Post on 09-Jul-2020
10/28/2019
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Advanced Computation:
Computational Electromagnetics
Plane Wave Expansion Method (PWEM)
Outline
• Formulation of the Basic Eigen‐Value Problem• 3D• 2D
• Implementation
• Calculation of Band Diagrams
• Calculation of Isofrequency Contours
• Example – Computing the band diagram for an EBG material with square symmetry
Slide 2
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Slide 3
Formulation of the Basic 3D Eigen‐Value Problem
Block Matrix Form
Slide 4
0 r
0 r
0 r
y z z y x
z x x z y
x y y x z
jk
jk
jk
K u K u s
K u K u s
K u K u s
0 r
0 r
0 r
y z z y x
z x x z y
x y y x z
jk
jk
jk
K s K s u
K s K s u
K s K s u
We start with the Fourier‐space Maxwell’s equations in matrix form.
These can be written in block matrix form as
r
0 r
r
z y x x
z x y y
y x z z
jk
0 K K u 0 0 s
K 0 K u 0 0 s
K K 0 u 0 0 s
r
0 r
r
z y x x
z x y y
y x z z
jk
0 K K s 0 0 u
K 0 K s 0 0 u
K K 0 s 0 0 u
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Compact Block Matrix Notation
Slide 5
These equations can be written even more compactly as...
0 rjk K u s
r
0 r
r
z y x x
z x y y
y x z z
jk
0 K K s 0 0 u
K 0 K s 0 0 u
K K 0 s 0 0 u
r
0 r
r
z y x x
z x y y
y x z z
jk
0 K K u 0 0 s
K 0 K u 0 0 s
K K 0 u 0 0 s
r
r r
r
0 0
0 0
0 0
0 rjk K s u
z y
z x
y x
0 K K
K K 0 K
K K 0
x
y
z
s
s s
s
x
y
z
u
u u
u
r
r r
r
0 0
0 0
0 0
Eliminate the Magnetic Field
Slide 6
Start with the following equations.
0 rjk K u s 0 rjk K s u
Solve for the magnetic field 𝐮
1
r0
1
jk
u K s
Substitute expression for 𝐮 into second equation.
1
r 0 r0
1jk
jk
K K s s
1 2r 0 rk
K K s s
Simplify
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The 3D Eigen‐Value Problem
Slide 7
The 3D eigen‐value problem in terms of the electric field is
1 2r 0 rk
K K s s
z y
z x
y x
0 K K
K K 0 K
K K 0
1
r
1 1
r r
1
r
0 0
0 0
0 0
x
y
z
s
s s
s
This has the form of a generalized eigen‐value problem
Ax Bx[V,D] = eig(A,B);
1
r
r
20k
A K K
B
x s
Notes: 1. It is possible to reduce this to a 2×2 block matrix
equation. See PWEM Extras lecture.2. It is more common to see this expressed in terms
of the magnetic field because it is an ordinary eigen‐value problem.
r
r r
r
0 0
0 0
0 0
Visualizing the Data
Slide 8
20kAx x
r
r
x
y
z
Inputs to PWEM
K
A
Intermediate Data
20k
1 2V x x
10k
20k
30k
0Nk
Eigen‐values are some property of the modes. Here, it is frequency.
Eigen‐vectors are pictures of the modes.
V
Outputs of PWEM
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Consequences of 𝑘 Being the Eigen‐Value
Slide 9
The quantity is really just frequency scaled by the speed of light.20k
00
kc
In the present formulation, k0 is the eigen‐value so it is the unknown quantity.
k0 is not known when constructing the eigen‐value problem.
Since frequency is not known, it is not possible to build frequency‐dependent material properties (i.e. dispersion) into the simulation without modifying the basic PWEM algorithm.
The basic PWEM cannot incorporate material dispersion.It must be modified to account for this. See PWEM Extras.
Slide 10
Formulation of Efficient 2DPlane Wave Expansion Method
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Two‐Dimensional Devices
Slide 11
For 2D problems, the device is uniform and infinite in the z‐direction and wave propagation is restricted to the x‐y plane.
x
y
z
infin
ite an
d unifo
rm
Representing Slab Photonic Crystals
Slide 12
eff,1n eff,2n
eff,1n eff,2n
3D Slab Photonic Crystal
2D Representation
1D Slab Waveguide Analysis
Step 1 – Analyze vertical cross sections of photonic crystal slab as slab waveguides. Calculate the effective refractive index of each cross section.
