SVL Short Course Module 3 -- SV latticesemlab.utep.edu/scSVL/SVL Short Course Module 3 -- SV... · 2015-06-10 · b) Solve for the Grating Phase 24 Construct Matrix Equation,, x xpq

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Pioneering 21st Century Electromagnetics and Photonics

Module 3 –Spatially-Variant Lattices

Course Website:http://emlab.utep.edu/scSVL.htm

EM Lab Website:http://emlab.utep.edu/

Short CourseLecture Outline

• Fourier decomposition of lattices

• Rotation matrices

• Algorithm for synthesis of spatially-variant lattices

• Improving efficiency of the algorithm

• Implementation tips and tricks

Slide 2SVL Short Course -- Module 3

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2

Fourier Decomposition

of Lattices

Short Course

Recall Complex Fourier Series

SVL Short Course -- Module 3 Slide 4

Periodic functions can be expanded into a Fourier series.

For 1D periodic functions, this is

22 2

2

1

px pxj j

p pp

f x a e a f x e dx


2 2 2 2

, ,

1, ,x y x y

px qy px qyj j

p q p qp q A

f x y a e a f x y e dAA


2 2 2 2 2 2

, , , ,

1, , , ,x y z x y z

px qy rz px qy rzj j

p q r p q rp q r V

f x y z a e a f x y z e dVV

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Short Course

Fourier Expansion in Terms of Grating Vectors


2 2

,, x y

px qyj

p qp q

f x y a e

, , ,,, ,x yj K p q x K p q y jK p q r

p qp q p q

f x y a e a p q e

2,

2,

xx

yy

pK p q

qK p q

,K p q

p

q

Short Course

Visualize the Terms


,, , jK p q r

p q

f x y a p q e

p

q

p

q

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Short Course

Lattices Can be Decomposed into a Set of Planar Gratings

7SVL Short Course -- Module 3

p

q

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 and a complex amplitude apq.pqK

Short Course

Two Parts to the Decomposition


,, , jK p q r

p q

f x y a p q e

Input

2 2ˆ ˆ,

x y

p qK p q x y

2D, FFT ,a p q f x y

Calculated using FFT. Calculated analytically.

p

q

Fourier Coefficients

Grating Vectors

p

q

x

y

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Short Course

The Grating Amplitudes


p

q

2D-FFT ,p qa

Short Course

Impact of Number of Planar Gratings

10

1×1

3×3

5×5

7×7

9×9

21×21

31×31

11×11

SVL Short Course -- Module 3

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Short Course

3D Decomposition

11

3D Unit Cell

3D Spatial Harmonics

ˆ ˆ ˆ

2 integer

2 integer

2 integer

x y z

xx

yy

zz

K K x K y K z

pK p

qK q

rK r


RotationMatrices

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Short Course

Definition of a Rotation Matrix

a

b ab R

Rotation matrix [R] is defined as

The rotation matrix should not change the amplitude. This implies that [R] is unitary.

1H

H H

R R

R R R R

b a

I

SVL Short Course -- Module 3 13

Short Course

2D Rotation Matrix

ab

cos sin

sin cosx

R

yy

xb

b

a

a


ab R

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Short Course

3D Rotation Matrices

1 0 0

0 cos sin

0 sin cosxR

Rotation matrices for 3D coordinates can be written directly from the previous result.

cos 0 sin

0 1 0

sin 0 cosyR

cos sin 0

sin cos 0

0 0 1zR

a

b

b b

a

a


Short Course

Composite Rotation Matrix

We can combine multiple rotation matrices into a single composite rotation matrix [R].

ab R


z y xR R R R Vector rotation using the composite rotation matrix is

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Algorithm for Synthesis of

Spatially-Variant Lattices

Short Course

Step 1 – Generate Input to Algorithm

18

Unit Cell Lattice SizeUnit Cell

OrientationLattice Spacing Fill Fraction

A grayscale unit cell allows easier adjustment of fill fraction.


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Short Course

Step 2 – Build a Grayscale Unit Cell


A grayscale unit cell allows for better control of fill fraction in the end.

Short Course

Step 3 – Decompose Unit Cell Into a Set of Planar Gratings


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Short Course

Step 4 – Truncate the Set of Planar Gratings

21

We must retain only a minimum number of spatial harmonics.

