Synthesizing Geometries for 21st Century Electromagnetics

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ECE 5322 21 st Century Electromagnetics Instructor: Office: Phone: E Mail: Dr. Raymond C. Rumpf A 337 (915) 747 6958 rcrumpf@utep.

1 ECE st Century Electromagnetics Instructor: Office: Phone: E Mail: Dr. Raymond C. Rumpf A 337 (915) rcrumpf@utep.edu Lecture #18b Synthesizing Geometries for 21st Century Electromagnetics Synthesis of Spatially Variant Lattices 1 Lecture Outline Review of spatially-variant planar gratings Decomposition of lattices into planar gratings Synthesis algorithm for spatially-variant lattices Improving efficiency Extras Deformation control Arrays of discrete elements Spatially-varying lattice symmetry Spatial variance over curved surfaces Slide 2 1

Review of Spatially-Variant Planar Gratings The

vector K is very similar to a wave vector k.

2 Review of Spatially-Variant Planar Gratings The Grating Vector in Two Dimensions y x x The grating vector K is very similar to a wave vector k. The direction of K is perpendicular to the grating planes. The magnitude of K is 2 divided by the period of the grating. 2 K Given the slant angle, it is calculated as 2 K aˆ cos ˆ x aysin It allows a convenient calculation of the analog grating. cos K r Slide 4 2

8 a r Kr cos b r 1 a r 2 a r r r r cos f r 5 Procedure

Define the grating vector 2 Kr aˆ ˆ xcos r aysin r

Calculate the grating phase r from Kr. r K r 3.

3 Analog Vs. Binary Gratings Analog Gratings r 0 Binary Gratings r 0.8 a r Kr cos b r 1 a r 2 a r r r r cos f r 5 Procedure for Generating Spatially-Variant Planar Gratings 1. Define the grating vector 2 Kr aˆ ˆ xcos r aysin r function Kr, or K function. r 2. Calculate the grating phase r from Kr. r K r 3. Calculate the analog grating from the grating phase. 4. Calculate the binary grating from the analog grating. a r cos r b 1 a r 2 a r r r r 6 3

4 Decomposition of Lattices Into Planar Gratings Lattices Can be Decomposed into a Set of Planar Gratings Using the concept of the complex Fourier series, we decompose the unit cell into a set of planar gratings. Each planar grating is described by a grating vector K p and a complex amplitude a p. p 8 4

Recall Complex Fourier Series Periodic functions can be expanded into a Fourier series.

functions, this is 2 px 2y 2 px 2y j j x y 1 x y, p, p,, f xy a e a f xye da A p A For 3D periodic

p r V Slide 9 Two Parts to the Decomposition Input,, f x y a p e p Fourier Coefficients jk p, r

5 Recall Complex Fourier Series Periodic functions can be expanded into a Fourier series. For 1D periodic functions, this is p 2 px 2 2 px j 1 j p p 2 f x a e a f x e dx For 2D periodic functions, this is 2 px 2y 2 px 2y j j x y 1 x y, p, p,, f xy a e a f xye da A p A For 3D periodic functions, this is 2 px 2y 2rz 2 px 2y 2rz j j x y z 1 x y z,, pr,, pr,,,, f xyz a e a f xyze dv V p r V Slide 9 Two Parts to the Decomposition Input,, f x y a p e p Fourier Coefficients jk p, r Grating Vectors y x, FFT, a p 2D f x y p Calculated using FFT. p 2 p 2 Kp, xˆ yˆ Calculated analytically. x y 10 5

6 Visualize the Terms,, f x y a p e p jk p, r p Slide 11 p Grating Vector Part of the Fourier Expansion 2 px 2y j x y p, p f x, y a e K K x y p, p, 2 p x 2 y K p, p j Kxp, x Kyp, y jkp, r p, p p f x, y a e a p, e Slide 12 6

