 Concept of Grting Phse Here we mke n nlogy with stndrd wve. A wve propgtes in the direction of the wve vector k. E r E cos tk r 0 As the wve propgtes, it ccumultes phse which is function tht increses in the direction of k. The phse results from the wve propgting with wve vector k nd we could lterntively reconstruct the wve just from the phse. E r E cos 0 r r tk r For grtings, the grting vector K serves similr purpose s the wve vector k does for wves. kr Kr rr E r r k r or K r or r r E r r or 9 Definition of Grting Phse We cn think of grting s wve becuse it hs lmost the sme mthemticl form. As wve propgtes it ccumultes phse. The wve cn be clculted from just the phse. Think of the grting phse this wy rcoskr cos r r Kr KKy y K X y r K K y K Y y + = cos Note: Unfortuntely, this intuitive definition of grting phse only holds when K is constnt nd not function of position. 10 5

6 Problem of Chirped Grting Suppose we wish to generte the following chirped grting with period (z). z z 1 z 2 0 This 1D grting is described by the following grting vector function. 2 2 Kz z z 0 Wht hppens when we try to clculte the grting using cos K r? 2 2 r zcos Kzz cos z cos z The rgument in the cosine is constnt so we do not even generte grting here. 11 Conclusion From the Chirped Grting When the grting vector is function of position, we cn no longer clculte the grting directly from it. r cos Kr r Insted, we must generte the grting through the intermedite prmeter of the grting phse. r cos r However, we must dopt more rigorous definition of grting phse. r K r Key eqution r K r r 12 6

7 Revised Solution for Chirped Grtings Now let s construct the chirped grting using the grting phse. K r d 2 dz z z z dz dz 0z 0z 2 z ln z1 0 The nlog grting is then 2 zcos z cos ln z1 0 1 z 1 This term is just constnt. Since it is phse, we re free to choose whtever is convenient. Here we choose zero. 2 2 dz ln z z 0 0 z 1 13 Generting Sptilly-Vrint Plnr Grtings 7

8 Procedure for Generting Sptilly-Vrint Plnr Grtings 1. Define the grting vector 2 Kr ˆ ˆ cos r ysin r function Kr, or K function. r 2. Clculte the grting phse r from Kr. r K r 3. Clculte the nlog grting from the grting phse. 4. Clculte the binry grting from the nlog grting. r cos r b 1 r 2 r r r r 15 Sptilly-Vrint Plnr Grtings r r r r r r No sptil vrince Sptilly Vrint Orienttion Sptilly Vrint Period Sptilly Vrint Threshold Sptilly Vrint Everything 16 8

9 Build Input Functions We need two things to build the K function: 1. Orienttion of the grting s function of position, r. 2. Period of the grting s function of position, r. r r tn y % GRATING PARAMETERS = 1; er1 = 2.5; er2 = 1.0; gth = 0; % GRID PARAMETERS S = 5*; Sy = S; N = 100; Ny = round(n*sy/s); % CALCULATE GRID d = S/N; dy = Sy/Ny; = [1:N]*d; y = [1:Ny]*dy; [Y,X] = meshgrid(y,); % SPATIALLY-VARIANT PARAMETERS PER = *ones(n,ny); THETA = tn2(y,x); Clculte the K-Function The K function is clculted directly from the input functions. K r K y r 2 K r cos r r 2 K y r sin r r % CALCULATE K-FUNCTION K = 2*pi./PER.*cos(THETA); Ky = 2*pi./PER.*sin(THETA);

10 Finite-Difference Solution to r K r We strt by epnding the governing eqution. K r r Kr y r Ky r z Kz r We pproimte this eqution using finite differences. K r D k y r Ky r Dy Φ k y z Kz r D z kz Neumnn boundry conditions cn be used in the derivtive opertors, but there my eist better boundry conditions to use. Φ column vector contining grting phse throughout grid k i column vector contining Ki throughout grid D bnded mtri tht clcultes prtil derivtive long ith is i 19 More Equtions Thn Unknowns Our liner lgebr problem hs more equtions thn it hs unknowns. D k Dy Φ k y D z kz DΦ k DΦ y k DΦk z y z It is usully not possible for the solution to simultneously stisfy ll of the equtions. For this reson, our solution must be best fit nd we cn solve it using lest squres

11 Solution in the Sense of Lest Squres (1 of 2) First, we cst our problem into the stndrd form A = b. D k A b A y,, D Φ b k D z k Net, we premultiplying both sides by A T. T T AAAb 21 Solution in the Sense of Lest Squres (2 of 2) We now hve new system of equtions with the sme number of equtions s there re unknowns. A b A b T A A T A b This is solved for using stndrd liner lgebr methods. 1 A b 22 11

