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9/22/2016
1
Lecture 11 Slide 1
EE 5303
Electromagnetic Analysis Using Finite‐Difference Time‐Domain
Lecture #11
Formulation of 2D‐FDTD Without a PML These notes may contain copyrighted material obtained under fair use rules. Distribution of these materials is strictly prohibited
Lecture Outline
Lecture 11 Slide 2
• Code development sequence for 2D
• Maxwell’s equations
• Finite‐difference approximations
• Reduction to two dimensions
• Update equations
• Boundary Conditions
• Revised FDTD Algorithm for 2D Simulations
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Lecture 11 Slide 3
Code Development Sequence
Lecture 11 Slide 4
Step 1 – Basic Update Equations
The basic update equations are implemented along with simple Dirichlet boundary conditions.
Dirichlet Boundary Condition
Dirichlet Boundary Condition
DirichletBoundary Condition D
irichlet
Boundary C
onditio
n
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Lecture 11 Slide 5
Step 2 – Incorporate Periodic Boundaries
Periodic boundary conditions are incorporated so that a wave leaving the grid reenters the grid at the other side.
Periodic Boundary Condition
Periodic Boundary Condition
Periodic Boundary Condition
Perio
dic B
oundary C
onditio
n
Lecture 11 Slide 6
Step 3 – Incorporate a PML
The perfectly matched layer (PML) absorbing boundary condition is incorporated to absorb outgoing waves.
y‐lo PML
y‐hi PML
x‐hi PMLx‐
lo PML
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4
Lecture 11 Slide 7
Step 4 – Total‐Field/Scattered‐Field
Most periodic electromagnetic devices are modeled by using periodic boundaries for the horizontal axis and a PML for the vertical axis. We then implement TF/SF at the vertical center of the grid for testing.
Periodic Boundary Condition
Perio
dic B
oundary C
onditio
n
Lecture 11 Slide 8
Step 5 – Calculate TRN, REF, and CON
We move the TF/SF interface to a unit cell or two outside of the top PML. We include code to calculate Fourier transforms and to calculate transmittance, reflectance, and conservation of power.
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Lecture 11 Slide 9
Step 6 – Model a Device to Benchmark
We build a device on the grid that has a known solution. We run the simulation and duplicate the known results to benchmark our new code.
Lecture 11 Slide 10
Summary of Code Development Sequence
Step 1 – Basic Update + Dirichlet
Step 2 – Basic Update + Periodic BC
Step 3 – Add PML Step 4 – TF/SF
Step 5 – Calculate Response Step 6 – Add a Device and Benchmark
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Lecture 11 Slide 11
Maxwell’s Equations
Lecture 11 Slide 12
Maxwell’s Equations
We start with Maxwell’s equations in the following form:
0
v
DH
t
BE
t
B
D
0
0
r
r
D t E t
B t H t
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7
Lecture 11 Slide 13
Normalize the Electric Fields
We will now adopt the more conventional approach in FDTD and normalize the electric field according to:
0
0 0
1E E E
To be consistent, we also need to normalize other parameters related to the electric field.
0
0 0
1D D c D
0
0 0
0
0 0
1
1
P P c P
J J c J
We won’t be using J and P.
Lecture 11 Slide 14
Normalized Maxwell’s Equations
Using the normalized fields, Maxwell’s equations become
0
0
1
r
r
HE
c t
DH
c t
D E
These equations are independent of r.
We can derive update equations from these expressions without complicating them with what might need to be done to model r.
This is a very simple equation that makes it much easier to model more sophisticated dielectrics that may be anisotropic, nonlinear, dispersive, all of the above, or something else altogether.
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Lecture 11 Slide 15
HDE Algorithm
To make our code more modular, we will modify the FDTD algorithm to what is called the HDE algorithm.
