8/5/2014
EE 5390 – Special Topics
Electromagnetic Analysis Using Finite‐Difference Time‐Domain
Lecture #15
Implementation of Two‐
Dimensional FDTD
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Lecture 15
Slide 1
Lecture Outline
• Review
– Update equations with PML
– Code development sequence
•
•
•
•
•
Numerical Boundary Conditions for 3D
Reduction to Two‐Dimensions
Calculating the PML Parameters
Implementation for Ez Mode
Total‐Field/Scattered‐Field Source
Lecture 15
Slide 2
1
8/5/2014
Review
Lecture 15
Slide 3
Uniaxial PML
Reflections can be prevented at all angles, all frequencies, and for all polarizations if we allow our absorbing material to be diagonally anisotropic.
        s 
x
y
        s 
Lecture 15
Slide 4
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8/5/2014
Derivation of Update Equation for Hx
1. Governing Equation in the Frequency‐Domain
 y  

  
c0  E z   E y   
 z 

j  1  x  1 


 1 
 H x    
j 0  
j 0  
j 0 
 xx  y
z 

1
F  g  t   G  
F ag  t   aG  
2. Prepare equation for conversion to time‐domain
j H x   
 da

a
F  a g  t     j  G  
dt


t
1
F  g   d 
G  

j
 y   z
c
1  y z
1 c0 x E
H x   
H x     0 C xE   
C x  
j  02
j  0  xx
0
 xx

3. Convert each term to the time‐domain
 y z
H x  t   y   z
H x t   2

0
0
t

t

H x   d  
c0
 xx
C xE  t  
c0 x
 0  xx

t

C xE   d
4. Approximate the equation for numerical implementation
i , j ,k
i , j ,k
H x t t 2  H x t  t 2
t

 yH
i , j ,k
  zH
0
i , j ,k
 H x i , j,kt  H x i , j,kt
t
t

2
2

2




H
 y



i , j ,k

H i , j ,k
z

 t  1  H
4
 
 02

c0
 xx
i , j ,k
C xE
i , j ,k
t
x
t
2
i , j ,k
t
H

i , j ,k
t
x
t
2
c0 t  xH
 0  xx
t
t

2

i , j ,k 
   Hx T 
 T 0

i , j ,k
i , j ,k
t
C
T 0
E i , j ,k
x T
5. Solve for H at the future time value
i , j ,k

H x t t 2  mHx1
i , j ,k
H
i , j ,k
x t t 2

 mHx 2
i , j ,k
C
E i , j ,k
x t

 mHx 3
i , j ,k
I
i , j ,k
CEx t

 mHx 4
i , j ,k
I
i, j,k
Hx t
Lecture 15
Slide 5
Code Development Sequence
Step 1 – Basic Update + Dirichlet
Step 2 – Basic Update + Periodic BC
Step 5 – Calculate Response
Lecture 15
Step 3 – Add PML
Step 4 – TF/SF
Step 6 – Add a Device and Benchmark
Slide 6
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Numerical Boundary Conditions for 3D
Lecture 15
Slide 7
Boundary Conditions
We have formulated our update equations so that the curl can be calculated separate from the update equation. Boundary condition problems occur only in these equations.
C xE
i , j ,k
C yE
i , j ,k
C
t
t
E i , j ,k
z t



E z

  E

 H


i , j 1, k
t
 E z
i , j ,k
t
y
i , j , k 1
i , j ,k
E x
 E x
t
t
z
i 1, j , k

Ey
 E y
t
x


i , j ,k
t

i , j , k 1
i , j ,k
E y
 E y
t
t
z
E z
i 1, j , k
t
 E z
t
y
Special boundary conditions are required at the high grid boundaries when computing the curl of the E field.
Lecture 15

i , j ,k

i , j ,k
t
x
i , j 1, k
i , j ,k

Ex
 E x
t
C
H i , j ,k
x t  t
2
Hz
i , j ,k
t  2t
 Hz
y
C
t  2t
H i , j ,k
z t  t
2

