A111DS 72TM1M

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

A1 1102729414

Wu, D. I/An Investigation of a ray-mode QC100 .U5753 N0.1312 1987 V198 C.1 NBS-P

NBS

PUBLICATIONS

NBS TECHNICAL NOTE 1312

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

Gaithersburg, Maryland 20899

Cluoo

An Investigation of a Ray-Mode %%?*

Representation of the Green's Function

in a Rectangular Cavity t

^

D.I. Wu

D.C. Chang*

Electromagnetic Fields Division

Center for Electronics and Electrical Engineering

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303-3328

fUniversity of Colorado

Boulder, Colorado 80309

w \

U.S. DEPARTMENT OF COMMERCE, Clarence J. Brown, Acting Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued September 1987
National Bureau of Standards Technical Note 1312

Natl. Bur. Stand. (U.S.), Tech Note 1312, 48 pages (Sept. 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

FOREWORD v

1 . Introduction 1

2 . Dyadic Green' s Function 3

3. Infinite Poisson Transformation 7

4. Finite Poisson Transformation 11

5 . Numerical Examples 16

6. Conclusion 20

References 21

APPENDIX: Summing Over Finite Intervals 23

in

FOREWORD

This report is a continuation of efforts and collaboration between the

staff of the University of Colorado at Boulder and the Electromagnetic

Fields Division of the National Bureau of Standards (NBS) to establish a theoretical basis for the design of a mode-stirred (reverberating) chamber.

This work is a part of the doctoral dissertation work undertaken by D . I. Wu. The project was sponsored by NBS under the technical supervision of

Professor David C. Chang of CU and Dr. Motohisa Kanda of NBS.

The goals of this project are to understand analytically the effect of

a rotating scatterer or a stirrer in a large rectangular cavity, and to

provide analytical tools usable in the design of an effective stirrer. In

treating the stirrer as a scatterer, the dyadic Green's function expressed

in the modal form as the kernel for the desired integral equation is

encountered. Due to the cumbersome nature of the dyad, numerical

computations are relied upon. However, since the convergence of the triple

summation of modes embedded in the dyad is known to be impractically slow,

the issue of evaluating the dyad in a numerically efficient manner arises.

This prompts our search for an alternate representation for the dyad which

would allow feasible numerical computation.

This report describes in detail the analytical methods used in

obtaining an efficient hybrid representation for the dyad. The

effectiveness of this hybrid representation is also illustrated in this

report. With this hybrid representation, any numerical computation

involving the dyad can now be carried out in a feasible manner.

Previous publications under the same effort include:

Tippet, J. C. ; Chang, D. C. Radiation characteristics of dipoles

sources located inside a rectangular coaxial transmission line.

Nat. Bur. Stand. (U.S.) NBSIR 75-829; 1976 January.

Tippet, J. C. ; Chang, D. C; Crawford, M. L. An analytical and

experimental determination of the cut-off frequencies of higher-order

TE modes in a TEM cell. Nat. Bur. Stand. (U.S.) NBSIR 76-841;

1976 June.

Tippet, J. C; Chang, D. C. Higher-order modes in rectangular coaxial

line with infinitely thin inner conductor. Nat. Bur. Stand. (U.S.)

NBSIR 78-873; 1978 March.

Sreenivasiah , I.; Chang, D. C. A variational expression for the

scattering matrix of a coaxial line step discontinuity and its

application to an over moded coaxial TEM cell. Nat. Bur. Stand. (U.S.)

NBSIR 79-1606; 1979 May.
Tippet, J. C; Chang, D. C. Dispersion and attenuation characteristics

of modes in a TEM cell with a lossy dielectric slab. Nat. Bur. Stand.

(U.S.) NBSIR 79-1615; 1979 August.

Sreenivasiah, I.; Chang, D. C; Ma, M. T. Characterization of

electrically small radiating sources by tests inside a transmission

line cell. Nat. Bur. Stand. (U.S.) Tech. Note 1017; 1980 February.

Wilson, P. F. ; Chang, D. C; Ma, M. T. Excitation of a TEM cell by a vertical electric Hertzian dipole. Nat. Bur. Stand. (U.S.) Tech Note

1037; 1981 March.

Sreenivasiah, I.; Chang, D. C; Ma, M. T. A method of determining

the emission and susceptibility levels of electrically small objects

using a TEM cell. Nat. Bur. Stand. (U.S.) Tech. Note 1040; 1981 April.

Wilson, P. F. ; Chang, D. C; Ma, M. T. Input impedance of a probe

antenna exciting a TEM cell. Nat Bur. Stand. (U.S.) Tech. Note

1054; 1982 April.

Liu, B. H. ; Chang, D. C; Ma, M. T. Eigenmodes and the composite

quality factor of a reverberating chamber. Nat. Bur. Stand. (U.S.)

Tech. Note 1066; 1983 August.

VI
An Investigation of a Ray-Mode Representation of

the Green's Function in a Rectangular Cavity

*

D . I . Wu and D . C . Chang

Electromagnetic Fields Division

National Bureau of Standards

Boulder, Colorado 80303

It is well known that a point-source excited field in a rectangular cavity can be represented either in terms of

summation of modes or in terms of rays produced by the equivalent

image sources. Both representations involve series that are slowly convergent; so computation of fields inside the cavity is

difficult. To obtain a numerically efficient scheme, a hybrid

ray-mode representation is developed here using the finite

Poisson summation formula. The modal representation is modified

in such a way that all the modes near resonance are retained

while the truncated remainder of the mode series is expressed in

terms of a weighted contribution of rays. For a large cavity,

the contribution of rays from far-away images becomes small;

therefore the ray sum can be approximated by one or two dominant

terms without a loss of numerical accuracy. To illustrate the

accuracy and the computational simplification of this ray-mode

representation, numerical examples are included with the

conventional mode series (summed at the expense of long

computation time) serving as a reference.

