FDFD
Transcript of FDFD
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Pioneering 21st Century Electromagnetics and Photonics
The Finite-Difference Frequency-Domain Method
Raymond C. Rumpf, Ph.D.
“I am always doing that which I cannot do,in order that I may learn how to do it.”
– Pablo Picasso
Short Course Outline
• Background Topics– Numerical methods, electromagnetics, linear algebra,
finite-difference approximations– Break
• The Finite-Difference Frequency-Domain Method– Yee grid, Maxwellmatrix, formulation, PML, TF/SF,
solution, post processing– Break
• Implementation– 3D2D, grids and materials, testing, convergence– Code examples: GMR, photonic crystal, and wire-grid
polarizer– Done!
Short Course on Finite-Difference Frequency-Domain
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Background
Background: Numerical MethodsGolden Rule #1
1. All numbers should equal 1
(1.234567…) + (0.0123456…) = Lost two digits of accuracy!!
Why?
Solution: NORMALIZE EVERYTHING!!!
0
0
E E
0
0
H H
or
0x k x
0y k y
0z k z
00 1 m
Short Course on Finite-Difference Frequency-Domain
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Background: Numerical MethodsGolden Rule #2
2. Never perform calculations
1. Golden Rule #1.2. Finite floating point precision introduces round-off errors.
Why?
Solution: MINIMIZE NUMBER OF COMPUTATIONS!!!
1. Take problems as far analytically as possible.2. Avoid unnecessary computations.
2 2
2exp
R x y
Rg R
2 2
2
2exp
r x y
rg r
Short Course on Finite-Difference Frequency-Domain
Background: Numerical MethodsGolden Rule #3
3. Follow rules 1 and 2
1. Improves accuracy by reducing numerical error.2. Enables codes to model larger and more intensive problems.
Why?
Solution
1. Normalize all parameters.2. Minimize computations.
Short Course on Finite-Difference Frequency-Domain
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Background: Numerical MethodsBenefits and drawbacks
Frequency-Domain Time-Domain
Semi-AnalyticalFully Numerical
Fourier-SpaceReal-Space
+ wideband simulations+ scales near linearly+ active & nonlinear devices+ easily locates resonances
- longitudinal periodicity- sharp resonances- memory requirements- oblique incidence
+ resolves sharp resonances+ handles oblique incidence+ longitudinal periodicity+ can be very fast
- scales at best NlogN- can miss sharp resonances- active & nonlinear devices
+ better convergence+ scales better than SA+ complex device geometry
- memory requirements- long uniform sections
+ very fast & efficient+ layered devices+ less memory
- convergence issues- scales poorly- complex device geometry
+ high index contrast+ metals+ resolving fine details+ field visualization
- slow for low index contrast + moderate index contrast+ periodic problems+ very fast and efficient
- field visualization- formulation difficult- resolving fine details
Unstructured GridStructured Grid
+ easy to implement+ rectangular structures+ easy for divergence free
- less efficient- curved surfaces
+ most efficient+ handles larger structures+ conforms to curved surfaces
- difficult to implement- spurious solutions
Short Course on Finite-Difference Frequency-Domain
Background: Numerical MethodsFinite-Difference Frequency-Domain
H j E
E j H
Maxwell’s Equations Matrix Equation
A x b
Numerical Solution
1x A b
Fields fit to a discrete grid
Short Course on Finite-Difference Frequency-Domain
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Background: ElectromagneticsGauss’s law in differential form
Electric fields diverge from positive charges and converge on negative charges.
-+
vD
yx zDD D
Dx y z
If there are no charges, electric fields must form loops.
Short Course on Finite-Difference Frequency-Domain
Background: ElectromagneticsNo magnetic charge
Magnetic fields always form loops.
0B
yx zBB B
Bx y z
Short Course on Finite-Difference Frequency-Domain
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Background: ElectromagneticsConsequence of Zero Divergence
The divergence theorems force the electric and magnetic fields to be perpendicular to the propagation direction of a plane wave.
k D
0
0jk r
D
de
d
no charges
0
0
jk d
k d
k
k
k B
0
0jk r
B
be
b
no charges
0
0
jk b
k b
k
k
Short Course on Finite-Difference Frequency-Domain
Background: ElectromagneticsAmpere’s law in differential form
DH J
t
ˆ ˆ ˆy yx xz zx y z
H HH HH HH a a a
y z z x x y
Circulating magnetic fields induce currents and/or time varying electric fields.Currents and/or time varying electric fields induce circulating magnetic fields.
Short Course on Finite-Difference Frequency-Domain
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Background: ElectromagneticsFaraday’s law in differential form
BE
t
ˆ ˆ ˆy yx xz zx y z
E EE EE EE a a a
y z z x x y
Circulating electric fields induce time varying magnetic fields.Time varying magnetic fields induce circulating electric fields.
Short Course on Finite-Difference Frequency-Domain
Background: ElectromagneticsConsequences of curl equations
The curl equations predict electromagnetic waves.
HE k
The curl equations force the electric and magnetic field components of a plane wave to be perpendicular.
k
Electric Field
Magnetic Field
Electric Field
Magnetic Field
Short Course on Finite-Difference Frequency-Domain
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Background: ElectromagneticsMaxwell’s equations
Divergence Equations
0
v
B
D
DH J
t
BE
t
Curl Equations
Constitutive Relations
D t t E t
B t t H t
What produces fields
How fields interact with materialsmeans convolution
Short Course on Finite-Difference Frequency-Domain
Background: ElectromagneticsSimplifying Maxwell’s equations
0
0
B
D
H D t
E B t
D t t E t
B t t H t
1. Assume no charges or current sources: 0v 0J
0
0
B
D
H j D
E j B
D E
B H
2. Transform Maxwell’s equations to frequency-domain:
0
0
H
E
H j E
E j H
3. Substitute constitutive relations into Maxwell’s equations:
Convolution becomes multiplication
Note: It is helpful to retain μ and ε and not replace with refractive index n.
Short Course on Finite-Difference Frequency-Domain
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Background: ElectromagneticsPhysical boundary conditions
Fields tangential to the interface are continuous across it.
1,TE 2,TE
1,TH 2,TH
1 1 and 2 2and
Fields normal to the interface are discontinuous across it.
