doubly periodic surfaces

8/10/2019 Doubly Periodic Surfaces

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732

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 41, NO.

6 .

JUNE 1993

Analysis

of

Electromagnetic Scattering

from Doubly Periodic Nonplanar

Surfaces Using a Patch-Current Model

Amir Boag,

Member,

IEEE,

Yehuda Leviatan,

Senior Member,

IEEE,

and Alona Boag

Abstract-A

novel solution is presented for the problem of

electromagnetic scattering

of

a time-harmonic plane wave from

a nonplanar doubly periodic surface separating two contrasting

homogeneous media. The reflected and the transmitted fields are

approximated by a linear combination of the

fields

due to sets

of

fictitious patch sources supporting surface currents of cross-

polarization. Spatially periodic and properly modulated, these

fictitious sources lie at some distance from the physical surface,

each in a plane parallel to the directions of the periodicity.

The fields radiated by these patch sources are computed by

summing a spectrum of Floquet modes. The complex amplitudes

of these fictitious sources are adjusted simultaneously to render

the tangential components

of

the electric and magnetic fields

continuousat a selected set of points on the surface. The suggested

procedure is simple to implement and is applicable to arbitrary,

smooth, doubly periodic surfaces. The accuracy

of

the method

has been demonstrated for doubly periodic sinusoidal surfaces.

Perfectly conducting surfaces have also been treated within the

general procedure as a reduced case.

I. INTRODUCTION

FFECTIVE computational techniques for analyzing elec-

E romagnetic scattering from periodic structures facilitate

the design of diffraction gratings often used as filters (fre-

quency selective surfaces), broadband absorbers, and polar-

izers. In this paper, we describe a novel solution for the

scattering problem of a time-harmonic plane wave from a non-

planar doubly periodic interface between two homogeneous

half spaces (see Fig. 1). The interface is of arbitrary smooth

shape and

it

is periodic ih two directions. The directions of

periodicity need not be constrained to be orthogonal. In the

proposed computational technique, the scattering problem is

formulated not in terms of equivalent source distributions

applying standard fornlulations, but in terms of fictitious

simple sources imple in the sense that their fields are

analytically derivable in the region of interest ocated on

suitably chosen mathematical supports which are displaced

from the physical boundaries. In many cases, these simple

sources are spatially impulsive sources [l]. However, in the

cases involving periodic structures there are preferences for

other sources, which are slightly more spatially diffused and

whose fields can also be derived analytically. In the latter class

of problems, this technique has been applied successfully to

analyze two-dimensional scattering from gratings of cylinders

Manuscript received March 24, 1992; revised December 22, 1992.

The authors are with the Department of Electrical Engineering, Tech-

IEEE Log Number

9210821.

nion-Israel Institute of Techn olog y, Haifa 32000, Israel.

HomogeneousHalfSpace

p,,

E )

@ n c , H i n c

Doublv-Periodic BoundarvS

Homogeneou s Half Space

VI , , E )

Fig.

1.

General problem of plane wave scattering from a doubly periodic

surface.

[ 2 ] ,

inusoidal gratings [3], and echelette gratings [4]. In these

works, a set of periodic strip-current sources is used to simulate

the periodic scattered field. Recently, the method has been

extended to treat scattering from linearly periodic and doubly

periodic arrays composed of finite-size disjoint bodies [ 5 ] , 6 ] .

In applying the above approach to the problem of plane

wave scattering from a doubly periodic surface, we set up

simulated equivalences for the two homogeneous half spaces,

one on either side of the surface. In this step, the reflected

and the transmitted fields are approximated by the fields of

the respective sets of doubly periodic and properly modulated

fictitious patch sources satisfying the Floquet periodicity con-

dition [7]. Each of the doubly periodic patch sources lies in

a plane parallel to the plane spanned by the directions of the

periodicity. They all are characterized by a common Fourier-

transfomlable current density profile, which is multiplied by

a unit direction vector and an adjustable constant amplitude

for each periodic source. The fictitious sources lie outside the

half-space in which they simulate the field at some distance

from the boundary. Usually, the centers of the fictitious sources

are placed on a surface of shape similar to that of the actual

boundary. They are assumed to radiate in an unbounded

homogeneous space filled with the same medium as that

in the half-space under consideration. Locating the sources

at some distance away from the surface allows us to use

doubly periodic patch sources with smooth current density

profile that lie in planes parallel to the plane spanned by the

0018-926)(/93 03.00 993 IEEE


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BOAG et al.: ANALYSIS OF ELECTROMAGNETIC SCATTERING

133

directions of the periodicity. This immediately facilitates the

presentation of the field produced by each doubly periodic

source by a uniformly convergent series of Floquet modes with

analytically known coefficients. Furthermore, the expansions

of the reflected and the transmitted fields in terms of Floquet

modes, known as the space-harmonic representations, are

analytically expressible in terms of the amplitudes of the

sources.

