FDTD Introduction

FDTD: An Introduction
Dr. Rakhesh Singh Kshetrimayum
2/25/2015FDTD by R. S. Kshetrimayum1
Dr. Rakhesh Singh Kshetrimayum

Background on commercial softwares andWho’sWho’s in
CEM
(a) EMSS software FEKO
Originated from Dr. Ulrich Jacobus.
Dr. Francs Meyer (a student of Prof. D. B. Davidson, a
Dr. Francs Meyer (a student of Prof. D. B. Davidson, a
Professor at University of Stellenbosch, South Africa) is one
of the main person behind this company

(b) IE3D by Zeland Software Inc.
Founded by Dr. Jian-Xiong Zheng
Did PhD under Prof. D. C. Chang (President Emeritus of
NYU Poly)
(c) REMCOM XFDTD is a product of Penn. State University
(c) REMCOM XFDTD is a product of Penn. State University
Founders are H. Scott Langdon, Dr. Raymond Lubbers and
Dr. Christopher Penny

(d) James C. Rautio is founder of SONNET
He did his PhD from Syracuse university, USA under Prof. R.
F. Harrington (considered as originator of MoM in EM area)
(e) Dr.WenhuaYu who was a member of Prof. Raj Mittra group
at Penn. State University, USA founded GEMS software
at Penn. State University, USA founded GEMS software
And list goes on

FDTD is one of the very popular techniques
in Computational electromagnetics (CEM)
What is CEM?
In CEM
Try to solve complex EM problems
Try to solve complex EM problems
which do not have tractable analytical solutions
Using numerical methods employing computers

Why CEM?
Possible to simulate a device/experiment/phenomenon any
number of times
Try to achieve the best or optimal result before actually
doing the experiments (preserving resources)
doing the experiments (preserving resources)
Some experiments are
dangerous to perform like lightning strike on aircraft or
not possible to conduct because of limited facilities/resources

How to learn CEM (including FDTD)?
Can be learnt by doing it
Once you learn the basic concepts
You should write or code simple programs in any language
Always start with simple problems to gain confidence
Always start with simple problems to gain confidence
and try for more difficult problems later

Many CEM methods are available
MoM
FEM
FDTD
FDTD (our choice)
FDTD (our choice)
Easy to implement
Can see fields and waves in real time (time domain techniques)
Expected outcome of this talk
Understand fundamentals of FDTD
Will explain and give 1-D, 2-D and 3-D FDTD programs
Can start your research works right way

Let us start with 4 Maxwell’s equations
2. 0B∇• =
r
ε
ρv
E =•∇
r
.1
3.
D
H J
t
∂
∇× = +
∂
r
r r
4.
B
E
t
∂
∇× = −
∂
ur
ur
Maxwell curl
equations (why?)

What is FDTD?
Finite DifferenceTime Domain
Basically, we replace spatial & time derivatives in the two
Maxwell’s curl equations
by central finite difference approximation
What is central finite difference approximation of derivatives?
Consider a function f(x), its derivative at x0 from central finite
difference approximation is given by
0 0
'0
0
( ) 2 2
( )
x x
f x f x
df x
f x
dx x
∆ ∆   
+ − −   
   = ≅
∆

The time-dependent and source free ( ) Maxwell’s curl
equations in a medium with ε=ε0εr and µ = µ0 µr are
0J =
r
0 0
1 1
;
r r
E H
H E
t tε ε µ µ
∂ ∂
= ∇× = − ∇×
∂ ∂
r r
r r
3 equations 3 equations
For Cartesian coordinate systems,
expanding the curl equations,
equating the vector components,
we have 6 equations

Expanding the first vector curl equation
we get 3 equations
0
1
r
E
H
t ε ε
∂
= ∇×
∂
r
r
0
0
0
1
1
1
yx z
r
y x z
r
y xz
r
HE H
t y z
E H H
t z x
H HE
t x y
ε ε
ε ε
ε ε
∂ ∂ ∂
= − 
∂ ∂ ∂ 
∂ ∂ ∂ 
= − 
∂ ∂ ∂ 
∂ ∂∂
= − 
∂ ∂ ∂ 