1 1,NE
1 1,NH
2 2,NE
2 2,NH
Note: normal components of D and B are continuous across an interface.
These are more complicated boundary condition that have numerical consequences.
Short Course on Finite-Difference Frequency-Domain
Background: ElectromagneticsSign Convention for Waves
Forward Propagation Along +x
SIGN CONVENTION #1
ikxeRefractive Index
N n i
00
0 0
oscillatory Decayingterm in exponential
ik n i xik Nxikx
ik nx k x
x
e e e
e e
Forward Propagation Along +x
SIGN CONVENTION #2
ikxeRefractive Index
N n i
00
0 0
oscillatory Decayingterm exponential
ik n i xik Nxikx
ik nx k x
x
e e e
e e
Use
d H
ere
Short Course on Finite-Difference Frequency-Domain
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Background: Linear AlgebraMatrices represent sets of equations
11 12 13 14 1
21 22 23 24 2
31 32 33 34 3
41 42 43 44 4
a w a x a y a z b
a w a x a y a z b
a w a x a y a z b
a w a x a y a z b
11 12 13 14 1
21 22 23 24 2
31 32 33 34 3
41 42 43 44 4
a a a a bw
a a a a bx
a a a a by
a a a a bz
A set of linear algebraic equations can be written in “matrix” form.
Short Course on Finite-Difference Frequency-Domain
Background: Linear AlgebraInterpretation of matrices
11 12 13 1a x a y a z b
21 22 23 2a x a y a z b
31 32 33 3a x a y a z b
11 12 13 1
21 22 23 2
31 32 33 3
a a a x b
a a a y b
a a a z b
11 12 13
21 22 23
31 32 33
a a a
a a a
a a a
Equation for x
Equation for y
Equation for z
EQUATION FOR… RELATION TO…
11 12 13
21 22 23
31 32 33
a a a
a a a
a a a
Short Course on Finite-Difference Frequency-Domain
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Background: Linear AlgebraMatrices have a compact notation
Ax b
11 12 13 14
21 22 23 24
31 32 33 34
41 42 43 44
a a a a
a a a a
a a a a
a a a a
A
w
x
y
z
x
1
2
3
4
b
b
b
b
b
Matrices and vectors can be represented and treated as single variables.
A x b
or
square matrixcolumnvector
columnvector
11 12 13 14 1
21 22 23 24 2
31 32 33 34 3
41 42 43 44 4
a a a a bw
a a a a bx
a a a a by
a a a a bz
Short Course on Finite-Difference Frequency-Domain
Background: Linear AlgebraMatrices require a special algebra
AB BA
A B B A
Commutative Laws
AB C A BC
A B C A B C
Associative Laws
A B A B
AB A B A B
Multiplication with a Scalar
A B C AB AC
A B C AC BC
Distributive Laws
11 1
1
?
n
m mn
a a
a a
A
I A
Addition with a Scalar
AB BA
Short Course on Finite-Difference Frequency-Domain
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Background: Linear AlgebraSpecial matrices
Zero Matrix
0 0
0 0
0
Identity Matrix
1
1
0
0
I
0 A A 0 0
I A A I A
0 A A 0 A
A A 0
Short Course on Finite-Difference Frequency-Domain
Background: Finite-DifferencesWhat is a finite-difference approximation?
1.5 2 1df f f
dx x
1f2f
df
dx
x
second-order accuratefirst-order derivative
Short Course on Finite-Difference Frequency-Domain
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Background: Finite-DifferencesTypes of finite-difference approximations
Backward difference
1.5 2 1df f f
dx x
Central difference
1 2 1df f f
dx x
Forward difference
2 2 1df f f
dx x
Short Course on Finite-Difference Frequency-Domain
Background: Finite-DifferencesGeneralized finite-difference
n
ni
ii
d fa
xf
d
Short Course on Finite-Difference Frequency-Domain
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Background: For more information…
• Electromagnetics– Matthew Sadiku, Elements of Electromagnetics, Saunders, 2000
– Constantine Balanis, Advanced Engineering Electromagnetics, Wiley, 1989
• Linear Algebra– Howard Anton, Elementary Linear Algebra 8th Ed., Wiley, New Jersey, 2000
– Gilbert Strang, Linear Algebra and Its Applications 4th Ed., Thomson, California, 2006
• http://web.mit.edu/18.06/www/Video/video-fall-99.html
Short Course on Finite-Difference Frequency-Domain
Finite-Difference Frequency-Domain
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FDFD Method: Yee GridWhat is a discrete grid?
A grid is constructed by dividing space
into discrete cells
Example physical
(continuous) field profile
Field is known only at discrete
points
Representation of what is actually
stored in memory
Short Course on Finite-Difference Frequency-Domain
FDFD Method: Yee GridWhat is a grid unit cell?
y
x
A field component is assigned to a specific point within the grid unit cell.
