lecture 3b -- diffraction gratings and the plane wave spectrum...kkn exy eek kkn phase matching into...
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Advanced Computation:
Computational Electromagnetics
Diffraction Gratings &the Plane Wave Spectrum
Outline
• Fourier Series
• Diffraction from gratings
• The plane wave spectrum
• Plane wave spectrum for crossed gratings
• Power flow from gratings
Slide 2
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Slide 3
Fourier Series
Jean Baptiste Joseph Fourier
Born:
Died:
March 21, 1768in Yonne, France.
May 16, 1830in Paris, France.
1D Complex Fourier Series
Slide 4
If a function f(x) is periodic with period x, it can be expanded into a complex Fourier series.
2
22
2
1
x
x
x
x
pxj
pp
pxj
px
f x a e
a f x e dx
Typically, only a finite number of terms are retained in the expansion.
2
x
pxP j
pp P
f x a e
x
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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 qy px qyj j
p q p qp q A
f x y a e a f x y e dAA
y
x
3D Complex Fourier Series
Slide 6
For 3D periodic functions, the complex Fourier series generalizes to
2 2 2 2 2 2
, , , ,
1, , , ,x y z x y z
px qy ry px qy ryj j
p q r p q rp q r V
f x y z a e a f x y z e dVV
xy
z
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Slide 7
Diffraction from Gratings
Fields are Perturbed by Objects
Slide 8
A portion of the wave front is delayed after travelling through the dielectric object.
inck
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Fields in Periodic Structures
Slide 9
Waves in periodic structures take on the same periodicity as their host.
k
1,trnk
2,trnk
3,trnk
4,trnk 5,trnk
x
Diffraction from Gratings
Slide 10
The field is no longer a pure plane wave. The grating “chops” the wave front and sends the power into multiple discrete directions that are called diffraction orders.
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Grating Lobes
Slide 11
If we were to plot the power exiting a periodic structure as a function of angle, we would get the following. The power peaks are called grating lobes or sometimes side lobes. The power minimums are called nulls.
Grating Lobe Level
Main Beam (Zero‐order mode)
Diffraction Configurations
Slide 12
Planar Diffraction from a Ruled Grating
• Diffraction is confined within a plane• Numerically much simpler than other cases
• E and H modes are independent
transmitted
Conical Diffraction from a Ruled Grating
• Diffraction is no longer confined to a plane
• Almost same analytical complexity as crossed grating case, but simpler numerically
• E and H modes are coupled
transmitted
reflected
Conical Diffraction from a Crossed Grating with Planar Incidence
• Diffraction occurs in all directions
• Almost same numerical complexity as next case
• E and H modes are coupled transmitted
reflected
Conical Diffraction from a Crossed Grating
• Diffraction occurs in all directions• Most complicated case numerically
• E and H modes are coupled• Essentially the same as previous case
transmitted
incidentreflected
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Grating Equation for Planar Diffraction
Slide 13
The angles of the diffracted modes are related to the wavelength and grating period through the grating equation.
The grating equation only predicts the directionsof the modes, not how much power is in them.
Reflection Region
0trn inc incsin sinm
x
n n m
0ref inc incsin sinm
x
n n m
Transmission Region
ref incn n
trnn
x
Effect of Grating Periodicity
Slide 14
0
avgx n
0 0
avg incxn n
0
incx n
0
incx n
SubwavelengthGrating
“Subwavelength”Grating
Low Order Grating High Order Grating
navg navg navg navg
ninc ninc ninc ninc
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Animation of Grating Diffraction (1 of 3)
Slide 15
Animation of Grating Diffraction (2 of 3)
Slide 16
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Animation of Grating Diffraction (3 of 3)
Slide 17
Slide 18
The Plane Wave Spectrum
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Periodic Functions Can Be Expanded into a Fourier Series
Slide 19
2
2
x
x
mxj
m
mxj
A x S m e
S m A x e dx
The envelop 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 x
Rearrange the Fourier Series (1 of 2)
Slide 20
x A x
A periodic field can be expanded into a Fourier series.
, j rE x y A x e
2
2
2
x
x
yx xz
mxj
j r
m
mxj
j r
m
mxj
j yj x j z
m
S m e e
S m e e
S m e e e e
Here the plane wave term 𝑒 · ⃗ is brought inside of the summation.
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Rearrange the Fourier Series (2 of 2)
Slide 21
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
xy xz
mj x
j y j z
m
S m e e e
Now let and , ,, y m y z m zk k
,
2x m x
x
mk
, jk m r
m
E x y S m e
2ˆ ˆ ˆx x y y z z
x
mk m a a a
The Plane Wave Spectrum
Slide 22
, jk m r
m
E x y S m e
Terms were rearranged to show that a periodic field can also be thought of as an infinite sum of plane waves at different angles. 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 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 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 kz
2. Purely imaginary kz.
Note: kx is always real.
incx
z
Visualizing Phase Matching into the Grating
Slide 24
The wave vector expansion for the first 11 modes can be visualized as…
inck
0xk 1xk 2xk 3xk 4xk 5xk 1xk 2xk 3xk 4xk 5xk
2n
1n
kz is real. A wave propagates into material 2.
kz is imaginary. The field in material 2 is evanescent.
kz 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 are not completely evanescent. They have a purely real kx so power flows 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 spatial harmonic (plane wave).• Since there are in infinite number of terms in the Fourier series, there are an infinite number of spatial harmonics. •Only a few of the spatial harmonics are actually propagating waves. Only these can carry power away from a device. Tunneling is an exception.
