<s>
Finite-difference	B-Algorithm
time-domain	I-Algorithm
(	O
FDTD	B-Algorithm
)	O
or	O
Yee	B-Algorithm
's	I-Algorithm
method	I-Algorithm
(	O
named	O
after	O
the	O
Chinese	O
American	O
applied	O
mathematician	O
Kane	O
S	O
.	O
Yee	O
,	O
born	O
1934	O
)	O
is	O
a	O
numerical	B-General_Concept
analysis	I-General_Concept
technique	O
used	O
for	O
modeling	O
computational	O
electrodynamics	O
(	O
finding	O
approximate	O
solutions	O
to	O
the	O
associated	O
system	O
of	O
differential	O
equations	O
)	O
.	O
</s>
<s>
Since	O
it	O
is	O
a	O
time-domain	O
method	O
,	O
FDTD	B-Algorithm
solutions	O
can	O
cover	O
a	O
wide	O
frequency	O
range	O
with	O
a	O
single	O
simulation	B-Application
run	O
,	O
and	O
treat	O
nonlinear	O
material	O
properties	O
in	O
a	O
natural	O
way	O
.	O
</s>
<s>
The	O
FDTD	B-Algorithm
method	O
belongs	O
in	O
the	O
general	O
class	O
of	O
grid-based	O
differential	O
numerical	B-Application
modeling	I-Application
methods	O
(	O
finite	B-Algorithm
difference	I-Algorithm
methods	I-Algorithm
)	O
.	O
</s>
<s>
The	O
time-dependent	O
Maxwell	O
's	O
equations	O
(	O
in	O
partial	O
differential	O
form	O
)	O
are	O
discretized	B-Algorithm
using	O
central-difference	B-Algorithm
approximations	O
to	O
the	O
space	O
and	O
time	O
partial	O
derivatives	O
.	O
</s>
<s>
The	O
resulting	O
finite-difference	B-Algorithm
equations	I-Algorithm
are	O
solved	O
in	O
either	O
software	O
or	O
hardware	O
in	O
a	O
leapfrog	B-Algorithm
manner	O
:	O
the	O
electric	O
field	O
vector	O
components	O
in	O
a	O
volume	O
of	O
space	O
are	O
solved	O
at	O
a	O
given	O
instant	O
in	O
time	O
;	O
then	O
the	O
magnetic	O
field	O
vector	O
components	O
in	O
the	O
same	O
spatial	O
volume	O
are	O
solved	O
at	O
the	O
next	O
instant	O
in	O
time	O
;	O
and	O
the	O
process	O
is	O
repeated	O
over	O
and	O
over	O
again	O
until	O
the	O
desired	O
transient	O
or	O
steady-state	O
electromagnetic	O
field	O
behavior	O
is	O
fully	O
evolved	O
.	O
</s>
<s>
Finite	B-Algorithm
difference	I-Algorithm
schemes	I-Algorithm
for	O
time-dependent	O
partial	O
differential	O
equations	O
(	O
PDEs	O
)	O
have	O
been	O
employed	O
for	O
many	O
years	O
in	O
computational	O
fluid	O
dynamics	O
problems	O
,	O
including	O
the	O
idea	O
of	O
using	O
centered	O
finite	B-Algorithm
difference	I-Algorithm
operators	O
on	O
staggered	O
grids	O
in	O
space	O
and	O
time	O
to	O
achieve	O
second-order	O
accuracy	O
.	O
</s>
<s>
The	O
novelty	O
of	O
Kane	O
Yee	O
's	O
FDTD	B-Algorithm
scheme	O
,	O
presented	O
in	O
his	O
seminal	O
1966	O
paper	O
,	O
was	O
to	O
apply	O
centered	O
finite	B-Algorithm
difference	I-Algorithm
operators	O
on	O
staggered	O
grids	O
in	O
space	O
and	O
time	O
for	O
each	O
electric	O
and	O
magnetic	O
vector	O
field	O
component	O
in	O
Maxwell	O
's	O
curl	O
equations	O
.	O
</s>
<s>
The	O
descriptor	O
"	O
Finite-difference	B-Algorithm
time-domain	I-Algorithm
"	O
and	O
its	O
corresponding	O
"	O
FDTD	B-Algorithm
"	O
acronym	O
were	O
originated	O
by	O
Allen	O
Taflove	O
in	O
1980	O
.	O
</s>
<s>
Since	O
about	O
1990	O
,	O
FDTD	B-Algorithm
techniques	O
have	O
emerged	O
as	O
primary	O
means	O
to	O
computationally	O
model	O
many	O
scientific	O
and	O