Key words: Green's function; hybrid representation; modal

representation; Poisson summation formula; ray-mode

representation; rectangular cavity.

1. Introduction

In analyzing fields due to scattering or excitation of a radiating

structure inside an electrically large and over-moded rectangular cavity

such as the NBS reverberating chamber used in EMI testing[ 1 , 2 ]

, we often

encounter the dyadic Green's function expressed in the modal form as the

kernel for the desired integral equation. One issue that often arises is

how to obtain a numerically efficient scheme for computing the dyad,

particularly when the observation point is close to the source point. The

^Department of Electrical Engineering, University of Colorado, Boulder,

CO 80309. 1
use of modal representation is clearly not practical, since the convergence

of the triple infinite sum of higher-order, nonresonant modes is

notoriously, if not impractically , slow.

In this report, a finite, three-dimensional Poisson summation formula

is utilized to obtain a hybrid representation for the Green's function.

This hybrid representation consists of two terms. The first term, called

the mode sum, consists of a finite number of modes near resonance. The

number of modes varies depending upon the summation bandwidth chosen. The

second term, referred to as the ray sum, consists of all the images produced

by the reflecting boundaries of the cavity. The bandwidth for the mode sum

is a mathematical quantity. A balancing effect exists between the two terms

in that as the bandwidth increases , the contribution from the mode sum increases while the contribution from the ray sum decreases. Though the

bandwidth is an arbitrary quantity, it does have a minimal requirement.

Below this minimal value the hybrid representation becomes a poor

approximation to the modal representation. As will be shown, this minimal

requirement stems from the approximation involved in transforming from the

rectangular coordinates to spherical coordinates in applying the finite

Poisson summation formulation.

This hybrid representation is especially effective when the source point is close to the observation point. For a large cavity, oftentimes the

second or the third layer images and beyond are far away from the

observation point; so the contribution from these images becomes very small.

Therefore, for the ray sum, we found that it is often sufficient to keep

just the self term and perhaps several adjacent images to obtain the desired

numerical accuracy.
2. Dyadic Green's Function

Consider a rectangular cavity with a perfectly conducting scatterer as

shown in figure 1. Fields incident on the scatterer induce a current J on

the surface of the scatterer, which in turn re-radiates fields inside the

cavity. The re-radiated or scattered fields can be expanded using two types

of orthogonal basis functions, of which one has a zero divergence and the

other a zero curl [3].

^s v v v J TE ztTE TM r>TM ^ . E = yy;ipE+pE+qFk (1)

a=(m,n,p)=0

where

-TE ->TM •* V • E - V • E - 0; V x F = 0. (la) a a a

A ->TE -+TM If we define TE and TM with respect to the z-axis, the modal fields E , E , a a

and F can be found using boundary conditions,

a

^TE „ a .mn . ,r\n . . ,p7r . E = V x cos( x)cos( y)sm(_ z)

Jab c

7+TM _ _ a ,mn . . ,r\n . ,-pn . E = V x V x sin( x)sm( y)cos(l z)

J ab(

e = V <| aF„ 1 a

sin(!!!L x)sin(^ y)sin(El_ z) }>, (2)

i~

T~ a b c

J abc

2 for m=0 , or n=0, or p=0

€ a =

4 -

J 8 for m,n,p * . TE TM The modal coefficients p , p , q can be found by substituting eq (2) into a a a the vector wave equation and applying orthogonality conditions, and they can be expressed as[4]
-> ->TE

i <J • E >

TE . . 1 N a

a

2 2 2 k - k k

a o mn

TM . , 1 , a

P = -iw/i ( ) , (3)

2 2 2 2 k - k k k

a o a mn

and

q

= iw/i

<J • F >

a a

2 2 k k

o a

where the time dependence is e , k = u>(u e ) is the wave number of the

o o o material in the cavityJ space, r and k and k are the normalized and the

a mn transverse cut-off frequencies of the cavity modes,

k = t(!!!L)

2 + (!!L)

2 + (ZIL)2

] 1/2

, a b c and

k = [{

TM ) 2 + ( n7r

) 2 ]V2 (4) mn —— J a b

The < • > notation implies the integral of the dot product of the two vector

functions enclosed, integrated over the surface of the scatterer.

The electric field can therefore be expressed as

E

S = -iw/j <J • §>, (5)

where

-+TE -+TE -+TM -TM + -

o co E[r)E (r') E (r)E(r') F (r)F (r')

G- I I l\ a - a a

- ° a I (5a) a=(m,n,p)-0 2

. k

2 2 2

. ^2 2 k

2

fc 2fc 2 a o mn a o mn a a o

Rearranging terms, we can expressed the dyadic Green's function in a modal

form,
00 00 00

- -W'Jl III • (r) • (r') (6)

k

2 m=0n=0p=0

R

2 _ R2 o a o cooooo A A AA A A + I I Z [4x (rU*(r')xx + <f>l(r)<f>l(r')yy + ^(r)^(r')zzj

,

m=0n=0p=0 ,2 ,2

where

<t>* (r)

e a

a

J abc

4> Y (r)

e a

J abc

4> Z (r) a

a

.nm . mr . pjr cos( x) sm( ~ y) sm( z) , a b c

sin( x) cos(~r y) sin( z)

,

(6a) a b c

sm( x) sin( r y) cos( z) , a b

y c

J abc

and

e ... q .nv7r . mr . pjr $ (r) = sin( — x) sin(T" y) sin( z) . a a b c

\ abc

The dyadic Green's function above is a solution for the dyadic differential

r< A A A A A A equation VxVxG-k G— l6(r-r'), I - xx + yy + zz , instead of the wave

equation (V + k ) G = -IcS(r-r'). It is a complete solution valid both in

and out of the source region. Although eq (6) does not have explicity a singular term in the form of fi(r-r') which normally arises when observing in

the source region, this singular term is in fact embedded in it. For our purpose we will retain the modal representation of eq (6) to utilize the

symmetry in the dyad.