Update H from E
0
r HE
c t
Update D from H
0
1 DH
c t
Update E from D
rD E
t
Lecture 11 Slide 16
Expand Maxwell’s Equations
0
r HE
c t
0
1 DH
c t
0
0
0
1
1
1
y xz
yx z
y x z
H DH
y z c t
DH H
z x c t
H H D
x y c t
0
0
0
1
1
1
y yxz zxx xy xz
yx xz zyx yy yz
y yx x zzx zy zz
E HHE H
y z c t t t
HE HE H
z x c t t t
E HE H H
x y c t t t
rD E
x xx x xy y xz z
y yx x yy y yz z
z zx x zy y zz z
D E E E
D E E E
D E E E
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Lecture 11 Slide 17
Assume Only Diagonal Tensors
0
r HE
c t
0
1 DH
c t
0
0
0
1
1
1
y xz
yx z
y x z
H DH
y z c t
DH H
z x c t
H H D
x y c t
0
1y yxzxx xy
E HHE
y z c t t
z
xz
H
t
0
1x xzyx
E HE
z x c t
y z
yy yz
H H
t t
0
1y x xzx
E E H
x y c t
y
zy
H
t
z
zz
H
t
rD E
x xx x xy yD E E xz zE
y yx xD E yy y yz zE E
z zx xD E zy yE
zz zE
Lecture 11 Slide 18
Governing Equations
These are the primary governing equations from which we will formulate the 2D and 3D FDTD method.
0
0
0
1
1
1
y xz
yx z
y x z
H DH
y z c t
DH H
z x c t
H H D
x y c t
0
0
0
y xx xz
yy yx z
y x zz z
E HE
y z c t
HE E
z x c t
E E H
x y c t
x xx x
y yy y
z zz z
D E
D E
D E
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Lecture 11 Slide 19
Final Analytical Equations
It will be beneficial in the formulation and implementation to calculate the curl terms separately. We start by expressing Maxwell’s equations in the following form.
0
0
0
1
1
1
H xx
yHy
H zz
DC
c t
DC
c t
DC
c t
0
0
0
E xx xx
yy yEy
E zz zz
HC
c t
HC
c t
HC
c t
x xx x
y yy y
z zz z
D E
D E
D E
yE zx
E x zy
yE xz
EEC
y z
E EC
z x
E EC
x y
yH zx
H x zy
yH xz
HHC
y z
H HC
z xH H
Cx y
Lecture 11 Slide 20
Finite‐Difference Approximations
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Lecture 11 Slide 21
Finite‐Difference Equations for H
0
0
0
E xx xx
yy yEy
E zz zz
HC
c t
HC
c t
HC
c t
2 2
2 2
2 2
, , , ,, ,, ,
0
, , , ,, ,
, ,
0
, , , ,, ,, ,
0
t t
t t
t t
i j k i j ki j ki j k x xt txxE
x t
i j k i j ki j ky yi j k t tyyE
y t
i j k i j ki j ki j k z zt tzzE
z t
H HC
c t
H HC
c t
H HC
c t
, 1, , , , , 1 , ,
, ,
, , 1 , , 1, , , ,
, ,
1, , , , , 1, , ,
, ,
i j k i j k i j k i j k
i j k z z y yE t t t tx t
i j k i j k i j k i j k
i j k x x z zE t t t ty t
i j k i j k i j k i j k
i j k y y x xE t t t tz t
E E E EC
y z
E E E EC
z x
E E E EC
x y
yE zx
E x zy
yE xz
EEC
y z
E EC
z x
E EC
x y
Curl Terms
Maxwell’s Equations
Lecture 11 Slide 22
Finite‐Difference Equations for D
0
0
0
1
1
1
H xx
yHy
H zz
DC
c t
DC
c t
DC
c t
2
2
2
, , , ,
, ,
0
, , , ,
, ,
0
, , , ,
, ,
0
1
1
1
t
t
t
i j k i j k
i j k x xH t t tx t
i j k i j k
i j k y yH t t ty t
i j k i j k
i j k z zH t t tz t
D DC
c t
D DC
c t
D DC
c t
2 2 2 2
2
2 2 2 2
2
2 2 2
2
, , , , 1, , , 1,
, ,
, , , , 1 , , 1, ,
, ,
, , 1, , ,
, ,
t t t t
t
t t t t
t
t t t
t
i j k i j ki j k i j ky yi j k z zt t t tH
x t
i j k i j k i j k i j k
i j k x x z zt t t tHy t
i j k i j k i jy yi j k xt t tH
z t
H HH HC
y z
H H H HC
z x
H H HC
x
2
, , 1,t
k i j k
x tH
y
yH zx
H x zy
yH xz
HHC
y z
H HC
z xH H
Cx y
Curl Terms
Maxwell’s Equations
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Lecture 11 Slide 23
Finite‐Difference Equations for E
x xx x
y yy y
z zz z
D E
D E
D E
, , , ,, ,
, , , ,, ,
, , , ,, ,
i j k i j ki j k
x xx xt t
i j k i j ki j k
y yy yt t
i j k i j ki j k
z zz zt t
D E
D E
D E
Lecture 11 Slide 24
Reduction to Two Dimensions
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13
Lecture 11 Slide 25
Real Electromagnetic Devices
All physical devices are three‐dimensional.