engineering	O
problems	O
dealing	O
with	O
electromagnetic	O
wave	O
interactions	O
with	O
material	O
structures	O
.	O
</s>
<s>
Current	O
FDTD	B-Algorithm
modeling	I-Algorithm
applications	O
range	O
from	O
near-DC	O
(	O
ultralow-frequency	O
geophysics	O
involving	O
the	O
entire	O
Earth-ionosphere	O
waveguide	O
)	O
through	O
microwaves	O
(	O
radar	O
signature	O
technology	O
,	O
antennas	O
,	O
wireless	O
communications	O
devices	O
,	O
digital	O
interconnects	O
,	O
biomedical	O
imaging/treatment	O
)	O
to	O
visible	O
light	O
(	O
photonic	O
crystals	O
,	O
nanoplasmonics	O
,	O
solitons	O
,	O
and	O
biophotonics	O
)	O
.	O
</s>
<s>
In	O
2006	O
,	O
an	O
estimated	O
2,000	O
FDTD-related	O
publications	O
appeared	O
in	O
the	O
science	O
and	O
engineering	O
literature	O
(	O
see	O
Popularity	O
)	O
.	O
</s>
<s>
As	O
of	O
2013	O
,	O
there	O
are	O
at	O
least	O
25	O
commercial/proprietary	O
FDTD	B-Algorithm
software	O
vendors	O
;	O
13	O
free-software/open	O
-source-software	O
FDTD	B-Algorithm
projects	O
;	O
and	O
2	O
freeware/closed	O
-source	O
FDTD	B-Algorithm
projects	O
,	O
some	O
not	O
for	O
commercial	O
use	O
(	O
see	O
External	O
links	O
)	O
.	O
</s>
<s>
An	O
appreciation	O
of	O
the	O
basis	O
,	O
technical	O
development	O
,	O
and	O
possible	O
future	O
of	O
FDTD	B-Algorithm
numerical	O
techniques	O
for	O
Maxwell	O
's	O
equations	O
can	O
be	O
developed	O
by	O
first	O
considering	O
their	O
history	O
.	O
</s>
<s>
Partial	O
chronology	O
of	O
FDTD	B-Algorithm
techniques	O
and	O
applications	O
for	O
Maxwell	O
's	O
equations.Adapted	O
with	O
permission	O
from	O
Taflove	O
and	O
Hagness	O
(	O
2005	O
)	O
.	O
</s>
<s>
year	O
event	O
1928	O
Courant	B-Algorithm
,	O
Friedrichs	O
,	O
and	O
Lewy	O
(	O
CFL	O
)	O
publish	O
seminal	O
paper	O
with	O
the	O
discovery	O
of	O
conditional	O
stability	O
of	O
explicit	O
time-dependent	O
finite	B-Algorithm
difference	I-Algorithm
schemes	I-Algorithm
,	O
as	O
well	O
as	O
the	O
classic	O
FD	O
scheme	O
for	O
solving	O
second-order	O
wave	O
equation	O
in	O
1-D	O
and	O
2-D	O
.	O
1950	O
First	O
appearance	O
of	O
von	O
Neumann	O
's	O
method	O
of	O
stability	O
analysis	O
for	O
implicit/explicit	O
time-dependent	O
finite	B-Algorithm
difference	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
1966	O
Yee	O
described	O
the	O
FDTD	B-Algorithm
numerical	O
technique	O
for	O
solving	O
Maxwell	O
's	O
curl	O
equations	O
on	O
grids	O
staggered	O
in	O
space	O
and	O
time	O
.	O
</s>
<s>
1975	O
Taflove	O
and	O
Brodwin	O
reported	O
the	O
first	O
sinusoidal	O
steady-state	O
FDTD	B-Algorithm
solutions	O
of	O
two	O
-	O
and	O
three-dimensional	O
electromagnetic	O
wave	O
interactions	O
with	O
material	O
structures	O
;	O
and	O
the	O
first	O
bioelectromagnetics	O
models	O
.	O
</s>
<s>
1980	O
Taflove	O
coined	O
the	O
FDTD	B-Algorithm
acronym	O
and	O
published	O
the	O
first	O
validated	O
FDTD	B-Algorithm
models	I-Algorithm
of	O
sinusoidal	O
steady-state	O
electromagnetic	O
wave	O
penetration	O
into	O
a	O
three-dimensional	O
metal	O
cavity	O
.	O
</s>
<s>
1981	O
Mur	O
published	O
the	O
first	O
numerically	B-General_Concept
stable	O
,	O
second-order	O
accurate	O
,	O
absorbing	O
boundary	O
