<s>
Sometimes	O
frequency	B-Algorithm
selective	I-Algorithm
surfaces	I-Algorithm
are	O
referred	O
to	O
simply	O
as	O
periodic	O
surfaces	O
and	O
are	O
a	O
2-dimensional	O
analog	O
of	O
the	O
new	O
periodic	O
volumes	O
known	O
as	O
photonic	O
crystals	O
.	O
</s>
<s>
Many	O
factors	O
are	O
involved	O
in	O
understanding	O
the	O
operation	O
and	O
application	O
of	O
frequency	B-Algorithm
selective	I-Algorithm
surfaces	I-Algorithm
.	O
</s>
<s>
This	O
technique	O
uses	O
the	O
method	B-Algorithm
of	I-Algorithm
moments	I-Algorithm
(	O
MoM	O
)	O
in	O
combination	O
with	O
a	O
Bloch	O
wave	O
expansion	O
of	O
the	O
electromagnetic	O
field	O
to	O
yield	O
a	O
matrix	O
eigenvalue	O
equation	O
for	O
the	O
propagation	O
bands	O
.	O
</s>
<s>
Bloch	O
wave	O
-	O
MoM	O
is	O
similar	O
in	O
principle	O
to	O
the	O
plane	B-Algorithm
wave	I-Algorithm
expansion	I-Algorithm
method	I-Algorithm
,	O
but	O
since	O
it	O
additionally	O
employs	O
the	O
method	B-Algorithm
of	I-Algorithm
moments	I-Algorithm
to	O
produce	O
a	O
surface	O
integral	O
equation	O
,	O
it	O
is	O
significantly	O
more	O
efficient	O
both	O
in	O
terms	O
of	O
the	O
number	O
of	O
unknowns	O
and	O
the	O
number	O
of	O
plane	O
waves	O
needed	O
for	O
good	O
convergence	O
.	O
</s>
<s>
Bloch	O
wave	O
-	O
MoM	O
is	O
the	O
extension	O
to	O
3	O
dimensions	O
of	O
the	O
spectral	O
domain	O
MoM	O
method	O
commonly	O
used	O
for	O
analyzing	O
2D	O
periodic	O
structures	O
such	O
as	O
frequency	B-Algorithm
selective	I-Algorithm
surfaces	I-Algorithm
(	O
FSS	O
)	O
.	O
</s>
<s>
However	O
,	O
it	O
can	O
also	O
be	O
readily	O
expanded	O
to	O
dielectric	O
structures	O
,	O
using	O
the	O
well-known	O
interior	O
and	O
exterior	O
equivalent	O
problems	O
commonly	O
employed	O
in	O
ordinary	O
spatial	O
domain	O
method	B-Algorithm
of	I-Algorithm
moments	I-Algorithm
formulations	O
.	O
</s>
<s>
Note	O
that	O
the	O
spatial	O
domain	O
convolution	O
has	O
been	O
transformed	O
into	O
a	O
simple	O
multiplication	O
in	O
the	O
spectral	O
domain	O
,	O
consistent	O
with	O
the	O
convolution	O
theorem	O
for	O
Fourier	B-Algorithm
transforms	I-Algorithm
.	O
</s>
<s>
As	O
is	O
usual	O
in	O
the	O
method	B-Algorithm
of	I-Algorithm
moments	I-Algorithm
,	O
the	O
source	O
currents	O
are	O
now	O
expanded	O
in	O
terms	O
of	O
a	O
sum	O
over	O
some	O
known	O
set	O
of	O
basis	O
functions	O
with	O
unknown	O
weighting	O
coefficients	O
Jj	O
:	O
</s>
<s>
Different	O
structures	O
will	O
have	O
different	O
sets	O
of	O
basis	O
functions	O
for	O
representing	O
the	O
currents	O
on	O
the	O
elements	O
and	O
as	O
in	O
the	O
ordinary	O
spatial	O
domain	O
method	B-Algorithm
of	I-Algorithm
moments	I-Algorithm
,	O
the	O
solution	O
(	O
in	O
this	O
case	O
,	O
the	O
band	O
diagram	O
)	O
is	O
a	O
function	O
of	O
the	O
set	O
of	O
basis	O
functions	O
used	O
.	O
</s>
<s>
The	O
electric	O
field	O
boundary/continuity	O
conditions	O
are	O
"	O
tested	O
"	O
(	O
or	O
enforced	O
)	O
by	O
being	O
integrated	O
against	O
electric	O
current	O
basis	O
functions	O
(	O
for	O
dielectric	O
structures	O
,	O
the	O
magnetic	O
field	O
continuity	O
conditions	O
are	O
additionally	O
tested	O
by	O
being	O
integrated	O
against	O
magnetic	O
current	O
basis	O
functions	O
)	O
,	O
and	O
this	O
is	O
how	O
the	O
electric	O
(	O
and	O
magnetic	O
)	O
field	O
boundary	O
conditions	O
are	O
converted	O
into	O
a	O
matrix	O
equation	O
via	O
the	O
method	B-Algorithm
of	I-Algorithm
moments	I-Algorithm
.	O
</s>
<s>
This	O
matrix	O
equation	O
is	O
easy	O
to	O
implement	O
and	O
requires	O
only	O
that	O
the	O
3D	O
Fourier	B-Algorithm
transform	I-Algorithm
(	O
FT	O
)	O
of	O
the	O
basis	O
functions	O
be	O
computed	O
,	O
