A 11 ID 2 b7TD33

NATL INST OF STANDARDS & TECH R.I.C.

A1 11 02679033

Marx, Egon/Electromagnetic scattering bv QC100 JU5753 NO.1236 1987 V198 C.T NBS-P NBS TECHNICAL NOTE 1236

U.S. DEPARTMENT OF COMMERCE/National Bureau of Standards

Electromagnetic Scattering by a

Thick Strip on a Half-Space

Egon Marx

QC

100

.U5753

No. 1236

1987

C2
TM he National Bureau of Standards 1 was established by an act of Congress on March 3, 1901. The Bureau's overall ^B goal is to strengthen and advance the nation's science and technology and facilitate their effective application for public benefit. To this end, the Bureau conducts research to assure international competitiveness and leadership of U.S.

industry, science arid technology. NBS work involves development and transfer of measurements, standards and related science and technology, in support of continually improving U.S. productivity, product quality and reliability, innovation

and underlying science and engineering. The Bureau's technical work is performed by the National Measurement

Laboratory, the National Engineering Laboratory, the Institute for Computer Sciences and Technology, and the Institute for Materials Science and Engineering.

The National Measurement Laboratory

Provides the national system of physical and chemical measurement;

coordinates the system with measurement systems of other nations and

furnishes essential services leading to accurate and uniform physical and

chemical measurement throughout the Nation's scientific community,

industry, and commerce; provides advisory and research services to other Government agencies; conducts physical and chemical research; develops,

produces, and distributes Standard Reference Materials; provides

calibration services; and manages the National Standard Reference Data

System. The Laboratory consists of the following centers:

• Basic Standards2 • Radiation Research

• Chemical Physics

• Analytical Chemistry

The National Engineering Laboratory

Provides technology and technical services to the public and private sectors

to address national needs and to solve national problems; conducts research

in engineering and applied science in support of these efforts; builds and

maintains competence in the necessary disciplines required to carry out this research and technical service; develops engineering data and measurement

capabilities; provides engineering measurement traceability services; develops test methods and proposes engineering standards and code

changes; develops and proposes new engineering practices; and develops

and improves mechanisms to transfer results of its research to the ultimate

user. The Laboratory consists of the following centers:

Applied Mathematics

Electronics and Electrical Engineering2 Manufacturing Engineering

Building Technology

Fire Research

Chemical Engineering3

The Institute for Computer Sciences and Technology

Conducts research and provides scientific and technical services to aid Federal agencies in the selection, acquisition, application, and use of

computer technology to improve effectiveness and economy in Government

operations in accordance with Public Law 89-306 (40 U.S.C. 759),

relevant Executive Orders, and other directives; carries out this mission by

managing the Federal Information Processing Standards Program,

developing Federal ADP standards guidelines, and managing Federal

participation in ADP voluntary standardization activities; provides scientific and technological advisory services and assistance to Federal agencies; and

provides the technical foundation for computer-related policies of the Federal Government. The Institute consists of the following divisions:

Information Systems Engineering

Systems and Software

Technology

Computer Security

Systems and Network

Architecture

Advanced Computer Systems

The Institute for Materials Science and Engineering

Conducts research and provides measurements, data, standards, reference

materials, quantitative understanding and other technical information

fundamental to the processing, structure, properties and performance of

materials; addresses the scientific basis for new advanced materials

technologies; plans research around cross-cutting scientific themes such as nondestructive evaluation and phase diagram development; oversees Bureau-wide technical programs in nuclear reactor radiation research and

nondestructive evaluation; and broadly disseminates generic technical information resulting from its programs. The Institute consists of the following Divisions:

• Ceramics

• Fracture and Deformation3 • Polymers

• Metallurgy Reactor Radiation

'Headquarters and Laboratories at Gaithersburg, MD, unless otherwise noted; mailing address

Gaithersburg, MD 20899.

2Some divisions within the center are located at Boulder, CO 80303.

'Located at Boulder, CO, with some elements at Gaithersburg, MD
Research information Center

JNationai t>ureau ot Standards .^NBS Technical Note 1236 Gaithersburg, Maryland 20899 ——

N ^sc

Ucioo

Electromagnetic no. ;* 2t

Scattering by a

Thick Strip on a

Half-Space

Egon Marx

Center for Manufacturing Engineering

National Engineering Laboratory

National Bureau of Standards

Gaithersburg, MD 20899

June 1987

rf*1

""»£AU Of *

U.S. Department of Commerce

Malcolm Baldrige, Secretary

National Bureau of Standards

Ernest Arnbler, Director
National Bureau of Standards Technical Note 1 236

Natl. Bur. Stand. (U.S.), Tech. Note 1236, 70 pages (June 1987)

CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1987

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington DC 20402
CONTENTS

Page

1. Introduction ..... . 2

2. Fields in the presence of cylindrical boundaries ... 4

2.1 Reduction to a scalar problem ..... 6

2.2 Solution of the two—dimensional Helmholtz equation 9

3. The homogeneous -fields 12

3.1 The perfect conductor in the TE mode 13

3.2 The perfect conductor in the TM mode 14

3.3 The dielectric general case ........... 15

3.4 The strip of a different material in the TE mode . 17

4. Scattering by a perfectly conducting strip on a

perfectly conducting half-space .... . . 19

4.1 The TE mode 19

4.2 The TM mode 23

4.3 The general case 25

5. Scattering by a strip on a half-space, both of the

same homogeneous material ...... ... 26

5. 1 The TE mode 28

5.2 The TM mode 30

5.3 The general case 31

5.4 Medium with a complex dielectric constant .... 34

6. Scattering by a homogeneous strip on a half—space of a different material 35

6.1 Two semi—inf ini te homogeneous regions separated by

a layer of finite thickness .... 36

ill
6.2 The homogeneous strip on a substrate of a

di-f-Ferent material 42

7. Interior resonances and sharp edges . 52

7.

1

Interior resonances ...... 53

7.2 Effects of sharp edges ....... 54

8. Computer programs ..... ..... 56

9. Concluding remarks ... ......... 61

10. Acknowledgements 61

11. References ............ .. 64

IV
ELECTROMAGNETIC SCATTERING BY A THICK STRIP ON A HALF-SPACE

Eg on Marx

National Engineering Laboratory

National Bureau of Standards

Gaithersburg, MD 20899

The problem of the determination of the scattered fields

from an incident plane monochromatic electromagnetic wave on a

strip of finite thickness on a half—space is reduced to the

numerical solution of integral equations for auxiliary fields

defined on the interfaces. These fields are chosen so as to

minimize their number. The derivation of the integral equations

is given for a perfectly conducting strip on a perfectly

conducting half—space, for a dielectric (or other homogeneous

material) on a dielectric half-space, and for a dielectric strip

on a half—space of a different dielectric material. The vector

FORTRAN programs written to carry out these calculations Are

briefly described and sample outputs a.r& shown.

Key words: dielectric strip; electromagnetic scattering;

integral equations; numerical solutions; resonance

region; perfect conductors.
1 . Introduction

The light scattered by a strip located on a substrate of the

same or of a different material can be used to determine the size

of the strip CI 3. Accurate linewidth measurements require a

precise evaluation of the scattered fields, especially in the

resonance region where the size of the strip is comparable to the

wavelength of the light used to do the measurement.

We present a method to determine numerically the

electromagnetic fields scattered by such an infinite strip of

finite cross section on an infinite half—space. This method is

exact in that it gives the fields with an accuracy limited only

by numerical considerations. The incident field is assumed to be

a plane monochromatic wave propagating through a medium with a

real dielectric constant. We consider different combinations of

materials for the strip and the region under the plane. Both

these regions can be perfectly conducting, both can be

homogeneous dielectrics possibly with complex dielectric

constants (to represent lossy media or conductors with finite

conductivities), or the strip and the lower half—space can be

different homogeneous media.

Because each region is filled with a homogeneous material,

the electromagnetic fields in the different regions of space can

be expressed in terms of surface fields defined on the

boundaries. These surface fields obey integral equations derived

from Maxwell's equations and the boundary conditions. The number

of these fields is kept as small as possible by a careful choice

of the jump conditions at the boundaries C2, 33-
The integral equations can be solved numerically by a pointmatching method, and we have written programs that do this for

the different problems. The determination of the fields

scattered by perfect conductors can be reduced to the solution of

integral equations of the second kind, and those for dielectrics

lead to integral equations of the first kind, but we found no

problems with i 1 1—posedness.

In Section 2 we recall the reduction of Maxwell's equations

to a pair of scalar Helmholtz equations for problems with

translational invariance in the z—direction and an incident plane

monochromatic wave. We then sketch the solution of the Helmholtz

equation by means of the Green function that satisfies an

outgoing wave condition (OWC) = In Section 3 we provide details

about the determination of the incident, reflected, and refracted

(if any) fields in the absence of the strip for the different

problems discussed in this paper. In Section 4 we show how we

reduce the problem of scattering by a perfectly conducting strip

on a perfectly conducting half—space to the solution of a single

integral equation for a function of two variables defined on the

boundary between the media both for the TE and TM modes. In

Section 5 we reduce the determination of the fields scattered by

a dielectric strip on a half-space of the same dielectric to the

solution of a system of coupled integral equations for four

surface fields for arbitrary direction of incidence and

polarization. In Section 6 we solve a similar problem with two

different dielectrics for the TE mode only, and we obtain a system of four integral equations. In section 7 we give a brief

.s
discussion of possible difficulties with the divergence of the

fields at sharp edges and with spurious numerical instabilities

in the integral equations at resonant frequencies of the interior

problem. In Section 8 we describe the computer programs used to

find the scattered fields, and we show some of the graphic

results in Section 9.

2» Fi.e^Lds 1Q the presence of cylindrical^ boundaries

The coordinate axes ^re shown in figure 1. We choose the z—

axis in the plane separating the two media away from the strip

and locate i t at the center of the strip parallel to the edges of

the strip. We choose the y—axis perpendicular to this plane

pointing out of the perfect conductor or other substrate.

The boundary between the media has a cross section that can

be given by an equation of the form

y = f (x)

.

