8/10/2019 Doubly Periodic - BLB
1/7
1712
J.
Opt.
Soc.
Am.
A/Vol.
7,
No.
9/September
1990
Boag
t
al.
Analysis
of
diffraction
from
doubly
periodic
arrays
of
perfectly
conducting
bodies
by
using
a
patch-current
model
Amir
Boag,
Yehuda
Leviatan,
and
Alona
Boag
Department
of
Electrical
Engineering,
Technion-Israel
Institute
of Technology,
Haifa
32000,
srael
Received
November
29,
1989;
accepted
May
10, 1990
A
novel
solution
is
presented
for
the
problem
of
three-dimensional
electromagnetic
scattering
of
aplane
wave
from
a
doubly
periodic
infinite
array
of
perfectly
conducting
bodies.
A set of
fictitious
spatially
periodic
and
properly
modulated
patches
of magnetic
current
is
used
to
simulate
the
scattered
field.
These
patch
currents
are
of
dual
polarization
and have
complex
amplitudes.
The
electromagnetic
field
radiated
by
each
of the
periodic
patch
currents
isexpressed
as
a double
series
of Floquet
modes.
The
complex
amplitudes
of the
fictitious
patch
currents
are
adjusted
to render
the
tangential
electric
field
zero
at
a selected
set
of points
on
the
surface
of
any
of the
scatterers.
The
procedure
is
simple
to
implement
and
is
applicable
to
arrays
composed
of
smooth
but
otherwise
arbitrary perfectly conducting scatterers. Results are givenand compared with an analytic approximation.
1.
INTRODUCTION
The
study
of
diffraction
of
a plane
wave
from
periodic
struc-
tures
is
long
standing.
It
has
been
motivated
by
academic
curiosity
as
well
as many
engineering
applications.
It is
of
practical
importance
in
designing
reflection
and
transmis-
sion
gratings
often
used
as
filters,
broadband
absorbers,
polarizers,
and
frequency
scanned
reflectors.
While
singly
periodic
gratings
have
been
treated
extensively,
doubly
peri-
odic
gratings,
being
in
general
more
difficult
not
only
to
analyzebut also to fabricate, have receivedconsiderably less
attention.
One
type of
doubly
periodic
structure
that
has
been
investigated
by many
researchers
comprises
infinitesi-
mally
thin
planar
doubly
periodic
screens
in various
configu-
rations,
often
referred
to
as
frequency-selective
surfaces.
-
3
Perfectly
conducting
screens
of
finite
thickness
consisting
of
doubly
periodic
arrays
of
apertures,
known
as inductive
grids,
have
also
been
extensively
studied.4
5
While
the
structures
in
Refs.
1-5
may
be
different,
a common
ingredi-
ent
of
essentially
all of
them
is
that
the
fields
in
the
various
regions
can
be
represented
by modal
expansions
relative
to
the axis
normal
to
the
grating.
These
modal
representa-
tions
are then
matched
by
using
appropriate
boundary
con-
ditions, and the unknown modal coefficients are readily de-
termined.
In
contrast
to
the
above
discussion,
rigorous
studies
of
diffraction
from
doubly
periodic
arrays
of
finite-
sized
perfectly
conducting
scatterers,
referred
to
as
capaci-
tive
grids,
have
not
been
reported
in
the
literature.
In this
latter
case
the fields
in
the
space
gap
between
the
scatterers
forming
the array
cannot
be
represented
in
terms
of
analyti-
cally
known
modal
functions.
5
Therefore
even
the
method
devised
in
Ref.
6 for
the
analysis
of
diffraction
by
doubly
periodic
surfaces
falls
short,
while
differential
method
pro-
cedures
might
be
infeasible
with
present-time
computer
storage
and
speed
limitations.
In
this
paper
we
present
a
new
method
for
analyzing
three-
dimensional electromagnetic scattering from doubly period-
ic
arrays.
The
technique
is
applicable
to
arrays
composed
of
perfectly
conducting
bodies
of smooth,
but
otherwise
arbi-
trary,
shapes.
An
example
of
an
array
is
depicted
in
Fig.
1.
