Lecture 8 -- Diffraction Gratings and the Plane Wave …emlab.utep.edu/ee5390cem/Lecture 8 --...
Transcript of Lecture 8 -- Diffraction Gratings and the Plane Wave …emlab.utep.edu/ee5390cem/Lecture 8 --...
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Lecture 8 Slide 1
EE 5337
Computational Electromagnetics
Lecture #8
Diffraction Gratings and the Plane Wave Spectrum These notes may contain copyrighted material obtained under fair use rules. Distribution of these materials is strictly prohibited
InstructorDr. Raymond Rumpf(915) 747‐[email protected]
Outline
Lecture 8 Slide 2
• Fourier Series
• Diffraction from gratings
• The plane wave spectrum
• Plane wave spectrum for crossed gratings
• Power flow from gratings
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Lecture 8 Slide 3
Fourier Series
Jean Baptiste Joseph Fourier
Born:
Died:
March 21, 1768in Yonne, France.
May 16, 1830in Paris, France.
Lecture 8 Slide 4
1D Complex Fourier Series
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, we retain only a finite number of terms in the expansion.
2
x
pxP j
pp P
f x a e
x
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Lecture 8 Slide 5
2D Complex Fourier Series
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
Lecture 8 Slide 6
3D Complex Fourier Series
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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Lecture 8 Slide 7
Diffraction from Gratings
Lecture 8 Slide 8
Fields are Perturbed by Objects
A portion of the wave front is delayed after travelling through the dielectric object.
inck
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Lecture 8 Slide 9
Fields in Periodic Structures
Waves in periodic structures take on the same periodicity as their host.
x
inck
Lecture 8 Slide 10
Diffraction From Gratings
The field is no longer a pure plane wave. The grating “chops” the wave front and sends the power into multiple discrete directions called diffraction orders.
inck
1,trnk
2,trnk
3,trnk
4,trnk 5,trnk
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Lecture 8 Slide 11
Grating Lobes
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)
Lecture 8 Slide 12
Diffraction ConfigurationsPlanar Diffraction from a Ruled Grating Conical Diffraction from a Ruled Grating
Conical Diffraction from a Crossed Grating with Planar Incidence
Conical Diffraction from a Crossed Grating
• Diffraction is confined within a plane
• Numerically much simpler than other cases
• E and H modes are independent
• Diffraction is no longer confined to a plane
• Almost same analytical complexity as crossed grating case
• E and H modes are coupled
• Diffraction occurs in all directions
• Almost same numerical complexity as next case
• E and H modes are coupled
• Diffraction occurs in all directions
• Most complicated case numerically
• E and H modes are coupled
• Essentially the same as previous case
transmittedtransmitted
reflected
transmitted
reflected
transmitted
incidentreflected
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Lecture 8 Slide 13
Grating Equation for Planar Diffraction
The angles of the diffracted modes are related to the wavelength and grating period through the grating equation.
The grating equation only predicts the directions of the modes, not how muchpower 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
Lecture 8 Slide 14
Effect of Grating Periodicity
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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Lecture 8 Slide 15
Animation of Grating Diffraction (1 of 2)
Lecture 8 Slide 16
Animation of Grating Diffraction (2 of 2)
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Lecture 8 Slide 17
The Plane Wave Spectrum
Lecture 8 Slide 18
Periodic Functions Can Be Expanded into a Fourier Series
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
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Rearrange the Fourier Series (1 of 2)
A periodic field can be expanded into a Fourier series.
x
, j rE x y A x e
2
2
2
x
x
yx x
mxj
j r
m
mxj
j r
m
mxj
j yj x
m
S m e e
S m e e
S m e e e
Lecture 8 Slide 19
Here the plane wave term ejxx is brought inside of the summation.