(2. 1)

A slightly more general representation of this curve is the set

of parametric equations

x = f (s) , y = f ^<s)

,

(2.2)

where s is the arclength, This form allows for curves where

there may be more than one value of y for a given x. When the incident wave is plane and monochromatic, the field

has a periodicity in the z-direction that is maintained by the

geometry and appears also in all other fields. The Maxwell

equations for the fields can be reduced to scalar Helmholtz
v.

Figure 1. Geometric configuration for the scattering o-f a wave

by a perfectly conducting strip on a perfectly conducting halfspace. In the region V we show long arrows that represent the

directions of propagation of the incident and reflected waves

and short ones that represent the scattered wave.
equations in the >;y-plane for the z-components of the -fields C4D,

and this reduction is shown in Subsection 2.1. We sketch the

solution o-f the two-dimensional Helmholtz equation in Subsection

2.2 by the methods in L21 and C33.

2. 1 Eduction to a scalar problem

An incident plane monochromatic wave o-f -frequency to with a

wave vector k of magnitude k = io/v, where v is the speed of light

in the medium, can be described by its electric field, which is

given at the point x and at the time t by the real part of the

complex vector

|

in (x,t> = l expLi(k-x - wt)3. (2.3)

The symmetry of the geometrical configuration in the z-direction

causes the z—dependence of all fields to be the same as that of

the incident field (2.3), that is, exp(ik^z). We henceforth .ji assume that this is the z—dependence of the fields, and that this

exponential is multiplied by functions of x and y only. Vectors

such as E, H, k, and y can be decomposed into a longitudinal part

along the z-axis and a perpendicular part. Then the fields E and

H can be written in the form

-» -+ -""- -± E(x,t) = CE_r (x,y)e_r + E. (x , y) 3expCi (k_z - wt)3,
H(x,t) = ChU(x,y)e_ + H, (x,y> 3expCi <kTz - cot) 1. <2,4)

.> -j ± --

It is then straightforward to show that the perpendicular

components of the fields E and H that obey the homogeneous

Maxwell equations,

$ x E = iwjifis 7xH = — iw«E, (2.5)

can be expressed in terms of the longitudinal fields E_, and H-, by

C4D

p. 1WM _ .. 3_, _ —, ^ v i = - -^e3

x Vj H + -2V ±E

3

, .2.6)

k k l

ik^

H = —^e, x v.E, + —-y h , (2.7)

k k

±

The fields E, and H-, satisfy the two—dimensional Helmholtz

equation

(V^ + k^)E, = 0, <vf + k^)l-U = 0. (2.8) 1 ±3 1 _L o>

The frequency w and the component k-, of the propagation vector of

the incident wave are real, but the permittivity e and the

perpendicular part of the propagation vector k. can be complex;

the permeability ju is usually assumed to be equal to u _ in free
space, but we make use of this assumption only occasionally when

we wish to simplify the notation. The magnitude of the

propagation vector and that of its perpendicular part are given

by

2 2 2 2 ** k = £{iu , k. = k - k_. (2.9)

The components of the perpendicular fields tangential to the

cross section (2.2) are then

E±t = 1rlaTr + -in*- (2 - 10)

k ik

±

3E, ik^ 3H, hoc -^ -J- -3 H.. = - ^

t—^ + £

- , <2.11) It . 2 3n , 2 ds

kl

kl

in terms of the normal and tangential derivatives of E_, and H_j. on

the curve, where the normal unit vector n and the tangential unit

vector t are related by n = e_j. x t, as seen in figure 1. There

is no general agreement in the literature about the definition of

the terms TE and TM; here we define the TE mode by the condition

H_, = and the TM mode by E-, = 0. Unless the scatterer is a

perfect conductor, we add the condition k_ = to the definition

of these particular modes.

For the incident wave (2.3), we have

A = k x I /««, (2. 12)

8
: = («u/k 2 )(- fc 2

S

1 + k

1 e

2 )H03 - <k 3 /k^)(k

1 e

1 + k 2

;

2 )E03 , (2.13)

a

0±

= ^/k^Kk^ - k

1 e

2

)E03

- (k 3 /k2)( ki S

i + k 2e

2 )H03 . (2.14)

The Poynting vector obtained from the -fields (2.4), (2.6),

and (2.7) is equal to half the real part of

I x H* = k~4

[u2|i* (V±

E*) x v ±

H^ + k 3«M j

v

1H

3 | 2e^

k,«« |v.E_| 2

e, + k?(7.E^) x v . H*l + k,

2

(- iwnH*7,hU

:> ± 3 1 :> o ± 3 1 -3 J 1 \ 3 1 O"

+ ik_H*v,E_ x e_ + iweE,v,E* + ik^E^v.H* x e„| . (2.15) 3 o J. c> o o. 1 3 o- 3 ± 3 3 /

The terms in the square brackets all point in the z —direction,

and they vanish in the TE and TM modes if k_, = 0.

2.2 Solution of the two-dimensiona^ He^rnhgitz eguation

We now find the solution of the Helmholtz equation (2.8) in

terms of the corresponding Green function and the jumps of the

function and its normal derivative across an arbitrary curve C.

The functions in this section a.r& restricted to the xy—plane and

we drop the subindex ±.

We have to find a function U that satisfies the homogeneous
Helmholtz equation in the xy-plane outside the curve C, that has

a j ump

AU = U+

- U_ = (2. 16)

across C, whose normal derivative has a jump

A (3U\ (3U\ {dU\

<2. 17)

across C, and that satisfies the OWC everywhere at infinity. The

curve C is parametrized by the arclength s and has a unit normal

n(s) = e_, x t(s), which points in the direction of the region

described by the subindex +,

The distribution C2H U then satisfies the equation

<V k

2

>U = a(|^)s(C) + y CnAUS (C) 3, (2.18)

where a distribution of the form f(s)S(C) is defined by its

operation on a test function a,

<f (s)« <C) a a(x f y) > = I dsf (s)aCx <s) ,y(s) 3. (2.19)

The solution of (2. 18) is given by the convolution of the right

side with the Green function & for the Helmholtz equation,

LI = - &*-n£ (C) - (38/an)**5(C), (2.20)

where & satisfies

(V

2 + k

2 )&(x,y) = - ff(x)ff(y). (2.21)

The distribution & is given by

10
S(x,y) = ^ H^ 1} (kp), p = (x

2

+ y

2

) 1/2

, (2.22)

where H is a Hankel function of the first kind. These

functions are defined in terms of the Bessel and Neumann

functions by H = J + iY . The solution (2.20) can be

n n n rewritten in terms of functionais of the jumps in the form

U = GW + f\Ktf>, (2.23)

where

C-n3 = -

Jj d5'n(x')Hj (y), (2.24)

if * ( 1

)

£ ds' A Cx' )H , 4jC 1 1 ±

NC*J = 7| nds'#(x')H; (k,R)k,n'-R, (2.25)

and ft = x - x', R = C (x - x

' ) 2

+ (y - y') 2

] 1/2

, R = ft/R.

The value of the function L) at either side of C is

U+ = ± | + uL, (2.26)

and, if the function is continuous across C, the value of the

normal derivative on either side of C is

ii
3. The homogeneous fields

The known functions in the integral equations are

combinations of fields that obey the homogeneous Maxwell

equations evaluated on the boundaries. They are the incident

field and the reflected and refracted fields that would be

produced at the plane interface in the absence of the strip.

These fields satisfy the correct boundary conditions on the xz —

plane and they are then assumed to be defined throughout all of

space.

For the perfect conductor, the TE and TM modes do not mix

even when k is not perpendicular to the z—axis. We give the

homogeneous fields needed in the solution of the scattering

problem for the TE mode in Subsection 3.1 and for the TM mode in

3.2. The most general case is obtained by superposition. The

situation is more complicated for the dielectric scatterer, and

we give the formulas for the most general case in Subsection 3.3.

In Subsection 3.4 we give the formulas needed for the case of the

strip of a different dielectric in the TE mode only.

For the perfect conductor, the reflected field is obtained

from the incident field by retaining the normal component of E

and the tangential component of H unchanged and changing the sign

of the tangential component of E and the normal component of H.

For the dielectric, the reflected and refracted fields are

obtained from the continuity of the tangential components of E

and H.

12
3. 1 The perfect conductor in the TE mode

The incident -field is assumed to be a plane monochromatic

wave described by <2.3). In terms of the z-component of the

amplitude of the incident electric field, ^-at' ^ ne amplitudes of

the perpendicular components of the incident electric and

magnetic fields are

E

01

= " (k

l k 3

/kl)E03' E02 = ~ (k2

k3

/kl

>E03> H

01

= <W* k2

/k±)E03'

HQ2

= - (M6k

1 /kpE03 . (3.1)

The reflected wave is given by

ire-f 1 + i -* -* E (x) = E"exp<ik"-x) , (3.2)

where

k " = k

i ei " k 2

e

2

+ k y*y.> (3 " 3)

F " = — F F // =F F " = — F C\ A) 01 01' 02 02' 03 03" u-^J

Consequently, the z—component of the total homogeneous field

outside the conductor is

cr 1 / \ rin r-ref 1 E_(x,y) = E_ + E_.

13
= E^expEi (k.x + k^y) 1 + E" expCi (k?x + k"y ) 3

= 2iE ^exp (i k x ) sin (k y)

(3.5)

This -Field, evaluated on the boundary C, appears in the

corresponding integral equation.

The perfect conductor iv\ the TM mode

The homogeneous field H^ now has to be expressed in terms of

the z-component of the incident magnetic field, H . The

perpendicular components of the incident fields are

H

Q1 = - (k

1 k3/k2)H03 , HQ2 = - (k2

k3 /k2)H03 , E

Q1 = - ( WMk2 /k^)H03 ,

EQ2 = (MMk

1 /k1

)H ^; . (3.6)

The components of the reflected field amplitude are

H

01

= H01' H02=

~ H

02' H03

= H03' <3 ' 7)

whence

i "03 K 1 2 H_(x,y) ; = 2HrtTexp ( i k , x ) cos (

y >

,

(3.8)

14
? ±H*(x,y) = 2HQ3eKp(ik

1 x) Cik1 cos(k2y)e

1 - k 2sin (k 2

y) e^Il , (3.9)

from which we -find 3H_j./5n on C to use in the integral equation.