Whole Grid
A Single Unit Cell
Short Course on Finite-Difference Frequency-Domain
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FDFD Method: Yee GridUnit cells of Yee grids
• Field components are in physically different locations• Field components may reside in different materials even if they are in the
same unit cell• Field components will be out of phase
xy
z
xEyE
zE
xHyH
zH
3D Yee Grid2D Yee Grids1D Yee Grid
zE
xHyH
xy
xy
zH yExE
Ez Mode
Hz Mode
z
xE
yH
yExH
Ey Mode
Ex Mode
z
Short Course on Finite-Difference Frequency-Domain
FDFD Method: Yee GridAnother interpretation of the Yee Grid
Short Course on Finite-Difference Frequency-Domain
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FDFD Method: Yee GridReasons for using a Yee grid
0E
0H
1. Divergence-free 3. Elegant arrangementto approximate curl equations
2. Physical boundary conditions are naturally satisfied
Short Course on Finite-Difference Frequency-Domain
FDFD Method: Yee GridExtended grids
y
x
ij
222 Grid
44 Grid (Ez Mode)
Short Course on Finite-Difference Frequency-Domain
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FDFD Method: Maxwell MatrixNormalize the magnetic field
E j H
H j E Standard Maxwell’s Curl Equations
Normalized Magnetic Field
377E
nH
0
0
H j H
Normalized Maxwell’s Equations
0 rE k H
0 rH k E
0 0 0k Note:
Short Course on Finite-Difference Frequency-Domain
FDFD Method: Maxwell MatrixExpand Maxwell’s equations
0 rE k H
0 rH k E
0
0
0
yzxx x xy y xz z
x zyx x yy y yz z
y xzx x zy y zz z
HHk E E E
y z
H Hk E E E
z x
H Hk E E E
x y
0
0
0
yzxx x xy y xz z
x zyx x yy y yz z
y xzx x zy y zz z
EEk H H H
y z
E Ek H H H
z xE E
k H H Hx y
Short Course on Finite-Difference Frequency-Domain
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FDFD Method: Maxwell MatrixNormalize the grid
Normalized Maxwell’s Equations
Normalized Grid
0x k x 0y k y 0z k z
0
0
0
0
0
0
yzxx x xy y xz z
x zyx x yy y yz z
y xzx x zy y zz z
yzxx x xy y xz z
x zyx x yy y yz z
y xzx x zy y zz
EEk H H H
y z
E Ek H H H
z xE E
k H H Hx y
HHk E E E
y z
H Hk E E E
z x
H Hk E E E
x y
z
yzxx x xy y xz z
x zyx x yy y yz z
y xzx x zy y zz z
yzxx x xy y xz z
x zyx x yy y yz z
y xzx x zy y zz
EEH H H
y z
E EH H H
z xE E
H H Hx y
HHE E E
y z
H HE E E
z x
H HE E E
x y
z
Short Course on Finite-Difference Frequency-Domain
FDFD Method: Maxwell MatrixAssume diagonal tensor functions
yzxx x
x zyy y
y xzz z
EEH
y z
E EH
z xE E
Hx y
yzxx x
x zyy y
y xzz z
HHE
y z
H HE
z x
H HE
x y
Short Course on Finite-Difference Frequency-Domain
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FDFD Method: Maxwell MatrixFinite-difference equation for Hx
yzxx x
EEH
y z
z
xyxE
, ,i j kyE
, ,i j kzE
, ,i j kxHyH
zH
, 1,i j kzE
, , 1i j kyE
, , 1 , ,, 1, , ,, , , ,
i j k i j ki j k i j ky yz i j k
xj k
xz i
x HE E
y
E
z
E
Short Course on Finite-Difference Frequency-Domain
x zyy y
E EH
z x
FDFD Method: Maxwell MatrixFinite-difference equation for Hy
z
xyyE
, ,i j kzE
xH, ,i j kyH
zH, ,i j kxE
1, ,i j kzE
, , 1 , , 1, , , ,, ,, ,
i j k i j k i j ki j
i j kx x z z k i k
yyj
yz x
E E E EH
, , 1i j kxE
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y xzz z
E EH
x y
FDFD Method: Maxwell MatrixFinite-difference equation for Hz
z
xy
zE
xHyH
, ,i j kzH
, ,i j kxE
1, ,i j kyE
1, , , , , 1, , ,, ,, ,
i j k i j k i j ki j
i j ky y x x k i k
zzj
zx y
E E E EH
, 1,i j kxE
, ,i j kyE
Short Course on Finite-Difference Frequency-Domain
yzxx x
HHE
y z
FDFD Method: Maxwell MatrixFinite-difference equation for Ex
, , , ,,
1, 1,, ,
,,
, i j k i j ki j
i j k i j ky yz k
xj
xxz i kH H
EH H
y z
z
xy
, ,i j kxE
yE
zE
xH, ,i j kyH
, ,i j kzH
, , 1i j kyH
, 1,i j kzH
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x zyy y
H HE
z x
FDFD Method: Maxwell MatrixFinite-difference equation for Ey
z
xy
xEyE
zE
yH, ,i j kxH
, ,i j kzH
, , , , 1 , , 1,,
,, ,,
i j k i j k i j k i j kx x z z i j i
yk
yjk
y
H H H H
z xE
, , 1i j kxH
1, ,i j kzH
Short Course on Finite-Difference Frequency-Domain
y xzz z
H HE
x y
FDFD Method: Maxwell MatrixFinite-difference equation for Ez
z
xy
xE
yE
, ,i j kzE
, ,i j kyH
, ,i j kxH
zH
, , 1, , , , , 1,
,, ,,
i j k i j k i j k i j ky y i jx
zi jxzz
kk EH H H H
x y
, 1,i j kxH
1, ,i j k
yH
Short Course on Finite-Difference Frequency-Domain
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FDFD Method: Maxwell MatrixExtended 2D Yee grid (Ez Mode)
12 , , 1,
y y yi j i j i j
x x
H H H
12, , , 1
x x xi j i j i j
H H
y
H
y
12 , 1, ,
z z zi j i j i jE
x x
E E