Expanding the second vector curl equation
We get 3 equations
0
1
r
H
E
t µ µ
∂
= − ∇×
∂
r
r
1 EH E∂ ∂ ∂
0
0
0
1
1
1
yx z
r
y xz
r
yxz
r
EH E
t z y
H EE
t x z
EEH
t y x
µ µ
µ µ
µ µ
∂ ∂ ∂
= − 
∂ ∂ ∂ 
∂ ∂∂ 
= − ∂ ∂ ∂ 
∂ ∂∂
= − 
∂ ∂ ∂ 

As we know that FDTD is a time-domain solver
The question is how do we solve those 6 equations above?
1. 1-D FDTD update equations
For 1-D case (a major simplification), we can consider
(a) Linearly polarized wave along x-axis
(a) Linearly polarized wave along x-axis
exciting an electric field which has Ex only (Ey = Ez = 0)
(b) propagation along z-axis
no variation in the x-y plane, i.e. ∂/∂x = 0 and ∂/∂y = 0

Then, the above 6 equations
reduce to 2 equations for 1-D FDTD
0
1 yx
r
HE
t zε ε
∂ ∂
= −  
∂ ∂ 
We want to solve these equations at different locations and
time in solution space
Hence we need to discretize both in time and space
0
1y x
r
H E
t zµ µ
∂ ∂ 
= −  ∂ ∂ 

2 3 4 5
∆z
1
space discretization size
running index k
Fig. 1(a) Discretizations in space
Fig. 1(b) Discretization in time
∆t
1 2 3 4 5
time discretization size
running index n

Notations:
n is the time index and
k is the spatial index,
which indexes
Locate positions z = k∆z
Locate positions z = k∆z
Usually ∆z is taken as 10-15 per wavelength gives accurate
results
times t = n∆t and
∆t (dependent on ∆z) chosen from Courant stability criterion
gives stable FDTD solution (will be discussed later)

for the first equation:
the central difference approximations for both the temporal
and spatial derivatives are obtained at
0
1 yx
r
HE
t zε ε
∂ ∂
= −  
∂ ∂ 
z = k∆z, (space discretization size and index)
t = n∆t (time discretization size and index)
0
1 1 1 1
, , , ,
12 2 2 2
x x y y
r
E k n E k n H k n H k n
t zε ε
       
+ − − + − −       
       = −
∆ ∆

for the second equation:
the central difference approximations for both the temporal
at (z + ∆z/2, t + ∆t/2)
0
1y x
r
H E
t zµ µ
∂ ∂ 
= −  ∂ ∂ 
at (z + ∆z/2, t + ∆t/2)
(increment all the time and space steps by 1/2):
0
1 1 1 1
, 1 , 1, ,
12 2 2 2
y y x x
r
H k n H k n E k n E k n
t zµ µ
       
+ + − + + + − +       
       = −
∆ ∆

The above equations can be rearranged as a pair of‘computer
update equations’,
which can be repeatedly updated in loop,
to obtain the next time values of Ex (k, n+1/2) and Hy (k + ½.
n+1)
n+1)
from the previous time values as follows
0
1 1 1 1 1
, , , ,
2 2 2 2
x x y y
r
t
zε ε
 ∆       
+ = − − + − −        ∆        
0
1 1 1 1 1
, 1 , 1, ,
2 2 2 2
y y x x
r
t
zµ µ
 ∆       
+ + = + − + + − +        ∆        

k-2 k-1 k k+1 k+2
2/1−n
xE
n
H
Fig. 2 Interleaving of E and H fields
k-2 k-1 k k+1 k+2
k-5/2 k-3/2 K-1/2 K+1/2K+3/2K+5/2
2/1+n
xE
n
yH

Interleaving of the E and H fields in space and time in the
FDTD formulation
to calculate Ex(k), the neighbouring values of Hy at k-1/2
and k+1/2 of the previous time instant are needed
Similarly, to calculate Hy(k+1/2), for instance, the
Similarly, to calculate Hy(k+1/2), for instance, the
neighbouring values of Ex at k and k+1 of the previous time
instant are needed