In computing the fields in a cavity, we resort to numerical computation

since G is not simple in form. However, since the summation indices of eq

(6) extend from to «>, numerical computation of G becomes more and more

tedious as the observation point approaches the source point. This
motivates our search for an alternate representation for the dyad which is more efficient from a computational viewpoint.

In searching for an alternate representation, our approach is to use

the Poisson summation transformation to obtain a hybrid ray-mode

representation for the dyad. This method of hybrid ray-mode reformulation

is not new, being first developed for guided electromagnetic and acoustic

fields [5]. For example, in [6,7], the equivalency between mode and ray

representations for guided propagation is illustrated by utilizing Poisson

summation formulation. Treatment of waveguide fields using hybrid

formulation can also be found in [8,9]. Different from the existing work

cited above, our treatment is a reformulation for a three-dimensional

cavity. We begin our treatment with a description and application of both

the infinite and the finite Poisson summation transformations.
3. Infinite Poisson Transformation

A one -dimensional Poisson summation formula can be expressed as[10]

CO 00 CO I f(2iMr) = __ I J f(r) e

1I/T

dr, (7) n=-co 2n i/=-co -co provided that

1) f(x) is a continuous and continuously dif ferentiable function of x, and

00 co 2) £ f (2n7r+t) and £ f ' (2n7r+t) converge absolutely for all t in the n= - 00 n= - 00 interval < t < 2n

For t=0 , the summation reduces to

CO CO CO I f(2iMT) - _ I J f(r)e

ll/T

dr. (8)

n=-00 2?T l/=-eo -co

Extending to three -dimens ions , we have

co co co I X I f(2m7r,2n7r > 2 P7r) = (9)

m=-eon=-oop=-co

1 00 00 co 00 co 00 I I I I S J" «r1 ..f 1 .r.i.

1 <"T > 4'T » +« T » ) dr I.dr 1 d,

3 ^ £, £ J J J *-A' 1 , ! ' 2 » ' 3/ c u 'i u '2 u, 3-

(27r) q;=-co/3=-co£=-co -co -co -co

The corresponding f(m,n,p) in the dyadic Green's function expression (eq 3)

consists of different combinations of sin() and cos(). For illustratative

purposes, we will consider only the case of a scalar Green's function

similar in form to the different components embedded in G. To generalize,

complex wave number k will be used to represent the cavity medium.

Consider

CO CO CO 0(r)<Kr') G(r.r') =1X1

m=0n=0p=0 r-2 ,2

o a
00 CO GO 1 Y v Y

^( r)^( r ') qqn

with

8 m=-°°n=-a>p=-oo _2 2 r k - k

$(r)= sin( — x) sin(~r y) sin(^~~ z). (10a)

SL D C

By letting k = —, k = —r , k = , and expanding ^(r)^(r'), we get

^(r)^(r') = _i_[cos(k (x-x'))-cos(k (x+x' ) ) ] [cos (k (y-y ' )

) -cos (k (y+y'))]»

o x x y y

[cos(k (z-z'))-cos(k (z+z'))]. (11)

By multiplying the bracket terms through, ^(r)^(r') can be decomposed into

eight o terms, each in the form of cosk X cosk Y cosk Z, where X=(x-x') or

x y z (x+x'), Y=(y-y') or (y+y'), Z=(z-z') or (z+z'). Each of these terms can be

further expanded into eight exponential terms in the form of (_

8 i(k X+k Y+k Z) x y z e

J

) . Consider each term separately in the generic form, let

i[k X+k Y+k Z]

f(m,n,p) = e X y Z. (12)

~2 2 64(k - k ) O Q

Applying the 3-D Poisson summation formula (eq 9) to the summation of

f (m,n,p) , we get

CO 00 00 III f(2m7r,2n7r,2P 7r) = (13)

m=0n=0p=0

, ^ co co co co co i(r 1 — X+r 2 ~ Y+7- 3 — Z)+i2?r(r 1 q+t 2 ^+t 3

c5) - I I II I I -* - - - dr ldr

2 d, 3

. 64a=-^=-^=-co -co -co -co [E2. ((riJ[

} 2 +(r ^

} 2 +(r3j } 2 }]
7T 7T 7T Using variable changes k = t, -

, k=T 9 ~

, k=r,~ , we can transform the b b x

x a y b z

d c integration to spherical coordinates. The integration with respect to 6

and <£, can be integrated directly, leaving

00 CO CO CO CO CO CO I I I f(2*r.2TMc,2piO = (^ ) I I X

^

J

k Sln(kR)

dk, (14)

m=0n=0p=0 ,.