condition	O
(	O
ABC	O
)	O
for	O
Yee	O
's	O
grid	B-Algorithm
.	O
</s>
<s>
1982	O
–	O
83	O
Taflove	O
and	O
Umashankar	O
developed	O
the	O
first	O
FDTD	B-Algorithm
electromagnetic	O
wave	O
scattering	O
models	O
computing	O
sinusoidal	O
steady-state	O
near-fields	O
,	O
far-fields	O
,	O
and	O
radar	O
cross-section	O
for	O
two	O
-	O
and	O
three-dimensional	O
structures	O
.	O
</s>
<s>
1984	O
Liao	O
et	O
al	O
reported	O
an	O
improved	O
ABC	O
based	O
upon	O
space-time	O
extrapolation	O
of	O
the	O
field	O
adjacent	O
to	O
the	O
outer	O
grid	B-Algorithm
boundary	O
.	O
</s>
<s>
1985	O
Gwarek	O
introduced	O
the	O
lumped	O
equivalent	O
circuit	O
formulation	O
of	O
FDTD	B-Algorithm
.	O
</s>
<s>
1986	O
Choi	O
and	O
Hoefer	O
published	O
the	O
first	O
FDTD	B-Algorithm
simulation	B-Application
of	O
waveguide	O
structures	O
.	O
</s>
<s>
1987	O
–	O
88	O
,	O
1992	O
Contour-path	O
subcell	O
techniques	O
were	O
introduced	O
by	O
Umashankar	O
et	O
al	O
to	O
permit	O
FDTD	B-Algorithm
modeling	I-Algorithm
of	O
thin	O
wires	O
and	O
wire	O
bundles	O
,	O
by	O
Taflove	O
et	O
al	O
to	O
model	O
penetration	O
through	O
cracks	O
in	O
conducting	O
screens	O
,	O
and	O
by	O
Jurgens	O
et	O
al	O
to	O
conformally	O
model	O
the	O
surface	O
of	O
a	O
smoothly	O
curved	O
scatterer	O
.	O
</s>
<s>
1988	O
Sullivan	O
et	O
al	O
published	O
the	O
first	O
3-D	O
FDTD	B-Algorithm
model	I-Algorithm
of	O
sinusoidal	O
steady-state	O
electromagnetic	O
wave	O
absorption	O
by	O
a	O
complete	O
human	O
body	O
.	O
</s>
<s>
1988	O
FDTD	B-Algorithm
modeling	I-Algorithm
of	O
microstrips	O
was	O
introduced	O
by	O
Zhang	O
et	O
al	O
.	O
</s>
<s>
1990	O
–	O
91	O
FDTD	B-Algorithm
modeling	I-Algorithm
of	O
frequency-dependent	O
dielectric	O
permittivity	O
was	O
introduced	O
by	O
Kashiwa	O
and	O
Fukai	O
,	O
Luebbers	O
et	O
al	O
,	O
and	O
Joseph	O
et	O
al	O
.	O
</s>
<s>
1990	O
–	O
91	O
FDTD	B-Algorithm
modeling	I-Algorithm
of	O
antennas	O
was	O
introduced	O
by	O
Maloney	O
et	O
al	O
,	O
Katz	O
et	O
al	O
,	O
and	O
Tirkas	O
and	O
Balanis	O
.	O
</s>
<s>
1990	O
FDTD	B-Algorithm
modeling	I-Algorithm
of	O
picosecond	O
optoelectronic	O
switches	O
was	O
introduced	O
by	O
Sano	O
and	O
Shibata	O
,	O
and	O
El-Ghazaly	O
et	O
al	O
.	O
</s>
<s>
1992	O
–	O
94	O
FDTD	B-Algorithm
modeling	I-Algorithm
of	O
the	O
propagation	O
of	O
optical	O
pulses	O
in	O
nonlinear	O
dispersive	O
media	O
was	O
introduced	O
,	O
including	O
the	O
first	O
temporal	O
solitons	O
in	O
one	O
dimension	O
by	O
Goorjian	O
and	O
Taflove	O
;	O
beam	O
self-focusing	O
by	O
Ziolkowski	O
and	O
Judkins	O
;	O
the	O
first	O
temporal	O
solitons	O
in	O
two	O
dimensions	O
by	O
Joseph	O
et	O
al	O
;	O
and	O
the	O
first	O
spatial	O
solitons	O
in	O
two	O
dimensions	O
by	O
Joseph	O
and	O
Taflove	O
.	O
</s>
<s>
1992	O
FDTD	B-Algorithm
modeling	I-Algorithm
of	O
lumped	O
electronic	O
circuit	O
elements	O
was	O
introduced	O
by	O
Sui	O
et	O
al	O
.	O
</s>
<s>
1993	O
Toland	O
et	O
al	O
published	O
the	O
first	O
FDTD	B-Algorithm
models	I-Algorithm
of	O
gain	O
devices	O
(	O
tunnel	O
diodes	O
and	O
Gunn	O
diodes	O
)	O
exciting	O
cavities	O
and	O
antennas	O
.	O
</s>
<s>
1993	O
Aoyagi	O
et	O
al	O
present	O