In the above equations,
ε0 and µ0 differ by several orders of magnitude,
Ex and Hy will differ by several orders of magnitude
Numerical error is minimized by making the following
change of variables as
change of variables as
xx EE
µ
ε
='

Hence,
' '1 1 1 1 1 1 1
, , , ,
2 2 2 2
x x y y
t
zεε ε
µ µ
 ∆       
+ = − − + − −        ∆        
' '
' '
1 1 1 1 1
, , , ,
2 2 2 2
1 1 1 1
, ,
2 2
x x y y
x x y
t
z
t
E k n E k n H k
z
ε
µ ε
µε
 ∆       
⇒ + = − − + − −        ∆        
∆   
⇒ + = − − +   
∆   
' '
1
, ,
2 2
1 1 1 1 1
, , , ,
2 2 2 2
y
x x y y
n H k n
t
zµε
    
− −    
    
 ∆       
⇒ + = − + − − +        ∆        

Similarly,
' '1 1 1 1 1 1
, 1 , 1, ,
2 2 2 2
1 1 1 1 1
y y x x
t
z
t
µε
µ
 ∆       
+ + = + − + + − +        ∆        
 ∆       
' '1 1 1 1 1
, 1 , 1, ,
2 2 2 2
1 1 1
, 1 ,
2 2
y y x x
y y
t
z
t
H k n H k n E
z
µε
µε
 ∆       
⇒ + + = + − + + − +        ∆        
∆   
⇒ + + = + +   
∆   
' '1 1
, 1,
2 2
x xk n E k n
    
+ − + +    
    

Courant stability criteria:
Courant stability criteria dictates the
relationship between the time increment ∆t with respect to
space increment ∆z
in order to have a stable FDTD solution of the electromagnetic
in order to have a stable FDTD solution of the electromagnetic
problems
In isotropic media, an electromagnetic wave propagates a
distance of one cell in time ∆t = ∆z/vp,
where vp is the phase velocity

This limits the maximum time step
This equation implies that an EM wave cannot be allowed
to move more than a space cell during a time step
Otherwise, FDTD will start diverging
If we choose ∆t > ∆z/v ,
If we choose ∆t > ∆z/vp,
the distance moved by the EM wave over the time interval ∆t
will be more than ∆z,
the EM wave will leave out the next node/cell and
FDTD cells are not causally interconnected and
hence, it leads to instability

cell
y∆
x∆
Fig. 3 (a) 2-D discretizations
x
y

∆
2
∆ Propagation at θ=450
Plane wave
front
∆
Propagation at θ=00
Fig. 3 (b) Plane wave front propagation along θ=00 andθ=450

Propagation at θ=450
Wavefront jumps from one row of nodes to the next row,
the spacing between the consecutive rows of nodes is
2
∆
Similarly for 3-D case
2
3
∆

In the case of a 2-D simulation,
we have to allow for the propagation in the diagonal
direction,
which brings the time requirement to ∆t = ∆z/√2vp
Obviously, three-dimensional simulation requires ∆t =
Obviously, three-dimensional simulation requires ∆t =
∆z/√3vp
We will use in all our simulations a time step ∆t of
2 p
z
t
v
∆
∆ =
⋅

where vp is the phase velocity,
which satisfies the requirements in 1-D, 2-D and 3-D for all
media (√2≈1.414<√3≈1.7321<2)
In 3D case, it may be more appropriate to modify the above
stability criteria as
stability criteria as
The above factor now can be simplified as
( )min , ,
2 p
x y z
t
v
∆ ∆ ∆
∆ =
⋅
1 1
2 2
p
p
p
v zt t
v
z z v zεµ
⋅∆∆ ∆
= ⋅ = =
∆ ∆ ⋅ ⋅∆

Making use of this in the above two equations,
we obtain the following equations
' '1 1 1 1 1
, , , ,
2 2 2 2 2
x x y yE k n E k n H k n H k n
        
+ = − + − − +        
        
' '1 1 1 1 1
, 1 , , 1,H k n H k n E k n E k n
        
+ + = + + + − + +        
Hence the computer update equations are
ex[k] = ex[k] + (0.5/er(k))*( hy[k-1] - hy[k] )
hy[k] = hy[k] + 0.5*( ex[k] - ex[k+1] )
' '1 1 1 1 1
, 1 , , 1,
2 2 2 2 2
y y x xH k n H k n E k n E k n
        