H x t  t  H x t  t
2
2
z
Hy
i , j ,k
t  2t

i , j , k 1
i , j ,k
C yH
i , j 1, k
t  2t
 Hy
x

i 1, j , k
t  2t
Hy
i , j ,k
t  2t
 Hy
t  2t
z
Hz
i, j ,k
t  2t
 Hz
i 1, j , k
t  2t
x
i , j 1, k
i , j ,k

i , j , k 1
H x t  t  H x t  t
2
2
y
Special boundary conditions are required at the low grid boundaries when computing the curl of the H field.
Slide 8
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8/5/2014
Handling the Boundary Conditions
Boundary conditions are handled by calculating the curl terms at the boundaries separate from the rest of the curl equation. This should be done explicitly without using ’if‘ statements.
For example,
C xE
i , j ,k
t
 E i , j 1,k  E i , j ,k E i , j ,k 1  E i , j ,k
z t
y t
y t
 zt


y
z
 i ,1,k
i , N y ,k
i , N y , k 1
i , N y ,k
 E z
 E z
E y
 E y
t
t
t
t


y
z

  i , j 1, N
i, j, Nz
i , j ,1
i, j, Nz
z



 Ez
 Ez
Ey
 E y
t
t
t
t


y
z


i ,1, N z
i,N y ,Nz
i , N y ,1
i,N y ,Nz
E y
 E z
 E y
 E z t
t
t
t


y
z

j  N y and k  N z
j  N y and k  N z
j  N y and k  N z
j  N y and k  N z
Calculating the curl terms separate from the update equations is an excellent way to isolate and modularize the boundary condition problem.
Lecture 15
Slide 9
MATLAB Code Implementation (1 of 3)
First, we blindly try to calculate the curl.
% Compute CEx
for nx = 1 : Nx
for ny = 1 : Ny
for nz = 1 : Nz
CEx(nx,ny,nz) = (Ez(nx,ny+1,nz) - Ez(nx,ny,nz))/dy ...
- (Ey(nx,ny,nz+1) - Ey(nx,ny,nz))/dz;
end
end
end
We see that we run into two problems at the y‐hi and z‐hi sides of the grid.
Lecture 15
Slide 10
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8/5/2014
MATLAB Code Implementation (2 of 3)
Next, we handle the z‐hi problem explicitly outside of the nz‐loop. We copy the code inside the nz‐loop and past it after the nz‐loop.
% Compute CEx
for nx = 1 : Nx
for ny = 1 : Ny
for nz = 1 : Nz-1
CEx(nx,ny,nz) = (Ez(nx,ny+1,nz) - Ez(nx,ny,nz))/dy ...
- (Ey(nx,ny,nz+1) - Ey(nx,ny,nz))/dz;
end
CEx(nx,ny,Nz) = (Ez(nx,ny+1,Nz) - Ez(nx,ny,Nz))/dy ...
- (Ey(nx,ny,1) - Ey(nx,ny,Nz))/dz;
end
end
We still have the problem at the y‐hi side of the grid, but now the problem occurs in two places.
Lecture 15
Slide 11
MATLAB Code Implementation (3 of 3)
Finally, we handle the y‐hi problem explicitly outside of the ny‐loop. We copy all of the code inside the ny‐loop and paste it after the ny‐
loop.
% Compute CEx
for nx = 1 : Nx
for ny = 1 : Ny-1
for nz = 1 : Nz-1
CEx(nx,ny,nz) = (Ez(nx,ny+1,nz) - Ez(nx,ny,nz))/dy ...
- (Ey(nx,ny,nz+1) - Ey(nx,ny,nz))/dz;
end
CEx(nx,ny,Nz) = (Ez(nx,ny+1,Nz) - Ez(nx,ny,Nz))/dy ...
- (Ey(nx,ny,1) - Ey(nx,ny,Nz))/dz;
end
for nz = 1 : Nz-1
CEx(nx,Ny,nz) = (Ez(nx,1,nz) - Ez(nx,Ny,nz))/dy ...
- (Ey(nx,Ny,nz+1) - Ey(nx,Ny,nz))/dz;
end
CEx(nx,Ny,Nz) = (Ez(nx,1,Nz) - Ez(nx,Ny,Nz))/dy ...
- (Ey(nx,Ny,1) - Ey(nx,Ny,Nz))/dz;
end
Lecture 15
Slide 12
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Reduction to Two Dimensions
Lecture 15
Slide 13
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.
absorbing BC
z
periodic BC
periodic BC
x
absorbing BC
y
Lecture 15
Slide 14
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2D Grids are Infinite in the 3rd Dimension
Anything represented on a 2D grid, is actually a device that is of infinite extent along the 3rd dimension.
absorbing BC
x
periodic BC
periodic BC
y
absorbing BC
Lecture 15
Slide 15
What is Different in Two Dimensions?
• We will assume it is the z direction that is uniform.
• All derivatives in the z direction are zero