preferably	O
in	O
closed	O
form	O
.	O
</s>
<s>
In	O
fact	O
,	O
computing	O
bands	O
of	O
a	O
3D	O
photonic	O
crystal	O
with	O
this	O
method	O
is	O
no	O
more	O
difficult	O
than	O
computing	O
reflection	O
and	O
transmission	O
from	O
a	O
2D	O
periodic	B-Algorithm
surface	I-Algorithm
using	O
the	O
spectral	O
domain	O
method	O
.	O
</s>
<s>
This	O
is	O
because	O
equation	O
(	O
)	O
is	O
identical	O
to	O
the	O
basic	O
EFIE	O
for	O
a	O
freestanding	O
PEC	O
FSS	O
(	O
see	O
Frequency	B-Algorithm
selective	I-Algorithm
surface	I-Algorithm
eq	O
.	O
</s>
<s>
(	O
)	O
is	O
Hermitian	B-Algorithm
.	O
</s>
<s>
As	O
a	O
matter	O
of	O
computational	O
convenience	O
,	O
whenever	O
the	O
matrix	O
is	O
larger	O
than	O
2×2	O
it	O
's	O
vastly	O
more	O
efficient	O
to	O
compute	O
the	O
determinant	O
either	O
by	O
reducing	O
the	O
matrix	O
to	O
upper	B-Algorithm
triangular	I-Algorithm
form	I-Algorithm
using	O
QR	O
decomposition	O
or	O
to	O
compute	O
the	O
determinant	O
by	O
reducing	O
to	O
echelon	O
form	O
using	O
Gaussian	B-Algorithm
elimination	I-Algorithm
,	O
rather	O
than	O
trying	O
to	O
actually	O
compute	O
the	O
determinant	O
of	O
the	O
matrix	O
directly	O
.	O
</s>
<s>
The	O
spectral	O
domain	O
method	O
is	O
known	O
at	O
Ohio	O
State	O
University	O
as	O
the	O
periodic	O
method	B-Algorithm
of	I-Algorithm
moments	I-Algorithm
(	O
PMM	O
)	O
.	O
</s>
<s>
To	O
solve	O
equations	O
(	O
)	O
and	O
(	O
)	O
for	O
a	O
free-standing	O
,	O
doubly	O
periodic	B-Algorithm
surface	I-Algorithm
,	O
we	O
consider	O
an	O
infinite	O
2D	O
periodic	B-Algorithm
surface	I-Algorithm
occupying	O
the	O
entire	O
x-y	O
plane	O
,	O
and	O
assume	O
a	O
discrete	O
plane	O
wave	O
expansion	O
for	O
all	O
currents	O
,	O
fields	O
and	O
potentials	O
(	O
Tsao	O
[1982],	O
Scott	O
[1989],	O
Fourier	O
optics	O
)	O
:	O
</s>
<s>
As	O
is	O
usual	O
in	O
the	O
method	B-Algorithm
of	I-Algorithm
moments	I-Algorithm
,	O
we	O
assume	O
an	O
expansion	O
for	O
the	O
source	O
currents	O
over	O
some	O
known	O
set	O
of	O
basis	O
functions	O
with	O
unknown	O
weighting	O
coefficients	O
Jj	O
(	O
Scott	O
 [ 1989 ] 	O
)	O
:	O
</s>
<s>
All	O
of	O
these	O
matrix	O
equations	O
are	O
very	O
simple	O
to	O
implement	O
and	O
require	O
only	O
that	O
the	O
2D	O
Fourier	B-Algorithm
transform	I-Algorithm
(	O
FT	O
)	O
of	O
the	O
basis	O
functions	O
be	O
computed	O
,	O
preferably	O
in	O
closed	O
form	O
.	O
</s>
<s>
(	O
)	O
above	O
,	O
and	O
the	O
Bloch	B-Algorithm
wave	I-Algorithm
-	I-Algorithm
MoM	I-Algorithm
method	I-Algorithm
eqn	O
.	O
</s>
<s>
As	O
an	O
example	O
of	O
how	O
to	O
use	O
FSS	O
equivalent	O
circuits	O
for	O
quick	O
and	O
efficient	O
design	O
of	O
a	O
practical	O
filter	O
,	O
we	O
can	O
sketch	O
out	O
the	O
process	O
that	O
would	O
be	O
followed	O
in	O
designing	O
a	O
5-stage	O
Butterworth	O
filter	O
(	O
Hunter	O
[2001],	O
Matthaei	O
 [ 1964 ] 	O
)	O
using	O
a	O
stack	O
of	O
5	O
frequency	B-Algorithm
selective	I-Algorithm
surfaces	I-Algorithm
,	O
with	O
4	O
air	O
spacers	O
in	O
between	O
the	O
FSS	O
sheets	O
.	O
</s>
<s>
First	O
,	O
the	O
series	O
inductors	O
will	O
be	O
replaced	O
by	O
their	O
equivalent	O
transmission	O
line	O
sections	O
,	O
and	O
then	O
the	O
shunt	O
capacitors	O
will	O
be	O
replaced	O
by	O
capacitive	O
frequency	B-Algorithm
selective	I-Algorithm
surfaces	I-Algorithm
.	O
</s>
<s>
Frequency	B-Algorithm
selective	I-Algorithm
surfaces	I-Algorithm
,	O
as	O
any	O
type	O
of	O
filter	O
,	O
may	O
also	O
be	O
classified	O
according	O
to	O
their	O
function	O
,	O
and	O
these	O
usually	O
fall	O
into	O
3	O
categories	O
:	O
low-pass	O
,	O
high-pass	O
and	O
bandpass	O
,	O
in	O
addition	O
to	O
band-stop	O
filters	O
.	O
</s>