+ + = + + + − + +        
        

k+1/2 and k-1/2 are replaced by k or k-1
Note that the n or n + 1/2 or n −1/2 in the superscripts do not
appear
FDTD Simulation of Gaussian pulse propagation in free space
(fdtd_1d_1.m)
(fdtd_1d_1.m)
FDTD Simulation of Gaussian pulse hitting a dielectric medium
(fdtd1_die.m)
From theoretical analysis of EM wave hitting a dielectric surface
21
21
12
12
εε
εε
ηη
ηη
+
−
=
+
−
=Γ
21
1
12
2
22
εε
ε
ηη
η
τ
+
=
+
=

For ε1=1.0 and ε2=4.0, we have
FDTD simulation of Absorbing Boundary Condition
(fdtd_1d_3.m)
In calculating the E field,
we need to know the surrounding H values;
3
2
;
3
1
=−=Γ τ
we need to know the surrounding H values;
this is the fundamental assumption in FDTD
At the edge of the problem space,
we will not have the value to one side,
but we know there are no sources outside the problem space

The wave travels ∆z/2 (=c0 · ∆ t) distance in one time step,
so it takes two time steps for a wave front to cross one cell
Suppose we are looking for a boundary condition at the end
where k=1
Now if we write the E field at k=1 as
Now if we write the E field at k=1 as
Ex (1,n) = Ex (2, n-2),
then the fields at the edge will not reflect
This condition must be applied at both ends

2. FDTD Solution to Maxwell’s equations in 2-D Space
In deriving 2-D FDTD formulation, we choose between one
of two groups of three vectors each:
(a)Transverse magnetic (TMz) mode, which is composed of
Ez, Hx, and Hy or
Ez, Hx, and Hy or
(b)Transverse electric (TEz) mode, which is composed of Ex,
Ey, and Hz

Unlike time-harmonic guided waves,
none of the fields vary with z so that
there is no propagation in the z-direction
But in general propagation along x- or y- directions or both is
possible
FDTD simulation ofTM mode wave propagation
(fdtd_2d_TM_1.m )
Expanding the Maxwell’s curl equations with
Ex = 0, Ey = 0, Hz = 0 and ∂/∂z = 0,
we obtain,
3 equations from the 6 equations

3 equations as follows
1 y xz
H HE
t x yε
∂ ∂∂
= − 
∂ ∂ ∂ 
1H E ∂ ∂
1
1
x z
y z
H E
t y
H E
t x
µ
µ
 ∂ ∂
=  
∂ ∂ 
∂ ∂ 
= − 
∂ ∂ 

Central finite difference approximations of the above 3
equations are as follows
1 1
, , , ,
2 2
1 1 1 1
z zE i j n E i j n
t
   
+ − −   
   
∆
        
1 y xz
H HE
t x yε
∂ ∂∂
= − 
∂ ∂ ∂ 
n+1/2 n+1 & j j+1/2
0
1 1 1 1
, , , , , , , ,
1 2 2 2 2
y y x x
r
H i j n H i j n H i j n H i j n
x yε ε
        
+ − − + − −        
        = −
∆ ∆ 
  
0
1 1 1 1
, , 1 , , , 1, , ,
12 2 2 2
x x z z
r
H i j n H i j n E i j n E i j n
t yµ µ
       
+ + − + + + − +       
       = −
∆ ∆

n+1/2 n+1 & i i+1/2
Rearranging the above equations, we can write the final 3
update equations’ expressions as
0
1 1 1 1
, , 1 , , 1, , , ,
12 2 2 2
y x z z
r
H i j n H i j n E i j n E i j n
t xµ µ
       
+ + − + + + − +       
       =
∆ ∆
update equations’ expressions as
0
1
, ,
2
1 1 1 1
, , , , , , , ,
1 2 2 2 2
, ,
2
z
y y x x
z
r
E i j n
t
E i j n
x yε ε
 
