5/6/2016
ECE 5322 21st Century Electromagnetics Instructor: Office: Phone: E‐Mail:
Dr. Raymond C. Rumpf A‐337 (915) 747‐6958
[email protected]
Lecture #18b
Synthesizing Geometries for 21st Century Electromagnetics Synthesis of Spatially Variant Lattices Lecture 18b
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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 – Spatial variance over curved surfaces
Lecture 18b
Slide 2
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Review of Spatially-Variant Planar Gratings
The Grating Vector in Two Dimensions
x
K is very similar to a • The grating vector wave vector k. • The direction of is perpendicular to K the grating planes. K • The magnitude of is 2 divided by the period of the grating . 2 K
• Given the slant angle , it is calculated as 2 K aˆx cos aˆ y sin
y
x Lecture 18b
• It allows a convenient calculation of the analog grating. cos K r
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Analog Vs. Binary Gratings Binary Gratings
Analog Gratings
r 0
a r cos K r
r 0.8
a r r a r r
b r 1 2
r cos f r Lecture 18b
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Procedure for Generating Spatially-Variant Planar Gratings 1. Define the grating vector function , or K‐function. K r
2 K r aˆ x cos r aˆ y sin r r
r 2. Calculate the grating phase K r from .
r K r
3. Calculate the analog grating from the grating phase.
a r cos r
4. Calculate the binary grating from the analog grating.
b r 1 2
Lecture 18b
a r r a r r
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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 pq and a complex amplitude apq.
q
p Lecture 18b
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Recall Complex Fourier Series Periodic functions can be expanded into a Fourier series. For 1D periodic functions, this is f x
ae
j
p
p
2 px
2
ap
2 px j 1 f x e dx 2
For 2D periodic functions, this is f x, y
p q
a p ,q e
2 px 2 qy j x y
a p ,q
2 px 2 qy x y
j 1 f x, y e A A
dA
For 3D periodic functions, this is f x, y , z
a p ,q ,r e
2 px 2 qy 2 rz j y z x
p q r
a p ,q ,r
1 V
f x, y , z e
2 px 2 qy 2 rz j y z x
dV
V
Lecture 18b
Slide 9
Two Parts to the Decomposition f x, y
a p, q e
jK p , q r
p q
q
y
x
q
p a p, q FFT2D f x, y
Calculated using FFT. Lecture 18b
Grating Vectors
Fourier Coefficients
Input
p 2 p 2 q K p, q xˆ yˆ x y
Calculated analytically. 10
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Visualize the Terms
f x, y
a p, q e
jK p , q r
p q
q
q
p
p
Lecture 18b
Slide 11
Grating Vector Part of the Fourier Expansion f x, y
a
p q
e
K x p, q
2 p x
K y p, q
2 q y
f x, y
p q
Lecture 18b
p ,q
2 px 2 qy j y x
a p ,q e
j K x p , q x K y p , q y
K p, q
q
p
a p, q e jK p ,q r
p q
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Complex Amplitude Part of the Fourier Expansion
a p ,q
2D‐FFT
q
p
Lecture 18b
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Impact of Number of Planar Gratings
Lecture 18b
1×1
9×9
3×3
11×11
5×5
21×21
7×7
31×31
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3D Decomposition 3D Unit Cell
K K x xˆ K y yˆ K z zˆ Kx
2 p x
p integer
Ky
2 q y
q integer
Kz
2 r z
r integer 3D Spatial Harmonics
Lecture 18b
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Algorithm for Synthesis of Spatially-Variant Lattices
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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.
Lecture 18b
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Step 2 – Build a Grayscale Unit Cell A grayscale unit cell allows for better control of fill fraction for the final lattice.