Rearrange the Fourier Series (2 of 2)
x can be combined with the last complex exponential.
x
2
, yx x
mxj
j yj x
m
E x y S m e e e
2
xy x
mj x
j y
m
S m e e
Lecture 8 Slide 20
Now let and y,m yk ,
2x m x
x
mk
, jk m r
m
E x y S m e
2
ˆ ˆx x y yx
mk m a a
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Lecture 8 Slide 21
The Plane Wave Spectrum
, jk m r
m
E x y S m e
We rearranged terms and now we see 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
Lecture 8 Slide 22
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
inc 0,inc 0 inc inc
sin,
cosx yj k x k y x
y
k k nE 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 y
y x
k m k m k n
k m k n k m
We have two possible solutions here.1. Purely real ky
2. Purely imaginary ky.
Note: kx is always real.
inc
x
y
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Lecture 8 Slide 23
Visualizing Phase Matching into the Grating
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
ky is real. A wave propagates into material 2.
ky is imaginary. The field in material 2 is evanescent.
ky 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
y
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.
Lecture 8 Slide 24
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Lecture 8 Slide 25
Plane Wave Spectrumfrom Crossed Gratings
Lecture 8 Slide 26
Grating Terminology
1D gratingRuled grating
Requires a 2D simulation
2D gratingCrossed grating
Requires a 3D simulation
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Lecture 9 Slide 27
Diffraction from Crossed Gratings
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
Lecture 8 Slide 28
Transverse Wave Vector Expansion (1 of 2)
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.
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Lecture 8 Slide 29
Transverse Wave Vector Expansion (2 of 2)
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
Lecture 8 Slide 30
Longitudinal Wave Vector Expansion (1 of 2)
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);
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Lecture 8 Slide 31
Longitudinal Wave Vector Expansion (2 of 2)
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
Lecture 8 Slide 32
Power Flow from Gratings
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Lecture 8 Slide 33
Electromagnetic Power Flow
The instantaneous direction and intensity of power flow at any point is given by the Poynting vector.
, , ,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
Lecture 8 Slide 34
Wave Expressions
To obtain an expression for the RMS Poynting vector, we first 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
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Lecture 8 Slide 35
Power Flow Away From Grating
To calculate the power flow away from the grating, we are only interested in the z‐component of the Poynting vector.
2
0 2Im
0 0 r
Re2
k rE ke
k
z
z
x
x
Note: power travelling in the xdirection 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
Lecture 8 Slide 36
RMS Power of the Diffracted Modes
Recall that the field scattered from a periodic structure can be decomposed into a Fourier series.
The term Sm 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
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Lecture 8 Slide 37
Z Directed Power
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
Lecture 8 Slide 38
Definition of Diffraction Efficiency
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.
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Lecture 8 Slide 39
Putting it All Together
So far, we have derived expressions for the incident power and power in the diffraction orders. At the edge of the grating, these are
inc
DE z
z
mm
We also defined the diffraction efficiency of the mth diffraction order as
We can now 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
Lecture 8 Slide 40
Diffraction Efficiency for Magnetic Fields
We just calculated the diffraction efficiency equations 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 we solve Maxwell’s equations for 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
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Lecture 8 Slide 41
Reflectance, Transmittance, and Absorptance
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
Lecture 8 Slide 42
Simplification for TMM
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
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Lecture 8 Slide 43
Diffraction Efficiency Equations in Lossy LHI Materials Right Against Device (i.e. z = 0)
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
ref
DE 2
inc
2trn
trn r,trn
DE 2 incr,incinc
Re
Re
z
z
ER R
E
E kT T
kE
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
ref
DE 2
inc
2trn
trn r,trn
DE 2 incr,incinc
Re
Re
z
z
HR R
H
H kT T
kH
DE DE m m
R R m T T m
Lecture 8 Slide 44
Diffraction Efficiency Equations in Lossless LHI Materials Right Against Device (i.e. z = 0)
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
ref
DE 2
inc
2trn
trnr,incDE 2 inc
r,trn inc
Re
Re
z
z
ER R
E
E kT T
kE
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
ref
DE 2
inc
2trn
trnr,incDE 2 inc
r,trn inc
Re
Re
z
z
HR R
H
H kT T
kH
DE DE m m
R R m T T m