a	O
hybrid	O
Yee	O
algorithm/scalar	O
-wave	O
equation	O
and	O
demonstrate	O
equivalence	O
of	O
Yee	O
scheme	O
to	O
finite	B-Algorithm
difference	I-Algorithm
scheme	I-Algorithm
for	O
electromagnetic	O
wave	O
equation	O
.	O
</s>
<s>
1994	O
Thomas	O
et	O
al	O
introduced	O
a	O
Norton	O
's	O
equivalent	O
circuit	O
for	O
the	O
FDTD	B-Algorithm
space	O
lattice	O
,	O
which	O
permits	O
the	O
SPICE	O
circuit	O
analysis	O
tool	O
to	O
implement	O
accurate	O
subgrid	O
models	O
of	O
nonlinear	O
electronic	O
components	O
or	O
complete	O
circuits	O
embedded	O
within	O
the	O
lattice	O
.	O
</s>
<s>
1994	O
Berenger	O
introduced	O
the	O
highly	O
effective	O
,	O
perfectly	B-Algorithm
matched	I-Algorithm
layer	I-Algorithm
(	O
PML	O
)	O
ABC	O
for	O
two-dimensional	O
FDTD	B-Algorithm
grids	O
,	O
which	O
was	O
extended	O
to	O
non-orthogonal	O
meshes	O
by	O
Navarro	O
et	O
al	O
,	O
and	O
three	O
dimensions	O
by	O
Katz	O
et	O
al	O
,	O
and	O
to	O
dispersive	O
waveguide	O
terminations	O
by	O
Reuter	O
et	O
al	O
.	O
</s>
<s>
1995	O
–	O
96	O
Sacks	O
et	O
al	O
and	O
Gedney	O
introduced	O
a	O
physically	O
realizable	O
,	O
uniaxial	B-Algorithm
perfectly	I-Algorithm
matched	I-Algorithm
layer	I-Algorithm
(	O
UPML	O
)	O
ABC	O
.	O
</s>
<s>
1998	O
Maloney	O
and	O
Kesler	O
introduced	O
several	O
novel	O
means	O
to	O
analyze	O
periodic	O
structures	O
in	O
the	O
FDTD	B-Algorithm
space	O
lattice	O
.	O
</s>
<s>
1998	O
Nagra	O
and	O
York	O
introduced	O
a	O
hybrid	O
FDTD-quantum	O
mechanics	O
model	O
of	O
electromagnetic	O
wave	O
interactions	O
with	O
materials	O
having	O
electrons	O
transitioning	O
between	O
multiple	O
energy	O
levels	O
.	O
</s>
<s>
1998	O
Hagness	O
et	O
al	O
introduced	O
FDTD	B-Algorithm
modeling	I-Algorithm
of	O
the	O
detection	O
of	O
breast	O
cancer	O
using	O
ultrawideband	O
radar	O
techniques	O
.	O
</s>
<s>
1999	O
Schneider	O
and	O
Wagner	O
introduced	O
a	O
comprehensive	O
analysis	O
of	O
FDTD	B-Algorithm
grid	B-Algorithm
dispersion	O
based	O
upon	O
complex	O
wavenumbers	O
.	O
</s>
<s>
2000	O
–	O
01	O
Zheng	O
,	O
Chen	O
,	O
and	O
Zhang	O
introduced	O
the	O
first	O
three-dimensional	O
alternating-direction	O
implicit	O
(	O
ADI	O
)	O
FDTD	B-Algorithm
algorithm	O
with	O
provable	O
unconditional	O
numerical	O
stability	O
.	O
</s>
<s>
2000	O
Rylander	O
and	O
Bondeson	O
introduced	O
a	O
provably	O
stable	O
FDTD	B-Algorithm
-	O
finite-element	O
time-domain	O
hybrid	O
technique	O
.	O
</s>
<s>
2002	O
Hayakawa	O
et	O
al	O
and	O
Simpson	O
and	O
Taflove	O
independently	O
introduced	O
FDTD	B-Algorithm
modeling	I-Algorithm
of	O
the	O
global	O
Earth-ionosphere	O
waveguide	O
for	O
extremely	O
low-frequency	O
geophysical	O
phenomena	O
.	O
</s>
<s>
2003	O
DeRaedt	O
introduced	O
the	O
unconditionally	O
stable	O
,	O
“	O
one-step	O
”	O
FDTD	B-Algorithm
technique	O
.	O
</s>
<s>
2004	O
Soriano	O
and	O
Navarro	O
derived	O
the	O
stability	O
condition	O
for	O
Quantum	O
FDTD	B-Algorithm
technique	O
.	O
</s>
<s>
2008	O
Ahmed	O
,	O
Chua	O
,	O
Li	O
and	O
Chen	O
introduced	O
the	O
three-dimensional	O
locally	O
one-dimensional	O
(	O
LOD	O
)	O
FDTD	B-Algorithm
method	O
and	O
proved	O
unconditional	O
numerical	O