Lecture 18b
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Step 3 – Decompose Unit Cell Into a Set of Planar Gratings
Lecture 18b
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Step 4 – Truncate the Set of Planar Gratings We must retain only a minimum number of spatial harmonics. ER128×128
ERF128×128
before truncation
ERF7×7
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 q0 = 1 + floor(Ny/2); %q position of zero-order harmonic p1 = p0 - floor(P/2); %start of p range p2 = p0 + floor(P/2); %end of p range q1 = q0 - floor(Q/2); %start of q range q2 = q0 + floor(Q/2); %end of q range ERF = ERF(p1:p2,q1:q2); %truncate harmonics Lecture 18b
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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
Lecture 18b
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Step 5 – For Each Grating a) Construct K-Function Grating Orientation uc(x,y)
Uniform Kpq Function
+
Intermediate Kpq
=
K pq
K uc p, q K uc p, q
Intermediate Kpq
Kpq-Function
Lattice Spacing a(x,y)
and uc p, q a K r r r r uc p, q K x r K r cos r K y r K r sin r
Kr
Lecture 18b
+
=
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Step 5 – For Each Grating b) Solve for the Grating Phase Construct Matrix Equation Dx A D y
k x , pq b k y , pq
Solve Using Least Squares A A T A
b A T b
Φ pq A b 1
Reshape Back to a 2D Grid Φ pq pq x, y
Interpolate to a Higher Resolution Grid Using interp2() pq x, y 2, pq x2 , y2 Lecture 18b
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Step 5 – For Each Grating
c) Add All of the Spatially-Variant Gratings Calculate the pqth spatially variant grating
pq r a pq exp j 2, pq r
Add this complex planar grating to the overall analog lattice
analog r analog r pq r
Lecture 18b
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Step 6 – Incorporate SpatiallyVariant 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.
analog r r analog r r
binary r 1 2
For analog lattices that have a smooth cosine looking profile, the threshold can be estimated as
r cos f r
Lecture 18b
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Summary of Spatially Variant Algorithm 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.
Lecture 18b
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Improving Efficiency of the Algorithm
Grid Strategy pq r
pq r
Lecture 18b
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Calculating the Grid Parameters • High Resolution Unit Cell Grid – Need enough points for the FFT to converge.
FFT a N P
• Low Resolution Lattice Grid – Need enough resolution to resolve the spatial variance. Usually this is N>10 grid cells per unit cell.
coarse a N • High Resolution Lattice Grid – Need enough resolution to resolve the shortest period planar grating.
fine
amin N
amin
2 max K pq
Lecture 18b
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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. K1
1 K1 We solve this numerically.
K 3 aK1
2 K 2 aK1
2 K1 a
2 a1 Lecture 18b
K3 bK1
We just scale the solution from . K1
3 b1 2 a
1
Again, we simply scale the solution from . K1
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Identifying Collinear Planar Gratings
121 spatial harmonics
40 “unique” spatial harmonics. The rest are collinear.
Lecture 18b
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Performance Gain by Eliminating Collinear Gratings
69% for 2D
59% for 3D Lecture 18b
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Eliminating Gratings According to Their Amplitude
We neglect all planar gratings with amplitude apq less than some threshold. a pq athreshold
athreshold 0.02 max a pq
A threshold that works in many cases is one that is around 2% of the maximum apq in the expansion. Lecture 18b
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Overall Efficiency Improvement Truncation by Grating Amplitude
Lecture 18b
Truncation by Coplanar K
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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. 2014. Lecture 18b
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Deformation Control (2 of 2)
R. C. Rumpf, et al, “Spatially‐variant periodic structures in electromagnetics,” accepted for publication in Phil. Trans. A, Dec. 2014. Lecture 18b
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Compensating for Deformations Lattice Spacing Deviation
Stretched unit cells
Compressed unit cells Lattice without any compensation. Lecture 18b
Lattice with fill fraction compensation 38
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Overall Algorithm
R. C. Rumpf, et al, “Spatially‐variant periodic structures in electromagnetics,” accepted for publication in Phil. Trans. A, Dec. 2014. Lecture 18b
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Arrays of Discontinuous Metallic Elements (1 of 2)
+
=
Spatially vary two planar gratings. Lecture 18b
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Arrays of Discontinuous Metallic Elements (2 of 2)
Place metallic elements at the intersections. Lecture 18b
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Arrays of Metallic Elements Over Curved Surfaces
Lecture 18b
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