Lecture 11 Slide 26
3D 2D (Exact)
Sometimes it is possible to describe a physical device using just two dimensions. Doing so dramatically reduces the numerical complexity of the problem and is ALWAYS GOOD PRACTICE.
periodic BC
perio
dic B
C
absorbing BC
absorbing BC
z
x
y
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Lecture 11 Slide 27
3D 2D (Approximate)Many times it is possible to approximate a 3D device in two dimensions. It is very good practice to at least perform the initial simulations in 2D and only moving to 3D to verify the final design.
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.
1,effn
2,effn
absorbing BC
absorbing BC
absorbing B
C
absorbing BC
Lecture 11 Slide 28
2D Grids are Infinite in the 3rd Dimension
periodic BC
perio
dic B
C
absorbing BC
absorbing BC
Anything represented on a 2D grid, is actually a device that is of infinite extent along the 3rd dimension.
0z
Assuming the z direction is uniform and of infinite extent, then the field is also uniform and of infinite extent in the z direction.
x
y
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15
Lecture 11 Slide 29
Assume Uniform in z Direction
For 2D devices,/z=0 and Maxwell’s equations reduce to
x xx x
y yy y
z zz z
D E
D E
D E
, , , ,, ,
, , , ,, ,
, , , ,, ,
i j k i j ki j k
x xx xt t
i j k i j ki j k
y yy yt t
i j k i j ki j k
z zz zt t
D E
D E
D E
yH zx
HHC
y z
H xy
HC
z
z
yH xz
H
x
H HC
x y
yE zx
EEC
y z
E xy
EC
z
z
yE xz
E
x
E EC
x y
0
0
0
E xx xx
yy yEy
E zz zz
HC
c t
HC
c t
HC
c t
0
0
0
1
1
1
H xx
yHy
H zz
DC
c t
DC
c t
DC
c t
, 1, , , , , 1 , ,
, ,
i j k i j k i j k i j k
i j k z z y yE t t t tx t
E E E EC
y z
, , 1 , ,
, ,
i j k i j k
i j k x xE t ty t
E EC
z
1, , , ,
1, , , , , 1, , ,
, ,
i j k i j k
z zt t
i j k i j k i j k i j k
i j k y y x xE t t t tz t
E E
x
E E E EC
x y
2 2
2 2
2 2
, , , ,, ,, ,
0
, , , ,, ,
, ,
0
, , , ,, ,, ,
0
t t
t t
t t
i j k i j ki j ki j k x xt txxE
x t
i j k i j ki j ky yi j k t tyyE
y t
i j k i j ki j ki j k z zt tzzE
z t
H HC
c t
H HC
c t
H HC
c t
2 2 2 2
2
, , , , 1, , , 1,
, , t t t t
t
i j k i j ki j k i j ky yi j k z zt t t tH
x t
H HH HC
y z
2 2
2
, , , , 1
, , t t
t
i j k i j k
i j k x xt tHy t
H HC
z
2 2
2 2 2 2
2
, , 1, ,
, , 1, , , , , 1,
, ,
t t
t t t t
t
i j k i j k
z zt t
i j k i j k i j k i j ky yi j k x xt t t tH
z t
H H
x
H H H HC
x y
2
2
2
, , , ,
, ,
0
, , , ,
, ,
0
, , , ,
, ,
0
1
1
1
t
t
t
i j k i j k
i j k x xH t t tx t
i j k i j k
i j k y yH t t ty t
i j k i j k
i j k z zH t t tz t
D DC
c t
D DC
c t
D DC
c t
Lecture 11 Slide 30
2D Finite‐Difference Equations
We eliminate the z‐derivative terms.