3H^(x,y)/dn = 2H exp(ik x)[- k n sin (k_y) + ik n.cos(k„y)].

(3. 10)

3.3 The dielectric general case

We assume that the incident wave is characterized by an

arbitrary wave vector k (with a negative k^) and an electric

field amplitude E perpendicular to k. The plane inter-face

produces a re-Fracted wave in medium 2 characterized by k" and E'

,

and a reflected wave in medium 1 characterized by k" and E".

The frequency of the fields is the same in the different

media, whence the magnitudes of the wave vectors a^re related by

2 2

k' /e^u^ = k /c.ji,. (3.11)

The wave vector of the reflected field is still given by

(3.3), and that of the refracted field is

%.' = k

1 e1 - C (u 2

e 2

/M

1 e

1 )k2 - k^ - k^] 1/2

e?

+ k.^. (3.12)

The magnetic intensity H.. is given by (2.12). The incident

fields, as well as the reflected and refracted fields, ^re plane

monochromatic waves that satisfy (2.13) and (2.14).

To find the amplitudes K E'_ and E''^ that determine the 03 03
refracted and reflected fields, we could decompose the fields in

the canonical components with parallel and perpendicular

polarization and obtain the z—components of the reflected and

refracted fields from the usual formulas. We prefer to obtain

the equations for these four unknowns directly from the boundary

conditions and solve them numerically. We have

E03

+ E53

= E

0Z> (3 - 13>

H03

+ H0Z = H03' (3 ' 14)

E

Q1

+ ( WM

1 k 2

/k

± )H-3

- (k

1 k 3 /k 1 )E-3

= - (MM_k'/k, >H'_ - (k.k T /kf)E' , (3.15)

H

Q1

- <«« k /kpE&- (k

1 k 3/kpH53

(w£ 2

k2 /k^')E^3

- (k

1 k 3/k^-)H^. (3.16)

Once the z-components of the amplitudes of the reflected and

refracted fields are determined, the fields on the curve C are

obtained from them by including the appropriate exponential

factors.

The equations for the TE or the TM mode can be obtained from

(3.13) to (3.16) by setting kL _ or E._ and k_ equal to 0.

0-> 03 -3

16
3.4 The stria of a different material in the TE mode

For the TE mode, (3.1), (3.13), and (3.16) allow us to -find

the amplitudes of the reflected and refracted fields, if we

substitute €_, for <e ^ as the permittivity of the half-space on

which the wave impinges. We set k^. = and H-, = in (3.16) and

use (3.11) to obtain

k (E_ - E" ) = k^E' . (3.17)

We solve (3.13) and (3.17) and find

EQ3

= kT^tL EQ3- (3 - 18)

k — If

T

E"fc = — — E . (^19) 03 k 2

+ k£ ^3" I^.IVJ

On the boundary plane we have y = and the incident field

reduces to

E^n (x,y) = E expUkjX), (3.20)

with similar formulas for the reflected and refracted fields,

whose amplitudes are given by (3.18) and (3.19). We assume that

U = ix = ju-j- = /i . We use (3.13) and (3.17) to derive

17
r0, . r-i n / ^ ,-re-fl , . ^refr,

E *x,y) = E_j. (x,y) + E_, (x,y> - E_, (x,y)

= (EQ3 + E"3

- EJ3)exp(ik

t x) = O, (3.21)

+

,v

' i n -* ref 1 + ref r IE_(x,y)/3n = ik»nE_ <x,y) + ik"-nE, (x,y) - ik'-nE., (x,y)

= i <k_E„ - k„E" - k'E'^expdk.x) = O, (3.22) 2 Uo> 2 03 Z 0-3 1

We also have to determine

Ele'fr (x,y) = E;.,exp«ik,x), (3.23)

o U0> 1

aElefr (x,y)/3n = i k'Ei ^exp (i k . x ) . (3.24)

Off the boundary plane we have y ^ and we have to use the full

expressions of the fields to find

E.\{x,y) = E expCi(k x + k y) 3 + E" expCiCk x - k^y)3, (3.25)

5E_r (x,y)/3n = if(k 1 n 1 + k n_j) E expCi <k x + k_y) 3

18
+ <k.n. - k^n_)E" expLi (k .x - k y)3>. (3.26) 11 JL. i U-J- 1 jL

4. Scattering by a perfectly conducting stele on a perfectly

conducting h^Lfzsgace

For the scattering by a perfectly conducting strip on a

perfectly conducting half-space, the total field is composed of

the incident field, the reflected field, and the scattered field.

The configuration of the problem is shown in figure 1. The

scattered field satisfies the DWC at large distances from the

strip. For the perfect conductor, the general case for arbitrary

direction of propagation and polarization of the incident wave

can be decomposed into separate TE and T!i modes. Thus, although

TE and TM modes a.re usually defined with the incident direction

in the plane perpendicular to the generator of the strip, we

allow for a nonvanishing k_, in this Section.

4. 1 The TE mode

In the TE mode we have H^. = 0, which implies that the z —

component of the reflected magnetic intensity also vanishes,

ref 1 H„ = 0. We will show that the boundary conditions can all be

satisfied if we further assume that the z -component of the

scattered magnetic field vanishes. The z-component of the total

electric field, E_,, satisfies the Helmholtz equation.

19
<VJ + »<2 )E3

= °> (4 - 1}

and the boundary condition

E_,. = on C, (4.2)

where E-? + is the value of this tangential component of the

electric field above the boundary C of the cross section of the

conductor. Since the value of E^ is constant (zero) on C, the

vector ^,Err * s perpendicular to C, which implies, from (2.6),

that E is normal to C and the electric field does not have a

tangential component. Also, from (2.7), we see that the

perpendicular component of the magnetic field, H , is tangential

to the surface, whence the normal component of H vanishes. Thus,

all the boundary conditions on the electromagnetic field at the

surface of a perfect conductor aire satisfied.

The z-component of the scattered electric field is defined

as the difference between the total and the homogeneous electric

fields,

Ef~ = E_ - E* (4.3)

where E is the sum of the incident and reflected fields, as

defined in (3.10).

We define a function U(x,y> that is equal to the field E^

in V , is continuous across C, and satisfies the Helmholtz

equation (4.1) also in V and the OWC for y -* -co. This function

is then determined by the discontinuity of its normal derivative

20
n across C. Since U satisfies the Helmholtz equation on the

whole plane and the OWC when y * ±®, we can write it in the -form

U = G€t»>-, (4.4)

where 6 is the functional (2.24). Then the boundary condition

(4.2) leads to the singular integral equation of the first kind.

B€n'> + e! = on C, (4.5)

for the unknown function v\. This equation is analogous to the

Petit equation C23.

To solve the equation numerically, we use the point—matching

method L21 . We cover the part of the curve C where we expect the

function is to differ significantly from with N patches or line

segments, and we seek to determine the values t\ of the function

•n at the center of these patches. For this purpose, we assume

that the integrand in (4.4) is constant on each patch, and we

obtain the system of linear equations by satisfying (4.5) at the

-t points x .

,

N

* AW\n = B * = " W' * = 1,2,..., N, (4.6)

m=l

where

A. = - ^A5 H.^Nk.R. ), R A = |x fl - x |, St * m, (4.7)

J?m 4 m 1 Am Sim J? m

and the coefficient of the contribution of the self—patch is

21
A

j?J? = 2^Cl0Q< 2

k

lAS^

) "

i- 11593 ^5

^ - 4A5jT (4 - 8)

This contribution is obtained -from the small-argument

approximation of the Hankel -function H^ . It is of the order of

log das) and is not necessarily negligible compared to the N—

1

terms of the order of As.

The field E_ is given by U in (4.4), and we can use the

large—argument approximation of H ,

H* U

<£) * (n.f/2)

1/2expCi(^ - n/4)D, (4.9)

if we have to determine the far fields only. Equations (2.6) and

(2.7) with E_, = E^ and H^. = give the perpendicular part of the

scattered electric and the scattered magnetic field. We expand,

in polar coordinates,

R % /d - .o'cos(# - #') (4.10)

and obtain the far-field approximation

E* * - expCKk.0 - n/4)](8nkj)

-:> i. 1

ds'n(x')expC-ik fl'cos(# - 0')3, (4.11)

and the intensity of the scattered field is proportional to

I(*) = I I ds'n(x')expE-ik jo'cosW - «4

' ) 1

| 2
p

ds' n (>;' ) expt-i k (x ' cosi + y'sintf)3 (4,12)

The radiation pattern for a symmetric scatterer is symmetric,

that is, I(n - #) = I(#) 5 if ^ (- x',y') = n<x',y')« This

condition is satisfied for normal incidence.

We obtain a different equation if we define a function

U'(x,y) which is similar to U except that the function has a jump

$ across C and the normal derivative is continuous across C C53«

Then (2.23) reduces to

U' = biitl <4. 13)

and the boundary condition leads to the integral equation of the

second kind

# 1 (4. 14)

This equation is similar to the Pavageau equation E23, but one is

not the precise analogue of the other.

4.2 The TM mode

In the TM mode we have E_, = 0, whence E_ = 0. We assume

that the z-components of all the electric fields vanish. The zcomponent of the total magnetic field, H_,, satisfies

CV? + k?)H, = 0, (4.15)

The boundary condition on the tangential electric field (2.10) is
satisfied when

(aHT/3n) +

= on C. <4. 16)

-t Equation (4.16) also implies that the normal component o-f H. in

(2.7) vanishes, and all the boundary conditions are satisfied.