Slide 25
Slide 26
Plane Wave Spectrumfrom Crossed Gratings
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Grating Terminology
Slide 27
1D gratingRuled grating
Requires a 2D simulation
2D gratingCrossed grating
Requires a 3D simulation
Diffraction from Crossed Gratings
Slide 28
Doubly‐periodic gratings, also called crossed gratings, can diffract waves into many directions.
They are described by two grating vectors, Kx and Ky.
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 29
Crossed gratings diffraction in two dimensions, x and y.
To quantify diffraction for crossed gratings, we must calculate an expansion 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
We will use this code for 2D PWEM, 3D RCWA, 3D FDTD, 3D FDFD, 3D MoL, and more.
Transverse Wave Vector Expansion (2 of 2)
Slide 30
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 31
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 center few 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.
,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);
Longitudinal Wave Vector Expansion (2 of 2)
Slide 32
The overall wave vector expansion can be visualized this way…
,ref ,zk m n ,tk m n
,ref ,zk m n ,trn ,zk m n
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Slide 33
Power Flow from Gratings
Electromagnetic Power Flow
Slide 34
The instantaneous direction and intensity of power flow at any point is given by the Poyntingvector.
, , ,r t E r t H r t
The RMS power flow is then
*1, Re , ,
2r E r H r
This is typically just written in the frequency‐domain as
*1Re
2E H
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Wave Expressions
Slide 35
To obtain an expression for the RMS Poynting vector, start by write expression for the electric and magnetic fields.
00, ,jk r jk rk E
E r E e H r e
Substituting these into the definition of RMS Poynting vector gives
*
**
** *
0 00 0 *
2* * *0 0 0* *
2
0 2Im
1 1Re Re
2 2
1 1Re Re
2 2
Re2
jk r jk r jk r jk r
j k k rjk r jk r
k r
k E k EE e e E e e
e e eE k E E k
E ke
2 2* *
0 02Im 2Im
* *0 0 r 0 0 r
Re Re2 2
k r k rE Ek ke e
k k
Power Flow Away From Grating
Slide 36
To calculate the power flow away from the grating, it is only the z‐component of the Poynting vector that is of interest.
2
0 2Im
0 0 r
Re2
k rE ke
k
z
z
x
x
Note: power travelling in the x direction does not transport power into or away from a device that is infinitely periodic along x. This simplifying statement is not true for devices of finite extent.
2
0 2Im
0 0 r
Re2
zk z zz
E kz e
k
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RMS Power of the Diffracted Modes
Slide 37
Recall that the field scattered from a periodic structure can be decomposed into a Fourier series.
The term 𝑆 𝑚 is the amplitude and polarization of the mth diffracted mode. Therefore, power flow away from the grating due to the mth diffraction order is
, x zjk m x jk m z
m
E x z S m e e
2 20
2x
x
z x
mk m
k m k n k m
22
0 2Im2Im
0 0 r 0 0 r
Re , Re2 2
zz k m zk z zzz z
S mE k mkz e z m e
k k
Z Directed Power
Slide 38
From the previous equation, the power flow of the incident, reflected, and transmitted waves into the grating are written as
inc
2inc
2Imincinc
0 0 r,inc
Re2
zk z zz
S kz e
k
refm
ref,z m
ref,x m
trnm
trn,z m
trn,x m
inc
incz
incx
ref
2ref
2Imrefref
0 0 r,ref
, Re2
zk m z zz
S m k mz m e
k
trn
2trn
2Imtrntrn
0 0 r,trn
, Re2
zk m z zz
S m k mz m e
k
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Definition of Diffraction Efficiency
Slide 39
Diffraction efficiency is defined as the power in a specific diffraction order divided by the applied incident power.
inc
DE z
z
mm
Despite the title “efficiency,” we don’t always want this quantity to be large. We often want to adjust how much power gets diffracted into each mode. So it is neither good nor bad to have high or low diffraction efficiency. It depends on how the grating is being used.