12, , 1 ,
z z zi j i j i j
y
E
y
E E
y
x
ij
Short Course on Finite-Difference Frequency-Domain
FDFD Method: Maxwell MatrixSummary of FD approximations
x
y
z
yz
x z
y
xx
yy
zz
xx
yy
zz
yz
x z
y
x
x
x
y
z
EE
E E
E E
E
E
E
y z
z x
x y
y z
H
H
H
HH
H H
H H
z x
x y
, , 1, ,
, ,
,
, ,
,
, ,, 1, , ,
, , 1 , , 1, , , ,
1, , , , , 1,
,
,
, ,
, , ,
, ,
1
,
i j kx
i j ky
i j kz
i j k i
i j k i j ki j ki j kxx
i j kyy
i j ky yz z
i j k i j k i j k i j kx x z z
i j k i j k i j k i j ky y x x i j k
j
z
kz z
z
y
E E
y z
E E
E E E E
H
H
H
HH
z x
x
E E E
y
H
y
E
, , , , 1
, , , , 1
, ,
, ,
, ,
, , 1, ,
, , 1, , , ,
, ,
, ,
,, 1
,,
i j kx
i j ky
i
i j k i j ky
i j k i j k i j k i j k
i j kxx
i jx x z z
i j k i j k i j k i j k
kyy
i j kyz
kz
xz
jy x
H
H H H H
H H
z
z x
x y
H
E
E
HE
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FDFD Method: Maxwell MatrixFields are put into column vectors
2-D Systems
1E 5E 9E 13E
2E 6E 10E 14E
3E 7E 11E 15E
4E 8E 12E 16E
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
1
2
3
4
5
E
E
E
E
E
E
1-D Systems
1E2E 3E 4E 5E
Short Course on Finite-Difference Frequency-Domain
FDFD Method: Maxwell MatrixConstruction of field column vectors
1E 5E 9E 13E
2E 6E 10E 14E
3E 7E 11E 15E
4E 8E 12E 16E
1E
2E
3E
4E
5E
6E
7E
8E
9E
10E
11E
12E
13E
14E
15E
16E
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
MATLAB ‘reshape’ commandE = E(:);E = reshape(E,Nx,Ny);
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FDFD Method: Maxwell MatrixPoint-by-point multiplication
,r i iE1E
2E 3E 4E 5E
1 1 1 1
2 2 2 2
3 3 3 3
4 4 4 4
5 5 5 5
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
r r
r r
r r r
r r
r r
E E
E E
E E
E E
E E
εE
1r 2r 3r 4r 5r
Short Course on Finite-Difference Frequency-Domain
FDFD Method: Maxwell MatrixPoint-by-point multiplication
,r i iE1E
2E 3E 4E 5E
1 1r E 1 1 1 1
2 2 2 2
3 3 3 3
4 4 4 4
5 5 5 5
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
r r
r r
r r r
r r
r r
E E
E E
E E
E E
E E
εE
1r 2r 3r 4r 5r
Short Course on Finite-Difference Frequency-Domain
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26
FDFD Method: Maxwell MatrixPoint-by-point multiplication
,r i iE1E
2E 3E 4E 5E
2 2r E
1 1r E 1 1 1 1
2 2 2 2
3 3 3 3
4 4 4 4
5 5 5 5
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
r r
r r
r r r
r r
r r
E E
E E
E E
E E
E E
εE
1r 2r 3r 4r 5r
Short Course on Finite-Difference Frequency-Domain
FDFD Method: Maxwell MatrixPoint-by-point multiplication
,r i iE1E
2E 3E 4E 5E
2 2r E
1 1r E
3 3r E
1 1 1 1
2 2 2 2
3 3 3 3
4 4 4 4
5 5 5 5
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
r r
r r
r r r
r r
r r
E E
E E
E E
E E
E E
εE
1r 2r 3r 4r 5r
Short Course on Finite-Difference Frequency-Domain
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27
FDFD Method: Maxwell MatrixPoint-by-point multiplication
,r i iE1E
2E 3E 4E 5E
2 2r E
1 1r E
3 3r E
4 4r E
1 1 1 1
2 2 2 2
3 3 3 3
4 4 4 4
5 5 5 5
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
r r
r r
r r r
r r
r r
E E
E E
E E
E E
E E
εE
1r 2r 3r 4r 5r
Short Course on Finite-Difference Frequency-Domain
FDFD Method: Maxwell MatrixPoint-by-point multiplication
,r i iE1E
2E 3E 4E 5E
2 2r E
1 1r E
3 3r E
4 4r E
5 5r E
1 1 1 1
2 2 2 2
3 3 3 3
4 4 4 4
5 5 5 5
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
r r
r r
r r r
r r
r r
E E
E E
E E
E E
E E
εE
1r 2r 3r 4r 5r
Short Course on Finite-Difference Frequency-Domain
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28
FDFD Method: Maxwell MatrixDerivative operators for Electric fields
1
12
i i
i
E E E
x x
1E2E 3E 4E 5E
1 1.5
2 2.5
3 3.5
4 4.5
5 5.5
1 1 0 0 0
0 1 1 0 01
0 0 1 1 0
0 0 0 1 1
0 0 0 0 1
x
xEx x
x
x
E E
E E
E Ex
E E
E E
DE
x
Short Course on Finite-Difference Frequency-Domain
FDFD Method: Maxwell MatrixDerivative operators for Electric fields
1
12
i i
i
E E E
x x
1E2E 3E 4E 5E
2 1E E
x
1 1.5
2 2.5
3 3.5
4 4.5
5 5.5
1 1 0 0 0
0 1 1 0 01
0 0 1 1 0
0 0 0 1 1
0 0 0 0 1
x
xEx x
x
x
E E
E E
E Ex
E E
E E
DE
x
Short Course on Finite-Difference Frequency-Domain
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29
FDFD Method: Maxwell MatrixDerivative operators for Electric fields
1
12
i i
i
E E E
x x
1E2E 3E 4E 5E
3 2E E
x
2 1E E
x
1 1.5
2 2.5
3 3.5
4 4.5
5 5.5
1 1 0 0 0
0 1 1 0 01
0 0 1 1 0
0 0 0 1 1
0 0 0 0 1
x
xEx x
x
x
E E
E E
E Ex
E E
E E
DE
x
Short Course on Finite-Difference Frequency-Domain
FDFD Method: Maxwell MatrixDerivative operators for Electric fields
1
12
i i
i
E E E
x x
1E2E 3E 4E 5E
3 2E E
x
2 1E E
x
4 3E E
x
1 1.5
2 2.5
3 3.5