This kind of approach offers a few attractive features.

First, the intensive field calculations involved are made simple

by avoiding the need to integrate over boundary quantities.

Second, the freedom in the choice of source locations makes

it possible to fit the actual fields on the boundaries as required

using a linear combination of smooth field functions. Further-

more, because of this freedom in the choice of source location,

a new internal consistency check becomes handy. Specifically,

one can consider two source locations, each providing a

check against the other. Finally, the inaccuracies

in

such an

approximate boundary field tend to be globally correlated.

Hence, no special testing procedure aimed at averaging out

these inaccuracies is usually needed and the amplitudes of the

sources can be adjusted for the continuity of the tangential

components of the electric and magnetic fields at a selected

set of points on the boundary.

The suggested procedure is simple to implement. It has

been applied here to doubly periodic sinusoidal surfaces. The

results have been shown to satisfy the boundary condition

and the energy conservation consistency checks within a very

low error. Perfectly conducting surfaces have also been treated

within the general procedure as a reduced case. The accuracy

of the proposed method has been further verified by comparing

the reflection efficiencies computed by this method for a

doubly periodic conducting surface of low corrugation with

the results obtained based on an analytic approximation.

The paper is organized in the following manner. The prob-

lem under study is specified in Section 11. The solution is

formulated in Section 111. Results of several numerical simu-

lations are presented

in

Section IV. Finally, a few conclusions

summarize the paper.

11. PROBLEMPECIFICATION

Considered is the problem of electromagnetic scattering

from a doubly periodic boundary surface of arbitrary smooth

shape separating two contrasting media. The periodicity of the

surface is described by two lattice vectors, designated dl and

dz ,

lying in the

xy

plane. The two lattice vectors are aligned

with the two directions of periodicity and their magnitudes

are specified by the respective periods. It is assumed that

the boundary surface is confined between the z = -h and

the z

= h

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 media above and below the boundary are

characterized by the constitdtive parameters

( P I ,

1)and(pr1,

E''),

respectively. For future convenience, we refer to the

upper region in Fig. 1as region I the lower region in Fig. 1

as region 11, nd to the periodic boundary surface between

these regions as S.

A plane wave given by ELnce-Jk""'.r's .ncident

on the

surface S from region

I .

Harmonic

e J wt

time dependence

is assumed and suppressed. Here, Einc and

t

denote,

respectively, the amplitude of the incident field and the wave

vector. The Floquet theorem [7] states that the field distribution

of a periodic structure illuminated by a plane wave remains

unchanged under a translation of the observation point through

a whole period

d , , p

=

1 , 2 .

Its amplitude, however, is multi-

plied by a complex constant

e - J k ' " ' . d p ,

which corresponds

to the variation of the incident field with this translation.

Our objective in general is to determine the field (E', H')

scattered by the boundary into region I and to determine

the field (E", HI') transmitted into region I I . These fields

should be source-free solutions of Maxwell s equations in their

respective regions. They should also satisfy the continuity

conditions across the boundary surface and obey both the

Floquet periodicity conditions and the radiation condition at

IzI --t

cc

111. FORMULATION

Applying the basic strategy of [11-[6] to the problem of this

paper, we set up two situations equivalent to the original ones

in

regions

I

and I I . In these equivalences, shown in Figs.

2

and 3, the scattered field

(E ' ,H ' )

in region

I

and the total

field (E", HI') in region I I are approximated by the fields of

respective sets of fictitious doubly periodic patch sources. For

conciseness, we use the character

a

to indicate parameters

associated with region a ,a = I ,11. n the equivalence for

region cy, a set of N , source points r ia, = 1 , 2 , . N e ,

is defined on a mathematical surface lying outside region

cy.

At each source point rg there are two sources of orthogonal

polarization, defined by two unit vectors

q = 1,2,

and adjustable constant amplitudes

I, ;.

The magnetic current

density of the ith source of the qth polarization,

JGpi,

entered

at a source point

rg

is described by

in which

J & ( r )

is a scalar function describing the spatial dis-

tribution of the sources simulating the field in the equivalence

for region

a.