0
z
• We do not need a PML to terminate the z axis
 z  z   0
sz  z   1
Lecture 15
Slide 16
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8/5/2014
Revisions for Update Equation for Hx
The update coefficients are computed before the main FDTD loop.
i , j ,k
mHx 0
i , j ,k
mHx 2

 H
1 y

t 


i , j ,k
2 0

 H
y




c0
1
mHx 0
i , j ,k
  zH
i , j ,k
 xx

i , j ,k
4
mHx3
i , j ,k
H i , j ,k
z
 t
mHx1
2
0
i , j ,k

c0 t 
1
mHx 0
i , j ,k

1
mHx 0
i , j ,k

1

 t

 H
y



H i , j ,k
x
mHx 4
 0  xx i , j ,k
i , j ,k
i , j ,k
  zH

 H
y




i , j ,k
2 0
i , j ,k

t
1
mHx 0
i , j ,k
 02
i , j ,k

4

H i , j ,k
y
H i , j ,k
z
2
0

 t 


H i , j ,k
z

The integration terms are computed inside the main FDTD loop, but before the update equation.
t
i , j ,k
I CEx t

C
t
E i , j ,k
x t
I
T 0
i , j ,k
t
Hx
t
2
t
2


i , j ,k
T t 2
Hx T
C
E i , j ,k
x t

E z
i , j 1, k
t
 E z
i , j ,k

t
y
E y
i , j , k 1
t
 E y
i , j ,k
t
z
The update equation is computed inside the main FDTD loop immediately after the integration terms are updated.

i , j ,k
H x t  t 2  mHx1
H
i , j ,k
i , j ,k
x t  t 2

C
i , j ,k
 mHx 2
E i , j ,k
x t

 mHx 3
i , j ,k
I
i , j ,k
CEx t

 mHx 4
i , j ,k
I
i , j ,k
Hx t
Lecture 15
Slide 17
2D Update Equation for Hx
The update coefficients are computed before the main FDTD loop.
mHx 0
i, j
mHx 2
i, j

 H i, j
1 y

t  2 0


mHx1
c0
1
mHx 0




i, j
 xx
mHx 3
i, j
i, j
i, j

1
i, j
mHx 0

  H i, j
y

 2 0




 
H i, j
c0 t  x
1
mHx 0

1
 t

i, j
 0  xx i , j
The integration terms are computed inside the main FDTD loop, but before the update equation.
t
i, j
I CEx t 
C
E i, j
x t
C
E i, j
x t

E z
i , j 1
t
T 0
 E z
i, j
t
y
The update equation is computed inside the main FDTD loop immediately after the integration terms are updated.
i, j

H x t t 2  mHx1
Lecture 15
i, j
H
i, j
x t t 2

 mHx 2
i, j
C
E i, j
x t

 mHx 3
i, j
I
i, j
CEx t
Slide 18
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8/5/2014
2D Update Equation for Hy
The update coefficients are computed before the main FDTD loop.
i, j
mHy 0
i, j
mHy 2