We
follow
the
approach
outlined
in
Ref.
7
for
analyzing
scattering
by
smooth
homogeneous
scatterers.
The
basic
idea
in
Ref.
7 is
as
follows:
Instead
of
employing
surface
integral
equations
in
solving
for
conventional
electric
and
magnetic
surface
currents,
we
solve
for
fictitious
source
cur-
rents
that
lie
a
distance
away
from
the
surface.
This
idea
has
been
applied
successfully
to
two-dimensional
diffraction
from
gratings
of
cylinders,
8
as
well
as
to
sinusoidal
and
ech-
elette
gratings.
9
-
10
In
Refs.
8-10,
an
expansion
of
periodic
strip
currents
is used
for
the
unknown
fictitious
currents
that simulate the periodic scattered field, and point match-
ing
is
used
for
testing.
Here,
we
employ
the
basic
strategy
of
Refs.
7-10
for
facili-
tating
the
analysis
of
three-dimensional
scattering
from
doubly
periodic
arrays
of
isolated
perfectly
conducting
scat-
terers.
We
set
up
a simulated
equivalent
situation
to
the
original
one
in
the
region
surrounding
the
scatterers.
The
scattered
field
must
be
a
source-free
Maxwellian
field
satis-
fying
the
radiation
condition
at
z
-
,
the
periodicity
conditions
of
the Floquet
theorem,
and
the
boundary
con-
dition
on
the
surfaces
of the
scatterers.
Instead
of express-
ing
the
scattered
field
as
a conventional
integral
in
terms
of
the
physical
surface
currents,
we
simulate
the
actual
field
by
the fields of fictitious sources of yet unknown amplitudes
that
lie
a distance
from
the
surface.
Hence,
in the
simulated
equivalence
the physical
bodies
are
removed
and
the
period-
ic
field
that
they
scatter
is
simulated
by
the
field
of
a set
of
fictitious
doubly
periodic
patches
of
currents
satisfying
the
Floquet
periodicity
conditions
and
situated
in the
region
originally
occupied
by
the
scatterers.
Each
periodic
patch
current
lies
in
a plane
parallel
with
the
xy plane
(the
plane
spanned
by
the
directions
of
the
periodicity).
All the
patch-
es
are
characterized
by a
common
Fourier-transformable
magnetic-current
density
profile,
which,
for
each
periodic
source,
is
multiplied
by
an as
yet
undetermined
constant
complex
amplitude.
They
are
assumed
to
radiate
in
an
unbounded homogeneousspace filledwith the same medium
as
that
surrounding
the
scatterers.
Patches
of
electric
cur-
rent
could
be
used
as
well.
Patches
of magnetic
current
were
chosen
because
the
electric
field
that
they
produce
is
0740-3232/90/091712-07$02.00
1990
Optical
Society
of America
8/10/2019 Doubly Periodic - BLB
2/7
Vol. 7,
No.
9/September
1990/J.
Opt.
Soc.
Am.
A
1713
Unbounded
Space
,
E)
magnitudes
are
specified
by
the
respective
periods.
It
is
assumed
that
any
inhomogeneity
is confined
between
the
z
=
-b
and
the
z = b
planes.
The problem
geometry
together
with
a
relevant
coordinate
system
is
shown
in
Fig.
1.
It
should
be noted
that,
according
to
our
convention,
the
z
axis
is oriented
downward.
The
medium
surrounding
the
scat-
terers is of permeability
As nd
permittivity
e.
The medium
can
be
dissipative;
thus
g and
e
are
allowed
to
be
complex.
A plane
wave
given
by
Einc(r)
=
EinC
xp(-jkinc
r),
(1)
Fig.
1.
General
problem
of plane-wave
scattering
periodic
grating
of
finite-sized
scatterers.
from
a doubly
easier
to
compute.
Locating
the
sources
some
distance
away
from
the
surface
permits
us
to
use
periodic
patch
currents
with smooth current density profile that lie in planes parallel
with
the
xy
plane
spanned
by the
two
directions
of
periodici-
ty.