x xx x
y yy y
z zz z
D E
D E
D E
, , , ,, ,
, , , ,, ,
, , , ,, ,
i j k i j ki j k
x xx xt t
i j k i j ki j k
y yy yt t
i j k i j ki j k
z zz zt t
D E
D E
D E
H zx
H zy
yH xz
HC
y
HC
xH H
Cx y
E zx
E zy
yE xz
EC
y
EC
x
E EC
x y
0
0
0
E xx xx
yy yEy
E zz zz
HC
c t
HC
c t
HC
c t
0
0
0
1
1
1
H xx
yHy
H zz
DC
c t
DC
c t
DC
c t
, 1, , ,
, ,
1, , , ,
, ,
1, , , , , 1, , ,
, ,
i j k i j k
i j k z zE t tx t
i j k i j k
i j k z zE t ty t
i j k i j k i j k i j k
i j k y y x xE t t t tz t
E EC
y
E EC
x
E E E EC
x y
2 2
2 2
2 2
, , , ,, ,, ,
0
, , , ,, ,
, ,
0
, , , ,, ,, ,
0
t t
t t
t t
i j k i j ki j ki j k x xt txxE
x t
i j k i j ki j ky yi j k t tyyE
y t
i j k i j ki j ki j k z zt tzzE
z t
H HC
c t
H HC
c t
H HC
c t
2 2
2
2 2
2
2 2 2 2
2
, , , 1,
, ,
, , 1, ,
, ,
, , 1, , , , , 1,
, ,
t t
t
t t
t
t t t t
t
i j k i j k
i j k z zt tHx t
i j k i j k
i j k z zt tHy t
i j k i j k i j k i j ky yi j k x xt t t tH
z t
H HC
y
H HC
x
H H H HC
x y
2
2
2
, , , ,
, ,
0
, , , ,
, ,
0
, , , ,
, ,
0
1
1
1
t
t
t
i j k i j k
i j k x xH t t tx t
i j k i j k
i j k y yH t t ty t
i j k i j k
i j k z zH t t tz t
D DC
c t
D DC
c t
D DC
c t
9/22/2016
16
Lecture 11 Slide 31
We No Longer Need Grid Index k
For 2D devices, we only retain grid indices i and j.
x xx x
y yy y
z zz z
D E
D E
D E
, ,,
, ,,
, ,,
i j i ji j
x xx xt t
i j i ji j
y yy yt t
i j i ji j
z zz zt t
D E
D E
D E
H zx
H zy
yH xz
HC
y
HC
xH H
Cx y
E zx
E zy
yE xz
EC
y
EC
x
E EC
x y
0
0
0
E xx xx
yy yEy
E zz zz
HC
c t
HC
c t
HC
c t
0
0
0
1
1
1
H xx
yHy
H zz
DC
c t
DC
c t
DC
c t
, 1 ,
,
1, ,
,
1, , , 1 ,
,
i j i j
i j z zE t tx t
i j i j
i j z zE t ty t
i j i j i j i j
i j y y x xE t t t tz t
E EC
y
E EC
x
E E E EC
x y
2 2
2 2
2 2
, ,,,
0
, ,,
,
0
, ,,,
0
t t
t t
t t
i j i ji ji j x xt txxE
x t
i j i ji jy yi j t tyyE
y t
i j i ji ji j z zt tzzE
z t
H HC
c t
H HC
c t
H HC
c t
2 2
2
2 2
2
2 2 2 2
2
, , 1
,
, 1,
,
, 1, , , 1
,
t t
t
t t
t
t t t t
t
i j i j
i j z zt tHx t
i j i j
i j z zt tHy t
i j i j i j i jy yi j x xt t t tH
z t
H HC
y
H HC
x
H H H HC
x y
2
2
2
, ,
,
0
, ,
,
0
, ,
,
0
1
1
1
t
t
t
i j i j
i j x xH t t tx t
i j i j
i j y yH t t ty t
i j i j
i j z zH t t tz t
D DC
c t
D DC
c t
D DC
c t
Lecture 11 Slide 32
Two Distinct Modes: Ez and HzMaxwell’s equations have decoupled into two distinct sets of equations.