Step 2 – Build a 2D representation of the photonic crystal slab using just the effective refractive indices from Step 1.
Be careful to consider the polarization. The alignment of the electric field must be consistent.
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0 r
0 r
0 r
y z x
x z y
x y y x z
jk
jk
jk
K s u
K s u
K s K s u
0 r
0 r
0 r
y z x
x z y
x y y x z
jk
jk
jk
K u s
K u s
K u K u s
Reduction to Two Dimensions
Slide 13
y z z yK u K u 0 r x
z x
jk s
K u
0 r
0 r
x z y
x y y x z
jk
jk
K u s
K u K u s
y z z yK s K s 0 r x
z x
jk u
K s
0 r
0 r
x z y
x y y x z
jk
jk
K s u
K s K s u
z K 0Maxwell’s equations reduce to
For the 2D devices described on the previous slide where the waves are restricted to the plane of the device, the wave has no vector components in the z‐direction.
0 rx y y x zjk K s K s u
0 ry z xjk K u s
Two Distinct Modes
Slide 14
E‐Mode
H‐Mode
After inspecting the remaining equations, it can be seen that Maxwell’s equations have split into two independent sets of equations. These correspond to two possible electromagnetic modes.
0 rx z yjk K u s
0 rx y y x zjk K u K u s
0 rx z yjk K s u
0 ry z xjk K s u
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0 rx y y x zjk K s K s u
0 ry z xjk K u s
Two Distinct Modes
Slide 15
0 r
0 r
0 r
x y y x z
y z x
x z y
jk
jk
jk
K s K s u
K u s
K u s
0 r
0 r
0 r
x y y x z
y z x
x z y
jk
jk
jk
K u K u s
K s u
K s u
E‐Mode
H‐Mode
After inspecting the remaining equations, it can be seen that Maxwell’s equations have split into two independent sets of equations. These correspond to two possible electromagnetic modes.
0 rx z yjk K u s
0 rx y y x zjk K u K u s
0 rx z yjk K s u
0 ry z xjk K s u
2D Eigen‐Value Problems
Slide 16
Two eigen‐value problems can now be derived for two dimensional photonic crystals.
0 r
0 r
0 r
x y y x z
y z x
x z y
jk
jk
jk
K u K u s
K s u
K s u
1 1 2r r 0 rx x y y z zk K K K K s s
H‐Mode
Note: For non‐magnetic materials, this mode calculates slower.
E‐Mode
1
r0
x y z
j
k u K s
1
r0
y x z
j
k u K s
1 1
r r 0 r0 0
x x z y y z z
j jjk
k k
K K s K K s s
0 r
0 r
0 r
x y y x z
y z x
x z y
jk
jk
jk
K s K s u
K u s
K u s
1 1 2r r 0 rx x y y z zk K K K K u u
1
r0
x y z
j
k s K u
1
r0
y x z
j
k s K u
1 1
r r 0 r0 0
x x z y y z z
j jjk
k k
K K u K K u u
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V
V
Solution in Homogeneous Unit Cell
Slide 17
For homogeneous unit cells, the A and Bmatrices of the generalized eigen‐value problem are diagonal. In this case, the modes have a closed form solution.
2 2r r
1r r
1
x y
Av BvV I
A K K
B I λ B A
When solved numerically, the columns of the eigen‐modes may be scrambled.
closed formnumerical
Interpreting the Eigen‐Vectors
Slide 18
An eigen‐vector contains the complex amplitudes of all the spatial harmonics (i.e. plane waves in the expansion) for that mode.
The numbers are complex to reflect both magnitude and phase of the spatial harmonics.
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v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v v1 v2 v3 v4 v5
v6 v7 v8 v9 v10
v11 v12 v13 v14 v15
v16 v17 v18 v19 v20
v21 v22 v23 v24 v25
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Slide 19
Implementation
PWEM from a User’s Perspective
Slide 20
Given the unit cell of an infinitely periodic lattice and the Bloch wave vector (i.e. wave direction and spatial period), the PWEM calculates all the modes with these conditions. The eigen‐values contain the frequencies and the eigen‐vectors contain the fields.