3 a=-oo^=-oo^=-oo R . r

2 . 2. r

647T (k -k ) o

where k=(k2 +k2 +k2

) 1/2 and R = R(a,B,0 = [ (X+2aa) +(Y+2b£) +(Z+2c£) ] 1/2

. x y z This integral is known exactly as

J

k Sln(kR)

dk = -! e ° . (15) 0-2 2 2 (k - k ) o Therefore, using the Poisson summation formula, we are able to obtain an alternate expression for the summations of f (m,n,p) , i.e.,

ooooco ooooooik R(a,/9,£)

I I I f(2mir.2nir,2pir) - ^

I I I

- e

° . (16)

m=0n=0p=0 8 a=-«>;9=-co£=-oo 4 tt R(a,/9,0

Since f(m,n,p) is only one of the decomposed component of ^(r)^(r'), we can perform the similar transformation to every f (m,n,p) , and recombine to yield

oo co co ^(r)^(r') abc co « <x> 8 £ ikR^a.0,0

G(r,r')= III -77 r- =-

^-

I I I I / R

/ « n

' m=0n=0p=0 k

2

- k

2 8 a—«v9—£-«i-l 4" R

i (a,/i,?)

o a where

R

i = [(X

i

+2aa)

2 +(Y

i

+2b^)

2+(Z

i

+2c0 2

] V2 , (17a)

,(x-x');i=l,2,3,4 ,(y-y');^=l,2,5,6 , (z-z ' ) ; JL-l , 4 , 6 , 7 i '(x+x');i«5,6,7,8 ' T ' (y+y ' )

; i-3 ,4 , 7 , 8 ' i ' (z+z ' )

; i-2 , 3 , 5 , 8
The right side of eq (17) is a summation of the free space Green's function

due to sources located at distance R. (see fig. 2). These sources

correspond to the image sources resulting from the reflection at the cavity

walls. Therefore by the use of the Poisson transformation we are able to

obtain a solution which has the physical interpretation of rays emanating

from the various image sources. A result similar to eq (17) was obtained by

M. Hamid and W. Johnson[ll]. Their approach was slightly different in that

they started with the right side of eq (17) by invoking the image theorm.

The Poisson summation formula was then applied to obtain a modal

representation of the Green's function.

10
4. Finite Poisson Transformation

For finite sums over arbitrary intervals, the Poisson summation formula

can be expressed as [12]

N oo N+l

X f(i) = X e J f(r+a-l/2)e dr

i=n i/=-« n N+a+1/2 . - I J f(r) e l2

^T dr, (18)

i/=-oo n+a-1/2

where f(x) is a function of real variable x such that f(x) possesses a Fourier series expansion over any interval in the range n-a-1/2 < x < N+l-a.

N and n are integers such that n < N, and a is any real number such that |a| < 1/2.

For a=0 , the formula in 3D form can be expressed as MNP III f(m,n,p) = (19) m=m n=n p=p

o o

r r o

co oo co M+l/2 N+l/2 P+l/2 i2n(ar 1 +^T 2 +^T 3 ) 1 I 1 S S S f(r 1 ,r 2 ,T 3 )e dr 1 dr 2 dr 3

. Q=.oofl=-oo£=-oo m -1/2 n -1/2 p -1/2

o o

r o

Applying eq (19) to the summation of our f(m,n,p) defined in eq (12), we encountered a difficulty in evaluating the triple integration on the

right side of eq (19). However, this triple integration can be done in the

spherical cooridantes . In transforming to the spherical coordinates, the

finite range of summation on the left side of eq (19) can no longer be

chosen arbitrarily.

Suppose we select a finite number of sets of (m,n,p) such that the

corresponding value of k for each mode falls within a spherical shell of

width (k

? -k ), where k < k < k (see fig. 3). Note that for this

selection, the summation limits for each sum on the left side of eq (19), as well as the integration limits on the right side, are no longer independent

11
of one another. As shown in the Appendix, the integration over this special

finite range can be approximated by transforming to the spherical

coordinates. Referring to the Appendix, the result of this integration is

oo oo oo 1 I I I f(m,n,p) m A*

I I I jg(M,Oi (20)

S Q=-oo/}=-co£=-co 8i?r o

ik R

&(*.£. O- e_!_[E

1 (iR(k

o -k1 ))-E1 (iR(ko -k2 ))+E

1 (iR(k

o+k1 ))-E1

(iR(k

o+k

2 ))] + R

-ik R

e LlE^-iRCCk +k,))-E (-iR(k +k ))+E 1 (-iR(£ -k1 ))-E.(-iR(k -k9 ))], (21)

1 OZi Oil Oil O L

K.

2 2 2 I/O

where R is again defined as R=[(X+2ao) + (Y+2b£) + (Z+2cf) ] /

, and E (x)

CO -t

1

is the exponential integral of order 1 defined as E

1 (x) = J" e dt . The

x t

spherical shell S under the summations represents all the modes (m,n,p)

such that k1 < k < k_ . Combining eqs (16) and (20), we can write

co oo oo oo oo oo I I I f(m,n,p)=I £ I f(m,n,p) + f^ III I<a,/3,£). (22)

m=0n=0p=0 S 8 a=-«>/9=-co£=-oo

where

ik R

i(<*,0,O = - e ° + J_ g(o,/9,|).

4ttR _ 2

07T

Performing the similar transformation to every decomposed component of

4>(r)(f>(r' ) and combining with eq (17), the Green's function can now be

expressed as

12
oo co co 4>{x)4>{r' ) £ i i ~2—~""

(23)

m=0n=0p=0 k - k

o a

<f>(r)<f>(r') co oo co 8 p lk

oRi , 1 III—2 — + T l l l l (>1) l^rtT T8(8

^

:R1^'

S k - k Q=.oo

yg=-co^=-coi=i WR

i 8tt i O O Q

where R. is defined in eq (17a), and g(a,/9,£;R ) is defined in eq (21) with

R replaced by R.. For ease of identification, we refer to the finite range

sum of eq (23) as the mode sum, the second sum involving (a,/3,£) as the ray

sum, and the sum on the left side of eq (23) as the triple sum. For the special case of real instead of complex wave number k , the

expression for g(a,/?,£;R) becomes

2icos(k R)

g(a,/3,0 - !_[Si(R(k -k_))-Si(R(k -k.))+Si(R(k +k.))-Si(R(k +k ))]