Putting it All Together
Slide 40
So far, expressions were derived for the incident power and power in the diffraction orders. Away from the edges of the grating, these are
inc
DE z
z
mm
The diffraction efficiency of the mth diffraction order was also defined as
It is now possible to derive expressions for the diffraction efficiencies of the diffraction orders by combining these expressions.
trn inc
2trntrn
2Imtrn r,trn
DE 2inc incr,incinc
Re
Rez zk m k z zz
z z
S m k mmT m e
kS
Note that: r,inc r,ref
inc ref
2refref
2Imref r,inc
DE 2inc incr,incinc
Re
Rez zk k m z zz
z z
S m k mmR m e
kS
inc
2inc
2Imincinc
0 0 r,inc
Re2
zk z zz
S kz e
k
ref
2ref
2Imrefref
0 0 r,ref
, Re2
zk m z zz
S m k mz m e
k
trn
2trn
2Imtrntrn
0 0 r,trn
, Re2
zk m z zz
S m k mz m e
k
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Diffraction Efficiency for Magnetic Fields
Slide 41
Diffraction efficiency equations were just derived from electric field quantities.
inc ref
trn inc
th
2refref
2Imref r,inc
DE 2inc incr,incinc
2trn
2Imtrn
DE 2inc
inc
Electric field amplitude of the diffraction order
Re
Re
Re
z z
z z
k k m z zz
z z
k m k z zz
z
S m m
S m k mmR m e
kS
S m kmT m e
S
trn
r,trn
incr,incRe z
m
k
Sometimes Maxwell’s equations in terms of the magnetic fields. In this case, the diffraction efficiency equations are
inc ref
trn inc
th
2refref
2Imref r,inc
DE 2inc incr,incinc
2trn
2Imtrn
DE 2inc
inc
Magnetic field amplitude of the diffraction order
Re
Re
Re
z z
z z
k k m z zz
z z
k m k z zz
z
U m m
U m k mmR m e
kU
U m kmT m e
U
trn
r,trn
incr,incRe z
m
k
Reflectance, Transmittance, and Absorptance
Slide 42
The reflectance R is the sum of the diffraction efficiencies of the reflected modes.
The transmittance T is the sum of the diffraction efficiencies of the transmitted modes.
DEm
R R m
DEm
T T m
The absorptance A is the fraction of power absorbed by the device.
1A R T 0 materials have loss
0 materials have no loss
0 materials have gain
A
A
A
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Simplification for TMM
Slide 43
For TMM, the layers are homogeneous so there is no diffraction. Therefore, only the m = 0 diffraction order exists.
inc ref
trn inc
th
2 refrefr,inc2Im 0ref
DE 2inc incr,incinc
2trn
2Im 0trn
DE 2inc
inc
0 Electric field amplitude of the 0 diffraction order
ˆRe 0000
ˆRe
0 Re00
z z
z z
zk k zz
z z
k k zz
z
S
kSR e
kS
ST e
S
trn
r,trn
incr,inc
0
Re
z
z
k
k
,ref ,incz zk k trn inc
2
ref
DE 2
inc
2trn
2Imtrn r,trn
DE 2 incr,incinc
Re
Rez zk k z z
z
ER R
E
E kT T e
kE
Since there is only one diffraction order,
inc inc
ref ref
trn trn
DE DE
DE DE
0
0
0
0
E S
E S
E S
R R
T T
Diffraction Efficiency Equations in Lossy LHI Materials Right Against Device (i.e. z = 0)
Slide 44
2ref
ref r,inc
DE 2 incr,incinc
2trn
trn r,trn
DE 2 incr,incinc
Re
Re
Re
Re
z
z
z
z
S m k mR m
kS
S m k mT m
kS
Equations for Multiple Diffraction Orders
TMM / Single Diffraction Order2 2
trnref trn r,trn
DE DE2 2 incr,incinc inc
Re
Re
z
z
E E kR R T T
kE E
2ref
ref r,inc
DE 2 incr,incinc
2trn
trn r,trn
DE 2 incr,incinc
Re
Re
Re
Re
z
z
z
z
U m k mR m
kU
U m k mT m
kU
2 2trn
ref trn r,trn
DE DE2 2 incr,incinc inc
Re
Re
z
z
H H kR R T T
kH H
DE DE m m
R R m T T m
43
44
![Page 23: Lecture 3b -- Diffraction Gratings and the Plane Wave Spectrum...kkn Exy Eek kkn Phase matching into the grating leads to ,inc 2 xx x km k m m ,2,1,0,1,2, Each wave must satisfy the](https://reader033.vdocument.in/reader033/viewer/2022042202/5ea24f3ab38982403f28b1a9/html5/thumbnails/23.jpg)
10/2/2019
23
Diffraction Efficiency Equations in Lossless LHI Materials Right Against Device (i.e. z = 0)
Slide 45
2ref
ref
DE 2 inc
inc
2trn
trnr,incDE 2 inc
r,trn inc
Re
Re
Re
Re
z
z
z
z
S m k mR m
kS
S m k mT m
kS
Equations for Multiple Diffraction Orders
TMM / Single Diffraction Order2 2
trnref trnr,inc
DE DE2 2 incr,trninc inc
Re
Re
z
z
E E kR R T T
kE E
2ref
ref
DE 2 inc
inc
2trn
trnr,incDE 2 inc
r,trn inc
Re
Re
Re
Re
z
z
z
z
U m k mR m
kU
U m k mT m
kU
2 2trn
ref trnr,incDE DE2 2 inc
r,trninc inc
Re
Re
z
z
H H kR R T T
kH H
DE DE m m
R R m T T m
45