x xx x
y y
z zz z
y y
D E
D E
D E
, ,,
,
, ,
,,
,
i j i ji j
x xx xt t
i j i ji j
y yy
i j i ji j
z zz zt t
yt t
D E
D E
D E
H z
yH xz
x
H zy
H HC
x
HC
y
HC
x
y
E zx
E
y
zy
E xz
E
EC
y
E
y
EC
Cx
x
0
0
0
E xx xx
yy yE
E zz z
y
z
HC
c t
HC
c
HC
c t
t
0
0
0
1
1
1
H z
xx
H
z
H
yy
DC
c t
DC
c
D
c t
t
C
, 1 ,
,
1, ,
,
1, , , 1 ,
,
i j i j
i j z zE t tx t
i j i j
i j z
i j i j i j i j
i j y y x xE t t t t
E
t
z
t
t
zty
E EC
y
E EC
x
E E E EC
x y
2
2 2
2 2
2
, ,,,
,
0
,,
,
0
, ,
,
0
,
t t
t t
t t
i j i ji ji j x xt txxE
x t
i j i ji jy yi j t tyyE
y
i j i ji ji j z zt t
t
zzEz t
H HC
c t
H HC
H HC
c
t
t
c
2 2 2 2
2
2 2
2
2 2
2
,
, , 1
,
,
1, , , 1
1
,
,
,
t t
t
t t t t
t
t t
t
i j i j
i j z zt tHx t
i j i
i j i j i j i jy
j
i j z zt
yi j x xt t t tH
t
tH
z
y t
H HC
y
H H H HC
x y
H HC
x
2
2
2
, ,
,
0
, ,
,
,
,
0
0
,
1
1
1
t
t
t
i j i j
i j x xH t t tx t
i j i j
i j y
i j i j
i j z zH t t tz t
yH t t ty t
D DC
c
D DC
c
t
D DC
c t
t
9/22/2016
17
Lecture 11 Slide 33
EzMode
z zz zD E , ,,i j i ji j
z zz zt tD E
yH xz
H HC
x y
E zx
E zy
EC
y
EC
x
0
0
E xx xx
yy yEy
HC
c t
HC
c t
0
1H zz
DC
c t
, 1 ,
,
1, ,
,
i j i j
i j z zE t tx t
i j i j
i j z zE t ty t
E EC
y
E EC
x
2 2
2 2
, ,,,
0
, ,,
,
0
t t
t t
i j i ji ji j x xt txxE
x t
i j i ji jy yi j t tyyE
y t
H HC
c t
H HC
c t
2 2 2 2
2
, 1, , , 1
, t t t t
t
i j i j i j i jy yi j x xt t t tH
z t
H H H HC
x y
2
, ,
,
0
1t
i j i j
i j z zH t t tz t
D DC
c t
Lecture 11 Slide 34
HzMode
x xx x
y yy y
D E
D E
, ,,
, ,,
i j i ji j
x xx xt t
i j i ji j
y yy yt t
D E
D E
H zx
H zy
HC
y
HC
x
yE xz
E EC
x y
0
E zz zz
HC
c t
0
0
1
1
H xx
yHy
DC
c t
DC
c t
1, , , 1 ,
,
i j i j i j i j
i j y y x xE t t t tz t
E E E EC
x y
2 2
, ,,,
0
t t
i j i ji ji j z zt tzzE
z t
H HC
c t
2 2
2
2 2
2
, , 1
,
, 1,
,
t t
t
t t
t
i j i j
i j z zt tHx t
i j i j
i j z zt tHy t
H HC
y
H HC
x
2
2
, ,
,
0
, ,
,
0
1
1
t
t
i j i j
i j x xH t t tx t
i j i j
i j y yH t t ty t
D DC
c t
D DC
c t
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Lecture 11 Slide 35
Update Equations
Lecture 11 Slide 36
Update Equations for EzMode
2 2
2 2
2
,, , 0,
,, ,0
,
, , ,
0
, ,
,
1
t t
t t
t
i ji j i j Ex x xi jt t t
xx
i ji j i j Ey y yi jt t t
yy
i j i j i jHz z z tt t t
i j i j
z zi jt t t tzz
c tH H C
c tH H C
D D c t C
E D
Solving the finite‐difference equations for the future time values of the fields associated with the Ezmode leads to:
2 2
2 2
2
, ,,,
0
, ,,
,
0
, ,
,
0
, ,,
1
t t
t t
t
i j i ji ji j x xt txxE
x t
i j i ji jy yi j t tyyE
y t
i j i j
i j z zH t t tz t
i j i ji j
z zz zt t
H HC
c t
H HC
c t
D DC
c t
D E
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Lecture 11 Slide 37
Update Equations for HzMode
2 2
2
2
,, , 0,