stability	O
.	O
</s>
<s>
2008	O
Taniguchi	O
,	O
Baba	O
,	O
Nagaoka	O
and	O
Ametani	O
introduced	O
a	O
Thin	O
Wire	O
Representation	O
for	O
FDTD	B-Algorithm
Computations	O
for	O
conductive	O
media	O
2009	O
Oliveira	O
and	O
Sobrinho	O
applied	O
the	O
FDTD	B-Algorithm
method	O
for	O
simulating	O
lightning	O
strokes	O
in	O
a	O
power	O
substation	O
2012	O
Moxley	O
et	O
al	O
developed	O
a	O
generalized	O
finite-difference	B-Algorithm
time-domain	I-Algorithm
quantum	O
method	O
for	O
the	O
N-body	O
interacting	O
Hamiltonian	O
.	O
</s>
<s>
2013	O
Moxley	O
et	O
al	O
developed	O
a	O
generalized	O
finite-difference	B-Algorithm
time-domain	I-Algorithm
scheme	O
for	O
solving	O
nonlinear	O
Schrödinger	O
equations	O
.	O
</s>
<s>
2014	O
Moxley	O
et	O
al	O
developed	O
an	O
implicit	O
generalized	O
finite-difference	B-Algorithm
time-domain	I-Algorithm
scheme	O
for	O
solving	O
nonlinear	O
Schrödinger	O
equations	O
.	O
</s>
<s>
2021	O
Oliveira	O
and	O
Paiva	O
developed	O
the	O
Least	O
Squares	O
Finite-Difference	B-Algorithm
Time-Domain	I-Algorithm
method	I-Algorithm
(	O
LS-FDTD	O
)	O
for	O
using	O
time	O
steps	O
beyond	O
FDTD	B-Algorithm
CFL	O
limit	O
.	O
</s>
<s>
This	O
results	O
in	O
the	O
basic	O
FDTD	B-Algorithm
time-stepping	O
relation	O
that	O
,	O
at	O
any	O
point	O
in	O
space	O
,	O
the	O
updated	O
value	O
of	O
the	O
E-field	O
in	O
time	O
is	O
dependent	O
on	O
the	O
stored	O
value	O
of	O
the	O
E-field	O
and	O
the	O
numerical	O
curl	O
of	O
the	O
local	O
distribution	O
of	O
the	O
H-field	O
in	O
space	O
.	O
</s>
<s>
Iterating	O
the	O
E-field	O
and	O
H-field	O
updates	O
results	O
in	O
a	O
marching-in-time	O
process	O
wherein	O
sampled-data	O
analogs	O
of	O
the	O
continuous	O
electromagnetic	O
waves	O
under	O
consideration	O
propagate	O
in	O
a	O
numerical	O
grid	B-Algorithm
stored	O
in	O
the	O
computer	O
memory	O
.	O
</s>
<s>
This	O
description	O
holds	O
true	O
for	O
1-D	O
,	O
2-D	O
,	O
and	O
3-D	O
FDTD	B-Algorithm
techniques	O
.	O
</s>
<s>
Kane	O
Yee	O
's	O
seminal	O
1966	O
paper	O
proposed	O
spatially	O
staggering	O
the	O
vector	O
components	O
of	O
the	O
E-field	O
and	O
H-field	O
about	O
rectangular	O
unit	O
cells	O
of	O
a	O
Cartesian	O
computational	O
grid	B-Algorithm
so	O
that	O
each	O
E-field	O
vector	O
component	O
is	O
located	O
midway	O
between	O
a	O
pair	O
of	O
H-field	O
vector	O
components	O
,	O
and	O
conversely	O
.	O
</s>
<s>
This	O
scheme	O
,	O
now	O
known	O
as	O
a	O
Yee	B-Algorithm
lattice	I-Algorithm
,	O
has	O
proven	O
to	O
be	O
very	O
robust	O
,	O
and	O
remains	O
at	O
the	O
core	O
of	O
many	O
current	O
FDTD	B-Algorithm
software	O
constructs	O
.	O
</s>
<s>
Furthermore	O
,	O
Yee	O
proposed	O
a	O
leapfrog	B-Algorithm
scheme	O
for	O
marching	O
in	O
time	O
wherein	O
the	O
E-field	O
and	O
H-field	O
updates	O
are	O
staggered	O
so	O
that	O
E-field	O
updates	O
are	O
conducted	O
midway	O
during	O
each	O
time-step	O
between	O
successive	O
H-field	O
updates	O
,	O
and	O
conversely	O
.	O
</s>
<s>
To	O
implement	O
an	O
FDTD	B-Algorithm
solution	O
of	O
Maxwell	O
's	O
equations	O
,	O
a	O
computational	O
domain	O
must	O
first	O
be	O
established	O
.	O
</s>
<s>
The	O
computational	O