In both of the equivalences

(cy

=

I

and

a = 11)

we use spatially periodic and properly modulated fictitious

patch-current sources of cross-polarization. This choice of

sources ensures that in region cy the simulated field (E",H " )

automatically obeys the Floquet periodicity conditions. The

patch currents are parallel to the zy plane and they have

smooth current profile. The use of these patch-current sources

in

the present problem is preferred to the use of elemental

dipoles. This is because the Floquet modal representation of

the fields arising from the patch currents converges every-

where. In contrast, the corresponding modal representation of

the fields from elemental dipoles does not converge in the

planes of the sources. The patch sources are of dimensions

s1 and

s2

in the directions of the reciprocal lattice vectors

I E ~= 27ri x

d2/ldl

x

d2I

and I E ~= 27~2 dl/ldl x

d21

respectively. It is assumed that

s 1

and sp are sufficiently

ry S . The current density of the periodic patch current is

small to ensure

that

the p tches

do nul

c u c

across th=

b und


3/7

~

734

Unbounded Homogeneous Space PI.l)

Mathematical Boundaty

S

( E i n C + E ' , H i " + H ' ) /

IEEE

TRANSACTIONS ON ANTENNAS AND PROPAGATION,

VOL. 41,

NO. 6, JUNE

1993

where

& " ( r )= 1 VQL r)

(44

Doubly-PeriodcPatch-Sources

Fig.

2.

Simulated equivalence for region I

Unbounded HomogeneousSpace PI[E l l )

Doubly-Periodic Patch-Sources

-

MathematicalBoundaty

S

(E" ,H" )

Fig. 3. Simulated equivalence for region 11

described by

Jg(r)= b ( z ) e - j k ~

r f ( t P n / s p ) ,

a = I 11

2 3 0

P n C

p = l n= 00

( 2 )

with pn =

( r -

nd,) K ~ / K . ~ .ere, 6 .) denotes the Dirac

delta function. The function f

.)

in ( 2 ) s a real-valued window

function of unit width characterized by a continuous profile

which is zero for all values of argument outside the interval

(-1/2, 1/2) and of piecewise continuous derivative on that

interval. Under these conditions, f( 0 or z < 0, respectively.

The coefficients C,l appearing in

( 5 )

are the Fourier series

coefficients of the current density profile in the two reciprocal

lattice directions. They are given by

Cpl = 2

1

f ( t / s B ) e j l K p F d [ ,

5 P P

p =

1 ,2 ; E

Z

- I 2

6)

Clearly, it is preferable to have a function f characterized by

Fourier series coefficients that decline rapidly in magnitude.

This will speed up the convergence of the double series in (5)

and thereby render the field calculations less computationally

intensive. A specific choice for f .) hat has been used in our

numerical solution is

f([)

=

0.35875+ 0.48829 COS ( 2 ~ t )

0.14128 cos ( 4 ~ < ) 0.01168

cos

(67r


4/7

BOAG et

al . :

ANALYSIS OF ELECTROMAGNETIC SCA'ITERING

135

V

Once the two simulated equivalent situations are set up, the

boundary conditions should be applied in some simple yet

adequate sense. Here, the boundary conditions are imposed at

M =

N I

=

N I I

selected points on S within the unit cell. The

result is a matrix equation which can be subsequently solved

for the complex amplitudes {I;}. Then, approximate values

for the fields anywhere in space can be readily computed. This

formally completes the solution procedure.

An alternative representation for the scattered field

E'

in

terms of Floquet modes, valid for observation points r in the

z h half-space, is obtained by

using the inequalities

z

< zf Vz and z > z '

V i ,

espectively.

The result is

- 0:

0 0 0 0

where

In (9) and (lo), the upper signs and a = I1 are used for the

transmitted field

E+

in the

z

>

h

half-space while the lower

signs and

a

= I correspond to the reflected field

E-

in the z 0 and

df'


5/7

736

IEEE

TRANSACTIONS ON ANTENNAS AND PROPAGATION,

VOL.

41 NO. 6, JUNE 1993

TABLE

I

EFFICIENCIESA,, /

Pint, POWER ONSERVATION

RROR P ,

AND BOUNDARYONDITION

RRORS

a X ( A ) AND

INCIDENT

T

VARIOUS NGLES

P n C N

THE Fnc

0

PLANE

m a x ( A H b c )