This
feature
is
attractive
because
it
enables
the
representation
of
the
field
produced
by
each
periodic
cur-
rent
patch
by
uniformly
convergent
series
of
z-directed
out-
going
and
decaying
plane
waves
known
as Floquet
modes.
It follows
hat
outside
the
grating
region
the
total
field
radi-
ated
by the
patches
can
also
be represented
analytically
by
means
of
these
Floquet
modes.
Thus
the
fields
can
be
deter-
mined
anywhere
by
summations
of
analytic
terms.
This
is
a
desirable
attribute
as
one
avoids
the
surface
integrations
associated
with
the
field
computation
at the
three
principal
stages
of
the
solution.
The
first
stage
is that
of
constructing
matrix equations for the problem, the second is that of test-
ing
the
solution
by
checking
the
degree
to
which
the
bound-
ary
conditions
are
satisfied
over
a denser
set
of points
on the
boundaries,
and
the
third
is
that
of
computing
the
scattered
field
and the
reflection
and
transmission
coefficients
of
vari-
ous
Floquet
modes
after
the
solution
has
been
established.
The
patch-current
sources
lying
a
distance
away
from
the
boundary
surfaces
produce
a set
of
smooth
field
functions
on
the
surfaces
that
may
be well
suited
for
spanning
the
actual
smooth
field
on
the
boundaries.
Furthermore,
since
we
are
actually
using
a
basis
of smooth
field
functions
for
repre-
senting
fields
on
the
boundary,
the
boundary
condition
can
be
enforced
by
a
simple
point-matching
testing
procedure
and the unknown source amplitudes are readily determined.
The
paper
is organized
in
the
following
manner.
The
problem
under
study
is
specified
in Section
2.
The
solution
is
formulated
in
Section
3.
Results
of several
numerical
simulations
are presented
in
Section
4 and
compared
with
an
analytic
approximation
in order
to
demonstrate
the
efficien-
cy
and
accuracy
of
the
proposed
technique.
Finally,
a few
conclusions
summarize
the
paper.
2.
PROBLEM
SPECIFICATION
Consider
a
doubly
periodic
array
of scatterers.
The
array
is
composed
of
an
infinite
set
of
identical
perfectly
conducting
scatterers arranged in a doubly periodic lattice. The lattice
is
described
by
two
vectors
d,
and
d
2
lying
in the
xy
plane.
The
vectors
d, and
d
2
are
referred
toas
lattice
vectors.
They
are
aligned
with
the
two
directions
of
periodicity,
and
their
with
harmonic
exp(jwt)
time
dependence
assumed
and
sup-
pressed,
is incident
on
the
grating.
Here,
kinc
and
E
0
c de-
note,
respectively,
the
wave
vector
and
the
amplitude
of
the
incident
field.
Our
objective
is
to
determine
the
field
scat-
tered
by
the
grating
(Es,
Hs)
(i.e.,
the
actual
field
minus
the
incident
field).
The
field
should
be
a source-free
solution
of
the
Maxwell
equations
and
obey
the
Floquet
periodicity
conditions
Es(r
+ dp) = exp(-jkinc -
dp)Es(r),
p = 1, 2.
(2)
In
addition,
(Es,
Hs) should
satisfy
the
boundary
condition
n
X ES
=
-h
X
Einc
(3)
where
S is
the boundary
of
an arbitrary
selected
scatterer
and
h is a
unit
vector
that
is
normal
to
S.
3.
FORMULATION
A.
Simulated
Equivalent
Situation
We
now
describe
how
the
simulated
equivalent
situation
to
the original one in the region surrounding the scatterers
is
set up.
According
to
our
general
idea,
in the
simulated
equivalence
that
is shown
in
Fig.
2, the
scattered
field
(Es,
Hs) is
simulated
by
a
field
of a
set
of
doubly
periodic
ficti-
tious
patches
of
magnetic
current
Mqi,
q =
1, 2, i
=
1,
2, ... ,
N.
These
sources
are
located
in
the
region
occupied
by the
scatterers
in
the original
situation
and
are treated
as
sources
Unbounded
Homogeneous
Space
(Einc
,Hinc)
kinc
(gia)
(Es+Einc
Hs
+Hinc)
Periodic
Patch
Currents
/
Mathematical
Boundary
C
Fig.