, , ,
0
, , ,
0
, ,
,
, ,
,
1
1
t t
t
t
i ji j i j Ez z zi jt t t
zz
i j i j i jHx x x tt t t
i j i j i jHy y y tt t t
i j i j
x xi jt t t txx
i j i j
y yi jt t t tyy
c tH H C
D D c t C
D D c t C
E D
E D
Solving the finite‐difference equations for the future time values of the fields associated with the Hzmode leads to:
2 2
2
2
, ,,,
0
, ,
,
0
, ,
,
0
, ,,
, ,,
1
1
t t
t
t
i j i ji ji j z zt tzzE
z t
i j i j
i j x xH t t tx t
i j i j
i j y yH t t ty t
i j i ji j
x xx xt t
i j i ji j
y yy yt t
H HC
c t
D DC
c t
D DC
c t
D E
D E
Lecture 11 Slide 38
Boundary Conditions
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Lecture 11 Slide 39
Where Are the Boundary Conditions?
All of the spatial derivatives appear only the curl calculations.
, 1 ,
,
1, ,
,
1, , , 1 ,
,
i j i j
i j z zE t tx t
i j i j
i j z
i j i j i j i j
i j y y x xE t t t t
E
t
z
t
t
zty
E EC
y
E EC
x
E E E EC
x y
2 2 2 2
2
2 2
2
2 2
2
,
, , 1
,
,
1, , , 1
1
,
,
,
t t
t
t t t t
t
t t
t
i j i j
i j z zt tHx t
i j i
i j i j i j i jy
j
i j z zt
yi j x xt t t tH
t
tH
z
y t
H HC
y
H H H HC
x y
H HC
x
Boundary conditions are handled in the curl computations.
We have “modularized” the boundary conditions by isolating the curl calculations.
(y‐hi)
(x‐hi)
(x‐hi and y‐hi)
(y‐lo)
(x‐lo)
(x‐lo and y‐lo)
Lecture 11 Slide 40
Hint About Implementing Boundary Conditions
DO NOT USEIF STATEMENTS!!!Your code will not be much shorter and it will run slower if you use if statements.
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Lecture 11 Slide 41
Dirichlet Boundary Conditions for CEx
, 1 ,
,
,
1.
0 2.
y
i j i j
z zt ty
i jEx i Nt
z ty
E Ej N
yC
Ej N
y
For CEx, the problem occurs at the y‐hi side of the grid.
We fix it explicitly.
% Compute CExfor nx = 1 : Nx
for ny = 1 : Ny-1CEx(nx,ny) = (Ez(nx,ny+1) - Ez(nx,ny))/dy;
endCEx(nx,Ny) = (0 - Ez(nx, Ny))/dy;
end
1
2 y‐hi
Lecture 11 Slide 42
Dirichlet Boundary Conditions for CEy
1, ,
,
,
1.
0 2.
i j i j
z zt txi jE
y i jt
z tx
E Ei N
xCE
i Nx
For CEy, the problem occurs at the x‐hi side of the grid.
We fix it explicitly.
% Compute CEyfor ny = 1 : Ny
for nx = 1 : Nx-1CEy(nx,ny) = - (Ez(nx+1,ny) - Ez(nx,ny))/dx;
endCEy(Nx,ny) = - (0 - Ez(Nx,ny))/dx;
end
1 2
x‐hi
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Lecture 11 Slide 43
Dirichlet Boundary Conditions for CHz
2 2 2 2
2 2 2
2 2 2
, 1, , , 1
1, 1, 1, 1
,
,1 1,12
1. for 1 and 1
0 2. for 1 and 1
t t t t
t t t
t t t
i j i j i j i jy y x xt t t t
j j jy x xt t t
i jHz i it t
y y xt t t
H H H Hi j
x y
H H Hi j
x yC
H H H
x
2 2
,1
1,1 1,1
0 3. for 1 and 1
0 0 4. for 1 and 1
t t
i
y xt t
i jy
H Hi j
x y
For CHz, the problem occurs at both the x‐lo side of the grid and y‐lo.