7 Complex Amplitude Part of the Fourier Expansion a 2D FFT p, p 13 Impact of Number of Planar Gratings

8 3D Decomposition 3D Unit Cell K KxˆKyˆKz ˆ x y z 2 p Kx K y x 2 y 2 r Kz z p integer integer r integer 3D Spatial Harmonics 15 Algorithm for Synthesis of Spatially-Variant Lattices 8

$fraction.$ $of fill fraction for the final$

9 Step 1 Generate Input to Algorithm Unit Cell Lattice Size Unit Cell Orientation Lattice Spacing Fill Fraction A grayscale unit cell allows easier adjustment of fill fraction. 17 Step 2 Build a Grayscale Unit Cell A grayscale unit cell allows for better control of fill fraction for the final lattice. 18 9

Step 3 Decompose Unit Cell Into a Set of Planar Gratings 19 Step 4 Truncate

COMPUTE 2D-FFT ERF = fftshift(fft2(er))/(nx*ny); % TRUNCATE SPATIAL HARMONICS

% position of zero-order harmonic p1 = p0 - floor(p/2); %start of p range p2

10 Step 3 Decompose Unit Cell Into a Set of Planar Gratings 19 Step 4 Truncate the Set of Planar Gratings We must retain only a minimum number of spatial harmonics. ER ERF ERF 7 7 before truncation after truncation (P=Q=7) % COMPUTE 2D-FFT ERF = fftshift(fft2(er))/(nx*ny); % TRUNCATE SPATIAL HARMONICS p0 = 1 + floor(nx/2); %p position of zero-order harmonic 0 = 1 + floor(ny/2); % position of zero-order harmonic p1 = p0 - floor(p/2); %start of p range p2 = p0 + floor(p/2); %end of p range 1 = 0 - floor(q/2); %start of range 2 = 0 + floor(q/2); %end of range ERF = ERF(p1:p2,1:2); %truncate harmonics 20 10

Step 5 Spatially Vary Each Planar Grating Individually and Sum the Results For each spatial

Interpolate grating phase into high resolution grid Compute spatially variant planar grating Add

Function Grating Orientation uc (x,y) Intermediate K p + = K p K p K p uc, uc, and p, Kr Kr r

11 Step 5 Spatially Vary Each Planar Grating Individually and Sum the Results For each spatial harmonic Construct spatially variant K function Compute grating phase on low resolution grid Interpolate grating phase into high resolution grid Compute spatially variant planar grating Add planar grating to overall lattice 21 Step 5 For Each Grating a) Construct K-Function Uniform K p Function Grating Orientation uc (x,y) Intermediate K p + = K p K p K p uc, uc, and p, Kr Kr r rruc p, K x r Kr cos r Kr Kr sin r y a uc Intermediate K p Lattice Spacing a(x,y) + = K p -Function 22 11

interp2() p x, y x, y 2, p 2 2 23 Step 5 For Each Grating c) Add All of the Spatially-Variant Gratings Calculate the p th

12 Step 5 For Each Grating b) Solve for the Grating Phase Construct Matrix Euation D k A x xp, b Dy k y, p Solve Using Least Suares T T A A A b A b p 1 Φ A b Reshape Back to a 2D Grid Φ p p x, y Interpolate to a Higher Resolution Grid Using interp2() p x, y x, y 2, p Step 5 For Each Grating c) Add All of the Spatially-Variant Gratings Calculate the p th spatially variant grating p r ap exp j2, p r Add this complex planar grating to the overall analog lattice r r r analog analog p 24 12

$analog r Re analog r From the analog lattice, we calculate the binary lattice with a spatially variant fill fraction.$

13 Step 6 Incorporate Spatially- Variant Fill Fraction At this point, we have the analog lattice. Set any imaginary components to zero. analog r Re analog r From the analog lattice, we calculate the binary lattice with a spatially variant fill fraction. binary r r 1 analogr 2 analogr cos f r r r For analog lattices that have a smooth cosine looking profile, the threshold can be estimated as 25 Summary of Spatially Variant Algorithm 1. Define the spatial variance: orientation, lattice spacing, fill fraction, Build the baseline unit cell. 3. Decompose unit cell into a set of planar gratings. 4. Truncate the set of planar gratings to a minimal set. 5. Loop over each spatial harmonic a. Construct K function that is uniform across the grid according to the grating vector of the spatial harmonic. b. Solve for grating phase from the K function. c. Add the planar grating to the overall lattice. 6. Incorporate spatially variant fill fraction