D

o z ol ol o z

2isin(k R) + l_[Ci(R(k -k.))-Ci(R(k -k_))+Ci(R(k +k_))-Ci(R(k +k1 ))] I (24)

D ol oz o/ ol

K

x . where Si and Ci are the sine and cosine integrals defined as Si(x)=J dt

,

t

CO Ci(x)=-f dt. If we choose S to be symmetrical about k so that k =k - J o

J o 1 o x t

7, and k_=k +7, where 27 is the summation shell width, eq (23) can be

simplified to

^(r)^(r')

G(r.r') « I I I — — + (25)

S k - k

O O Q

co co co 8 cos(k R )

8 Q=-oo^=-co^=-coi=l l

13
where

cos(k R.)

A„= f~

JL- [Si(R.(2k +7 ))-Si(R.(2k -7))] - (25a)

x , Z- x o x o

sin(k R ) [Ci(R„(2k +7 ))-Ci(R.(2k -7))].

i

L v r o

(// v r o 4* R

i

For 7 « k , A. contributes only a negligible amount. When k R„ is small,

o x

J ox

^ -l cos(k R n ) , n

. the ray sum is dominated by o x , and when 7R. is large, the image

2 terms are oscillatory and are weighted by (l/R.7)

.

A balancing effect exists between the mode sum and the ray sum. When

we increase 7, i.e. increase the bandwidth for the mode sum, the number of

modes that fall within the band will be increased. Therefore the

contribution from the mode sum will be increased at the same time the

2 quantity (l-_Si(R_7)) decreases, thus reducing the contribution from the ray

7T sum. When the dimensions of the cavity are large and the observation point

is close to the source point (but not close to the wall) , the ray sum will

cos(k

Q

R

o

) 2 2 2 1/2 be dominated by the self term, , R = [(x-x') +(y-y') +(z-z') ]

. 4ttR ° o For a very large cavity, 7R. can be large for all image terms if 7 is not

too small, so the contribution from all the images is of the order ( ) R

i^

smaller. Therefore for a source located not near the wall, we can approximate eq (25) further to yield

14
oo oo oo y y y

<j> (?) <j> (r ' ) _ yy y

4>(r)4>{x')

_ ^^

m=0n=0p=0 ,2,2 S .2 ,2 ^ k

o

- k

a ok-ko a

, cos(k R ) !^< If. (i-£si(,R

o

>) - Aj,

8 4ttR n

where the correction,

cos(k R ) A = ° ° [Si(R (2k +7 ))-Si(R (2k - 7 ))] - (26a)

O 2 O O O O

4tt R

o sin(k R ) _!L.[Ci(R

o (2ko+7 ))-Ci(Ro (2ko

- 7 ))]

2 4tt R

O

contributes a negligible amount. Numerical data showing the closeness of

this approximation will be presented in the next section, along with the

criterion for choosing a minimum 7.

Although eq (26) is only an approximation to the exact expression,

addition of a few or more images will not necessarily improve the

approximation. This is so because while the higher order images are

2 decaying at the rate of 1/R , the number of images is increasing at the rate

2 proportioned to R . Therefore the summation of the remaining terms is

likely to be a slow but bounded oscillatory term of order 0(1) . This

contribution is small only because the mode sum usually has a large

2 2

- 1 amplitude, i.e. (k - k ) » 1 near resonance. As will be seen in the

o a numerical examples in the next section, retaining only those images with

k R « 1 is sufficient to yield satisfactory agreement with the numerically

"exact" answe r

For the case when the cavityJ is lossy,J k is complex, i.e. k =k +i7 , o 000

where 7 is usually associated with the loss in the cavity. Expressions for

15
g(a,/3,£ ;R.) for | -y R | » 1 and |k R| « 1 are given respectively in the

Appendix

5. Numerical Examples

To simplify computations, we assume k is real and the cavity is cubic.

The frequency of operation is fixed at 1 GHz. Two cases will be considered.

In case 1 we fix the source point near the center of the cavity and vary the

observation distance. In case 2 we vary the source point along a vertical

axis and show that when the source point is close to the wall, the first

image must be included in eq (26) to achieve a good approximation.

co co oo In evaluating the triple sum, Z Z Z is first reduced to a m=0n=0p=0 2 ,2

o a double sum using the known summation result[13],

jjj cos(nx) = . _1_ . *cos ( x -*)e

; < x < 2tt. (27)

n=0 2 2

2 /l 2£ sin7r£

2 2 2 The variable £ in eq-i (27) becomes imaginary o when (k + k ) > k . For j xyo

imaginary £, the summation function becomes exponential functions that decay

rapidly as either k or k approaches infinity. Thus the resulting sums can

in fact be truncated. At the expense of long computation time, the reduced

summations are summed with indices extending from to a large number M. To

minimize error, care is taken in the determination of M to ensure that the

remaining sum from M to » is neglibible compared to the sum from to M.

Typically 90000 terms are needed to yield an error of less than 1 percent

for k R > 0.15*.

o o The length of the cubic cavity is arbitrarily chosen to be 15.23A.

Unless indicated otherwise, the ray sum consists of only the self term (eq

26).