4 4.5
5 5.5
1 1 0 0 0
0 1 1 0 01
0 0 1 1 0
0 0 0 1 1
0 0 0 0 1
x
xEx x
x
x
E E
E E
E Ex
E E
E E
DE
x
Short Course on Finite-Difference Frequency-Domain
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30
FDFD Method: Maxwell MatrixDerivative operators for Electric fields
1
12
i i
i
E E E
x x
1E2E 3E 4E 5E
3 2E E
x
2 1E E
x
4 3E E
x
5 4E E
x
1 1.5
2 2.5
3 3.5
4 4.5
5 5.5
1 1 0 0 0
0 1 1 0 01
0 0 1 1 0
0 0 0 1 1
0 0 0 0 1
x
xEx x
x
x
E E
E E
E Ex
E E
E E
DE
x
Short Course on Finite-Difference Frequency-Domain
FDFD Method: Maxwell MatrixDerivative operators for Electric fields
1
12
i i
i
E E E
x x
1E2E 3E 4E 5E
3 2E E
x
2 1E E
x
4 3E E
x
5 4E E
x
56E E
x
1 1.5
2 2.5
3 3.5
4 4.5
5 5.5
1 1 0 0 0
0 1 1 0 01
0 0 1 1 0
0 0 0 1 1
0 0 0 0 1
x
xEx x
x
x
E E
E E
E Ex
E E
E E
DE
x
Short Course on Finite-Difference Frequency-Domain
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31
FDFD Method: Maxwell MatrixDerivative operators for Magnetic fields
0.51
1.52
2.53
3.54
4.55
1 0 0 0 0
1 1 0 0 01
0 1 1 0 0
0 0 1 1 0
0 0 0 1 1
x
xHx x
x
x
HH
HH
HHx
HH
HH
D H
1H2H 3H 4H 5H
x 1
12
i i
i
H H H
x x
Short Course on Finite-Difference Frequency-Domain
FDFD Method: Maxwell MatrixDerivative operators for Magnetic fields
0.51
1.52
2.53
3.54
4.55
1 0 0 0 0
1 1 0 0 01
0 1 1 0 0
0 0 1 1 0
0 0 0 1 1
x
xHx x
x
x
HH
HH
HHx
HH
HH
D H
1H2H 3H 4H 5H
x 1
12
i i
i
H H H
x x
01H H
x
Short Course on Finite-Difference Frequency-Domain
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32
FDFD Method: Maxwell MatrixDerivative operators for Magnetic fields
0.51
1.52
2.53
3.54
4.55
1 0 0 0 0
1 1 0 0 01
0 1 1 0 0
0 0 1 1 0
0 0 0 1 1
x
xHx x
x
x
HH
HH
HHx
HH
HH
D H
1H2H 3H 4H 5H
x 1
12
i i
i
H H H
x x
2 1H H
x
01H H
x
Short Course on Finite-Difference Frequency-Domain
FDFD Method: Maxwell MatrixDerivative operators for Magnetic fields
0.51
1.52
2.53
3.54
4.55
1 0 0 0 0
1 1 0 0 01
0 1 1 0 0
0 0 1 1 0
0 0 0 1 1
x
xHx x
x
x
HH
HH
HHx
HH
HH
D H
1H2H 3H 4H 5H
x 1
12
i i
i
H H H
x x
2 1H H
x
01H H
x
3 2H H
x
Short Course on Finite-Difference Frequency-Domain
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33
FDFD Method: Maxwell MatrixDerivative operators for Magnetic fields
0.51
1.52
2.53
3.54
4.55
1 0 0 0 0
1 1 0 0 01
0 1 1 0 0
0 0 1 1 0
0 0 0 1 1
x
xHx x
x
x
HH
HH
HHx
HH
HH
D H
1H2H 3H 4H 5H
x 1
12
i i
i
H H H
x x
2 1H H
x
01H H
x
3 2H H
x
4 3H H
x
Short Course on Finite-Difference Frequency-Domain
FDFD Method: Maxwell MatrixDerivative operators for Magnetic fields
1
12
i i
i
H H H
x x
2 1H H
x
01H H
x
3 2H H
x
4 3H H
x
5 4H H
x
0.51
1.52
2.53
3.54
4.55
1 0 0 0 0
1 1 0 0 01
0 1 1 0 0
0 0 1 1 0
0 0 0 1 1
x
xHx x
x
x
HH
HH
HHx
HH
HH
D H
1H2H 3H 4H 5H
x
Short Course on Finite-Difference Frequency-Domain
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34
FDFD Method: Numerical BC’sSimplest boundary conditions
1 11
1 12
1 13
1 14
15
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0 0
x x
x x
Ex xx
x x
x
E
E
E
E
E
DE
Dirichlet Boundary Conditions
6Assume 0E
Periodic Boundary Conditions
6 1Assume E E
1E2E 3E 4E 5E
6E
1E2E 3E 4E 5E 1E
1 11
1 12
1 13
1 14
15
1
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
x x
x x
Ex xx
x x
xx
E
E
E
E
E
DE
x
x
51E E
x
Short Course on Finite-Difference Frequency-Domain
FDFD Method: Numerical BC’sPseudo-periodic boundary conditions
6 1Assume x xjkEE e
1E2E 3E 4E 5E 6E
1 11
1 12
1 13
1 14
15
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0jkx x
x x
x x
Ex
ex
xx
x x
x
E
E
E
E
E
DEx
Short Course on Finite-Difference Frequency-Domain
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35
FDFD Method: Maxwell MatrixDerivative operators on a 4x4 grid
1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0
0 0 0 0 0 0 0 0
0
0
0 0 1 1 0
1Ex x
D
0 0 0
0 0 0 0 0 0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1
0 0 0 0 0 0 0 0 0 0 0 1
0
0 0 0 0
1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0
0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 0 1 0
1Ey y
D
0 0 1 0
0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0
0
0 1 1 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 1 1 0 0
0
0
1Hx x
D
0 0
0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1
0
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0
0 0 0 0 0 0 1 0 0 0 1 0 0 0 0
1Hy y
D
0
0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0
0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1
Short Course on Finite-Difference Frequency-Domain
FDFD Method: Maxwell MatrixFinally in matrix form!!!