12 Clculte Grting Phse in 2D We clculte the grting phse from the K-function s best fit. r K D k Φ D k y y r % CONSTRUCT DERIVATIVE OPERATORS NS = [N Ny]; RES = [d dy]; BC = [1 1]; [DX,~,DY,~] = fdder(ns,res,bc); % COMPUTE GRATING PHASE A = [ DX ; DY ]; b = [ K(:) ; Ky(:) ]; PHI = (A. *A)\(A. *b); PHI = reshpe(phi,n,ny); 23 Clculte Anlog Grting The nlog grting is clculted directly from the grting phse. r r cos r % COMPUTE ANALOG GRATING ERA = cos(phi); 24 12

13 Clculte Binry Grting The binry grting is clculted directly from the nlog grting using the threshold technique. b r b r 1 r 2 r r r % COMPUTE BINARY GRATING ERB = er1*(era <= gth) + er2*(era > gth); 25 Controlling Fill Frction Through the Threshold Function Anlog Grting r b r 1 r 2 r r r We cn estimte the threshold vlue in order to relize given fill frction f of 1. r cos f r f = 0 f = 0.2 f = 0.4 f = 0.6 f = 0.8 f = = 1.0 = 0.8 = 0.3 = -0.3 = -0.8 =

14 More Efficient Grid Strtegy Low-Res / High-Res Grids Observe how smooth the grting phse function is compred to the finl grting. We cn usully get wy with much corser grid up to the point where the grting phse is clculted. r Corse Grid PER, THETA, K, PHI b r Fine Grid FF, PHI2, ERA, ERB 28 14

15 Deformtion Control Typicl Input Functions r f r K r K r K y r 30 15

16 Typicl Results r b r 31 Typicl Problems Grting orienttion is not correct here. b r Grting period is stretched here. Grting period is compressed here. Grting orienttion is not correct here

17 Wht Grting Did We Relly Get? r We strted the process with K function tht we will cll the trget K function. r 2 K ˆ cos ˆ T r r ysin r From this, we clculted the grting phse s best fit. r K r T Given the grting phse, we cn clculte the ctul K function tht we got from the grdient of the grting phse. K r r % CALCULATE ACTUAL K-FUNCTION KA = [DX;DY]*PHI(:); M = N*Ny; K = KA(1:M); Ky = KA(M+1:2*M); K = reshpe(k,n,ny); Ky = reshpe(ky,n,ny); 33 Comprison of K-Functions KT, r KT,y r K r T K, r K,y r K r 34 17

18 Quntifying the Problems (1 of 2) The two prmeters quntified by the K function re the grting period nd the grting orienttion. We cn clculte the ctul grting period nd ctul grting orienttion from the ctul K function. 2 K r r r K r r 1 y tn K r r r r r r % CALCULATE PROBLEMS KM = sqrt(bs(k).^2 + bs(ky).^2); PERA = 2*pi./KM; THETAA = tn2(ky,k); DPER = PER - PERA; DTHETA = THETA - THETAA; 35 Quntifying the Problems (2 of 2) r r r T T r r r 36 18

19 Four Approches to Improve the Lttice 1. Improve the grting period t the cost of the grting orienttion. 2. Improve the grting orienttion t the cost of the grting period. 3. Improve the grting in some prts t the cost of the other prts. 4. Hybrids nd weighted combintions of the bove pproches. 37 Improve Grting Period t the Cost of Grting Orienttion (1 of 2) Step 1 Compute the trget K function 2 K ˆ ˆ T r cos T r ysin T r T r Step 2 Solve for the grting phse r K T r Step 3 Compute the ctul functions 2 Ky r K r 1 Kr r r r tn K r Step 4 Enforce the grting period in new K function 2 Knew r ˆ cos ˆ r ysin r T r Step 5 Compute the grting phse from the new K function r K new r Step 6 If grting hs not converged, go bck to Step 3. for iter = 1 : NITER % CALCULATE K-FUNCTION K = 2*pi./PER.*cos(THETA); Ky = 2*pi./PER.*sin(THETA); % SOLVE FOR GRATING PHASE A = [ DX ; DY ]; b = [ K(:) ; Ky(:) ]; b = A.'*b; A = A.'*A; PHI = A\b; % COMPUTE ACTUAL K-FUNCTION KA = [DX;DY]*PHI(:); K = KA(1:M); Ky = KA(M+1:2*M); % COMPUTE PERIOD AND ANGLE THETA = tn2(ky,k); THETA = reshpe(theta,n,ny); end 38 19