2.
Simulated
equivalence
for
the
region
surrounding
the
scat-
terers.
(EH )
\k
kin'
Boag
et
al.
8/10/2019 Doubly Periodic - BLB
3/7
1714
J.
Opt.
Soc.
Am.
A/Vol.
7,
No.
9/September
1990
radiating
in
an
unbounded
space
filled
with
homogeneous
material
that is
identical
to that
surrounding
the
scatterers
in
the
original
situation.
They
have
constant
dimensionless
amplitudes
Kqj
that
are
yet
to be
determined.
The
electric
field
Es
at
observation
point
r due
to these
sources
is given
by
2
N
Es(r)
=
Z
KqjEqj(r),
q=1
i=1
(4)
where
Eqi
describe
the
field
due
to
a source
Mqj of
unit
amplitude
(Kq =
1).
Obviously,
since
these
periodic
patch-
es
produce
fields
satisfying
the
Floquet
periodicity
condi-
tions,
the
simulated
scattered
field
[Eq.
(4)] also
satisfies
them.
It is
important
to note
that
the
location
of
the
sources
in
the
simulated
equivalence
has
not
been
specified
yet.
As
far
as
the
formulation
is concerned,
their
location
can
be arbi-
trary.
The
question
of
selecting
source
locations
that are
suitable
for
a numerical
solution
is
an important
one.
From
the numerous geometries considered in our earlier research
with
perfectly
conducting
and
penetrable
scatterers,
7
-'
0
we
have
concluded
that
the sources
should
be placed
on
surfaces
of
a shape
similar
to
that
of the
actual
boundary.
We
will
give
this
issue
further
attention
in
Section
4.
B.
Evaluation
of the
Unknown
Amplitudes
Kqj}
By
the construction,
the
simulated
scattered
field
Es
satis-
fies
the radiation
and
the periodicity
conditions.
Evidently,
if a
set
of
periodic
patch
currents
Mqij
could
be found
such
that
the
boundary
condition
[Eq.
(3)]
was
strictly
satisfied,
then
Es
would
be
the
exact
field
scattered
by
the grating.
To
obtain
an
approximate
solution,
the
boundary
condition
is
imposed at M selected points
on
the
boundary
S.
This
reduces
the
functional
relation
[Eq.
(3)]
to
the
matrix
equa-
tion
[Z]K
= V,
(5)
where
=
[Z1
1
(6)
ZI
1
22
6
is a 2N
by
2M
generalized
impedance
matrix,
[]
(7)
is
a 2N-element
generalized
unknown-current
column
vec-
tor, and
V2]
(8)
is a 2M-element
generalized
voltage-source
column
vector.
In
Eq. (6),
the
matrices
[Zpq]
p,
q =
1,
2)
denote
M
by N
matrices
whose
(m,
n)
element
is the
tpm component
of
the
electric
field
at
observation
point
r
on
S due
to a patch
current
Mqn of
unit
amplitude
(Kqn
=
1).
Here,
pm p
= 1,
2)
are
orthogonal
unit vectors tangential to S at observation
point
rm
on
S.
In Eq.
(7),
the
vectors Kq
(q
=
1,2) denote
N-
element
column
vectors
whose
nth
element
is
Kqn.
Finally,
in
Eq.
(8), the
vectors
V (p
=
1,
2) denote
M-element
column
vectors
whose
mth
element
is the
negative
of
the
tpm
component
of
Einc
at
observation
point
r
on
S.
Having
formulated
the
matrix
equation
[Eq. (5)],
the
unknown
cur-
rent
vector
can
be
found
in
a simple
manner.
If the
bound-
ary
condition
is
imposed
at
M
= N
points
on
S,
then
the
exact solution
to
Eq. (5)
will
be
K = [Zl-1V. (9)
If, on
the
other
hand,
the
boundary
condition
is forced
at M
> N
points
on
S, then
the
solution,
in a
least-square
sense,
will be
K
= [ZIt[Z1I-[Z]t
V.
(10)
This
completes
the
solution
of the
matrix
equation
[Eq.
(5)].