 H i, j
1 x

t  2 0

mHy 0
i, j
 yy
i, j
mHy1
c0
1





mHy 3
i, j
i, j


1
 t

1
mHy 0

i, j
  H i, j
x

 2 0




 
H i, j
c0 t  y
1
 0  yy i , j
i, j
mHy 0
The integration terms are computed inside the main FDTD loop, but before the update equation.
i, j
I CEy
t
t

C
E i, j
y t
C
E i, j
y t

E z
i 1, j
t
 E z
i, j
t
x
T 0
The update equation is computed inside the main FDTD loop immediately after the integration terms are updated.
Hy
i, j
t
t
2

i, j
 mHy1
H

i, j
 mHy 2
y t  t
2
i, j
C
E i, j
y t

 mHy 3
i, j
I
i, j
CEy t
Lecture 15
Slide 19
2D Update Equation for Hz
The update coefficients are computed before the main FDTD loop.
mHz 0
i, j
mHz 2
i, j
 H i, j   H
1 x
y


t 
2 0



  xH



i, j
  t

 H i, j   H
y
 1 x
mHz1 

i, j 
2 0
mHz 0  t 


1 t H i , j
i, j
mHz 4  
x
 yH
i, j
2
mHz 0  0
H i, j
y
i, j
 zz
1
i, j
4 02

c0
1
mHz 0
i, j
i, j

i, j
i, j

  xH



i, j
  t 
H i, j
y
4 02




The integration terms are computed inside the main FDTD loop, but before the update equation.
t
I Hz
i, j
t
t
2

t
2

T t 2
i, j
CzE
Hz T
i, j
t

E y
i 1, j
t
 E y
i, j
t
x

i , j 1
i, j
E x
 E x
t
t
y
The update equation is computed inside the main FDTD loop immediately after the integration terms are updated.
Hz
Lecture 15
i, j
t
t
2

 mHz1
i, j
H
z
i, j
t
t
2

 mHz 2
i, j
C
E i, j
z t

 mHz 4
i, j
I
Hz
i, j
t
2
t
Slide 20
10
8/5/2014
2D Update Equation for Dx
The update coefficients are computed before the main FDTD loop.
D i, j
mDx 0
i, j
mDx 2
i, j
1 y

t
2 0

c0

mDx 0
i, j
mDx1
i, j
mDx 3
i, j

i, j

D 
1  y 
 t
2 0 


1
mDx 0
i, j
c0 t  xD
1

mDx 0
i, j
0
i, j
The integration terms are computed inside the main FDTD loop, but before the update equation.
t
I
i, j
t
CHx
t
2
t
2


CxH
i, j
i, j
CxH
T
t  2t
T t 2

Hz
i, j
i , j 1
 Hz
t  2t
t  2t
y
The update equation is computed inside the main FDTD loop immediately after the integration terms are updated.




i, j
i, j
i, j
i, j
D x
 mDx1
D x  mDx 2
CxH
t t
t

i, j
t
 mDx 3
t
2
i, j
I
CHx
i, j
t
2
t
Lecture 15
Slide 21
2D Update Equation for Dy
The update coefficients are computed before the main FDTD loop.
mDy 0
mDy 2
i, j
i, j
D i, j


1 x

t
2 0
mDy1
c0
mDy 0
mDy 3
i, j
i, j
i, j


1
mDy 0
i, j
c0 t  yD
1
mDy 0
i, j

D 
1  x 
 t
2 0 


i, j
i, j
0
The integration terms are computed inside the main FDTD loop, but before the update equation.
t
I
i, j
CHy t  t
2

t
2

C yH
i, j
C yH
T
T t 2
i, j
t  2t

Hz
i, j
t  2t
 Hz
i 1, j
t  2t
x
The update equation is computed inside the main FDTD loop immediately after the integration terms are updated.
D y
Lecture 15
i, j
t t

 mDy1
i, j
 D
i, j
y t

 mDy 2
i, j
C
H i, j
y t  t
2

 mDy 3
i, j
I
i, j
CHy t  t
2
Slide 22
11
8/5/2014
2D Update Equation for Dz
The update coefficients are computed before the main FDTD loop.
D i, j
mDz 0
i, j
mDz 2
i, j


