<s>
A	O
perfectly	B-Algorithm
matched	I-Algorithm
layer	I-Algorithm
(	O
PML	O
)	O
is	O
an	O
artificial	O
absorbing	O
layer	O
for	O
wave	O
equations	O
,	O
commonly	O
used	O
to	O
truncate	O
computational	O
regions	O
in	O
numerical	B-Algorithm
methods	I-Algorithm
to	O
simulate	O
problems	O
with	O
open	O
boundaries	O
,	O
especially	O
in	O
the	O
FDTD	B-Algorithm
and	O
FE	B-Application
methods	O
.	O
</s>
<s>
In	O
particular	O
,	O
PMLs	O
were	O
shown	O
to	O
correspond	O
to	O
a	O
coordinate	O
transformation	O
in	O
which	O
one	O
(	O
or	O
more	O
)	O
coordinates	O
are	O
mapped	O
to	O
complex	O
numbers	O
;	O
more	O
technically	O
,	O
this	O
is	O
actually	O
an	O
analytic	B-Algorithm
continuation	I-Algorithm
of	O
the	O
wave	O
equation	O
into	O
complex	O
coordinates	O
,	O
replacing	O
propagating	O
(	O
oscillating	O
)	O
waves	O
by	O
exponentially	O
decaying	O
waves	O
.	O
</s>
<s>
where	O
we	O
have	O
taken	O
a	O
planewave	O
propagating	O
in	O
the	O
+x	O
direction	O
(	O
for	O
)	O
and	O
applied	O
the	O
transformation	O
(	O
analytic	B-Algorithm
continuation	I-Algorithm
)	O
to	O
complex	O
coordinates	O
:	O
,	O
or	O
equivalently	O
.	O
</s>
<s>
for	O
vacuum	B-General_Concept
)	O
because	O
of	O
the	O
dispersion	O
relation	O
between	O
and	O
k	O
.	O
However	O
,	O
this	O
frequency-dependence	O
means	O
that	O
a	O
time	O
domain	O
implementation	O
of	O
PML	O
,	O
e.g.	O
</s>
<s>
in	O
the	O
FDTD	B-Algorithm
method	O
,	O
is	O
more	O
complicated	O
than	O
for	O
a	O
frequency-independent	O
absorber	O
,	O
and	O
involves	O
the	O
auxiliary	O
differential	O
equation	O
(	O
ADE	O
)	O
approach	O
(	O
equivalently	O
,	O
i/	O
appears	O
as	O
an	O
integral	O
or	O
convolution	B-Language
in	O
time	O
domain	O
)	O
.	O
</s>
<s>
Perfectly	B-Algorithm
matched	I-Algorithm
layers	I-Algorithm
,	O
in	O
their	O
original	O
form	O
,	O
only	O
attenuate	O
propagating	O
waves	O
;	O
purely	O
evanescent	O
waves	O
(	O
exponentially	O
decaying	O
fields	O
)	O
oscillate	O
in	O
the	O
PML	O
but	O
do	O
not	O
decay	O
more	O
quickly	O
.	O
</s>
<s>
One	O
caveat	O
with	O
perfectly	B-Algorithm
matched	I-Algorithm
layers	I-Algorithm
is	O
that	O
they	O
are	O
only	O
reflectionless	O
for	O
the	O
exact	O
,	O
continuous	O
wave	O
equation	O
.	O
</s>
<s>
Once	O
the	O
wave	O
equation	O
is	O
discretized	B-Algorithm
for	O
simulation	O
on	O
a	O
computer	O
,	O
some	O
small	O
numerical	O
reflections	O
appear	O
(	O
which	O
vanish	O
with	O
increasing	O
resolution	O
)	O
.	O
</s>
<s>
In	O
general	O
,	O
any	O
absorber	O
,	O
whether	O
PML	O
or	O
not	O
,	O
is	O
reflectionless	O
in	O
the	O
limit	O
where	O
it	O
turns	O
on	O
sufficiently	O
gradually	O
(	O
and	O
the	O
absorbing	O
layer	O
becomes	O
thicker	O
)	O
,	O
but	O
in	O
a	O
discretized	B-Algorithm
system	O
the	O
benefit	O
of	O
PML	O
is	O
to	O
reduce	O
the	O
finite-thickness	O
"	O
transition	O
"	O
reflection	O
by	O
many	O
orders	O
of	O
magnitude	O
compared	O
to	O
a	O
simple	O
isotropic	O
absorption	O
coefficient	O
.	O
</s>
<s>
Another	O
important	O
limitation	O
of	O
PML	O
is	O
that	O
it	O
requires	O
that	O
the	O
medium	O
be	O
invariant	O
in	O
the	O
direction	O
orthogonal	O
to	O
the	O
boundary	O
,	O
in	O
order	O
to	O
support	O
the	O
analytic	B-Algorithm
continuation	I-Algorithm
of	O
the	O
solution	O
to	O
complex	O
coordinates	O
(	O
the	O
complex	O
"	O
coordinate	O
stretching	O
"	O
)	O
.	O
</s>