domain	O
is	O
simply	O
the	O
physical	O
region	O
over	O
which	O
the	O
simulation	B-Application
will	O
be	O
performed	O
.	O
</s>
<s>
The	O
permittivity	O
of	O
dispersive	O
materials	O
in	O
tabular	O
form	O
cannot	O
be	O
directly	O
substituted	O
into	O
the	O
FDTD	B-Algorithm
scheme	O
.	O
</s>
<s>
Once	O
the	O
computational	O
domain	O
and	O
the	O
grid	B-Algorithm
materials	O
are	O
established	O
,	O
a	O
source	O
is	O
specified	O
.	O
</s>
<s>
In	O
the	O
last	O
case	O
FDTD	B-Algorithm
can	O
be	O
used	O
to	O
simulate	O
light	O
scattering	O
from	O
arbitrary	O
shaped	O
objects	O
,	O
planar	O
periodic	O
structures	O
at	O
various	O
incident	O
angles	O
,	O
and	O
photonic	O
band	O
structure	O
of	O
infinite	O
periodic	O
structures	O
.	O
</s>
<s>
Since	O
the	O
E	O
and	O
H	O
fields	O
are	O
determined	O
directly	O
,	O
the	O
output	O
of	O
the	O
simulation	B-Application
is	O
usually	O
the	O
E	O
or	O
H	O
field	O
at	O
a	O
point	O
or	O
a	O
series	O
of	O
points	O
within	O
the	O
computational	O
domain	O
.	O
</s>
<s>
The	O
simulation	B-Application
evolves	O
the	O
E	O
and	O
H	O
fields	O
forward	O
in	O
time	O
.	O
</s>
<s>
Processing	O
may	O
be	O
done	O
on	O
the	O
E	O
and	O
H	O
fields	O
returned	O
by	O
the	O
simulation	B-Application
.	O
</s>
<s>
Data	O
processing	O
may	O
also	O
occur	O
while	O
the	O
simulation	B-Application
is	O
ongoing	O
.	O
</s>
<s>
While	O
the	O
FDTD	B-Algorithm
technique	O
computes	O
electromagnetic	O
fields	O
within	O
a	O
compact	O
spatial	O
region	O
,	O
scattered	O
and/or	O
radiated	O
far	O
fields	O
can	O
be	O
obtained	O
via	O
near-to-far-field	O
transformations	O
.	O
</s>
<s>
Every	O
modeling	O
technique	O
has	O
strengths	O
and	O
weaknesses	O
,	O
and	O
the	O
FDTD	B-Algorithm
method	O
is	O
no	O
different	O
.	O
</s>
<s>
FDTD	B-Algorithm
is	O
a	O
versatile	O
modeling	O
technique	O
used	O
to	O
solve	O
Maxwell	O
's	O
equations	O
.	O
</s>
<s>
FDTD	B-Algorithm
is	O
a	O
time-domain	O
technique	O
,	O
and	O
when	O
a	O
broadband	O
pulse	O
(	O
such	O
as	O
a	O
Gaussian	O
pulse	O
)	O
is	O
used	O
as	O
the	O
source	O
,	O
then	O
the	O
response	O
of	O
the	O
system	O
over	O
a	O
wide	O
range	O
of	O
frequencies	O
can	O
be	O
obtained	O
with	O
a	O
single	O
simulation	B-Application
.	O
</s>
<s>
Since	O
FDTD	B-Algorithm
calculates	O
the	O
E	O
and	O
H	O
fields	O
everywhere	O
in	O
the	O
computational	O
domain	O
as	O
they	O
evolve	O
in	O
time	O
,	O
it	O
lends	O
itself	O
to	O
providing	O
animated	O
displays	O
of	O
the	O
electromagnetic	O
field	O
movement	O
through	O
the	O
model	O
.	O
</s>
<s>
The	O
FDTD	B-Algorithm
technique	O
allows	O
the	O
user	O
to	O
specify	O
the	O
material	O
at	O
all	O
points	O
within	O
the	O
computational	O
domain	O
.	O
</s>
<s>
FDTD	B-Algorithm
allows	O
the	O
effects	O
of	O
apertures	O
to	O
be	O
determined	O
directly	O
.	O
</s>
<s>
FDTD	B-Algorithm
uses	O
the	O
E	O
and	O
H	O
fields	O
directly	O
.	O
</s>
<s>
Since	O
most	O
EMI/EMC	O
modeling	O
applications	O
are	O
interested	O
in	O
the	O
E	O
and	O
H	O
fields	O
,	O
it	O
is	O
convenient	O
that	O
no	O
conversions	O
must	O
be	O
made	O
after	O
the	O
simulation	B-Application
has	O
run	O
to	O
get	O
these	O
values	O
.	O
</s>