1 x

t
  yD
i, j
2 0

   t
D i, j
x
D i, j
y
4 02
c0
mDz 0
i, j
mDz1
mDz 4
i, j
i, j



i, j
i, j

D i, j
D i, j
 xD
 yD t 
 1 x  y



i, j 

2 0
4 02
mDz 0  t



1 t D i , j
D i, j



x
y
i, j
2
mDz 0  0

1



The integration terms are computed inside the main FDTD loop, but before the update equation.
I Dz
i, j
t
t
  D z
T 0
i, j
C
T
H i, j
z t  t
2

Hy
i, j
 Hy
t  2t
i 1, j
t  2t
x
i , j 1
i, j

H x t  t  H x t  t
2
2
y
The update equation is computed inside the main FDTD loop immediately after the integration terms are updated.
D z
i, j
t  t

 mDz1
i, j
 D
i, j
z t

 mDz 2
i, j
C
H i, j
z t  t
2

 mDz 4
i, j
I
i, j
Dz t  t
Lecture 15
Slide 23
2D Update Equations for Ex, Ey, and Ez
The update coefficients are computed before the main FDTD loop.
mEx1
i, j

1
 xx
mEy1
i, j
i, j

1
 yy
i, j
mEz1
i, j

1
 zz
i, j
The update equations are computed inside the main FDTD loop.


i, j
i, j
i, j
E x
 mEx1
D x
t t
E y
i, j
E z
i, j
t t
t t

 mEy1

 mEz1
i, j
i, j
 D
 D
t t
i, j
y t t
i, j
z t  t
No changes here!
Lecture 15
Slide 24
12
8/5/2014
MATLAB Code for Update Equations
The update equation for Hx was
i, j

H x t t 2  mHx1
i, j
H
i, j
x t t 2

 mHx 2
i, j
C
E i, j
x t

 mHx 3
i, j
I
i, j
CEx t
This could be implemented in MATLAB as follows
for ny = 1 : Ny
for nx = 1 : Nx
Hx(nx,ny) = mHx1(nx,ny)*Hx(nx,ny) + mHx2(nx,ny)*CEx(nx,ny) + mHx3(nx,ny)*ICEx(nx,ny);
end
end
However, it is much simpler and faster to implement it using “vectorized” MATLAB commands.
Hx = mHx1.*Hx + mHx2.*CEx + mHx3.*ICEx;
Lecture 15
Slide 25
Calculating the PML Parameters
Lecture 15
Slide 26
13
8/5/2014
MATLAB Code to Calculate PML Parameters
To calculate the PML parameters x and y we employ the 2× grid concept.
First, we calculate x and y on the 2× grid.
This ‘2’ is here because we are calculating the fictitious conductivity terms on the 2× grid, but NPML contains the size of the PML on the 1× grid.
% COMPUTE PML PARAMETERS
Nx2 = 2*Nx;
Ny2 = 2*Ny;
sigx = zeros(Nx2,Ny2);
for nx = 1 : 2*NPML(1)
nx1 = 2*NPML(1) - nx + 1;
sigx(nx1,:) = (0.5*e0/dt)*(nx/2/NPML(1))^3;
end
for nx = 1 : 2*NPML(2)
nx1 = Nx2 - 2*NPML(2) + nx;
sigx(nx1,:) = (0.5*e0/dt)*(nx/2/NPML(2))^3;
end
sigy = zeros(Nx2,Ny2);
for ny = 1 : 2*NPML(3)
ny1 = 2*NPML(3) - ny + 1;
sigy(:,ny1) = (0.5*e0/dt)*(ny/2/NPML(3))^3;
end
for ny = 1 : 2*NPML(4)
ny1 = Ny2 - 2*NPML(4) + ny;
sigy(:,ny1) = (0.5*e0/dt)*(ny/2/NPML(4))^3;
end
 x  x  
 y  y  
0  x 
3
 