The z-component of the scattered magnetic field is given by

an expression similar to (4,3). We define an auxiliary field

U(x,y) that is equal to H-;. in V , satisfies the same Helmholtz

equation and the OWC in V , and is continuous across C. The

function U can then be expressed in terms of the discontinuity of

its normal derivative, Tf>, across C by (4.4) by means of the

functional 6 defined in (2.24). The gradient of the function U

on (C) is then expressed by

v,UL = ^Lds'n(x')H«a) (k.R)k.R. (4.17) 1|C 4JC ' 1 1 1

The boundary condition (4.16) then reduces to the singular

integral equation of the second kind

5D + N'-CfiJ + ani/Sn = on C, (4.18)

where N' is the functional

N'OijJ = |n- I ds'n(«' )H* 1} (k R)k

±

R; (4.19)

this functional differs from N in (2.25) in that the normal n at

the field point replaces the normal n' at the source point. We

find the approximate values of n at the center of the patches

from the system of linear equations (4.6), where now the

24
coefficients and constants ar<E

A-: = ^45 H« <k,R, )k.n" -<jL - x )/R„ , A # m, (4.20)

jfm 4 m 1 1 in 1 J J? m J?m'

A' = k + T~k^s,, (4.21)

Jfjf 2 4n J? J?'

b; = - dnlix .) /an. (4.22)

J? 3 JJ

The curvature k „ of the cross section of the strip at the point

x„ is positive when the curve is concave in the direction of n,

the normal defined as going from medium 2 to medium 1. The

curvature vanishes for a segment of a straight line. The selfpatch correction, which is proportional to as, can be neglected

compared to 1/2 if the curvature is small enough.

The field H_, is given by U in (4.4), the other component of

the scattered magnetic intensity can be obtained from (2.7), and

the electric field can be obtained from (2.6). The far-field

expressions for the scattered magnetic field and intensity Sire

essentially those in (4=11) and (4.12).

4.3 The general case

The linearity of Maxwell's equation and that of the boundary

conditions allow us to use superposition to find the fields

scattered by a perfectly conducting strip on a perfectly

conducting half—space when neither E_, nor H_ vanishes. The

25
incident -field is separated into TE and Tli modes based on the

in in respective values of E^. and H-;. . The scattered -far fields and

intensities have to be computed from the sums of expressions

obtained in Subsections 4=1 and 4.2.

5- Scattering by. a strip QD sL bs:i:f_Z=.Ei*£*?j- both of the same

homogeneous (Osi^Li§li

We now consider the scattering of a plane monochromatic wave

by a strip on a half-space of the same homogeneous material, as

shown in figure 2=

The fields that obey the homogeneous Maxwell eguations and

that would exist in the absence of the strip are the incident and

the reflected fields in V and the refracted field in V . We

define the scattered fields as the additional fields in both

regions that together with the homogeneous fields give the

physical total fields. The constants of the region V are £

i and

H and those in the region V,-, Are e ,_, and .u^. The magnitudes of

the corresponding two propagation constants, k and k', are given

by (2.9). The frequency w and the z-component of the propagation

vector, k_,, remain the same in both media. No current density or

charge density can exist on the surface of a dielectric, whence

the tangential components of E and H have to be continuous across

the boundary. If we have H_ = 0, the reflected and refracted

magnetic fields will have a z—component unless k^ = 0, and the

same happens for the electric field. Thus we include the

condition k_ = in the definition of the TE mode discussed in

26
/

sc

V2

6 2 , /^2

Figure 2. Geometric configuration -for the scattering of a wave

by a homogeneous strip on a half-space of the same material.

In addition to the incident, reflected, and scattered waves in

V we show the refracted and scattered waves in V^,.

27
Subsection 5.1 and the TM mode discussed in Subsection 5.2. We

consider the scattering of an incident wave with an arbitrary

direction of propagation and polarization in Subsection 5.3, and

we extend the results to a medium with a complex dielectric

constant in Subsection 5.4.

5. 1 The TE mode

We follow the procedure and notation developed in Subsection

4=1, but add an index to the operators to make clear which

constants associated with a particular medium Are used to

determine the propagation constant. Since we include the

condition k^ = in the definition of the TE mode, we can assume

-* that hU = 0, and (2.6) then implies that E. vanishes. The

-j- -L boundary condition (4.2) is replaced by

E^_ = E^+ on C, (5. 1)

and the continuity of the tangential component of the magnetic

intensity gives a second boundary condition.

<dE,/3n)_ = aidE^/dn) + on C, (5.2)

where

ct = M_/ji i. (5.3)

We now define two auxiliary functions, U and U9, The

function U is equal to the z—component of the scattered electric

field in the region V , satisfies the same Helmholtz equation in

V^ (with the constants of medium 1), is continuous across C, and

28
also satisfies the OWC when y * —ffl • Again we call i\ the

discontinuity of its normal derivative across C, and Li is then

given by (4.4) using an operator G . The function U vanishes in

1 jL V and is equal to the z-component of the scattered electric

field in V^. We compute the values of U^ and tJU^/cJn at C in the

medium 2 from the boundary conditions (5.1) and (5.2),

U„ = E*C = E, - E!I efr

= E,+

- EZ

Gir

= E2 + B.tnJ, (5.4) 2— o>— •->— 3 o>+ 3 3 1

faU^/3n)_ = 3E^/dn + a f | + NJC?i>J, (5.5)

where

in refl refr

E = E + E - E , (5.6)

3 3 3 3

-0 Lin rrefl\ _refr ,_ _. E_ = a(E_ + E^ I - E_ . (5.7)

-:> \ 3 3 / 3

The function U^ then satisfies the same Helmholts equation in V

and V? , satisfies an OWC at infinity, and the jumps in U^, and

3U^/«3n are equal to minus the values given in (5.4) and (5.5).

Consequently, U^ can be written in the form (2.23),

LL, = G^CA (3LU/3n)} + N?

{4UJ = - G^i idU^/dn) _> - N9 (U^_}. (5.8)

The integral equation for t> is then obtained by imposing the

condition that the function U_ vanish at C in V , 2 1
LU. = k*U„ + U. r = O. (5.9)

Substituting from (5.4) and (5.8) we obtain

(|g i + |G^, + «G2NJ + N^^C-nJ + |e^ + G

2^dEi/da^ + N^E^V = O

on C, (5. 10)

where the meaning of a composite operator such as N^G is

N^G^'nJ = N^CG,^}}. (5.11)

Equation (5.10) is an integral equation of the first kind and it

can be reduced to a system of linear equations to find the

approximate values of t\ at the center of the patches. The

evaluation of the functionals is reduced to matrix

multiplication, where the matrix elements are essentially those

in (4.7), (4,8), (4.20), and (4.21). The products of the

operators in (5.11) corresponds to the matrix product, and (5.10)

-0 also involves functionals of the known functions EZ. and EJ^.

o> -.> The scattered electric field is then obtained from t\ by

means of (4.4) in V and by means of (5.4), (5.5), and (5.S) in

V^, with the corresponding expressions for the far fields.

5.2 The TM mode

We now assume that E^ = 0, which together with k_ =

•j -3

-t 5C implies that H, = from (2.7). The determination of H_ follows

sc

a path similar to the determination of ET in the previous

30
subsection if we interchange E and H. The boundary conditions

are

H, = H_ on C, (5. 12)

(3H,/3n) = «(dH_/3n) on C, (5.13) .\ — o +

where

a = e^/€ . (5, 14)

The definitions of U and U^, are the same as before, and the

integral equation for n has the same form as (5. 10) , with a defined by (5=14) instead of (5.3) and the homogeneous magnetic

fields replacing the corresponding electric fields.

5.3 The general case

If the propagation vector of the incident wave is

perpendicular to the strip and neither H-, nor E_, vanishes, we can

use superposition of the fields discussed in Subsections 5.1 and

5.2 to find the scattered fields.

If the propagation vector is not perpendicular to the strip,

the z-components of the reflected, refracted, and scattered

fields do not vanish even when E_ or H^ vanishes. Thus, there

is no point in defining separate TE or TM modes for an arbitrary

direction of propagation of the incident wave.

The tangential fields are continuous across the interface on

C and (5.1) and (5.12) are still valid. The continuity of E_

31
and H^ on C implies the continuity of the tangential derivatives

SE^/cJs and clH^/ds, and we use the continuity of the components

of the perpendicular fields tangential to C, given by (2.10) and

(2.11), to replace (5.2) and (5.13) with

(3E,/3n)_ = aOE^/dn) +

+ £ (<9hU/3s) , (5.15)

(3HT /an)_ = a' OH-r/dn) + + $' idE^/ds) , (5.16)

where

r> ^

a = € k' /t 2k 1

, a' = Mjk^ /M,,k, , (5.17)

£ = - <k,/w*_) <k' /kt - 1), £' = ik^/iou^) (k' /kt -1). (5.18)

O .£ J. JL -3 j£ J. J.

To reduce Maxwell's equations to integral equations for

surface fields we define two sets of two auxiliary fields and one

surface field. We define U , Li—, and rt in the same way as in

Subsection 5.1, and U' , U', and t\' as the corresponding functions

for the magnetic intensity, as in Subsection 5.2. We then have

two equations of the form (4.4) for LL and U' . The jumps in U^

and U^ are still determined by (5.4) and its analogue for the

magnetic field. The jumps in t?U /<9n and <3U^/t?n are determined

from (5.15) and (5.16), which lead to

32
'3U^\

an

—3

an

+ a 2

+ H

i i,Ay

( du: + N" £»>'} (5. 19)

fau^'

an~

dH2

an

+ oc i- + N

i<*' } ^'fji

1 * NJft>), (5. 20)

where N" is the -functional

N"{7\} = |t- | ds'n(x' )H^ 1} (k R)k R, (5.21)

E_ and l-C Are de-fined as in (5.7) with the appropriate values of

o» 3

a and a' , and

si = Ein + E:

efl

, h£ = Hin + H*: Gfl

Oi Oi Oi 3 -j- 3 (5.22)

We impose the conditions that U^ and U;L, vanish in V^ at the

boundary C by equations like (5.9)? we obtain two integral

equations for n and -n',

(iGi + i

G

2

+ aG2

N

i + Vi) tlJ fiG2Nfin'y

|e^ + G

2

-/aE^/an + #aH*/asV + im^/e^V = o. (5.23)

•JO
0'GJi^W + (|g i + |M32

+ ff 'G2NJ + N^jJtu'J

+ |h^ + G^aH^/dn + 0'aE*/3sV + N

2{

HS} = °- (5.24)

Once n and v\ ' are computed from (5.23) and (5.24), the

scattered fields can be obtained by integration from U , LI' , U^,

and UL, from equations similar to (4.4) and (5.8). The radiated

energy is obtained from the Poynting vector determined by (2.15)

where we introduce the far fields in terms of the approximation

(4.11). Then only the first factor on the right side of (4.11)

contributes to the gradient term, so that we may write

symbolically that y £ ik.fi when the gradient is applied to a

field, and minus that if it is applied to the complex conjugate

of a field. Equation (2.15) then reduces to

§ = -Re ilk x A*) * oidhrtfikp' 1 Lei (tf) + ul ' <tf) 3 <k_,e^+ k , 2) , (5.25)

where 1(0) is given by (4.12) and I' (tf) is the corresponding

quantity in which t\ has been replaced by y\' .