Once
the
unknown
current
vector
is
derived,
either
from
Eq.
(9)or
(10),
one
can
readily
proceed
in evaluating
an
approxi-
mate scattered
field
(Es,
Hs)
and,
of course,
any
other
field-
related
quantity
of
interest.
C.
Fields of Doubly Periodic Magnetic Patch-Current
Sources
In the
simulated
equivalence
for
the
region
surrounding
the
scatterers,
the periodic
scattered
field
is simulated
by
the
field
of
a set
of 2N
spatially
periodic
and
properly
modulated
fictitious
patch-current
sources
placed
outside
that
region.
These
patches
lie
in
planes
parallel
with the
xy
plane.
They
are
of
dimensions
s
by s
2
in the
directions
of the
reciprocal
lattice
vectors
K = 2
X d
2
/Id
1
X
d
2
1
and
K2
= 2
X d/Id
X
d
2
1, espectively.
It is
assumed
that si
and
S2
re
sufficiently
small
compared
with
the
dimensions
of
the
bodies
so
that
the patches
can
be completely
enclosed
inside
the bodies.
The
current
density
of
the ith
periodic
patch
current
Mqj
(q
= 1, 2,
i =
1, 2,...,
N)
centered at a point
r
inside S is
described
by
2
Mqj
= tqjKqj(z
-
z)exp[j
k c
(r-r)]
J
fPSP)
p=1 n=-'
(11)
with {ipn
=
(r
-
ri-
ndp) Kp/Kp
and
hasa
constant
complex
amplitude
Kqi
that
is
yet
to
be determined.
Here,
denotes
the Dirac
delta
function,
z
is
the z
component
of r,
and
azqi
(q
=
1,
2) are
two
unit
vectors
defining
the directions
of the
sources
centered
at
ri. The
function
f(-)
in
Eq. (11)
s a
real-
valued
window
function
of
unit
width
characterized
by
a
continuous
profile
that
is zero
for
all
values
of argument
outside
the interval
(/2,
/2)
and
of
piecewise
continuous
derivative
on
that
interval.
Under
these conditions
f/sp)
as
a function
of
can be
represented
by
a
Fourier
series
whose
convergence
to
fS/sp)
on
the
period
interval
(-rl/Kp,
JrIKp)
s absolute
and
uniform.
It
should
be noted
that
the
above
continuity
requirements
on
f -)
are
sufficient
in
order
to ensure
uniform
convergence
of
the
Fourier
series.
How-
ever,
a
smoother
function
f -)
should
be
preferred
since
its
Fourier
series
converges
faster.
A
specific
choice
forf(.)
that
has been
used
in
our numerical
solution
is
f() =
0.35875
+ 0.48829
cos(27rt)
+ 0.14128
cos(47rt)
+ 0.01168
cos(67rt),
(12)
which
is
known
in
signal
processing
as
the
Blackman-Harris
window.
2
This
window
function
and
its
Fourier
transform
are
shown,
respectively,
in
Figs.
3(a) and
3(b).
As seen
in
Boag et al.
8/10/2019 Doubly Periodic - BLB
4/7
Vol.
7, No.
9/September
1990/J.
Opt. Soc.
Am.
A 1715
(16)
Tmn
=
kT
+
Ml
1
+
nK
2
and
kZmn
=
(k
2
-
kTmn
kTmn)'
1 2
,
(17)
which
are subject
to
the
requirements
Re(kzmn)
2 0
and
Im(kzmn)
0 for
all m
and
n,
which stem
directly
from the
radiation
condition
at
IzI
Cow. Here,
k
is the
intrinsic
wave
number
in
the surrounding
medium,
and
k iTc
and
kTmn de-
note
the transversal
to z components
of
the wave
vectors
of
the incident
field
and
of the mnth
Floquet
mode,
respective-
ly.
Also,
z is the
unit
vector
in
the z direction,
and
kZmnand
-kZmn
are,
respectively,
the z
components
of the
wave
vec-
0.50
tors
of
the
z and
-z traveling
mnth Floquet
modes.
Thus,
in
Eq. (14),
k
or k
are
used depending
on whether
z
> zi or
z