14 Improving Efficiency of the Algorithm Grid Strategy p r p r 28 14

15 Calculating the Grid Parameters High Resolution Unit Cell Grid Need enough points for the FFT to converge. FFT a NP Low Resolution Lattice Grid Need enough resolution to resolve the spatial variance. Usually this is N>10 grid cells per unit cell. coarse an High Resolution Lattice Grid Need enough resolution to resolve the shortest period planar grating. amin 2 fine amin N max K p 29 Collinear Spatial Harmonics If a spatial harmonic is collinear to another (i.e. their grating vectors are parallel), we can calculate its grating phase by scaling the grating phase of the other. Therefore, we only have to solve for one of them. K 1 K3 ak1 K bk K 1 2 K2 ak1 We solve this numerically. 2 2 K1 1 a a a We just scale 2 1 the solution from. K 1 b 3 1 Again, we simply scale the solution from K

16 Identifying Collinear Planar Gratings 121 spatial harmonics 40 uniue spatial harmonics. The rest are collinear. 31 Performance Gain by Eliminating Collinear Gratings 69% for 2D 59% for 3D 32 16

17 Eliminating Gratings According to Their Amplitude We neglect all planar gratings with amplitude a p less than some threshold. ap a threshold a a threshold 0.02max p A threshold that works in many cases is one that is around 2% of the maximum a p in the expansion. 33 Overall Efficiency Improvement Truncation by Grating Amplitude Truncation by Coplanar K 34 17

Extras Deformation Control (1 of 2) To control deformations of a lattice, we simply control the deformations of the constituent planar gratings in the

18 Extras Deformation Control (1 of 2) To control deformations of a lattice, we simply control the deformations of the constituent planar gratings in the desired manner. R. C. Rumpf, et al, Spatially variant periodic structures in electromagnetics, accepted for publication in Phil. Trans. A, Dec

Rumpf, et al, Spatially variant periodic

publication in Phil. Trans. A, Dec. 2014.

19 2/26/2018 Deformation Control (2 of 2) R. C. Rumpf, et al, Spatially variant periodic structures in electromagnetics, accepted for publication in Phil. Trans. A, Dec Compensating for Deformations Lattice Spacing Deviation Stretched unit cells Compressed unit cells Lattice without any compensation. Lattice with fill fraction compensation 38 19

20 2/26/2018 Overall Algorithm R. C. Rumpf, et al, Spatially variant periodic structures in electromagnetics, accepted for publication in Phil. Trans. A, Dec Arrays of Discontinuous Metallic Elements (1 of 2) + = Spatially vary two planar gratings

41 Spatially Varying Lattice Symmetry K 2 x, y 1 x, y 2 x, y SVL 1 2 K1 x, y

21 2/26/2018 Arrays of Discontinuous Metallic Elements (2 of 2) Place metallic elements at the intersections. 41 Spatially Varying Lattice Symmetry K 2 x, y 1 x, y 2 x, y SVL 1 2 K1 x, y K 2 x, y 1 x, y 2 x, y SVL 1 2 K1 x, y K 2 x, y 1 x, y 2 x, y SVL 1 2 S Hex Hex S Suare to Hex Hexagonal Suare K1 x, y 42 21

22 2/26/2018 Arrays of Metallic Elements Over Curved Surfaces 43 22

Electromagnetic Waves & Polarization

Course Instructor Dr. Raymond C. Rumpf Office: A 337 Phone: (915) 747 6958 E Mail: rcrumpf@utep.edu EE 4347 Applied Electromagnetics Topic 3a Electromagnetic Waves & Polarization Electromagnetic These