5.4 Medium wi^th a complex d^el^ectric constant

A lossy medium, such as a conductor with a finite

conductivity, can often be represented as a dielectric medium

with a complex dielectric constant which is a function of

frequency. For instance, for a conductor with conductivity <r we

34
may set

6 = € + iff/w. (5,26)

r

The complex propagation constant in the lossy medium is obtained

from

k

2

= k

2

+ k

2

+ k

2

= k? + k

2

= ejiw2

, (5.27)

1 jC -' J- ->

which determines k since we know a and k^., as well as k , from

the incident wave.

In the case of a wave in a medium with a real dielectric

constant incident on a half—space with a complex dielectric

constant bounded by the xz —plane, the wave vector of the

refracted wave is still given by (3.12) where k^ is now a complex

number determined from (5.27). The equations that lead to the

integral equations (5.23) and (5,24) remain unchanged.

6. Scattering by a homogeneous strip on a haif -space of a different m^teriai

We now consider the scattering by a strip composed of a

dielectric or other homogeneous material located on a half-space

of a different homogeneous material. We first present in

Subsection 6. 1 the general solution of the scattering of an

electromagnetic wave at a boundary where two media that are

infinite in the y—direction Are separated by a layer of another

medium C21. We then show in Subsection 6.2 how the solution

changes when the intermediate layer is reduced to a finite strip.

35
We restrict ourselves to the TE mode with k T = to keep the

notation simple, but the generalization to an arbitrary direction

o-f propagation for the incident wave can be carried out following

the procedure shown in Subsection 5.3.

6.1 Two sejm^—£nf :ini.te homogeneous regions separated by a

I^YsX Qf iiQite thickness

We first show how the solution of the problem of coated

dielectric gratings C21 is modified when there is no periodicity

in the x-directi on. The two regions V and V_, are separated by a

layer V^ 5 as shown in figure 3. The cross section of the

boundary between V and V 9 is the curve C , and that between V^

and V_ is C_,. We assume that C 1 and C^ are straight parallel

lines except in a finite region, which allows us to solve the

problem of reflection and refraction at plane boundaries in the

usual manner C63.

Here we assume that k_, = and that H^ = 0, and we express

all the fields in terms of the 2 -component of the electric field.

The scattered fields in V f W,, and V_ are defined as the

1 *- —• differences between the total fields in those regions and the

fields that correspond to plane interfaces. These fields are the

incident and reflected fields in v\ , the up and down fields on V

and the refracted field in V_, . In this manner the scattered

fields satisfy the OWC in all three regions.

The boundary conditions that arise from the continuity of

the tangential components of the electric and magnetic fields at

the interfaces give the equations

36
V:

V3

v3

e3 . fa

Figure 3. Geometric configuration -for the scattering of a wave

by a local deviation in the boundaries between three

homogeneous media separated by parallel planes. The up and

down homogeneous fields in V Are represented by the long

arrows in that region, where there also is a scattered field.

37
5+ -3_ - <6.1)

(5E_/an) = a.<3E^/<3n> on C. , i = 1, 2, (6.2)

where

a, = m./m^s a = u^/ji-^ (6.3)

We now define the auxiliary functions U 1? U^, and U^. The

sc function U is equal to E_ in V and vanishes in V and V_,.

The function U is equal to E_, in Vo3, vanishes in V_j. ? satisfies

the Helmholtz equation for medium 2 in V , is continuous across

C , has a discontinuous normal derivative with a jump n across

C , and satisfies the OWC as y -* cd. The function U.^ is equal to

E~ in V_, satisfies the Helmholtz eqution for medium 3 in V and j -

j

1 V^, is continuous everywhere, has a discontinuous normal

derivative with a jump rirr across C^, and satisfies the OWC as

y * Qj .

We express U^ in terms of the jump t^_, of its normal

derivative across C^ by

U^ = G32 Cn 3

>, (6.4)

where G^^, is defined as in (2.24). The first index indicates

that we have to use the constants of medium 3 in the propagation

constant, and the second index that we have to carry out the

integration over the curve C^.

The jumps of U„ and 3U„/3n across C are equal to the values

of these functions at the boundary in V_,, which &re

38
n - tr sc - it ir uP r-down _ up down

j£.~«~ •->+ -_>+ -_> --> •_:

-_:• -_:>

rsc rrefr ,_up ndown ,_0

y „ r , . . _, E_ + E_ - E_H - E„ = E_ + G_„C^_>, (6.5)

(aU2

/fln) +

= (aE*C

/*3n) +

= (aE3

/an)

+

- 3E^P /an - 5E°°Wn /3n

= a 2

(5E3/an)_ - 3E^P /an - 3 Ef

Wn /5n

= a 2 [(aE5C /3nj + dE^r /Bn] - 3^/dn - 3E^°Wn /3n

= 3E

(i /an + a^(- t^/2 + N' (ul, (6.6)

Ci —O ' where N' is defined in (4.19), and E^ and E^ Are the

combinations of homogeneous fields in (6.5) and (6.6) evaluated

on C^. We express the function U^ in terms of its jump across C_,

and the jumps of its normal derivative across C and C^5 it is

1 .£. given by

U

2 = °2l"V + G22{{dU2 /dn )+} + N22"(U2+}* <6 ' 7)

where N is defined in (2.25), t^ is the jump of its normal

derivative at C. , and the other arguments of the functionals Are

39
the jumps at C. , which are functions of n^. as defined in (6.5)

and (6. 6)

.

The jumps of LL and at) /an across C are equal to their

values at the boundary in V , which are

m _ E

xn _ Erefr = _ £

1 £sc

+ Eup

+ E

down _ E

l 1+ 3+ o o> !•— -' j>— 3 3 3

G

21 C^ 2

> + «3 22{(au2 /3n)

+ } + N22{U2+ } + E°, (6.8)

Uu /3n\ = [aE^/3nj

+

- 3E^/3n = otj faE^/an) + aE*/an

= ^[(aE^/an) + aE;;p /an + aE^own /an] - aE*/an

= aJ-^2 + N

2i C V + N22{(dU2 /3n )+} + M22"(U2+}] " dEpdn > (6 - 9)

where M'' iril- is the normal derivative of N-CnJ

M' C-nJ = - |n- I ds'Ti(K' )k rH

1<l>

(k R)R 1 (2n / RR - n ' )

- H* 1} <k R)k n'

Rr] , (6.10)

and E_,, E^, and E^ are combinations of homogeneous fields in

(6.8) and (6.9). We need not be concerned here with the

2

40

singularity of the operator M^, because the functional is
evaluated at a point on C and the integration is over C . We

use (6.8) and (6.9) to express U in terms of the jumps in U and

3U 1 /<3n as

u

i = g ii-(KH+ }

+ N u{u

i + }> <6 - iij

The integral equations are obtained by requiring that U 1 vanish in V and that U vanish in V.,,. In particular, they

vanish at C and C^,, respectively, which gives

U

4 _ = - ^AUj + U^ = 0, (6.12)

h*1 \ c

Substitutions -from (6.5), (6.7), (6.8), and (6.11) trans-form

(6.12) and (6.13) into

(- |G21

- |« 1 G

11 + « lGll N^ + N llG21 )£V

(|« 2G22 - |« 2G22N^2

- |N22G32 - |« 1 «2

B

11 NJ|2 + «

1 « 2

G

11

N22N32

+ a

i Gll

M22G32 " 2a 2N

ll

G 22

+ a 2N

ll

S22N32

+ N

l

1

N22G

32) *V

41
!6

22{3E3' /an}

" h

N

Z2{4'} " 5

E

3

+ a

l Gll

N

i2{SEZ

/3n }

a

l G

ll

M22-(E3'} + a

l G

llidEl /Sn }

+ M

li

B2z{iE

S' /in}

+ N

11

N22K'} + MilK} " °' <6 - 14)

G {*„} + ('- 1g - \*JS + a.,6„.,N:U + N,^S„) Cti,}

i o' B^AsB^ /dn\ + N^-Ze" V = 0. (6.15)

These two integral equations determine the unknown -fields i\^ and

n^ f which determine the functions U , U9 , and U_j. that are

equal to the scattered fields in the three regions.

6.2 The homogeneous string on a substrate of a different material

For this geometry, shown in figure 4, the region V,, is

finite and lies between regions V and V-,. The region V. is

separated from V^, by C and from V_, by C^, with the normal

pointing into V . The region V_ is separated from V^, by C^,, with

the normal pointing out of V^. The homogeneous fields in the

region V are the incident field and the field reflected at the

plane boundary of the region V^ in the absence of the region V .
V33 ^32 ^2

/ 1

sc

Figure 4. Geometric con-figuration for the scattering of a wave

by a homogeneous strip on a half—space composed of a different

homogeneous material. We only define a total field in the

region V «

43
The homogeneous field in the region V_, is the field refracted

under those conditions. Since none of the homogeneous fields

satisfies the Helmholtz equation with the constants of the medium

2, we work with the total field in V . We now define three

auxiliary functions, U. , (_!_, and U_. The function LL equals E^C

1 ' 2 3 1^3

in V , satisfies the same Helmholtz equation as in medium 1 in V

and VV, is continuous everywhere, has a discontinuous normal

derivative with a jump v\ . . across C., a jump n . _ across C_, and a

11 1

1-3 -j- zero jump across C^, and satisfies the OWC as y -* —oo, The

function U? equals E_j. in V^, and vanishes in V and V_,. The

function U_ equals E_ in V_, satisfies the same Helmholtz

-3 3 3

equation as in medium 3 in V and V^, is continuous everywhere,

has a discontinuous normal derivative with a jump ts^^ across C ,

a jump t^_^ across C_, and a zero jump across C., and satisfies

3 3 3 1

the OWC as y -» od.

