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