16
5.1 Case I: Centered source point

With the source point at the center, figure 4 shows the variations of

the mode sum, the hybrid sum, and the triple sum with k R varying from 1

On

to 107r, and figure to 5 has k R varying from 0.2n to 1 . On- . R is the distance

o o

J ° o between the source and the observation point. 7 is chosen to be (0.01)k ,

which corresponds to 831 modes for the size of cavity chosen. Within the

range of small k R , especially at k R < 1

to On , the self term plays an oo rv oo r j

important role. With just the self term included, the hybrid sum provides a very good approximation to the triple modal sum. As k R increases, the self term loses its dominant effect. At large 00 ° kR0000 , i.e. kR > 5n , every term in the ray sum, including the self term,

becomes very small. Therefore the contribution on the right side of eq (26) comes directly from the finite mode sum as is evident in figure 4.

At the intermediate range of k R (0.4tt < k R < 57r) , the contribution 00 00

from the self term is losing its dominant effect, but it is not quite small

enough to be totally negligible. To further close the gap between the

hybrid sum representation and the triple sum, one either has to sum all of

the image terms or increase the bandwidth to increase the contribution from

the mode sum. Figure 6 shows the effect of increasing bandwidth for k R in

the range . 9n < k R < 1

On . As can be seen, the process of increasing

bandwidth produces a very slow converging effect; at the same time the

computation time for the mode sum increases proportionally to the number of

modes involved. Since neither summing more images, as discussed earlier, nor increasing the bandwidth provides a feasible way to close the gap, we may have to accept the slight deviation from the exact value in exchange for

long computation time for k R in this intermediate range.

5.2 Case II: Off -centered source point

17
Figures 7 and 8 show the variations of the mode sum, the hybrid sum,

and the triple sum as the source point r is varied. The self term k R

o o remains fixed at 0.27T throughout, while k R.. is varied from . 3n to 107r (see

fig. 9). With the first image term included, eq (26) becomes

00 00 00 ^(r)^(r')

I I I

' (r) * (r '> « III

m=0n=0p=0 r ,2,2 k-k S ok-k ,2,2

(28)

O Q

, cos(k R ) I (-D \ (l-_Si(R 7)) - A \, 8 i-0,1 4wR 7T

cos(k R ) sin(k R ) A^ ° [Si(R

i (2ko+7 ))] - !__[Ci(Ri

(2k

o+7))-Ci(R

i

(2k

o

- 7 ))]

4tt R

i

4tt R^

R

q= [(x-x')

2 + (y-y')2 + (z-z')2

] 1/2 (28a)

2 2 2 1/2 R1= [(x-x') + (y-y') + (z- (2c-z ' ) T] 1/Z

When the source is close to the wall, i.e. when k R.. is small, the

o 1 addition of the first image term becomes essential. Figure 8 shows that,

without the addition of the first image term, the self term alone is not

enough for the equality to hold in eq (26). As the source is moved away

from the wall, i.e. k Rn is increased, fieure 7 shows that the contribution

o 1 ° from the first image in eq (28) becomes small (almost negligible at k R.. >

6n) . With large k R,, we revert back to case I where the self term is

o i

dominant

In the above computations, we have chosen 7 to be (0.01)k . Increasing

7 increases the number of modes in the band, which may increase the

18
computation time if the number of modes in the mode sum is large. However,

while decreasing the shell bandwidth may decrease the computation time, it

may also introduce an approximation error which may cause eq (20) to become

invalid if 7 is too small. To show this we must go back to the derivation

of eq (20) in the Appendix.

In figure A- 2 we show the approximation made for the shaded grid area with the spherical shell. The shaded area represents the range of

integration for the function F(a, /?,£)• Although we can arbitrarily choose

the number of grids to match the shell, each grid has a minimum width since

the minimum increment of (m,n,p) must be 1, i.e. the minimum summation

interval must be from (m ,n ,p ) to (m +l,n +l,p +1). The minimum grid

o o

r o 00*0

width is therefore 7r/a where a corresponds to the smallest dimension of the

cavityJ size. For our choice of a-=15.23A and 7=0. 01k , we have a shell width

o to minimum grid width ratio of approximately 0.6, i.e.

width ratio -= 7 « 0.6. (29)

7r/a

This width ratio of 0.6 was determined numerically to be adequate in

approximating the grid area with the shell. We need not go to a full width

ratio of 1.0 because the integrand F(a,/3,£) has the greatest value at k near k , and it decays down as k is moved away from k . As the width ratio is

decreased below 0.6, the approximation illustrated in figure A- 2 becomes

poor, and eq (20) (and thus eq 26) becomes a poor approximation.

To illustrate the effect of different shell widths, figure 10 shows the

variation of the hybrid sum for 0.3;r < k R < 0.4tt as the width ratio is J 00

decreased below 0.6. With a very small width ratio, the deviation between

the hybrid sum and the triple sum is indeed not acceptable. A width ratio

of 0.6 represents a conservative choice in that a smaller width ratio of 0.4

or 0.5 may work just as well as 0.6. For width ratios greater than 0.6, we get into the region of slow convergence and increasing computation time.

This tradeoff does not seem to be worthwhile for choosing a width ratio

greater than 0.6.

19
6. Conclusion

In this report we have shown that the Green's function for a rectangular cavity in the modal representation form can be transformed into

a hybrid representation consisting of a finite mode sum and a sum over all

the images. This hybrid sum was effective because it allowed for the

disposal of all the distance image terms without suffering an unacceptable

loss of numerical accuracy. When the observation point was very close to

the source point, and when the source was not close the wall, we have shown

numerically that retaining just the self term in the ray sum was sufficient

to yield a good approximation of the triple modal sum. When the source was close to the wall the first image term became important. The hybrid

representation developed in this report is valid for either real or complex

k . Except for the requirement that the bandwidth chosen for the mode sum

is not too small (width ratio above 0.6), the hybrid representation is in

general a good approximation of the modal representation, and it possesses

unique properties that allow for feasible numerical evaluation.