, , 1 , ,, 1, , ,
, , 1 , , 1, , , ,
1, , ,
, ,
, ,
,
,
, , 1,
,
, ,
, ,, ,
,
i j k i j ki j k i j ky yz z
i j k i j k i j k
i j
i j
i j kx
i j ky
i j k
kx x z z
i j k i j k
kxx
i j k
i j k i j ky y x
yy
i j kzz z
x
y z
z x
HE EE E
E
x y
H
H
E E E
E E E E
z y
x z
y x
E Ey z xx
E Ez x yy
E Ex y
y
zz
x
z
E E
E E
E E
D D μ
D D μ
D D μ
H
H
H
, , , , 1, , , 1,
, , , , 1 , , 1, ,
,
, ,
, ,
, 1, , , , , 1,
, ,
,
, ,,
,
,
i j k i j ki j k i j ky yz z
i j k i j k i j k i j kx x z z
i j k
i j kxx
i j kyy
i j k i j k i ji j kzz
k
i j kx
i j k
y y x
y
i j kz
x
H HH H
H
y z
z x
H H H
H H
y
HE
H
x
E
E
H Hy z xx
H Hz x yy
H Hx y
z y
x
x
z
y zx zz
y
H H
H H
ED D ε
D D E
ED D εH H
ε
xx
yy
x
y
zy x z
z
z
y
x z
y z
z x
x y
H
H
E E
E E
E E H
xx
yy
zz
z y
x z
y x
x
y
z
y z
z
E
E
E
H H
Hx
yH H
x
H
Short Course on Finite-Difference Frequency-Domain
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36
FDFD Method: Maxwell MatrixSummary
E Ey z z y xx x
E Ez x x z yy y
E Ex y y x zz z
H Hy z z y xx x
H Hz x x z yy y
H Hx y y x zz z
D E D E μ H
D E D E μ H
D E D E μ H
D H D H ε E
D H D H ε E
D H D H ε E
E j H
H j E
0
0
r
r
E k H
H k E
No charges
yzxx x
x zyy y
y xzz z
yzxx x
x zyy y
y xzz z
EEH
y z
E EH
z xE E
Hx y
HHE
y z
H HE
z x
H HE
x y
Normalized H
Normalized Grid Finite-Difference Approximation, , 1 , ,, 1, , ,
, , , ,
, , 1 , , 1, , , ,, , , ,
1, , , , , 1, , ,, , , ,
, , , 1,
i j k i j ki j k i j ky y i j k i j kz z
xx x
i j k i j k i j k i j ki j k i j kx x z zyy y
i j k i j k i j k i j ky y i j k i j kx x
zz z
i j k i j kyz z
E EE EH
y z
E E E EH
z x
E E E EH
x y
HH H
y
, , , , 1, , , ,
, , , , 1 , , 1, ,, , , ,
, , 1, , , , , 1,, , , ,
i j k i j ky i j k i j k
xx x
i j k i j k i j k i j ki j k i j kx x z zyy y
i j k i j k i j k i j ky y i j k i j kx x
zz z
HE
z
H H H HE
z x
H H H HE
x y
Matrix Form
Short Course on Finite-Difference Frequency-Domain
FDFD Method: Maxwell MatrixHints for other formulations
E Ey z z y xx x
E Ez x x z yy y
E Ex y y x zz z
H Hy z z y xx x
H Hz x x z yy y
H Hx y y x zz z
D E D E μ H
D E D E μ H
D E D E μ H
D H D H ε E
D H D H ε E
D H D H ε E
Fully Numerical Methods BPM, Waveguides, & Semi-Analytical Methods
Ey z y xx x
Ex x z yy y
E Ex y y x zz z
Hy z y xx x
Hx x z yy y
H Hx y y x zz z
d
dzd
dz
d
dzd
dz
D E E μ H
E D E μ H
D E D E μ H
D H H ε E
H D H ε E
D H D H ε E
Short Course on Finite-Difference Frequency-Domain
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37
FDFD Method: Maxwell MatrixBlock Matrix Form
E Ey z z y xx x
E Ez x x z yy y
E Ex y y x zz z
H Hy z z y xx x
H Hz x x z yy y
H Hx y y x zz z
D E D E μ H
D E D E μ H
D E D E μ H
D H D H ε E
D H D H ε E
D H D H ε E
Fully Numerical Methods
E Ez y x xx x
E Ez x y yy yE Ey x z zz z
0 D D E μ 0 0 H
D 0 D E 0 μ 0 H
D D 0 E 0 0 μ H
H Hz y x xx x
H Hz x y yy yH Hy x z zz z
0 D D H ε 0 0 E
D 0 D H 0 ε 0 E
D D 0 H 0 0 ε E
E C E μH
H C H εE
E H
H E
Short Course on Finite-Difference Frequency-Domain
FDFD Method: Formulation3D FDFD method
Block Matrix Form
E
H
C E μH
C H εE
x x
y y
z z
xx xx
yy yy
zz zz
H Hz y
H H H Ez xH Hy x
H E
H H E E
H E
μ 0 0 ε 0 0
μ 0 μ 0 ε 0 ε 0
0 0 μ 0 0 ε
0 D D 0 D
C D 0 D C
D D 0
E Ez y
E Ez xE Ey x
D
D 0 D
D D 0
Matrix Wave Equations
1
1
E H
H E
C ε C μ H 0
C μ C ε E 0
AE 0
For 3D analysis, A is usually too big to solve by simple means.
For information on 3D analysis, see
R. C. Rumpf, A. Tal, S. M. Kuebler, “Rigorous electromagnetic analysis of volumetrically complex media using the slice absorption method,” J. Opt. Soc. Am. A 24, 3123-3134 (2007).
Short Course on Finite-Difference Frequency-Domain
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38
FDFD Method: FormulationReduction to two dimensions
H Hx y y x zz z
Ey z xx x
Ex z yy y
D H D H ε E
D E μ H
D E μ H
For problems uniform along the z-direction,
E Hz z D D 0
Maxwell’s equation decouple into two distinct modes:
Ez Mode Hz Mode
E Ex y y x zz z
Hy z xx x
Hx z yy y
D E D E μ H
D H ε E
D H ε E
z
Short Course on Finite-Difference Frequency-Domain
FDFD Method: Formulation2D matrix wave equations
H Hx y y x zz z
Ey z xx x
Ex z yy y
D H D H ε E
D E μ H
D E μ H
Ez Mode Hz ModeE Ex y y x zz z
Hy z xx x
Hx z yy y
D E D E μ H
D H ε E
D H ε E
1
1
Ex xx y z
Ey yy x z
H μ D E
H μ D E
1
1
Hx xx y z
Hy yy x z
E ε D H
E ε D H
1 1H E H Ex yy x z y xx y z zz z
D μ D E D μ D E ε E 1 1E H E Hx yy x z y xx y z zz z
D ε D H D ε D H μ H
1 1
E z
H E H EE x yy x y xx y zz
A E 0
A D μ D D μ D ε 1 1
H z
E H E HH x yy x y xx y zz
A H 0
A D ε D D ε D μ
AE = DHX/URyy*DEX + DHY/URxx*DEY + ERzz; AH = DEX/ERyy*DHX + DEY/ERxx*DHY + URzz;
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FDFD Method: FormulationThese cannot yet be solved
1 Ax b x A b
General solution procedure
Solution of matrix wave equation
1 E z z E A E 0 E A 0 0 trivial solution!!
A source must be incorporated.