20, Eigen-values
Eigen-vectors
1
i
i
k
S
i
𝛽
2𝜋𝜆
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Choosing the Number of Spatial Harmonics
Slide 21
The only true way to determine the correct number of spatial harmonics is to test for convergence.
There are, however, some rules of thumb to follow in order to make a good initial guess.
min
10aM
For each direction.
Here are some examples…
M = 1N = 7
M = 1N = 15
M = 7N = 1
M = 15N = 1
M = 15N = 15
M = 7N = 7
M = 11N = 7
M = 15N = 7
M = 1N = 25
M = 7N = 7
M = 21N = 21
M = 21N = 21
M = 31N = 31
M = 21N = 13
Normalizing the Frequency
Slide 22
The eigen‐value problem was derived so that 𝑘 (frequency) is the eigen‐value.
1
r2
r 0z z k K K s s
Think of 𝑘 as frequency because it is frequency divided by a constant c0.
00
kc
It is very useful when designing devices to scale the eigen‐value to 𝑎/𝜆 .
n0 0
202 2k
a a a
c
Given a/0, a design can be easily scaled to operate at any frequency that is desired.
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Block Diagram of 2D Analysis
Slide 23
Define Problem
r r,r r
Compute Convolution Matrices
's
Calculate Wave Vector Expansion
1 2
, ,, diag meshgrid ,
pq
y x y pq x pq
k pT qT
k k
K K
1 1
r r r
1 1
r r r
E Mode: ,
H Mode: ,
x x y y
x x y y
A K K K K B
A K K K K B
Build Eigen‐Value Problem 𝐀𝐱 𝑘 𝐁𝐱
Solve Generalized Eigen‐Value Problem
Start
Finish
Compute List of Bloch Wave Vectors
' s
Scale Eigen‐Value k0
Record Data
20
0 2n
a ak
Build Unit Cell on High Resolution Grid
r r, and ,x y x y
ERC = convmat(ER,P,Q);URC = convmat(UR,P,Q); [V,D] = eig(A,B)
Slide 24
Calculation of Photonic Band Diagrams
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Band Diagrams (1 of 2)
Slide 25
Band diagrams are a compact, but incomplete, means of characterizing the electromagnetic properties of a periodic structure. Along the horizontal axis is a list of Bloch wave vectors (direction and period of the Bloch wave). Vertically above each Bloch vector are all of the frequencies which have a mode with that Bloch wave vector.
Band Diagrams (2 of 2)
Slide 26
To construct a band diagram, make small steps around the perimeter of the irreducible Brillouin zone (IBZ) and compute the eigen‐values at each step. Plot all these eigen‐values as a function of and the points line up to form continuous “bands.”
CAUTION:When solved numerically, the order of the modes will be different for different values of . For this reason, it is difficult to plot the bands as continuous lines.
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Animation of the Construction of a Band Diagram
Slide 27
List of Bloch Wave Vectors
Slide 28
To calculate a band diagram, it is necessary to generate an array of Bloch wave vectors that march around the perimeter of the IBZ.
2T
1T
X
M
0 0.01 3.13 3.14 3.14 3.14 3.14 3.13 0.01 0
0 0 0 0 0.01 3.13 3.14 3.13 0.01 0
X M
• Use more points in parts of this array that cover longer distances.• Do not repeat adjacent points.
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Animation of Band Calculation & Bloch Waves
Slide 29
Slide 30
Calculation of Isofrequency Contours
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The Band Diagram is Missing Information
Slide 31
x
y
y
x
M
X
Direct lattice: This lattice has an array of air holes in a dielectric with 𝑛 3.0.
Reciprocal lattice: Construct the band diagram by marching around the perimeter
of the irreducible Brillouin zone.
The band extremes “almost” always occur at the key points of symmetry.
Information is missing from inside the Brilluoin zone. Perhaps something important is being missed?
The Complete Band Diagram
Slide 32
02
a
c
aa
a
a0
0
y x
The Full Brillouin Zone
yk
a
a
0
a 0 a
xkThere is an infinite set of eigen‐frequencies associated with each point in the Brillouin zone. These form “sheets” as shown at right.
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IFCs From Second‐Order Band
Slide 33Index ellipsoids are “isofrequency contours” in k‐space.