+ 
 
        
+ − − + − −        ∆          = − + − 
∆ ∆   
  

0
1 1
, 1, , ,
1 1 2 2
, , 1 , ,
2 2
z z
x x
r
E i j n E i j n
t
H i j n H i j n
yµ µ
   
+ + − +   ∆       + + = + −   
∆   
0
1 1
1, , , ,
1 1 2 2
, , 1 , ,
2 2
z z
y x
r
E i j n E i j n
t
H i j n H i j n
xµ µ
   
+ + − +   ∆       + + = + +   
∆   

Computer update equations are (k+1/2 and k-1/2 are
0
1
, ,
2
1 1 1 1
, , , , , , , ,
1 2 2 2 2
, ,
2
z
y y x x
z
r
E i j n
t
E i j n
x yε ε
 
+ 
 
        
+ − − + − −        ∆          = − + − 
∆ ∆   
  
Computer update equations are (k+1/2 and k-1/2 are
replaced by k or k-1)
ez(i,j)=ez(i,j)+{dt/(eps_0*eps_r)}*[{Hy(i,j)-Hy(i-
1,j)}/dx-{Hx(i,j)-Hx(i,j-1)}/dy]
Hx(i,j) = Hx(i,j)-{dt/(mu_0*dy)}*{Ez(i,j+1)-Ez(i,j)}
Hy(i,j) = Hy(i,j)+{dt/(mu_0*dx)}*{Ez(i+1,j)-Ez(i,j)}

Some new MATLAB commands:
image: IMAGE(C) displays matrix C as an image
imagesc: data is scaled to use the full color map
Colorbar: appends a vertical color scale to the current axis
Colormap: a matrix may have any number of rows but it
Colormap: a matrix may have any number of rows but it
must have exactly 3 columns
Each row is interpreted as a color, with the first element
specifying the intensity of red, the second gree and the third
blue
Color intensity is specified on the interval of 0.0 to 1.0

For example, [0 0 0] is black
[1 1 1] is white
[1 0 0] is red
[0.5 0.5 0.5] is gray
[127/255 1 212/255] is aquamarine
[127/255 1 212/255] is aquamarine

FDTD simulation of TM wave hitting a dielectric surface (right side),
source at the center (2-D) (fdtd_2d_TM_2.m)
Modify the above program as follows:
Specify the position of the dielectric surface (id=..,jd=..)
Choose a value of eps_r=10.0..
Choose a value of eps_r=10.0..
Modify the constant value C1 accordingly
Run the program
In free space wave is propagating freely and
on the other side reflection and transmission of the wave at
the interface

Wave propagates more slowly on the medium on the right
and
the pulse length is shorter
FDTD simulation of TM wave propagation due to multiple sources
(fdtd_2d_TM_multi_source.m)
(fdtd_2d_TM_multi_source.m)
3. 3-D FDTD
From two Maxwell’s curl equations,
we need to consider all 6 equations for 3-D case
0 0
1 1
;
r r
E H
H E
t tε ε µ µ
∂ ∂
= ∇× = − ∇×
∂ ∂
r r
r r

Yee’s FDTD grid
Electric fields are calculated at “integer” time-steps
and magnetic field at “half-integer” time-steps
Electric field components are placed at mid-points of the
corresponding edges
corresponding edges
E.g.
Ex is placed at midpoints of edges oriented along x-direction
Ex is on half-grid in x and n the integer grids in y & z

The magnetic field components are
placed at the centers of the faces of the cubes and
oriented normal to the faces
E.g.
Hx components are placed at the centers of the faces in the
Hx components are placed at the centers of the faces in the
yz-plane
Hence Hx is
on the integer grid in x and
on the half-grid in y & z

Fig. 4Yee’s FDTD grid

( ) ( )
( ) ( )
( ) ( )








∆
−+−++
−






∆
−+−++
=
∆
+−+
++
++
+
z
rqpHrqpH
y
rqpHrqpH
t
rqpErqpE
n
y
n
y
n
z
n
z
r
n
x
n
x
2/1,,2/12/1,,2/11
,2/1,2/1,2/1,2/11
,,2/1,,2/1
2/12/1
2/12/1
0
1
εε
εε