<s>
Since	O
FDTD	B-Algorithm
requires	O
that	O
the	O
entire	O
computational	O
domain	O
be	O
gridded	O
,	O
and	O
the	O
grid	B-Algorithm
spatial	O
discretization	B-Algorithm
must	O
be	O
sufficiently	O
fine	O
to	O
resolve	O
both	O
the	O
smallest	O
electromagnetic	O
wavelength	O
and	O
the	O
smallest	O
geometrical	O
feature	O
in	O
the	O
model	O
,	O
very	O
large	O
computational	O
domains	O
can	O
be	O
developed	O
,	O
which	O
results	O
in	O
very	O
long	O
solution	O
times	O
.	O
</s>
<s>
Models	O
with	O
long	O
,	O
thin	O
features	O
,	O
(	O
like	O
wires	O
)	O
are	O
difficult	O
to	O
model	O
in	O
FDTD	B-Algorithm
because	O
of	O
the	O
excessively	O
large	O
computational	O
domain	O
required	O
.	O
</s>
<s>
Methods	O
such	O
as	O
eigenmode	O
expansion	O
can	O
offer	O
a	O
more	O
efficient	O
alternative	O
as	O
they	O
do	O
not	O
require	O
a	O
fine	O
grid	B-Algorithm
along	O
the	O
z-direction	O
.	O
</s>
<s>
Space	O
and	O
time	O
steps	O
must	O
satisfy	O
the	O
CFL	B-Algorithm
condition	I-Algorithm
,	O
or	O
the	O
leapfrog	B-Algorithm
integration	I-Algorithm
used	O
to	O
solve	O
the	O
partial	O
differential	O
equation	O
is	O
likely	O
to	O
become	O
unstable	O
.	O
</s>
<s>
FDTD	B-Algorithm
finds	O
the	O
E/H	O
fields	O
directly	O
everywhere	O
in	O
the	O
computational	O
domain	O
.	O
</s>
<s>
Far-field	O
extensions	O
are	O
available	O
for	O
FDTD	B-Algorithm
,	O
but	O
require	O
some	O
amount	O
of	O
postprocessing	O
.	O
</s>
<s>
Since	O
FDTD	B-Algorithm
simulations	O
calculate	O
the	O
E	O
and	O
H	O
fields	O
at	O
all	O
points	O
within	O
the	O
computational	O
domain	O
,	O
the	O
computational	O
domain	O
must	O
be	O
finite	O
to	O
permit	O
its	O
residence	O
in	O
the	O
computer	O
memory	O
.	O
</s>
<s>
In	O
many	O
cases	O
this	O
is	O
achieved	O
by	O
inserting	O
artificial	O
boundaries	O
into	O
the	O
simulation	B-Application
space	O
.	O
</s>
<s>
Most	O
modern	O
FDTD	B-Algorithm
implementations	O
instead	O
use	O
a	O
special	O
absorbing	O
"	O
material	O
"	O
,	O
called	O
a	O
perfectly	B-Algorithm
matched	I-Algorithm
layer	I-Algorithm
(	O
PML	O
)	O
to	O
implement	O
absorbing	O
boundaries	O
.	O
</s>
<s>
Because	O
FDTD	B-Algorithm
is	O
solved	O
by	O
propagating	O
the	O
fields	O
forward	O
in	O
the	O
time	O
domain	O
,	O
the	O
electromagnetic	O
time	O
response	O
of	O
the	O
medium	O
must	O
be	O
modeled	O
explicitly	O
.	O
</s>
<s>
The	O
most	O
commonly	O
used	O
grid	B-Algorithm
truncation	O
techniques	O
for	O
open-region	O
FDTD	B-Algorithm
modeling	I-Algorithm
problems	O
are	O
the	O
Mur	O
absorbing	O
boundary	O
condition	O
(	O
ABC	O
)	O
,	O
the	O
Liao	O
ABC	O
,	O
and	O
various	O
perfectly	B-Algorithm
matched	I-Algorithm
layer	I-Algorithm
(	O
PML	O
)	O
formulations	O
.	O
</s>
<s>
FDTD	B-Algorithm
does	O
not	O
require	O
a	O
matrix	O
inversion	O
.	O
</s>
<s>
Being	O
a	O
fully	O
explicit	O
computation	O
,	O
FDTD	B-Algorithm
avoids	O
the	O
difficulties	O
with	O
matrix	O
inversions	O
that	O
limit	O
the	O
size	O
of	O
frequency-domain	O
integral-equation	O
and	O
finite-element	O
electromagnetics	O
models	O
to	O
generally	O
fewer	O
than	O
109	O
electromagnetic	O
field	O
unknowns	O
.	O
</s>
<s>
FDTD	B-Algorithm