2t  Lx 
0  y 
3
 
2t  Ly 
Lecture 15
Slide 27
Visualizing the PML Parameters
If calculated correctly, the PML conductivity terms should look like:
x
 y  y 
 x  x 
y
e0 = 8.8542e-12
dt = 1.6678e-10
Lecture 15
0.0265
0.0194
0.0136
0.0091
0.0057
0.0033
0.0017
0.0007
0.0002
0.0000
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.0000
0.0002
0.0007
0.0017
0.0033
0.0057
0.0091
0.0136
0.0194
0.0265
Slide 28
14
8/5/2014
MATLAB Code to Calculate Update Coefficients
Next, we overlay the PML functions onto the 1× grid to calculate the update coefficients containing PML terms.
% COMPUTE UPDATE COEFFICIENTS
sigHx = sigx(1:2:Nx2,2:2:Ny2);
sigHy = sigy(1:2:Nx2,2:2:Ny2);
mHx0 = (1/dt) + sigHy/(2*e0);
mHx1 = ((1/dt) - sigHy/(2*e0))./mHx0;
mHx2 = - c0./URxx./mHx0;
mHx3 = - (c0*dt/e0) * sigHx./URxx ./ mHx0;
sigHx
sigHy
mHy0
mHy1
mHy2
mHy3
=
=
=
=
=
=
sigx(2:2:Nx2,1:2:Ny2);
sigy(2:2:Nx2,1:2:Ny2);
(1/dt) + sigHx/(2*e0);
((1/dt) - sigHx/(2*e0))./mHy0;
- c0./URyy./mHy0;
- (c0*dt/e0) * sigHy./URyy ./ mHy0;
sigDx = sigx(1:2:Nx2,1:2:Ny2);
sigDy = sigy(1:2:Nx2,1:2:Ny2);
mDz0 = (1/dt) + (sigDx + sigDy)/(2*e0) ...
+ sigDx.*sigDy*(dt/4/e0^2);
mDz1 = (1/dt) - (sigDx + sigDy)/(2*e0) ...
- sigDx.*sigDy*(dt/4/e0^2);
mDz1 = mDz1 ./ mDz0;
mDz2 = c0./mDz0;
mDz4 = - (dt/e0^2)*sigDx.*sigDy./mDz0;
mHx 0
i, j
mHx 2
i, j
mHy 0
mHy 2
i, j
i, j