∂
∂
−
∂
∂
=
∂
∂
z
H
y
H
t
E yz
r
x
0
1
εε




 ∆zr 0εε
( ) ( )
( ) ( )
( ) ( )






∆
+−−++
−






∆
−+−++
=
∆
+−+
++
++
+
z
rqpHrqpH
z
rqpHrqpH
t
rqpErqpE
n
z
n
z
r
n
x
n
x
r
n
y
n
y
,2/1,2/1,2/1,2/11
2/1,2/1,2/1,2/1,1
,2/1,,2/1,
2/12/1
0
2/12/1
0
1
εε
εε






∂
∂
−
∂
∂
=
∂
∂
x
H
z
H
t
E zx
r
y
0
1
εε

( ) ( )
( ) ( )
( ) ( )






∆
+−−++
−








∆
+−−++
=
∆
+−+
++
++
+
y
rqpHrqpH
x
rqpHrqpH
t
rqpErqpE
n
x
n
x
n
y
n
y
r
n
z
n
z
2/1,2/1,2/1,2/1,1
2/1,,2/12/1,,2/11
2/1,,2/1,,
2/12/1
2/12/1
0
1
εε
εε






∂
∂
−
∂
∂
=
∂
∂
y
H
x
H
t
E xy
r
z
0
1
εε



 ∆
−
yr 0εε
( ) ( )
( ) ( )
( ) ( )






∆
+−++
−








∆
+−++
=
∆
++−++ −+
y
rqpErqpE
z
rqpErqpE
t
rqpHrqpH
n
z
n
z
n
y
n
y
n
x
n
x
2/1,,2/1,1,1
,2/1,1,2/1,1
2/1,2/1,2/1,2/1,
0
0
2/12/1
µ
µ






∂
∂
−
∂
∂
−=
∂
∂
z
E
y
E
t
H yzx
0
1
µ

( ) ( )
( ) ( )
( ) ( )






∆
+−++
−






∆
+−++
=
∆
++−++ −+
z
rqpErqpE
x
rqpErqpE
t
rqpHrqpH
n
x
n
x
n
z
n
z
n
y
n
y
,,2/11,,2/11
2/1,,2/1,,11
2/1,,2/12/1,,2/1
0
2/12/1
µ
µ 





∂
∂
−
∂
∂
−=
∂
∂
x
E
z
E
t
H zxy
0
1
µ



 ∆z0µ
( ) ( )
( ) ( )
( ) ( )








∆
+−++
−






∆
+−++
=
∆
++−++ −+
x
rqpErqpE
y
rqpErqpE
t
rqpHrqpH
n
y
n
y
n
x
n
x
n
z
n
z
,2/1,,2/1,11
,,2/1,2/1,2/11
,2/1,2/1,2/1,2/1
0
0
2/12/1
µ
µ






∂
∂
−
∂
∂
−=
∂
∂
y
E
x
E
t
H xyz
0
1
µ

FDTD simulation of cubical cavity (fdtd_3D_demo.m)
We will use FDTD to find
the resonant frequencies of an air-filled cubical cavity with
metal walls
Peaks in the frequency response indicate
Peaks in the frequency response indicate
the presence of resonant modes in the cavity
An initial Hz field excites only
theTE modes
The first four eigenfrequencies
TE101,TE011/TE201,TE111 andTE102 modes

4. Perfectly Matched Layers
Berenger’s PML is an artificial material
that is theoretically designed to create no reflections
regardless of the
frequency,
frequency,
polarization and
angle of incidence of a plane wave upon its interface

PML could be analyzed in stretched coordinate systems
Source-free Maxwell’s equations
( ) ( )HE
EjHHjE
ss
ss
=•∇=•∇
=×∇−=×∇
;0;0
;;
rr
rrrr
µε
ωεωµ
Consider plane waves
( ) ( )
( ) ( ) ( ) zzs
z
yys
y
xxs
x
zyx
s
ss
∂
∂
+
∂
∂
+
∂
∂
=∇
1
ˆ
1
ˆ
1
ˆ
zkykxkrk
eHHeEE
zyx
rkjrkj
++=•
== •−•−
rr
rrrr rrrr
00 ;