1 E z z E A E f E A f
Short Course on Finite-Difference Frequency-Domain
FDFD Method: Boundary Cond’sPerfectly matched layer (1 of 2)
PML
PML
PM
L
PM
L ProblemSpace
1ys
1xs 1x y zs s s
1ys
1xs
Perfectly Matched Layer (PML)
Short Course on Finite-Difference Frequency-Domain
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FDFD Method: Boundary Cond’sPerfectly matched layer (2 of 2)
0
0
r
r
E k s H
H k s E
0 0
0 0
0 0
y z
x
x z
y
x y
z
s s
s
s ss
s
s s
s
0
0
0
0
0
0
1
1
1
x x x
y y y
z z z
s x a x xjk
s y a y yjk
s z a z zjk
max
max
max
1
1
1
p
x x
p
y y
p
z z
a x a x L
a y a y L
a z a z L
2max
2max
2max
sin2
sin2
sin2
xx
yy
zz
xx
L
yy
L
zz
L
Maxwell’s eqs. with PML Computing PML Parameters
max
max
0 5
3 5
1
a
p
Short Course on Finite-Difference Frequency-Domain
FDFD Method: TF/SF SourceTotal-field / scattered-field framework
Problem rows
tota
l-fie
ldsc
atte
red-
field
Short Course on Finite-Difference Frequency-Domain
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FDFD Method: TF/SF SourceCompute source field
1,1
src
,
exp
x y
x y
N N
f
j k k
f
f x y
Compute source as it would exist in a completely homogeneous grid.
incsrc
jk rE r e
This is NOT the “f” term in AE=f
Unit amplitude plane wave
Short Course on Finite-Difference Frequency-Domain
FDFD Method: TF/SF SourceCompute SF masking matrix, Q
tota
l-fie
ldsc
atte
red-
field 1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
1
1
0
0
0
Q
Short Course on Finite-Difference Frequency-Domain
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FDFD Method: TF/SF SourceCompute TF/SF source vector
Source isolated to the scattered-field:
Source isolated to the total-field:
Quantity that must be subtracted from TF terms:
…but only from SF equations:
Quantity that must be added to SF terms:
…but only from TF equations:
“Corrected” matrix problem:
…or:
scat srcf Qf
tot src f I Q f
totAf
totQAf
scatAf
scatI Q Af
tot scat AE QAf I Q Af 0
src AE QA AQ f
src AE = f f QA AQ f
Corrects SF Equations
Corrects TF Equations
Short Course on Finite-Difference Frequency-Domain
FDFD Method: TF/SF SourceExample simulation (1 of 2)
Materials Field Q
Materials Field Q
PML
PML
PML
PML
PML
PML
PML
PML
PML
PML
PML
PML
Scattered-FieldScattered-Field
Scattered-FieldScattered-Field
Total-FieldTotal-Field
Total-FieldTotal-Field
1 1.0n
1 1.0n
2 3.0n
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FDFD Method: TF/SF SourceExample simulation (2 of 2)
Materials Field Q
Materials Field Q
PML
PML
PML
PML
PML
PML
PML
PML
PML
PML
PML
PML
Scattered-FieldScattered-Field
Scattered-FieldScattered-Field
Total-FieldTotal-Field
Total-FieldTotal-Field
1 1.0n
1 1.0n
2 3.0n
Short Course on Finite-Difference Frequency-Domain
FDFD Method: Formulation Summary
1 1H E H EE x yy x y xx y zz
A D μ D D μ D ε
1 1E H E HH x yy x y xx y zz
A D ε D D ε D μ
Matrix Wave Equations
src f QA AQ f
Source Vector
FDFD Matrix Problem
E EA E = f
H HA H = f
Short Course on Finite-Difference Frequency-Domain
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FDFD Method: SolutionMatrix division in MATLAB
1E = A f
Matrix Division
WARNING: Do not compute inverse of A
MATLAB Implementation
E = A\f; WARNING: Do not perform
E = inv(A)*f;
Short Course on Finite-Difference Frequency-Domain
FDFD Method: SolutionIterative algorithms
• Good for very large and sparse matrices
• Many algorithms exist
• Arguably more accurate for large matrices
• Usually do not have to explicitly compute matrix A
• Popular iterative algorithms for FDFD– Generalized minimum residue (GMRES)
– Biconjugate gradient (BCG)
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FDFD Method: SolutionSlice absorption method
1 0 1 1 1 2 1 1 0 1 1 1 3 1
2 1 2 2 2 3 2
3 2 3 3 3 4 3 3 1 3 3 3 4 3
a E b E c E f a E b E c E f
a E b E c E f
a E b E c E f a E b E c E f
R. C. Rumpf, A. Tal, S. M. Kuebler, “Rigorous electromagnetic analysis of volumetrically complex media using the slice absorption method,” J. Opt. Soc. Am. A 24, 3123-3134 (2007).
Short Course on Finite-Difference Frequency-Domain
FDFD Method: Post ProcessingDiffraction by periodic structures
Reflected Power
Transmitted Power
Diffraction Efficiency
inc
DE mPmP
Short Course on Finite-Difference Frequency-Domain
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FDFD Method: Post ProcessingCompute spatial harmonics
refE
trnE
Step 1: Extract Eref and Etrn Step 2: Remove phase tilt
ref refxjk xE x E x e
trn trnxjk xE x E x e
Step 3: Compute FFT
ref refFFTS m E x
trn trnFFTS m E x
Note: Some FFT algorithms require that you divide by the number of points and shift after calculation.