The functions U and U-;. are given in terms of the jumps of

their normal derivatives by

U. = G..^..} + G -C^-J-, (6.16)

1 11 11 13 1 -

3

lu = g_ -c^ > + G_<:n-^3. <6.i7) 3 32 Ji 33 3 3

The values of U and SU^/dn at C 1 are now determined from the

continuity of the total fields across C , they are

U^ = eI + gJ«€ii,«J + G?^U H ^3, (6.18) '2- " "3 ^ L

ll in ll

y T °13 LQ 13J

44
(<3U2 /dn)_ = a^aE^n + ^t^ + N i\^n y + N

l3 t,»l3> )» (6.19)

where E_, is the sum of the z—components of the incident and

reflected fields evaluated on C , and we have added a superindex

to the operators to indicate on which curve they are evaluated.

Similarly, the values of U.-, and 3U^/5n at C-, are

LU^ = EZ-eir

+ B^„C^_„J + G^{*„}, (6.20) 2+ -> -5Z c-2 33 ->o

IdU^/dn)^ = QL^

1 (sEr

l eir /dn - ±*„, + N^tii-^J + N^? {*,_}} , (6.21)

\ 2 / + Z \ o 2 32 -->.*- ^>2 J'J oo /

"

r ef r where E_, is the z -component of the refracted field evaluated

at C . These equations can then be used to express U^ in terms

of its discontinuities across C and C^, which in turn are

1 £. functions of "n H « , "n*^, **\-~^% and Tft-^. We have

11 l-j>" o>2 -j 3

U

2 = -

G2l{(aU2

/e ")-} - N2l{U2-} + MKHJ + MU2+ }-

(6.22)

Two integral equations are obtained by setting the field U? equal

to outside V on C and C . On C we have

£. 1 jL 1

U

2+ " 54U2

+ U

2 | Cl " " !U2- + "zlc, " °" <6 - 23>

and on C we have

45
U2- = " 2AU2

+ ^Ic = "

2U2+ + U2|c = °' (6.24)

which, substituting from (6,18) through (6.22), become

Mi + l«i e

2i +«i s2i N h

+ N

2 i G iiKi}

+

(k

B U

+ "AAi + "zl^s)*"^

- (*2

iG

22N33 + n22G33 )ctw3

>

- " H " -I

&2iK"n}

" N2l{E

3>

+ "^K^"")

+ N^/e^"*1"^ on Cj, <6.25)

(l«l S

21

+

«l S

21 N h

+M

21 G!l) C "ll

46
- («1 G

21 N!3

+ N

21 Gl3) { "l3:

+

/ 1 2 -122 22\

(" 2

G33

+ «2 G

22N33 * N22G33 ) £ *33

:

i._ref r " 2E3

H

i G2l{aE^ /5n }

+ H2l{Ey} ~

a 2 1S22"(5E^efr/an

}

- H%2^Erl^

fr

^ on C

2

. (6.26)

We obtain two more integral equations from the continuity of the

total electric and magnetic fields on C^. These equations are

11 11 1 -_> 1 -J> -->xl OjC -j-_> -Ji-j jf -J>

"

N li^ll 3 +

(I + N li)^13J - «3N32^32> + «3 (1 " N

33) <*33>

3E^/dn on C^, (6.28)

where

47
£=£ ^i n p-refl rrefr . , _._,, «„ = u 1 /,«!_, E_ = E_ + E_ - a_E^ , (6.29)

and the homogeneous -fields have to be evaluated at C^.. When the

curves C , C_, and C_ are covered with patches, (6.25), (6.26),

(6=27), and (6.28) provide the right number of equations for the

unknown values of n 1 i

, if\ 1 ^, n —,* and in^-, at the centers of these

patches.

Since the normal on C_ and C_, is perpendicular to the

vectors R that go from one point on this line to another, in

(6.21) and, consequently, in (6.25) and (6.26) we have

Ni^X = W^5 = 0. (6.30)

Similarly we have

N;L = 0, (6.31)

in (6.25) only, since N^,., vanishes when it is evaluated on C , but not on C , and in (6.28) we have

W't = N't = Nirt - 0, (6.32) lO •-•^1 -J>0>

Since C_ is part of the plane boundary, we have from (3.21) and

(3.22) that

E'i = 0, 3EL/3n = on C_, (6.33)

and (6.27) and (6.28) reduce to the homogeneous equations

G'?.^..} + BT-tn.-i - &*<.*--* - B^£in„> = 0, (6.34)

11 11 lo lo c>^ j'^ -> J Jj

48
W

l I

C<n 11" + 2*13 + 2^33 = °" (6,35)

We can use (6.34) and (6.35) to eliminate rn, T and t\__ in terms of

in and t^-,, reducing the use of computer memory. We find

11 -J> jC

'

n,, = h^li-n^y + M^c-n,^, (6.36)

10> lo 11 1-3 ->Z

n^ = Mlif.^.} + M^H^-J, (6.37)

where

_> . „•_> M?i = - CG* + GI^) 1 <G~' + 2g;,n: ' ) , 1-i 13 o--> 11 ->_> 11

MT^ = (6* + GlL) 1 G^, (6.39) lo lo OO -_>Jl

M^i = - mH - 2N^', (6.40) jo 1 o 11"

Pitt = - MTt. (6.41)

-j-J> 1 o

Substituting of (6.30), (6.31), (6.36), and (6.37) into

(6.25) and (6.26) we obtain equations of the form

A

n

{D (< } + A?2 {*„} = B, i = 1, 2, (6.42)

l 11 l o-2 l

49
where B and B^ are the right sides o-f (6=25) and (6.26),

respectively, and

ii i i . i _i A _i M i, kl i _i

1 = 2Gii + 2

K

1 G21

+ a

l b2i Nll + N

2i Gii

(*G* + o^G^nJ' + N*G* lllji - Ni^G^hii, (6.43)

A~^ = (-6 + a G N ' + N G I M

_Vi" 1 \2 13 1 21 13 21 13/ 13

-

f- ia^^L + N* G^L } - N^GfLli^. (6.44)

A = - l-a R^" +aGJ^N / +MJ:"G t 2 2 1 21 1 21 11 21 11 )

~

( a

l G21 N13 + N21 G

1 3 K3 " lG33M33' <6 " 45>

^2 / 2 1 2 1 \ "^2

A

2 " "

( a

l Sil Nli

+ N

il G 13K3

12 1-12 12 ^2 -G^ " ^B, B* - ^G^M'". (6.46)

Three limiting cases that can be used to check the program

Are those in which £ = e (no strip), e = £_, (strip embedded in

50
a uniform dielectric), and t^ = e^. <strip and half-space of the

same dielectric).

By setting k_ = and H_, = in (2.15), the Poynting vector

reduces to

± 1 /i *\ S = ^Rel^-E^w ,E,h (6,47)

where we have to substitute (6. 16) for the scattered field in

region V to find the energy flux of the scattered field. We can

use this expression to find the energy flux at a given distance

from the plane. If we assume that there is no interference

between the scattered and the incident plus reflected fields,

the vertical component of the Poynting vector is given by

(x,y) =

* Im/fL ds'n, , <x' )h! 1} <kR) + L ds' n . _ (x ' )f-0 * ' <kR) 1

r r * * ( i ) * ds'u, (x')H (kR)k(y - y')/R

+

J c

ds'Ti*7(x' )H* 1) *<kR)k(y - y' ) /r] V. (6.48)

3

If we are in a region so close to the scatterer that the

incident and reflected fields have to be taken into account, the

energy flux must be computed by using the z—component of the

total field in (6.47). Then (6.48) has to be replaced by

51
S (x,y) = - Re< exp (ik .x ) CE._exp (ik^y) + E" exp (-i k^y) 1 2 • 2toju \ L 1 <-*-- 2 03 2

| c

ds'Tt

11 ii')HQ1} <kR) -

|| c

ds'fi 13 (M')H^ 1}

(kR)J 4jC

t

* * exp (-i k.x ) k_EE-_exp (-i k_y) - E" exp (i k„y) 3 ^ 1 ^ 03 2 Uo 2

|J C

dB'r,

11 (x')H

1 (kR)k(y - y')/R

+

|| c

ds'?> 1T ix ' )H (kR>k(y - y')/R]>. (6.49)

The scattered fields may be much larger than the homogeneous

fields, in which case the difference between (6.48) and (6.49) is

negl igible.

7. Interior H^sgnances and sharp edges

For a perfect conductor, there may be resonant modes of the

interior problem that have no physical effect on the exterior

problem, but that disturb the numerical computations and give

rise to instabilities on the solution of the exterior problem

C7D. We examine the importance of this problem to our methods of
solution in Subsection 7.1.

Another potential problem with numerical calculations is

related to the divergent -fields that occur at sharp edges C81.

We discuss this problem in Subsection 7.2.

7.1 Interior resonances

Instabilities have been predicted and observed C71 in a

number of calculations of fields scattered by objects at

frequencies that corresponds to resonances of the interior

problem.

We have not had any of these problems in calculations with

the programs described in this report.

For the perfect conductor, we have specifically calculated

the scattering by a square strip of a side equal to half a

wavelength and to a full wavelength, for which we might expect

problems. For normal incidence, the scattered fields are

negligibly small for the TM case, and there are no problems for

incidence at 60° or for the TE case.

For the problem with two different dielectrics, we can set

the values of e and £_, equal and choose the dimensions of the

strip so that the cross section is a square. We then compute the

wave scattered by a square cylinder in free space, and we have

found no problems either for these related values of wavelength

and size.

53
7.2 Effects of sharp edges

Fields near the edge of a wedge have been determined C8D to

diverge with a power law that depends on the angle of the wedge.

Locally, the same type of divergence is expected near any sharp

edge.

We have used two different approaches to this problem in

numerical calculations. Either we replace the sharp edge by a

curve matched to the sides or we stay away from the edge itself

in our selection of points.