With this alternate representation, the effect of a scatterer in a large

rectangular cavity can now be examined numerically. The scattered field can

be computed once the induced current on the scatterer is known. As

illustrated in [14], this hybrid Green's function is most useful in the

numerical computations of the induced current J and the corresponding

scattered field for a simple -structured scatterer in a rectangular cavity.

20
References

[I] Corona, P.; Latmiral, G.; Paolini, E.; and Piccioli, L. Use of a reverberating enclosure for measurements of radiated power in the

microwave range. IEEE Trans, on EMC, EMC-22: 54-59; 1976.

[2] Liu, B. H.; Chang, D. C. ; and Ma, M. T. Eigenmodes and the composite

quality factor of a reveberating chamber. Nat. Bur. Stand. (U.S.) NBS

Tech Note 1066; 1983 August.

[3] Ghose, R. N. Microwave circuit theory and analysis. New York, NY:

McGraw Hill; 1963.

[4] Harrington, R. F. Time-harmonic electromagnetic fields. New York, NY:

McGraw Hill; 1961.

[5] Felsen, L. B. Hybrid formulation of wave propagation and scattering.

Dordrecht, The Netherlands: Martinus Nijhoff; 1984. 3-21.

[6] Gao, T. F. ; Shang, E. C. The transformation between the mode

representation and the generalized ray representation of a sound field.

J. Sound Vib. Vol. 80: 105-115; 1982.

[7] Kamel, A.; Felsen, L. B. On the ray equivalent of a group of modes. J.

Acoust. Soc. Am., Vol. 71: 1447-1452; 1982.

[8] Felsen, L. B.; Kamel, A. Hybrid ray-mode formulation of parallel

waveguide Green's functions. IEEE Trans. AP, AP-29: 637-649; 1981

[9] Ishihara, T.; Felsen, L. B. ; Green, A. High frequency fields excited

by a line source on a perfectly conducting concave cylindrical surface.

IEEE Trans. AP, AP-26: 757-767; 1978.

[10] Courant, R. ; Hilbert, D. Methods of mathematical physics. Vol. 1 New

York, NY: Interscience ; 1962.

[II] Hamid, M. A. K. ; and Johnson, W. A. Ray-optical solution for the dyadic

Green's function in a rectangular cavity. Electronics Letters, Vol. 6,

No. 10: 317-319; 1970 May.

[12] Crowley, B. J. B. Some generalisation of the Poisson summation

formula. J. Phys . A: Math. Gen., Vol. 12, No. 11: 1951-1955; 1979.

[13] Collin, R. E. Field theory of guided waves. New York, NY: McGraw

Hill; 1960.

21
14] Wu, D. I. On the effect of a rotating scatterer in an over-moded

rectangular cavity. Ph.D. dissertation, Univ. of Colorado, Boulder, CO,

May 1987.

22
APPENDIX

Summing Over Finite Intervals

Consider a finite range triple summation defined as MNP MNP H— X+^f Y+^ Z)

Q = I I I f(m,n,p) =111 _e

&

_ °

C

_ . (Al) m=m n=n p=p m=m n=n p=p ,~2 , 2 V o o

r r o o o

r r o (k - k ) o a

7T 7T Applying the finite Poisson formula (eq 19), and letting k =t x — , k -r 2 ~

k =t 3 ~ , eq (Al) becomes

oo oo oo x V Z Q-<_)X I I /

2 /

2 /2 F(a,/8,0 dk dk dk , (A2)

where

647r

3 a—/J—*— X]

_ yx Z]

_ x

y

z

x1= (m

o -l/2)^ , x

2= (M+l/2)* , a a

y - (n -1/2)1 , Y9 = (N+l/2)* , (A3)

b b

z^ ( Vq -1/2)Z , z

3= (P+l/2)* ,

i[k (X+2aa)+k (Y+2b£)+k (Z+2c|)]

F(a,/3,0 = e X

I \ . (A4)

r2 „ 2 .2 . 2 . k - (k + k + k ) o x y z

The summation intervals on the left side transform directly into integration

limits on the right side with a small amount of shifting given by ±1/2. A

and

23
typical range of integration corresponds to the shaded area in figure A-l

with each grid on the figure corresponding to a different set of summation

interval (m ,n ,p ) to (M , N ,P ). o o o o o o Suppose we now arbitrarily select a finite set of grids (or summation

intervals) such that they are clustered around k as shown in figure A- 2.

Then

MNP

rrr+-+rn -O rO rO

MNP

i Si Si m n p

o o o mi n

i P i

J

f(m,n,p) (A5)

CO CO 00 (_5L) X I I

647T

3 Q=-00^3=-00|= gridl

+ J J s

gridi

F(a )y9,C)dk dk dk , x y z

We now make the approximation,

[/ I /+•••+/ J J]F(a,)8,e)dk dk dk « J X jF(a,/9,Odk dk dk , (A6)

gridl gridi x y z V

x y z

where V is the volume of the spherical shell bounded by kn and k~ , and

o L Z MNP

irrr -OvOvO+ ... + Wirrrifon.n.p) - j j s f(«.n.p>. (A7) m n p

o o o miniPi

where the S notation represents all the modes, (m,n,p), that fall within

the spherical shell. Combining eqs (A6) and (A7) above, we get

I I Ef(m,n,p).