We fix both of these explicitly.
12
34 y‐lo
x‐lo
Lecture 11 Slide 44
MATLAB Code for CHz (1 of 3)
First, we just blindly implement the curl calculation…
% Compute CHzfor ny = 1 : Ny
for nx = 1 : NxCHz(nx,ny) = (Hy(nx,ny) - Hy(nx-1,ny))/dx ...
- (Hx(nx,ny) - Hx(nx,ny-1))/dy;end
end
As expected, this will produce an error when trying to access Hy(0,ny)or Hx(nx,0)
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Lecture 11 Slide 45
MATLAB Code for CHz (2 of 3)
Second, we handle the problem at nx = 1 explicitly by copying the code inside the nx loop, pasting it above, and handling the problem.
% Compute CHzfor ny = 1 : Ny
CHz(1,ny) = (Hy(1,ny) - 0)/dx ...- (Hx(1,ny) - Hx(1,ny-1))/dy;
for nx = 2 : NxCHz(nx,ny) = (Hy(nx,ny) - Hy(nx-1,ny))/dx ...
- (Hx(nx,ny) - Hx(nx,ny-1))/dy;end
end
We still will have an error at Hx(1,0) and Hx(nx,0)
Lecture 11 Slide 46
MATLAB Code for CHz (3 of 3)
Third, we handle the problem at ny = 1 explicitly by copying the code inside the ny loop, pasting it above, and handling the problem.
% Compute CHzCHz(1,1) = (Hy(1,1) - 0)/dx ...
- (Hx(1,1) - 0)/dy;for nx = 2 : Nx
CHz(nx,1) = (Hy(nx,1) - Hy(nx-1,1))/dx ...- (Hx(nx,1) - 0)/dy;
endfor ny = 2 : Ny
CHz(1,ny) = (Hy(1,ny) - 0)/dx ...- (Hx(1,ny) - Hx(1,ny-1))/dy;
for nx = 2 : NxCHz(nx,ny) = (Hy(nx,ny) - Hy(nx-1,ny))/dx ...
- (Hx(nx,ny) - Hx(nx,ny-1))/dy;end
end
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Lecture 11 Slide 47
Revised FDTD Algorithm
for 2D Simulations
Lecture 11 Slide 48
FDTD Algorithm for EzMode
, 1 , 1, ,
, ,
, ,
0 0
y
i j i j i j i jz z z zt t t t
y xi j i jE Ex yi N i jt t
z zt ty x
E E E Ej N i N
y xC CE E
j N i Ny x
2 2 2 2 2
, ,, ,, , 0 0, , t t t t t
i j i ji j i ji j i j E Ex x x y y yi j i jt t t tt t
xx yy
c t c tH H C H H C
2 2 2 2
2 2 2
2 2 2
, 1, , , 1
1, 1, 1, 1
,
,1 1,12 ,1
for 1 and 1
0 for 1 and 1
t t t t
t t t
t t t
i j i j i j i jy y x xt t t t
j j jy x xt t t
i jHz i it t i
y y xt t t
H H H Hi j
x y
H H Hi j
x yC
H H H
x
2 2
1,1 1,1
0 for 1 and 1
0 0 for 1 and 1
t ty xt t
i jy
H Hi j
x y
, , ,
0 2
i j i j i jHz z z t tt t tD D c t C
, ,
,
1i j i j
z zi jt t t tzz
E D
time
src src src src, ,i j i j
z zt t t tD D g T
x & y
x & y
x & y
x & y
x & y
Simple soft source
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Lecture 11 Slide 49
Animation of Basic 2D FDTD Algorithm
Dirichlet Boundary Condition
Dirichlet Boundary Condition
DirichletBoundary Condition D
irichlet
Boundary C
onditio
n