Finite-Difference Time-Domain

THE UNIVERSITY OF TEXAS AT EL PASO

Pioneering 21st Century Electromagnetics and Photonics

APPROXIMATING THE TIME-DERIVATIVES

The finite-difference time-domain (FDTD) method simulates electromagnetic devices by evolving the fields over time. It can model a device over an enormous band of frequencies in a single simulation making it well suited to broadband and transient analysis. The model scales near linearly making it the dominant method for modeling electrically large structures with complex geometries. It accommodates parallel processing very well.

FLOW OF MAXWELL’S EQUATIONS

To approximate the time derivatives using central finite-differences, the E and H fields are staggered in time.

H t

E tt

E t

H tt

2 2t t t t

t

H HE

t

2

t t t

t t

E EH

t

The update equations are derived by solving the above equations for the fields at the future time value.

2 2t t t t t

tH H E

2t t t t t

tE E H

UNIAXIAL PERFECTLY MATCHED LAYER

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

1

1

1

x

x

y

y

z

z

xs x

j

ys y

j

zs z

j

3

0

3

0

3

0

2

2

2

x

x

y

y

z

z

xx

t L

yy

t L

zz

t L

? length of the PML in the ? directionL

We wish to incorporate loss at the outer boundaries of the grid to absorb outgoing waves, but this must be done in a manner with perfectly matched impedance to prevent reflections from the lossy regions themselves.

The PML is a frequency-domain concept and is incorporated into Maxwell’s equations much like it was a constitutive parameter.

0 r sE j H

H E j Ds

0 rD E

TRANSMITTANCE AND REFLECTANCE

TOTAL-FIELD/SCATTERED-FIELD FRAMEWORK

We excite the simulation with an impulse and record the impulse response wherever we are interested.

We must normalize our spectra according to the source spectrum.

We calculate the diffraction efficiency of all the spatial harmonics separately.

ref2, ,ref ref

, ,

,inc

trn2, , ,reftrn

,

,inc ,trn

trn,

DE Re

DE Re

z m n

m n m n

z

z m n r

m n

z r

m n

k ff S f

k f

k ff S f

k f

, , , , , ,

, ,

, ,

x m x m n y n y m n

z m n

z m n

k S k SS

k

, ,

, , , ,FFT 2 FFT 2i j i j

x m n x y m n xS E S E