FOR THE CASE

OF A m - P O L A R I Z E D

PLANE

WAVE

g O j

00

__

0.0185

0.0104

0.0090

0.1091

3.6350

74.2871

5.3867

0.1647

3.6350

0.1091

0.0090

0.0104

0.0185

20

0.0056

0.001 8

0.0255

0.0681

0.0009

0.1562

3.2584

74.7177

6.2008

0.1738

3.0550

0 1551

0.0142

0.0485

40

0.0004

0.0023

0.0006

0.0034

0.0502

0.0062

0.3387

3.1808

69.5950

7.2062

0.2416

6.6736

0.2812

60

0.0004

0.0019

0.0002

0.0159

0.0601

0.0129

0.801

3.0010

64.2109

11.6033

0.1815

5.6482

0.1363

n

-3

-3

-2

2

-2

1

1

1

0

0

0

1

1

1

2

2

-2

2

1

1

0

0

1

1

-

fl

0

f

fl

0

f

fl

0

0

1

f

0

*1

f

0

l

0.1

0.2 0 3

0.4 0.5

l

0

*1

0

0

1

0

*1

0.0194

0.1912

0.3605

1.1499

0.9964

0.6255

0.0196

0.1637

0.1865

0.8031

0.1953

0.0761

0.4970

1.1

598

1.3177

0.9222

Fig. 5.

Plots

o f m a x ( A E b , ) , m a x ( A H b , ) , an d

A P

versus s = SI = s2

for the case of Fig. 4obtained with

N

= 100 and d , = 1.125h.

0.2933

1.1335

1.2139

1.0595

1.1335

0.2933

0.0004

0.60

__

P

[h]

0.0016

3.13

2.34

0.0067

2.06

4 0022

6.13

ax AEb,) [A]

15:

-30:

-45:

0.804 3.46 5.49

The w ave is incident upon a dou bly sinusoidal surface define by

(11)

with

periods d l =

d:

=

d

=

1.5X and roughness amplitude h

=

0.2X bounding

a dielectric half-space

of

permittivity 11 =

3i1

obtained with N = 100 ,

d , = 1 .125 h, and

SI

= ~2 =

0 . 4 5 d .

a

\

complete backscatter pattern versus angle can easily

be

done

on conventional computers.

In the remainder of this section, we compare the results

of our numerical solution with those obtained on the basis

of an approximate analytic solution. This comparison pro-

vides an independent external check on the accuracy of the

numerical solution. A first-order perturbation analysis of the

scattering from a doubly periodic perfectly conducting surface

is presented in the Appendix. This first-order approximation

is expected to be accurate provided that (a) the surface is

smooth, (b) the maximum roughness is small compared with

the wavelength, and (c) the maximum slope is small compared

with unity. This approximation might not

be

good enough,

though, when a grazing mode is excited since a grazing mode

amplitude may not be negligible compared with that of the

incident wave. We consider the case of the normal incidence

of an 2-polarized plane wave on a perfectly conducting doubly

sinusoidal surface with orthogonal lattice of periods d l =

d 2 = 1.5X. Fig. 7 shows plots of the reflection efficiency

P&JPinc

versus

h

obtained by the method of this paper

(solid line) and by the approximate perturbation analysis of

the Appendix (dashed line). The former is computed based on

- 6 O k

10

25 40 55

70

85 100

N

Fig. 6.

Plots

of m a x ( A E b , ) , m a x ( A H b , ) , and

A P

versus N for the case

of

Fig. 4obtained with

SI =

s2

= 0 .45d

and

d , =

1 . 1 2 5 h .

The decay of the errors with increasing number of sources

clearly demonstrates the fast convergence of the procedure.

Table

I

presents the efficiencies Pmfn/Pincn the various

spectral orders

for

several incident angles. Here,

P;

and

PA

are the reflected and transmitted mnth Floquet mode

power flows per unit area in the negative and positive z

directions, respectively. Pinc s the incident power flow per

unit area in the positive

z

direction. Also shown are the power

conservation error A P and the maximum boundary condition

errors max(AEbc) and max(AHbc). The computation time

is quite reasonable. An analysis of the scattering for a single

angle of incidence required about 800 s

of

CPU time. Hence, a


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BOAG et al.: ANALYSIS OF ELECTROMAGNETIC SCATTERING 737

60

:

45

:

-

E

I

0.00 0 05 0 10

0.15

0.20

Fig. 7. Reflection efficiency P ~ l o / P 1 n cersus h/X for the case

of

normal

incidence

of

an z-polarized plane wave upon a perfectly conducting doubly

sinusoidal surface given by 1 1 ) with periods d l =

d l

= 1.5X computed

by the current model method (solid line) and by the first-order perturbation

method (dashed line).

the Floquet mode amplitudes given by (10) while the latter is

based on the approximate Floquet mode amplitudes calculated

using A12). Note that for

h

doubly periodic surfaces

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