IFCs From First‐Order Band
Slide 34
02
a
c
aa
a
a0
0
y x
02
a
c
aa
a
a0
0
y x
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Calculating and Visualizing the IFCs
Slide 35
Full Band Data wn Isofrequency Contours from Full Band Data wn
y
x
0
a
0
a
y
xfor nby = 1 : NBY
for nbx = 1 : NBXb = [ Bx(nbx,nby) ; By(nbx,nby) ];k0 = pwem2d(...);wn(nbx,nby) = a*k0/(2*pi);
endend
pcolor() contour()
Standard View of IFCs
Slide 36
xa
ya
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Slide 37
Example –Band Diagram for a 2D EBG with Square Symmetry
Define the Lattice
Slide 38
rr
a
r
1
0.35
9.0
a
r a
Extended Lattice 2D Unit Cell
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Build Lattice on a High‐Resolution Grid
Slide 39
rr
a
r
1
0.35
9.0
a
r a
Unit Cell High Resolution Grid
9’s
1’s
512
512
Construct Convolution Matrices
Slide 40
9’s
1’s
1’s
r(x,y) or UR
r(x,y) or ER
URC = convmat(UR,P,Q)
ERC = convmat(ER,P,Q)
or URC
or ERC
r
r
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Compute the Reciprocal Lattice
Slide 41
Direct Lattice Reciprocal Lattice
a
2 a
1 xt a
2 yt a
1
2ˆT x
a
2
2ˆT y
a
Construct the Brillouin Zone
Slide 42
2T
1T
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Identify the Irreducible Brillouin Zone
Slide 43
1. Start with the full Brillouin zone. 2. Lattice has left/right symmetry.
3. Lattice has up/down symmetry. 4. Lattice has 90 rotational symmetry.
IBZ
Identify the Key Points of Symmetry
Slide 44
2T
1T
X
M
1
1 2
0
0
0.5
0.5 0.5
X T
M T T
The key points of symmetry are calculated from a linear combination of the reciprocal lattice vectors.
Formulas for calculating the key points of symmetry along with their naming convention can be found in [M. Lax, Symmetry Principles in Solid State and Molecular Physics, (Dover, New York, 1974). See supplemental notes for Lecture 6 “Periodic Structures.”
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The Numbers
Slide 45
0 3.14 3.14
0 0 3.14X M
Typically the lattice constant is normalized to the value of 1.0.
1a
The direct and reciprocal lattice vectors are then
1
2
1
0
0
1
t
t
1
2
6.28
0
0
6.28
T
T
From these, the key points of symmetry are calculated to be
Generate List of ’s
Slide 46
X M
Next, an array of Bloch wave vectors is generated that march around the perimeter of the IBZ.
2T
1T
X
M
0 0.01 3.13 3.14 3.14 3.14 3.14 3.13 0.01 0
0 0 0 0 0.01 3.13 3.14 3.13 0.01 0
X M
• Use more points in parts of this array that cover longer distances.• Do not repeat adjacent points.
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For Each , Construct KX and KY
Slide 47
Compute kx and ky wave vector expansions along the x and y axes respectively.
,
,
2 2 , , 1,0,1, , 2
2 2 , , 1,0,1, , 2
x x i
y y i
pk p p P P
aq
k q q Q Qa
Compute wave vector meshgrid() expansion.
KX = diag(sparse(kx(:)));KY = diag(sparse(ky(:)));
,xk p q
Form diagonal matrices KX and KY.
kx = bx - 2*pi*p/a;ky = by - 2*pi*q/a;
[ky,kx] = meshgrid(ky,kx);
x yK K
,yk p q
Solve Eigen‐Value Problem for Each
Slide 48
The eigen‐values calculated from this problem are:
0.00 0.01 3.13 3.14 3.14 3.14 3.14 3.13 0.01 0.00
0.00 0.00 0.00 0.00 0.01 3.13 3.14 3.13 0.01 0.00
X M
20k
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Plot Eigen‐Values Vs.
Slide 49
20k
Array index of Bloch wave vector
Generate a Professional Looking Plot
Slide 50
Bloch wave vector
The frequency axis has been normalized for easy scaling.
Horizontal axis is labeled with the key points of symmetry. Notice the spacing between key points is consistent with the segment lengths around the perimeter of the IBZ.
A picture of the lattice unit cell, Brillouin zone, IBZ, and key points of symmetry should be shown.
Band Diagram of a Square Lattice
Given a meaningful title
n0
0
20
how it is labeled2
how it is used
how it is calculated2
a
c
a
ak
12
X M
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50