We have replaced the profile that has a discontinuous

derivative with a smoother curve by selecting a circular arc of a

given radius tangential to the sides of the trapezoid. This

procedure corresponds to the physical assumption that an edge is

never absolutely sharp and it presents fewer numerical problems

at the price of increased complexity in the calculations and a

degree of arbitrariness in the choice of the curve. We recall

that the contribution of the self-patch to the operator N'

depends on the curvature, as shown in (4.21). We have used this

approach for the homogeneous strip on a half—space of a different

material, and we have found that the values of the surface fields

at the top of the strip Are smaller in magnitude than those that

arise from computations with sharp edges, but this is not so for

the bottom of the strip, where three different media meet. It is

thus best to smooth the edge at the top of the strip and keep the

sharp edges at the bottom, which does not affect much the values

of the fields near the top of the strip. A better approach would

be to determine the analytical behavior of the unknown fields
near the edges and to take it into account in the numerical

calculations.

We have not smoothed the edges of the strip in calculations

of the intensity in the radiation zone. We have chosen our

patches so that the edge corresponds to the point where two

patches meet; the matching is carried out at the center of the

patches. For the trapezoidal strips, the curvature at the center

of a patch always vanishes.

We have had no difficulties in our numerical calculations of

the energy flow in the radiation zone that can be attributed to

the divergence of the fields at edges. The surface fields

increase sharply in the vicinity of an edge and they often change

sign at the edge. We have also verified that getting closer to

the edge with a smaller patch does not change appreciably the

value of the far field. In an example, a change of a factor of

10 in the surface field near the edge was associated with a

change of only 1 percent in the maximum value of the radiated

energy flux.

The fields near the top surface depend strongly on the

values of the nearby surface fields. This dependence is enhanced

because the Green function diverges when the argument vanishes.

To compute the integrals of the surface fields for points near

the top surface the integration over the nearby patches has to be

broken up into smaller segments, and we have interpolated the

values of the surface fields from the three closest patches. We

have modified the edges at the top of the strip by using circular

arcs.

55
8. Computer programs

We have written a group of four computer programs to solve

the problems of scattering by a perfectly conducting strip on a

perfectly conducting half-plane in the TE mode (STRTE) or in the

TM mode (STRTM) , and by a dielectric strip on a dielectric halfplane when the dielectrics are the same (STRDL) or different

(STR2DL) , One version of STR2DL uses rounded edges for the top

of the strip.

The input parameters include the dimensions of the strip,

the radius at the edges, the number of patches in the different

sections of the interface, the wavelength, direction of

propagation, and polarization of the incident wave, the

dielectric constants, and the locations where the output is

computed.

The output is composed of graphs that show the scatterer and

a polar diagram of the far—field intensity or of the energy flux

density (the magnitude of the Poynting vector S at large

distances from the scatterer) or the vertical energy flux density

(S^) at a given distance from the plane. Optionally a

superimposed graph shows the absolute value of the surface

function (s). These graphs are scaled to the size of the strip

for display purposes.

Each program has a subroutine PATCHES where the coordinates

of the points and the components of the unit normals on the top

and sides of the trapezoid, as well as the x-coordi nates on the

base, if needed, and on the boundary between the two infinite

56
media out to a given distance (usually a few wavelengths), are

determined. By default, the patches are of equal size on each

sector of the boundary. If we want to get closer to the edges

without increasing the number of patches, we can specify the size

of the patch closest to the edge and compute the sizes of the

other patches so that they form a geometric progression and still

cover the section with the given number of patches. The point

distribution is symmetric with respect of the midpoint of the top

of the trapezoid, and also on each side of the trapezoid with

respect to its midpoint. The sizes of the sides of the trapezoid

are appropriately modified when the edges are rounded.

The height of the trapezoid can be negative to represent a

depression in the lower medium, the width of the top of the

trapezoid can be zero to represent a triangular strip, and the

width of the base can be equal to or smaller than the width of

the top to represent a rectangular strip or an "upside down"

trapezoid. A different subroutine PATCHES could be written to

represent more general shapes of strips; the computations should

then include the curvature to take into account the self -patch

contribution in (4.21). The curvature at points of the circular

arc are equal to the inverse of the radius.

Each program has a subroutine INFL where the homogeneous

fields Are determined on the boundaries as required in each case.

These will be used to determine the known values in the system of

linear equations. Then a subroutine MATRIX is used to find the

required values of the matrices representing the operators at the

given points and to determine the coefficients and constants on
the system of linear equations. The subprograms used to compute

the values of the Hankel functions are BESJO, BESJ1, BESYO, and

BESY1 if the arguments are real, and CJYHBS if the arguments are

complex. These subroutines are in CMLIB C93.

The systems of linear equations are solved using the

subroutines CGECO and CGESL C93, which also provide a

conditioning parameter that can be checked for difficulties

related to i 1 1—posedness. These subroutines lead to excessive

paging when the sizes of the arrays exceed the available computer

memory, and we had to replace them by CGEL from the MAGEV

1 ibrary.

The intensity of the scattered field in medium 1 is computed

from the appropriate surface fields in a subroutine FARFL- A

similar computation could be carried out for the lower medium if

there is no absorption. The fields at a given distance above the

plane are computed in a subroutine NRFL; we have done this for

the two dielectrics and we assume that the distance is larger

than the height of the trapezoid.

These programs take advantage of the vector processing on

the CYBER 205, although not all of the subroutines in CMLIB are

vector i zed.

In figure 5 we show samples of outputs from the programs

STRTE and STRTM for perfect conductors. The total number of

patches for the triangular strip was 420, and for the trapezoidal

strip, 600. The running times were approximately 9 and 20 cpu—

s

for the TE mode and 13 and 29 cpu-s for the TM mode.

In figure 6 we show samples of the output of STRDL for

different directions of propagation and polarization of the

58
I

.2> 1

-6.0

(a)

/ \

iiH

___s^4kk—

_

J

I 6

e 5

-S- >• 4

COc& 3

_co 2

'(1) X

-4.8 -3.6 -2.4 -1.2 1.2 2.4 3.6 4.8 6.0

Position on Plane (/*m)

-6-3036

Position on Plane (/xm)

J

I] [I

l

r II i'iii

pill ', i

j

:

\

'< i

i 1

-6.0-4.8-3.6-2.4-1.2 1.2 2.4 3.6 4.8 6.0

Position on Plane (/xm)

(d) [ fc ^

:\lf/

7

:

20 -16 -12 -8-4048

Position on Plane (ttm)

12 16 20

Figure 5. Polar diagrams of the intensity of the -field

scattered by a triangular or trapezoidal perfectly conducting

strip. The dotted lines show the strips, which in <a) and (b)

have a triangular cross section with a base equal to 4X and

height 2X, and in (c) and (d) are trapezoids with top 2X, base

3X, and height X/2. The dashed lines represent the magnitude

o-f the surface field. The incidence is normal and (a) and (c)

correspond to the TE mode and (b) and (d) to the TM mode.

59
CO

w

c£

o

I

-7.5-6.0-4.5-3.0-1.5 1.5 3.0 4.5 6.0 7.5

Position on Plane (/*m)

6

(b) A

5 / \

4

-

3 I / /

2 \ \ j / /

1 \\ Ki \ ,\- ^

| /

/pC^

-7.5-6.0-4.5-3.0-1.5 1.5 3.0 4.5 6.0 7.5

Position on Plane (/im)

/

6

(d)

\

5 / \ ' 4 /

-

3 f~~~^ i

2

1t

*

1

1 —,—

.

rf

, -,-,..-.-.----------- ---.-- — -

l—

/ 1

Li 1

-6.0-4.5-3.0-1.5 1.5 3.0 4.5 6.0 7.5

Position on Plane (/xm)

-7.5-6.0-4.5-3.0-1.5 1.5 3.0 4.5 6.0 7.5

Position on Plane (/im)

Figure 6. Polar diagram of the intensity of the field scattered

by a trapezoidal dielectric strip (e = 2.14, top X, base 2X,

and height X/2) on a half—space of the same dielectric for

different incident waves, (a) TE mode, (b) TM mode, (c)

direction of propagation of the incident wave with polar, angle

q = 60°, azimuthal angle $ = 60°, polarized so that E_, = 0,

and (d) direction, of propagation with 8 = 60°, = 90°,

i -J 4-1-4. rr ln cin polarized so that E^ = E .

60
incident wave. For a total of 460 patches, the running time was

approximately 110 CPU-s, In figure 7 we show the output of the

same program for the scattering of a wave by square or

rectangular glass cylinder with a side equal to the wavelength.

In figure S we show the vertical component of the Poynting

vector for the field above a silver strip on a glass substrate at

different heights above the strip.

9. Concluding remarks

We have presented the derivation of the integral equations

that can be used to find the fields scattered by strips on a

half -space for different configurations and materials. These are

singular integral equations designed to minimize the number of

unknown fields on the interfaces between the media. We have also

pointed out some possible difficulties with sharp edges and

spurious resonances that require further investigation.

The theory is exact, well suited to the resonance regime,

where the wavelength is comparable to the size of the scatterer.

We have briefly described the programs that were written to

implement these calculations, and we have shown sample outputs

from these programs.

10. Acknowledgements

I thank Dr. Daniel Maystre, Dr. Ralph E. Kleinman, Dr. A. George

Lieberman, Dr. David Gilsinn, and Dr. Jay F. Marchiando for

helpful comments and suggestions.

61
CO

CD I

-6.0

COc

i—O

JO

2>

CD I

(a)

1 ..-,--—. -,.,-r ;=];,. ..

r

- -,-V- 4cSal --,,... LIT ";-; ., nl , , .

(c)

7

0)c 6

3

.Q 5

CO

>. 4

</> c

0) 3

co

2

_^J .cen 1

CD I

4.8-3.6-2.4-1.2 1.2 2.4 3.6 4.8-6.0

Position on Plane {(im)

CO

Jd

CD I

(b)

/'

-6.0-4.8-3.6-2.4-1.2 1.2 2.4 3.6 4.8 6.0

Position on Plane {fim)

1

6

(d)

5 /^

4

-

3 i

2 W"-- .'-

I

'"'--.;

1 i

\

I 'f. !----<{ v

-..„. -7.5-6.0-4.5-3.0-1.5 1.5 3.0 4.5 6.0 7.5

Position on Plane (/xm)

-7.5-6.0-4.5-3.0-1.5 1.5 3.0 4.5 6.0 7.5

Position on Plane (/xm)

Figure 7. Polar diagram of the intensity of the field scattered

by a square glass cylinder of side X in free space in the TE

mode, e = 2.14, incident direction (a) perpendicular to top (b)

at = 45° and for a rectangular cylinder of height X and width

4X, with (c) e = 2.14, and <d) e = 2. 14 + 0.5i. The maximum

intensity in (d) is about 1/3 of that in (c).