% COMPUTE 2D-FFTERF = fftshift(fft2(ER))/(Nx*Ny);

% TRUNCATE SPATIAL HARMONICSp0 = 1 + floor(Nx/2); %p position of zero-order harmonicq0 = 1 + floor(Ny/2); %q position of zero-order harmonicp1 = p0 - floor(P/2); %start of p rangep2 = p0 + floor(P/2); %end of p rangeq1 = q0 - floor(Q/2); %start of q rangeq2 = q0 + floor(Q/2); %end of q rangeERF = ERF(p1:p2,q1:q2); %truncate harmonics

ERF128×128before truncation

ER128×128 ERF7×7after truncation (P=Q=7)

ERF128×128before truncation

ER128×128 ERF7×7after truncation (P=Q=7)


Short Course

Step 5 – Spatially Vary Each Planar Grating Individually and Sum the Results

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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

For each spatial harmonic…


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Short Course

Step 5 – For Each Gratinga) Construct K-Function

23

Unit Cell Orientation (x,y)

Lattice Spacing a(x,y)

Uniform Kpq Function

+ =

Intermediate Kpq

+ =

Intermediate Kpq K-Function

pqK


Short Course

Step 5 – For Each Gratingb) Solve for the Grating Phase

24

Construct Matrix Equation

,

,

x x pq

y y pq

D kA b

D k

Solve Using Least Squares

1

pq

Φ A b

Reshape Back to a 2D Grid

,pq pq x yΦ

Interpolate to a Higher Resolution Grid Using interp2()

T T A A A b A b

2, 2 2, ,pq pqx y x y


All of this is performed by

svlsolve(Kx,Ky,dx,dy)

or

svlsolve(Kx,Ky,Kz,dx,dy,dz)

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Short Course

Step 5 – For Each Gratingc) Sum All of the Spatially-Variant Gratings

25

Generate the pqth spatially variant grating

Add this analog planar grating to the overall analog lattice

2,exppq pq pqr a j r

analog analog pqr r r


Short Course

Step 6 – Incorporate Spatially-Variant Fill Fraction

26

At this point, we have the analog lattice. Set any imaginary components to zero.

cosr f r


1 analogbinary

2 analog

r rr

r r

analog analogRer r

From the analog lattice, we calculate the binary lattice with a spatially-variant fill fraction.

From analog lattices that have a smooth cosine looking profile, the threshold can be estimated as

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Short Course

Summary of Spatially Variant Algorithm

27

1. Define the spatial variance: orientation, lattice spacing, fill fraction,...

2. 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.


Improving Efficiency of the

Algorithm

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Short Course

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.

1K

2 1K aK

3 1K bK

1 1K

2 2 1

2 21 1

2 1

K aK

Ka a

a

3 1b

We solve this numerically.

We just scale the solution from .1K

Again, we simply scale the solution from .1K

Short Course

Identifying Collinear Planar Gratings


121 spatial harmonics 40 “unique” spatial harmonics.

The rest are collinear.

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Short Course

Performance Gain by Eliminating Collinear Gratings


69% for 2D

59% for 3D

Short Course

Eliminating Gratings According to Their Amplitude


thresholdpqa a

We neglect all planar gratings with amplitude apq less than some threshold.

threshold 0.02max pqa a A threshold that works in many cases is one that is around 2% of the maximum apq in the expansion.

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Short Course

Overall Efficiency Improvement


Truncation by Grating Amplitude

Truncation by Coplanar K

ImplementationTips and Tricks

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Short Course

Grid Strategy


Short Course

Calculating the Grid Parameters

36

• High Resolution Unit Cell Grid– Need enough points for the FFT to converge.

• Low Resolution Lattice Grid– Need enough resolution to resolve the spatial variance. Usually this is N>10

grid cells per unit cell.

• High Resolution Lattice Grid– Need enough resolution to resolve the shortest period planar grating.

coarse a N

FFT a N P

minfine min

2

max pq

aa

N K


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Short Course

Arrays of Discontinuous Metallic Elements (1 of 2)


Place metallic elements at the intersections.

37

Short Course

Arrays of Discontinuous Metallic Elements (2 of 2)


When the lattice is composed of an array of discontinuous metallic elements, it is not efficient to decompose the unit cells into planar gratings.

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Short Course

Arrays of Metallic Elements Over Curved Surfaces


Documents

SVL Short Course Module 3 -- SV latticesemlab.utep.edu/scSVL/SVL Short Course Module 3 -- SV... · 2015-06-10 · b) Solve for the Grating Phase 24 Construct Matrix Equation,, x xpq