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4/17/2020
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Advanced Electromagnetics:
21st Century Electromagnetics
The Plane Wave Spectrum
Lecture Outline
•Complex Fourier series
• The plane wave spectrum•Plane wave spectrum for crossed gratings
Slide 2
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Complex Fourier Series
Jean Baptiste Joseph Fourier
Born:
Died:
March 21, 1768 in Yonne, France.
May 16, 1830 in Paris, France.
Slide 3
1D Complex Fourier Series
Slide 4
If a function 𝑓 𝑥 is periodic with period Λ , it can be expanded into a complex Fourier series.
2
22
2
1
x
x
x
x
mxj
m
mxj
x
f x a m e
a m f x e dx
In computer analysis, only a finite number of terms can be retained in the expansion.
2
x
mxM j
m M
f x a m e
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2D Complex Fourier Series
Slide 5
For 2D periodic functions, the complex Fourier series generalizes to
2 2
2 2
, ,
1, ,
x y
x y
px qyj
p q
px qyj
A
f x y a p q e
a p q f x y e dxdyA
Λ
Λ
The Plane Wave Spectrum
Slide 6
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Periodic Functions Can Be Expanded into a Fourier Series
Slide 7
2
2
x
x
mxj
m
mxj
A x S m e
S m A x e dx
The envelope A(x) is periodic along x with period x so it can be expanded into a Fourier series.
Waves in periodic structures obey Bloch’s equation.
, j rE x y A x e
x A xExamine a cross section
2
x
mxj
j r
m
S m e e
Rearrange the Fourier Series (1 of 2)
Slide 8
x A x
Plane wave term 𝑒 · ⃗ is brought inside of the summation. 2
x
mxj
j r
m
S m e e
2
yx xz
mxj
j yj x j z
m
S m e e e e
Envelope term 𝐴 𝑥 is expanded into a complex Fourier series.
The dot product 𝛽 · 𝑟 is expanded.
Waves in periodic structures obey Bloch’s equation.
, j rE x y A x e
Examine a cross section
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Rearrange the Fourier Series (2 of 2)
Slide 9
x A x
x can be combined with the last complex exponential.
2
, yx xz
mxj
j yj x j z
m
E x y S m e e e e
2
xyx z
mj x
j y j z
m
S m e e e
Now let 𝑘 𝑚 𝛽 , 𝑘 𝑚 𝛽 and 𝑘 𝑚 𝛽 .
, jk m r
m
E x y S m e
2ˆ ˆ ˆx x y y z z
x
mk m a a a
Examine a cross section
The Plane Wave Spectrum
Slide 10
x A x
, jk m r
m
E x y S m e
Examine a cross section
Terms were rearranged to show that a periodic field can also be expressed as an infinite sum of plane waves. This is the “plane wave spectrum” of a periodic field.
FieldPlane Wave Spectrum
=
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Longitudinal Wave Vector Components of the Plane Wave Spectrum
The wave incident on a 1D grating can be written as
,inc ,inc
,inc 0 inc inc
,incinc 0
,inc 0 inc inc
sin
0,
cos
x z
xj k x k z
y
z
k k n
kE x y E e
k k n
Phase matching into the grating leads to
,inc
2x x
x
k m k m
, 2, 1,0,1, 2,m
Each plane wave must satisfy the dispersion relation.
22 20 grat
2 20 grat
x z
z x
k m k m k n
k m k n k m
There are two possible solutions here.1. Purely real 𝑘 𝑚2. Purely imaginary 𝑘 𝑚
Note: 𝑘 𝑚 is always real.
incx
z
Visualizing Phase Matching into the Grating
Slide 12
The wave vector expansion for the first 11 diffraction orders can be visualized as…
inck
0xk 1xk 2xk 3xk 4xk 5xk 1xk 2xk 3xk 4xk 5xk
2n
1n
𝑘 𝑚 is real. A wave propagates into material 2.
𝑘 𝑚 is imaginary. The field in material 2 is evanescent.
𝑘 𝑚 is imaginary. The field in material 2 is evanescent.
Each of these is phase matched into material 2. The longitudinal component of the wave vector is calculated using the dispersion relation in material 2.
Note: The “evanescent” fields in material 2 do not have a completely imaginary wave vector. They have a purely real 𝑘 𝑚 so power does flow in the transverse direction.
x
z
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Conclusions About the Plane Wave Spectrum
• Fields in periodic media take on the same periodicity as the media they are in.
• Periodic fields can be expanded into a Fourier series.• Each term of the Fourier series represents a diffraction order.
• Since there are in infinite number of terms in the Fourier series, there are an infinite number of diffraction orders.
•Only a few of the diffraction orders are propagating waves and only these transport power to and from a device.
Slide 13
Plane Wave Spectrum from
Crossed Gratings
Slide 14
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Grating Terminology
Slide 15
1D gratingRuled grating
Requires a 2D simulation
2D gratingCrossed grating
Requires a 3D simulation
Diffraction from Crossed Gratings
Slide 16
Doubly‐periodic gratings, also called crossed gratings, can diffract waves into many directions.
Square gratings are described by two grating vectors, 𝐾 and 𝐾 .
Two boundary conditions are necessary here.
,inc
,inc
..., 2, 1,0,1, 2,...
..., 2, 1,0,1,2,...
x x x
y y y
k m k mK m
k n k nK n
inck
xK
yK
2ˆ
2ˆ
xx
yy
K x
K y
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Transverse Wave Vector Expansion (1 of 2)
Slide 17
Crossed gratings can diffract in all directions.
To quantify diffraction for crossed gratings, an expansion must be calculated for both kx and ky.
% TRANSVERSE WAVE VECTOR EXPANSIONM = [-floor(Nx/2):floor(Nx/2)]';N = [-floor(Ny/2):floor(Ny/2)]';kx = kxinc - 2*pi*M/Lx;ky = kyinc - 2*pi*N/Ly;[ky,kx] = meshgrid(ky,kx);
,inc
,inc
2 , , 2, 1,0,1, 2,
2 , , 2, 1,0,1, 2,
ˆ ˆ,
x xx
y yy
t x y
mk m k m
nk n k n
k m n k m x k n y
This code is used in many numerical methods including 2D PWEM, 3D RCWA, 3D FDTD, 3D FDFD, 3D MoL, and more.
Transverse Wave Vector Expansion (2 of 2)
Slide 18
The vector expansions can be visualized this way…
xk m yk n
ˆ ˆ,t x yk m n k m x k n y
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Longitudinal Wave Vector Expansion (1 of 2)
Slide 19
The longitudinal components of the wave vectors are computed as
2 2 2,ref 0 ref
2 2 2,trn 0 trn
,
,
z x y
z x y
k m n k n k m k m
k m n k n k m k m
The lowest‐order modes will have real kz’s. These correspond to propagating waves. The others will have imaginary kz’s and correspond to evanescent waves that do not transport power away from the diffraction grating.
,ref ,zk m n ,tk m n
% Compute Longitudinal Wave Vector Componentsk0 = 2*pi/lam;kzinc = k0*nref;kzref = -sqrt((k0*nref)^2 - kx.^2 - ky.^2);kztrn = +sqrt((k0*ntrn)^2 - kx.^2 - ky.^2);
propagating waves
evanescent fields
Longitudinal Wave Vector Expansion (2 of 2)
Slide 20
The overall wave vector expansion can be visualized this way…
,ref ,zk m n ,tk m n
ref ,k m n trn ,k m n
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