62
CO % 2

i

1

-2- £•

"coc

ffi -1 c

fe -2h

"5X

- -^ -

_

co

x

D)

<D I

(a)

f\A

1 1 1 1

\r^\

coc 1 3

n

a> -1

C/5 c

CD -2

C-O -3

.C

'<x> -4

-1.25-1.0-0.75-0.5-0.25 0.25 0.50 0.75 1.0 1.25

Position on Plane (/mi)

1

(c)

-

-1 -

-2 -

-3 - r\ r\ -

-4 - \y \j \J \y -

-5

(b)

(. VA|

-•aJ Ia- -

:

-1.25-1.0-0.75-0.5-0.25 0.25 0.50 0.75 1.0 1.25

Position on Plane (fim) 2

'tr 1

ZS x>m

>, -1

c/5 c

CD -2

c_^O -3 X

r-n 0) -4 X

(d)

/-^

\

-1.25-1.0-0.75-0.5-0.25 0.25 0.50 0.75 1.0 1.25

Position on Plane (/im)

-1.25-1.0-0.75-0.5-0.25 0.25 0.50 0.75 1.0 1.25

Position on Plane (/xm)

Figure 8. Vertical component of Poynting vector for the total

fields above a silver strip (e = - 11.68 + 1 . 39i , h = O. 1 urn, w = 0.3 um) on a glass substrate (e = 2.14) for X = 0.6328 urn. The energy flux density is shown for a distance above the strip

of <a) 0.0005 urn, (b) 0.001 urn, (c) 0.01 urn, and (d) 0.1 urn.

63
11. References

Ell Nyyssonen, Diana. Theory of optical edge detection and

imaging of thick layers. J. Opt. Soc. Am. 72: 1425-1436;

1982.

C2U Maystre, Daniel. Integral methods, chapter 3 in

Electromagnetic theory of gratings. R. Petit, ed. Berlin:

Springer Verlag; 1980. 63-100.

Maystre, Daniel. Electromagnetic scattering from perfectly

conducting rough surfaces in the resonance region. IEEE

Trans. Antennas Propag. AP-31: 885-895; 1983.

C33 Marx, Egon. Single integral equation for wave scattering.

J. Math. Phys. 23: 1057-1065; 1982.

Marx, Egon. Integral equation for scattering by a

dielectric. IEEE Trans. Antennas. Propag. AP-32: 166-172;

1984.

Glisson, Allen W. An integral equation for electromagnetic

scattering from homogeneous dielectric bodies. IEEE Trans.

Antennas Propag. AP-32: 173-175; 1984.

E43 Panofsky, Wolfgang K. ; Phillips, Melba. Classical

electricity and magnetism, 2nd ed. Reading: Addi son-Wesley

Publishing Co.; 1962. p. 219 ff.

Jackson, John D. Classical electrodynamics, 2nd ed. New

York: John Wiley & Sons; 1975. p. 339 ff.

C51 Willers, Achim. The Helmholtz equation in disturbed halfspaces. Preprint.

C61 Sommerfeld, Arnold. Optics. New York: Academic Press;

1964. p. 40 ff.

64
C7D Mitra, R. ; Klein, C. R. Stability and convergence of moment

method solution, chapter 5 in Numerical and asymptotic

techniques in electromagnetics. R. Mittra, ed. New York:

Springer-Verlag? 1975. 129-163.

Glisson, Allen W. ; Sholy, Beshara I. Numerical solution of

electromagnetic scattering problems involving homogeneous

dielectric bodies via a single integral equation. 1986

National Radio Science Meeting; 1986, June 8-13?

Philadelphia, PA. p. 100.

CS13 Meixner, Josef. The behavior of electromagnetic fields at

edges. IEEE Trans. Antennas Propag. AP-20: 442-446; 1972.

Van Blade!, J. Electromagnetic fields. New York: McGraw-Hill

Book Company; 1964. p. 382 ff.

Bach Andersen, J.; Solodukhov, V. V. Field behavior near a

dielectric wedge. IEEE Trans. Antennas Propag. AP-26: 598-

602? 1978.

C91 Boisvert, R. F. ; Howe, S. E. ; Kahaner, D. K. Guide to

available mathematical software. Nat. Bur. Stand. (U.S.)

NBSIR 84-2824; 1984 January.

NBS-114A (rev. 2-bc) U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBS/TN-1236

2. Performing Organ. Report No 3. Publication Date

June 1987

4. TITLE AND SUBTITLE

Electromagnetic Scattering by a Thick Strip on a Half-Space

5. AUTHOR(S)

Egon Marx

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions)

NATIONAL BUREAU OF STANDARDS

U.S. DEPARTMENT OF COMMERCE

GAITHERSBURG, MD 20899

7. Contract/Grant No.

8. Type of Report & Period Covered

Final

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City, State, ZIP)

Same as item #6.

10. SUPPLEMENTARY NOTES

; ]] Document describes a computer program; SF-185, FlPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

The problem of the determination of the scattered fields from an incident plane

monochromatic electromagnetic wave on a strip of finite thickness on a half-space

is reduced to the numerical solution of integral equations for auxiliary fields

defined on the interfaces. These fields are chosen so as to minimize their number,

The derivation of the integral equations is given for a perfectly conducting strip

on a perfectly conducting half-space, for a dielectric (or other homogeneous

material) on a dielectric half-space, and for a dielectric, strip on a half-space

of a different dielectric material. The vector FORTRAN programs written to carry

out these calculations are briefly described and sample outputs are shown.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

dielectric strip; electromagnetic scattering, integral equations; numerical

solutions; resonance region; perfect conductors

13. AVAILABILITY

P^l Unlimited

| | For Official Distribution. Do Not Release to NTIS

[XT] Order From Superintendent if Documents, U.S. Government Printing Office, Washington, D.C.

20402.

[

~] Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

70

15. Price

USCOMM-DC 6043-P80

U.S. Government Printing Office: 1987 0- 181-076 (60104)

NBSTechnical Publications

Periodical

Journal of Research—The Journal of Research of the National Bureau of Standards reports NBS research and development in those disciplines of the physical and engineering sciences in which the Bureau is active. These include physics, chemistry, engineering, mathematics, and computer sciences. Papers cover a broad

range of subjects, with major emphasis on measurement methodology and the basic technology underlying

standardization. Also included from time to time are survey articles on topics closely related to the Bureau's

technical and scientific programs. Issued six times a year.

Nonperiodicals

Monographs—Major contributions to the technical literature on various subjects related to the Bureau's scientific and technical activities. Handbooks—Recommended codes of engineering and industrial practice (including safety codes) developed in cooperation with interested industries, professional organizations, and regulatory bodies.

Special Publications—Include proceedings of conferences sponsored by NBS, NBS annual reports, and other

special publications appropriate to this grouping such as wall charts, pocket cards, and bibliographies.

Applied Mathematics Series—Mathematical tables, manuals, and studies of special interest to physicists,

engineers, chemists, biologists, mathematicians, computer programmers, and others engaged in scientific and

technical work.

National Standard Reference Data Series—Provides quantitative data on the physical and chemical properties

of materials, compiled from the world's literature and critically evaluated. Developed under a worldwide pro- gram coordinated by NBS under the authority of the National Standard Data Act (Public Law 90-396).

NOTE: The Journal of Physical and Chemical Reference Data (JPCRD) is published quarterly for NBS by

the American Chemical Society (ACS) and the American Institute of Physics (AIP). Subscriptions, reprints,

and supplements are available from ACS, 1155 Sixteenth St., NW, Washington, DC 20056.

Building Science Series—Disseminates technical information developed at the Bureau on building materials,

components, systems, and whole structures. The series presents research results, test methods, and perfor- mance criteria related to the structural and environmental functions and the durability and safety

characteristics of building elements and systems. Technical Notes—Studies or reports which are complete in themselves but restrictive in their treatment of a

subject. Analogous to monographs but not so comprehensive in scope or definitive in treatment of the subject

area. Often serve as a vehicle for final reports of work performed at NBS under the sponsorship of other government agencies.

Voluntary Product Standards—Developed under procedures published by the Department of Commerce in Part 10, Title 15, of the Code of Federal Regulations. The standards establish nationally recognized re- quirements for products, and provide all concerned interests with a basis for common understanding of the

characteristics of the products. NBS administers this program as a supplement to the activities of the private

sector standardizing organizations.

Consumer Information Series—Practical information, based on NBS research and experience, covering areas of interest to the consumer. Easily understandable language and illustrations provide useful background

knowledge for shopping in today's technological marketplace.

Order the above NBS publications from: Superintendent of Documents, Government Printing Office,

Washington, DC 20402. Order the following NBS publications—FIPS and NBSIR 's—from the National Technical Information Service, Springfield, VA 22161.

Federal Information Processing Standards Publications (FIPS PUB)—Publications in this series collectively constitute the Federal Information Processing Standards Register. The Register serves as the official source of

information in the Federal Government regarding standards issued by NBS pursuant to the Federal Property

and Administrative Services Act of 1949 as amended, Public Law 89-306 (79 Stat. 1127), and as implemented

by Executive Order 11717 (38 FR 12315, dated May 11, 1973) and Part 6 of Title 15 CFR •(Code of Federal

Regulations).

NBS Interagency Reports (NBSIR)—A special series of interim or final reports on work performed by NBS

for outside sponsors (both government and non-government). In general, initial distribution is handled by the sponsor; public distribution is by the National Technical Information Service, Springfield, VA 22161, in paper copy or microfiche form.
U.S. Department of Commerce

National Bureau of Standards

Gaithersburg, MD 20899

Official Business

Penalty for Private Use $300