Substituting these into Maxwell’s equations, we have,
( ) ( )
ˆˆˆ
;0;0
;;
++=
=•=•
−=×=×
zyx
s
ss
ss
s
k
z
s
k
y
s
k
xk
HkEk
EHkHEk
r
rrrr
rrrrrr
µε
εωµ
Solution to the above equation is given by
222








+








+





=•
y
z
y
y
x
x
ss
zyx
s
s
k
s
k
s
k
kk
sss
rr
θφθφθ cos,sinsin,cossin zzyyxx kskkskksk ===

If sx is a complex number with a negative imaginary part,
the wave will be attenuated in the x direction
Wave impedance is independent of coordinate stretching
( ) 0,1, '''cossincossin '''
>≥== −−−−
sseee xjssjkxjksxjk xx φθφθ
Wave impedance is independent of coordinate stretching
ε
µωµ
η ===
kH
E
rr
r

'''
,1 jsssss yzx −===
==
'''
,1:4
jssss
scorners
yx
z
−==
=
y
Object
Fig. 5 Computational domain truncated using the conductor backed PMLs
'''
,1 jsssss yzx −===
'''
,1
jsss
ss
x
zy
−=
=='''
,1
jsss
ss
x
zy
−=
==
x

( ) ( ) ( )
( ) ( ) ωµωµ
ωµ
sy
y
sx
x
zyx
s
HjEy
ys
HjEx
xs
HjEz
zs
Ey
ys
Ex
xs
E
−=×
∂
∂
−=×
∂
∂
−=×
∂
∂
+×
∂
∂
+×
∂
∂
=×∇
;ˆ
1
;ˆ
1
;ˆ
1
ˆ
1
ˆ
1
rrrr
rrrrr
( )
ωε
σ
ωε
σ
ωε
σ
ωµ
z
z
y
y
x
x
szsysxsz
z
yx
jsjsjsgChoo
HHHHHjEz
zs
ysxs
−=−=−=
++=−=×
∂
∂
∂∂
1,1,1sin
;;ˆ
1 rrrrrr

( )
( )
( ) sz
zsz
szz
sy
ysy
syy
sx
xsx
sx
x
H
t
H
HsjEz
z
H
t
H
HsjEy
y
H
t
H
HjjEx
x
r
r
rr
r
r
rr
r
r
rr
ωε
µσ
µωµ
ωε
µσ
µωµ
ωε
µσ
µ
ωε
σ
ωµ
−
∂
∂
−=−=×
∂
∂
−
∂
∂
−=−=×
∂
∂
−
∂
∂
−=





−−=×
∂
∂
ˆ
ˆ
1ˆ
( )
( )
( )
( ) szz
sz
syy
sy
sxx
sx
szszz
E
t
E
Hz
z
E
t
E
Hy
y
E
t
E
Hx
x
tz
r
r
r
r
r
r
r
r
r
σε
σε
σε
ωε
+
∂
∂
=×
∂
∂
+
∂
∂
=×
∂
∂
+
∂
∂
=×
∂
∂
∂∂
ˆ
ˆ
ˆ

5. References
1. S. D. Gedney, Introduction to the Finite-DifferenceTime-
Domain Method for Electromagnetics, Morgan &
Claypool, 2011
2. R. Garg,Analytical and Computational Methods in
Electromagnetics,Artech House, 2008
Electromagnetics,Artech House, 2008
3. A.Taflove and S. C. Hagness, Computational
electrodynamics the Finite-DifferenceTime-Domain
Method,Artech House, 2005
4. T. Rylander, P. Ingelstrom and A. Bondeson,
Computational Electromagnetics, Springer, 2013

5. J. M. Jin,Theory and Computation of Electromagnetic
Fields, IEEE Press, 2010
6. A. Elsherbeni andV. Demir,The Finite-DifferenceTime-
Domain Method for Electromagnetics with MATLAB
Simulations, Scitech Publishng, 2009
Simulations, Scitech Publishng, 2009
7. D. B. Davidson, Computational Electromagnetics for RF
and Microwave Engineering, Cambridge University Press,
2011
8. U. S. Inan and R.A. Marshall, Numerical Electromagnetics
The FDTD Method, Cambridge University Press, 2011

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