models	I-Algorithm
with	O
as	O
many	O
as	O
109	O
field	O
unknowns	O
have	O
been	O
run	O
;	O
there	O
is	O
no	O
intrinsic	O
upper	O
bound	O
to	O
this	O
number	O
.	O
</s>
<s>
FDTD	B-Algorithm
is	O
accurate	O
and	O
robust	O
.	O
</s>
<s>
The	O
sources	O
of	O
error	O
in	O
FDTD	B-Algorithm
calculations	O
are	O
well	O
understood	O
,	O
and	O
can	O
be	O
bounded	O
to	O
permit	O
accurate	O
models	O
for	O
a	O
very	O
large	O
variety	O
of	O
electromagnetic	O
wave	O
interaction	O
problems	O
.	O
</s>
<s>
FDTD	B-Algorithm
treats	O
impulsive	O
behavior	O
naturally	O
.	O
</s>
<s>
Being	O
a	O
time-domain	O
technique	O
,	O
FDTD	B-Algorithm
directly	O
calculates	O
the	O
impulse	O
response	O
of	O
an	O
electromagnetic	O
system	O
.	O
</s>
<s>
Therefore	O
,	O
a	O
single	O
FDTD	B-Algorithm
simulation	B-Application
can	O
provide	O
either	O
ultrawideband	O
temporal	O
waveforms	O
or	O
the	O
sinusoidal	O
steady-state	O
response	O
at	O
any	O
frequency	O
within	O
the	O
excitation	O
spectrum	O
.	O
</s>
<s>
FDTD	B-Algorithm
treats	O
nonlinear	O
behavior	O
naturally	O
.	O
</s>
<s>
Being	O
a	O
time-domain	O
technique	O
,	O
FDTD	B-Algorithm
directly	O
calculates	O
the	O
nonlinear	O
response	O
of	O
an	O
electromagnetic	O
system	O
.	O
</s>
<s>
This	O
allows	O
natural	O
hybriding	O
of	O
FDTD	B-Algorithm
with	O
sets	O
of	O
auxiliary	O
differential	O
equations	O
that	O
describe	O
nonlinearities	O
from	O
either	O
the	O
classical	O
or	O
semi-classical	O
standpoint	O
.	O
</s>
<s>
One	O
research	O
frontier	O
is	O
the	O
development	O
of	O
hybrid	O
algorithms	O
which	O
join	O
FDTD	B-Algorithm
classical	O
electrodynamics	O
models	O
with	O
phenomena	O
arising	O
from	O
quantum	O
electrodynamics	O
,	O
especially	O
vacuum	O
fluctuations	O
,	O
such	O
as	O
the	O
Casimir	O
effect	O
.	O
</s>
<s>
FDTD	B-Algorithm
is	O
a	O
systematic	O
approach	O
.	O
</s>
<s>
With	O
FDTD	B-Algorithm
,	O
specifying	O
a	O
new	O
structure	O
to	O
be	O
modeled	O
is	O
reduced	O
to	O
a	O
problem	O
of	O
mesh	O
generation	O
rather	O
than	O
the	O
potentially	O
complex	O
reformulation	O
of	O
an	O
integral	O
equation	O
.	O
</s>
<s>
For	O
example	O
,	O
FDTD	B-Algorithm
requires	O
no	O
calculation	O
of	O
structure-dependent	O
Green	O
functions	O
.	O
</s>
<s>
FDTD	B-Algorithm
scales	O
with	O
high	O
efficiency	O
on	O
parallel-processing	O
CPU-based	O
computers	O
,	O
and	O
extremely	O
well	O
on	O
recently	O
developed	O
GPU-based	O
accelerator	O
technology	O
.	O
</s>
<s>
While	O
this	O
trend	O
positively	O
influences	O
all	O
numerical	O
techniques	O
,	O
it	O
is	O
of	O
particular	O
advantage	O
to	O
FDTD	B-Algorithm
methods	O
,	O
which	O
generate	O
time-marched	O
arrays	O
of	O
field	O
quantities	O
suitable	O
for	O
use	O
in	O
color	O
videos	O
to	O
illustrate	O
the	O
field	O
dynamics	O
.	O
</s>
<s>
the	O
dominant	O
computational	O
electrodynamics	O
techniques	O
(	O
as	O
well	O
as	O
potentially	O
other	O
multiphysics	B-Algorithm
problems	O
)	O
.	O
</s>
<s>
There	O
are	O
hundreds	O
of	O
simulation	B-Application
tools	O
(	O
e.g.	O
</s>
<s>
that	O
implement	O
FDTD	B-Algorithm
algorithms	O
,	O
many	O
optimized	O
to	O
run	O
on	O
parallel-processing	O
clusters	O
.	O
</s>