 H i, j
1 y

t  2 0



i, j
 xx
i, j
i, j
mDx1
i, j
mDz 2
i, j
mDz 4
i, j



1 x

t
1
mDx 0
i, j




 yy
D i, j
mDz 0
mHx 3
i, j
mHy 3
i, j
  yD
i, j
2 0

i, j
i, j
mHy1
c0
1
mHy 0
i, j
mHx1
c0
1
mHx 0
 H i, j
1 x

t  2 0






i, j

1
mHx 0


i, j
i, j
i, j
 0  xx i , j

1
 t

  H i, j
x

 2 0




 
H i, j
c0 t  y
1
mHy 0



 
H i, j
1

  H i, j
y

 2 0

c0 t  x
1
mHx 0
mHy 0

1
 t

i, j
 0  yy i , j
   t
D i, j
x
D i, j
y
4

2
0


i, j
i, j

D i, j
D i, j
 xD
 yD
t 
 1 x  y


2
 t 

2 0
4 0




c0
mDz 0

i, j
t
1
mDz 0
i, j
 02
  
D i, j
x
D i, j
y
Lecture 15
Slide 29
Implementation
Ez Mode
Lecture 15
Slide 30
15
8/5/2014
Block Diagram for Ez Mode (1 of 2)
Define Device Parameters
Initialization…
Define FDTD Parameters
Compute Grid Parameters
Build Device on Grid
Compute Update Coefficients
mHx1, mHx2, mHx3
mHy1, mHy2, mHy3
mDz1, mDz2, mDz4
mEz1
Compute Time Step
Initialize Fields
Compute Source
Hx, Hy, Dz, Ez
Compute PML Parameters
Initialize Curl Arrays
CEx, CEy, CHz
Initialize Integration Arrays
ICEx, ICEy, IDz
Lecture 15
Slide 31
Block Diagram for Ez Mode (2 of 2)
Done?
yes
Main loop…
Finished!
no
BC’s
Compute Curl of E
CEx, CEy
Update H Integrations
ICEx = ICEx + CEx
ICEy = ICEy + CEy
Update H Field
Hx = mHx1.*Hx + mHx2.*CEx + mHx3.*ICEx;
Hy = mHy1.*Hy + mHy2.*CEy + mHy3.*ICEy;
Compute Curl of H
CHz
Update D Integrations
IDz = IDz + Dz
Update Dz
Dz = mDz1.*Dz + mDz2.*CHz + mDz4.*IDz;
Inject Source
Dz(nxs,nys) = Dz(nxs,nys) + g(T);
Update Ez
Ez = mEz1.*Dz;
Visualize Fields
Lecture 15
Slide 32
16
8/5/2014
Block Diagram for Hz Mode (1 of 2)
Initialization…
Define Device Parameters
Define FDTD Parameters
Compute Update Coefficients
Compute Grid Parameters
Compute Time Step
Compute Source
mHz1,
mDx1,
mDy1,
mEx1,
mHz2, mHz4
mDx2, mDx3
mDy2, mDy3
mEy1
Initialize Fields
Hz, Dx, Dy, Ex, Ey
Compute PML Parameters
Initialize Curl Arrays
CEx, CEy, CHz
Initialize Integration Arrays
ICHx, ICHy, IHz
Lecture 15
Slide 33
Block Diagram for Hz Mode (2 of 2)
Done?
yes
Finished!
Main loop…
no
BC’s
Compute Curl of E
CEz
Compute Curl of H
CHx, CHy
Update H Integrations
IHz = IHz + Hz
Update H Field
Update D Integrations
ICHx = ICHx + CHx;
ICHy = ICHy + CHy;
Hz = mHz1.*Hz + mHz2.*CEz + mHz4.*IHz;
Update D Field
Inject Source
Hz(nxs,nys) = Hz(nxs,nys) + g(T);
Dx = mDx1.*Dx + mDx2.*CHx + mDx3.*ICHx;
Dy = mDy1.*Dy + mDy2.*CHy + mDy3.*ICHy;
Update E Field
Visualize Fields
Lecture 15
Ex = mEx1.*Dx;
Ey = mEy1.*Dy;
Slide 34
17
8/5/2014
Total‐Field/Scattered‐Field Plane Wave Source
Lecture 15
Slide 35
Total‐Field / Scattered‐Field
scattered‐field
• The total‐field/scattered‐field (TF/SF) condition is a technique to inject a “one‐way” source.
• Benefits
– Eliminates backward propagating power
– Ensures waves at the boundaries are only travelling outward
– 100% of power injected by the source is incident on the device being simulated.
Lecture 15
total‐field
• Needed for calculation of reflection
• Minimizes power incident on PML
Slide 36
18
8/5/2014
Typical 2D FDTD Grid Layout For Modeling Periodic Structures
PML
TF/SF Interface

20cells
2 cells
x
Periodic Boundary
The position of the TF/SF source really can be anywhere above the device. It cannot slice through the device.
Periodic Boundary
y

PML
20cells
Lecture 15
Slide 37
Typical 2D FDTD Grid Layout For Modeling General Scatterers
20 cells
20 cells
20cells
PML
2 cells
PML

2 cells
x

PML
y
2 cells


2 cells
TF/SF Interface
PML
Lecture 15
20cells
Slide 38
19
8/5/2014
The Total‐Field/Scattered‐Field Framework
scattered‐field
2D FDTD Grid
x
jsrc  1
jsrc
total‐field
y
Problem Points!
Lecture 15
Slide 39
Correction to Finite‐Difference Equations at the Problem Cells (1 of 2)
On the scattered‐field side of the TF/SF interface, the finite‐difference equation contains a term from the total‐field side. Due to the staggered nature of the Yee grid, this only occurs in the update equation for a magnetic field. In fact, this only occurs in the computation of the curl of E used in the H field update equations.
C
E i , jsrc 1
x t