20 3/20/2018 Improve Grting Period t the Cost of Grting Orienttion (2 of 2) Grting orienttion is wy off here. We observe convergence fter bout 10 itertions. Notice tht orienttion hs been scrificed. 39 Improve Grting Orienttion t the Cost of Grting Period (1 of 2) Step 1 Compute the trget K function KT r for iter = 1 : NITER 2 ˆ cos T r ˆ y sin T r T r % CALCULATE K-FUNCTION K = 2*pi./PER.*cos(THETA); Ky = 2*pi./PER.*sin(THETA); Step 2 Solve for the grting phse r KT r Step 3 Compute the ctul functions K r r 2 r K r Ky r K r r tn 1 Step 4 Enforce the grting orienttion in new K function K new r % COMPUTE ACTUAL K-FUNCTION KA = [DX;DY]*PHI(:); K = KA(1:M); Ky = KA(M+1:2*M); 2 ˆ cos T r ˆ y sin T r r Step 5 Compute the grting phse from the new K function % COMPUTE PERIOD AND ANGLE KM = sqrt(bs(k).^ bs(ky).^2); PER = 2*pi./KM; PER = reshpe(per,n,ny); r K new r Step 6 If grting hs not converged, go bck to Step 3. % COMPUTE GRATING PHASE A = [ DX ; DY ]; b = [ K(:) ; Ky(:) ]; b = A.'*b; A = A.'*A; PHI = A\b; end 40 20

21 Improve Grting Orienttion t the Cost of Grting Period (2 of 2) Grting period is wy off here. We observe convergence fter bout 50 itertions. Notice tht period hs been scrificed. 41 Improve Grting In Some Regions t the Cost of Other Regions (1 of 2) Step 1 Compute the trget K function 2 K ˆ ˆ T r cos T r ysin T r T r Step 2 Solve for the grting phse r K T r Step 3 Compute the ctul functions 2 K 1 y r Kr r r r tn K r K r Step 4 Enforce the grting orienttion in new K function KT r inside of regions to optimize Knew r K r outside of regions to optimize Step 5 Compute the grting phse from the new K function r K new r Step 6 If grting hs not converged, go bck to Step 3. % CALCULATE TARGET K-FUNCTION KT = 2*pi./PER.*cos(THETA); KTy = 2*pi./PER.*sin(THETA); 42 K Ky = KT; = KTy; for iter = 1 : NITER % COMPUTE GRATING PHASE A = [ DX ; DY ]; b = [ K(:) ; Ky(:) ]; b = A.'*b; A = A.'*A; PHI = A\b; % COMPUTE ACTUAL K-FUNCTION KA = [DX;DY]*PHI(:); KA = KA(1:M); KAy = KA(M+1:2*M); % ENFORCE CRITICAL REGION R KA = reshpe(ka,n,ny); KAy = reshpe(kay,n,ny); K = KT.*R + KA.*(1 - R); Ky = KTy.*R + KAy.*(1 - R); end 21

22 3/20/2018 Improve Grting In Some Regions t the Cost of Other Regions (2 of 2) Period nd orienttion optimized in this circle region. Grting period nd orienttion re wy off here. We observe convergence fter bout 3 itertions. Notice the lttice is most distorted just outside the criticl region. 43 Block Digrm for Controlling Deformtions 44 22

23 Another Emple Grting generted from stndrd lgorithm. Grting generted fter enforcing the orienttion t the epense of period. Grting generted fter enforcing the period t the epense of orienttion. 45 Sptilly-Vrint Plnr Grtings on Curved Surfces Rymond C. Rumpf, Jvier J. Pzos, Jennefir L. Digum, Stephen M. Kuebler, Sptilly Vrint Periodic Structures in Electromgnetics, ccepted for publiction in Phil. Trns. A, December

24 The Problem Suppose we wish to put plnr grting onto curved surfce without deforming the orienttion or period of the grting. In this cse, we must sptilly vry the grting in order not to sptilly vry it! 47 Curved Surfce Mesh b â On curved surfce, our sense of nd y directions must be modified to conform to the curvture. We will cll these directions â nd b nd they re defined to lwys be tngentil to the surfce

25 Modified Grting Phse Eqution We must modify our grting phse eqution to operte on the curved mesh. r K r ˆ ˆ b b 49 Modified Finite-Difference Approimtion r r ˆ b r bˆ r, ˆ pq r r r r rp1, q rp1, q rp, q1 rp, q1 p1, q p1, q pq, 1 pq, 1 bˆ 50 25

26 Emple Plnr Grtings Grting #1 Grting #2 Perspective view Perspective view top view top view 51 26