Eref = fftshift(fft(Fref))/Nx;Etrn = fftshift(fft(Ftrn))/Nx;
SF
TF
Short Course on Finite-Difference Frequency-Domain
FDFD Method: Post ProcessingCompute diffraction efficiencies
Step 2: Compute DE
ref
2 ,ref inc
Re z m
z
kR m S m
k
Step 1: Compute wave vectorcomponents
, ,inc
2x m x
mk k
*
2ref 2, 0 ref ,z m x mk k n k
*
2trn 2, 0 trn ,z m x mk k n k
trn
2 , reftrn inc
trn
Re z mE
z
kT m S m
k
trn
2 , reftrn inc
trn
Re z mH
z
kT m S m
k
Note: These equations assume the source has unit amplitude
,..., 2, 1,0,1,2,...,2 2x xN N
m
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FDFD Method: Post ProcessingCompute total power
totm
R R mStep 1: Compute total reflected power
totm
T T mStep 2: Compute total transmitted power
Step 3: Verify conservation of energy
tot tot 1R T
Note: This conservation condition is only obeyed when no materials have loss or gain.
Short Course on Finite-Difference Frequency-Domain
FDFD Method: Algorithm
1. Construct FDFD Problema. Define your problemb. Choose a gridc. Assign materials to the grid
2. Implement PMLa. Compute sx, sy, and sz
b. Incorporate into r and r
3. Construct Matrix Problema. Compute derivative matricesb. Construct diagonal materials
matricesc. Compute Ad. Compute source
i. compute source fieldii. compute Qiii. compute source vector f
4. Solve Matrix Problem: E = A-1f;
5. Post Process Dataa. Extract Eref and Etrn
b. Remove phase tiltc. FFT the fieldse. Compute wave vector termsd. Compute diffraction efficienciesf. Verify conservation of energy
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FDFD Method: For more information…
• Literature– W. Sun, K. Liu, C. A. Balanis, “Analysis of Singly and Doubly Periodic
Absorbers by Frequency-Domain Finite-Difference Method,” IEEE Trans. Ant. and Prop. 44, 798-805 (1996)
– S. Wu, E. N. Glytsis, “Finite-number-of-periods holographic gratings with finite-width incident beams: analysis using the finite-difference frequency-domain method,” J. Opt. Soc. Am. A 19, 2018-2029 (2002)
• Raymond Rumpf’s Ph.D. Dissertation– R. C. Rumpf, “Design and optimization of nano-optical elements by
coupling fabrication to optical behavior,” Ph.D. dissertation, University of Central Florida, 2006.
– See chapter 3, pp. 60—81
– http://purl.fcla.edu/fcla/etd/CFE0001159
Short Course on Finite-Difference Frequency-Domain
Implementation
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Implementation: 3D2DUniform devices
perio
dic
boun
dary
periodic boundary
Absorbing boundary
Absorbing boundary
Short Course on Finite-Difference Frequency-Domain
Implementation: 3D2DEffective index method
1,effn
2,effn
1,effn
2,effn
Effective indices are best computed by modeling the vertical cross section as a slab waveguide.
A simple average index can also produce good results.
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Implementation: Grid & Materials(1) Choose initial grid resolution
Must resolve the minimum wavelength
0min max , 10
n x yN
N
x
Must resolve the minimum structural dimension
min 1d d
d
dN
N
Initial grid resolution is the smallest number computed above
min ,x y d
y
Short Course on Finite-Difference Frequency-Domain
Implementation: Grid & Materials(2) “Snap” grid to critical dimensions
Decide what dimensions along each axis are critical
Compute how many grid cells comprise dc, and round UP
ceil
ceil
x x x
y y y
M d
M d
Adjust grid resolution to fit this dimension in grid EXACTLY
Typically this is a lattice constant or grating period along x Typically this is a film thickness along y
and x yd d
x x x
y y y
d M
d M
initi
al g
rid
critical dimension
adju
sted
grid
critical dimension
Short Course on Finite-Difference Frequency-Domain
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Implementation: Grid & Materials(3) Compute total grid size
x
y
yN
xN
• Don’t forget to add cells for PML!
• Must often add “space” between PML and device.
PML space2 2
xx
x
yy
y
N
N N N
Problem Space
Buffer Space
Buffer Space
PML
PML
Note: This is particularly important when modeling devices with large evanescent fields.
Short Course on Finite-Difference Frequency-Domain
Implementation: Grid & Materials(4) Compute 2X grid
1X Grid 2X Grid
Short Course on Finite-Difference Frequency-Domain
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Implementation: Grid & Materials(5) Assign materials
Direct Averaged
Short Course on Finite-Difference Frequency-Domain
Implementation: Grid & Materials(6) Extract materials onto 1X grid
2X Grid
x
yzE
zH
,x yH E
,y xH E
zz
zz
,xx yy
,yy xx
Field and materials assignments
zE
xHyH
xy
Ez Mode
xy
zH yExE
Hz Mode
I II
III IV
1X GridsI II
III IV
Short Course on Finite-Difference Frequency-Domain
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Implementation: Grid & Materials(7) Oh yeah, metals!
Perfect Electric Conductors
10000r
1 1
00 0 1 0 0 m
M M
E f
E
E f
or
Include Tangential Fields at Boundary (TM modes!)
0mE
xy
zH yExE
Hz Mode
Bad placement of metals Good placement of metals
Short Course on Finite-Difference Frequency-Domain
Implementation: TestingGuided-mode resonance filters
Both a diffraction grating and a waveguide
Incident
Reflected
Transmitted
1
2
134 nm
314 nm
1.0
1.52
2.0
2.1
0.5
L
L
T
n
n
n
n
f
2n
Ln Hn
1n
Guided-mode resonance filters are excellent devices for benchmarking and testing numerical codes because they are so sensitive to their geometry, refractive indices, and dispersion.
Hz Ez
S. Tibuleac R. Magnusson, “Reflection and transmission guided-mode resonance filters,” J. Opt. Soc. Am. A 14, 1617-1626 (1997)
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Implementation: TestingConvergence
Grid Resolution
Ans
wer
Probably “good enough”
Short Course on Finite-Difference Frequency-Domain
Simulation Examples
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Simulation Examples:Guided-Mode Resonance Filter
Short Course on Finite-Difference Frequency-Domain
Simulation Examples:Photonic Crystal
Short Course on Finite-Difference Frequency-Domain
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Simulation Examples:Broadband Polarizer
Short Course on Finite-Difference Frequency-Domain