E z
i , jsrc
t
i , jsrc 1
 E z
This is an equation in the scattered‐field, but E zjsrc
is a total‐field quantity.
t
y
Ezj
We must subtract the source from to make it look like a scattered‐field quantity.
src
C
E i , jsrc 1
x t
C
E i , jsrc 1
x t
 E


E z
i , jsrc
z t
 Ezsrc
  E
i , jsrc 1
z t
y
i , jsrc
t
 E z
i , jsrc 1
t
y
standard curl equation
Lecture 15
i , jsrc
t

1 src i , jsrc
Ez
t
y
This is a correction term that can be implemented after calculating the curl to inject a source.
Slide 40
20
8/5/2014
Correction to Finite‐Difference Equations at the Problem Cells (2 of 2)
On the total‐field side of the TF/SF interface, the finite‐difference equation contains a term from the scattered‐field side. Due to the staggered nature of the Yee grid, this only occurs in the update equation for an D field. In fact, this only occurs in the computation of the curl of H used in the D field update equations.
C
H i , jsrc
z t  t
2

Hy
i , jsrc
t  2t
 Hy
i 1, jsrc
t  2t
x
i , j 1
i, j

H x t  srct  H x t  srct
2
This is an equation in the scattered‐field, but H xjsrc 1
is a total‐field quantity.
2
y
H xj 1 make it look like a total‐
We must add the source to to
field quantity.
src
C
H i , jsrc
z t  t
2
C
H i , jsrc
z t  t
2


Hy
i , jsrc
t  2t
 Hy
i 1, jsrc
t  2t
x
Hy
i , jsrc
t  2t
 Hy
x

i 1, jsrc
t  2t

i , jsrc 1
i, j
i , j 1
H x t  srct  H x t  srct  H xsrc t
2
2

y
i , j 1
i, j

t
2
H x t  srct  H x t  srct
2
2
y
standard update equation

1  src i , jsrc 1
H x t
t 2
y
This is a correction term that can be implemented after calculating the curl to inject a source.
Lecture 15
Slide 41
The Two Source Terms
From the previous slides, we now know that we need to calculate two source functions before the main FDTD loop. These are:
i , jsrc 1
H xsrc t
t
2
Ezsrc
i , jsrc
t
We need to make a few observations that must be accounted for before we can calculate these source functions correctly.
1. The amplitude of these functions can be different as E and H are related through the material impedance.
2. These functions are a half grid cell apart and have a small time delay between them
3. These functions exist at different time steps.
Lecture 15
Slide 42
21
8/5/2014
Calculation of the Source Functions (Ez Mode)
We calculate the electric field as
Ezsrc
jsrc
t
 g t 
We calculate the magnetic field as
H xsrc
jsrc 1
t  2t

 r  nsrc y t 
g t 
 
2c0
2 
r 
Amplitude due to Maxwell’s equations
Delay through one half of a grid cell
Half time step difference
Lecture 15
Slide 43
TF/SF Block Diagram for Ez Mode
Done?
yes
Finished!
Main loop…
no
Compute Curl of E
Compute Curl of H
Inject TF/SF Source into curl of E
Inject TF/SF Source into curl of H
Update H Integrations
Update D Integrations
Update H Field
Update Dz
Update Ez
Visualize Fields
Lecture 15
Slide 44
22
8/5/2014
TF/SF Block Diagram for Hz Mode
Done?
yes
Finished!
Main loop…
no
Compute Curl of E
Compute Curl of H
Inject TF/SF Source into curl of E
Inject TF/SF Source into curl of H
Update H Integrations
Update H Field
Update D Integrations
Update D Field
Update E Field
Visualize Fields
Lecture 15
Slide 45
23