S (A8)

oo oo oo

647T

! Q=- 00^3= -oo£ = -oo V

i[k (X+2aa)+k (Y+2b£)+k (Z+2c£)]

kox- (k + k + k ) y z

dk dk dk . x y z

Transforming to spherical coordinates, and letting

24
k

2= k2 +k2 +k2

, dk dk dk = k

2 xyz xyz sin0, dkdfl, d^ , k kk

k = ksinfl. cos<i. , k = ksinfl. sin^. , k = kcosfl. , (A9) x k k y k k z k

2 2 1/2 -1 Y+2biS 2 2 1/2 p=[(X+2aa) +(Y+2b£) ] L/

, <f> - tan U

Z °P

) . R=[p + (Z+2cO ] X, ap X+2aa

we obtain

I I If(m,n,p)

S

abc ,k2 k

2 dk ,* ,2?r ipksin* cos(tf +0 )+ikcos0 (Z+2cO

«<_>/ k dk

I sin*,d*, J e k k a£ k

, . 3 k, r

2 . 2 K K K

647r 1 k -k

o c

abc u^f

k2

!^<g) dk

3 k ~? 2 64tt x R(k -k ) o

-( ab° )g(a,l,P (A10)

2 64i7r

where

ik R

z(a.8.n=e ° fE.fiRfk - k,) ) -E. (iR(k -k. ) )+E. (iR(k +k ) ) -E . (iR(k +k ) ) ] +

D

1 o Z 1 oil oil o Z K

-ik R

e ^_[E.(-iR(k +k_))-E1 (-iR(k +k1 ))+E 1 (-iR(K -k

1 ))-E.(-iR(k -k9 ))], (All)

D

1 o z 1 oil oil o Z

or equivalently

2icos(k R)

z(a.B.n= ° [Si(R(kft -k-))-Si(R(ko -k1 ))+Si(R(k +k-))-Si(R(k +k9 ))]-f o Z ol ol o Z R

2isin(k R)

l_[Ci(R(k -k.))-Ci(R(k -k.))-Ci(R(k +k.))+Ci(R(k +k_))]. (A12) Ol OZ Ol O A K

25
CO r -t

E is the exponential integral function defined as E- (z)=J e dt . Si and Ci

z t

z . . smt

are the sine and cosine integrals defined as Si(z) = J" dt , and

" t

cost Ci(z) = -J dt.

z t

Therefore using Poisson summaton formula, we can write

I I Xf(m,n,p)«(

abC

) J I I g(a,fi.Z). (A13)

S ,. . 2 a=-oofl=-oo£=-

O 64l7T r *

00

If we let k =k +i7 , kn =k -7. , k~=k +7. , then for small argument, i.e. Ik Rl

o o o 1 o 1 2 o 1 b ' o

' « 1, E- (z)=-7-ln(z) where 7 is the Euler constant, g(a,£,£) becomes

isin(k R) 2k +7 (1+7../7 ) g(a.£.0=__!_[-2i7r+4tan" i (7 H. )+21n( ° °

° )]. (A14)

R 2k +7 (i-7,/7 ) o o 1 o

n . 1 • TM,^i^./x 7r cosz _.. . smz , For large argument, i.e. |7 R| » 1, Si(z)=_ - , Ci(z)= , and using

°

2 z z expression A12, g(a >yi3,0 becomes

. . ... ik R

S (a '^°

g - e ° + (A15)

. cos(R(k +7,)) cos(R(k -k.)) cos(R(7 1 -k )) cos(R(7n +k )) . x

f °

1 + °

x L ° +

1 °

i + o(JL>.

4tt R

2 n2 7, - i7 7i + 17 2k + i7 + 7. 2k + i7 - 7.. D3 R

x o 1 'o o o 1 o o 1 R

26
Figure 1: A rectangular cavity with a perfectly conducting

scatterer

.

27
J.

I

I 4-

1

J

I "T

I -I.

- ^

I

I Y^x'.y'.z')

I

T

__l

--T--+-

I I "T 1"

-4.--X--J

• I • I • ' • --! r-"i —

x-y PLANE

Figure 2: Equivalent image sources in free space,

28
kj k k 2

Figure 3: A summation shell

29
CENTERED SOURCE POINT

Li

D

_l

<r>

_j

U

M

a:u

z

D

Z

10 1 1 1 1 1 1 1 1 1

5 -

-5

-10

n "\ Mr ^®v //

* -

\ \ -

-15

-20

* * TRIPLE SUM

# # MODE SUM

HYBRID SUM

-

-25 1 t I I 1 I I ! 1 Itt 2ir 3w 4tt 5 7r 6tt 7 it 8tt 9tt 10tt

^0^0

Figure 4: Comparison of the mode sum and the hybrid sum with

G(r,r') for centered source point with 1.0* < k R < 10*

o o

30
CENTERED SOURCE POINT

Ld

D

_1

CE>

_l

CEU

M

q:

u

r

D

z

10

-10

-20-

-30

*- * TRIPLE SUM

# # MODE SUM

O O HYBRID SUM

0. Itt 0.2w 0.3tt 0.4tt 0.5ir 0.67T 0.7* 0.8*r 0.9* U

Figure 5: Comparison of the mode sum and the hybrid sum with

G(r,r') for centered source point with 0.2w < k R < I.Ojt. o o

31
Id

Z)

_J

CE

>

_l

CE

U

M

Ld

z

Z)

z

-1 -

-2

-3 -

-4

-* TRIPLE SUM

HYBRID,

-a HYBRID,

-b HYBRID,

BW=0. 02K,

BW=0. 03K.-

BW = 0. 05K,

c HYBRID, BW-0. IK,

-5

-G

0. 8rr 0. 977- llT 1 . lTT knR0^0

Figure 6: Comparison of the hybrid sum with G(r,r') for different

summation bandwidths.

32
u

D

_J

GE

>

CE

U

M

u

z

D

OFF-CENTERED SOURCE POINT

10 1 l l l l l I I 1 # --# MODE SUM @ @ HYBRID (SELF)

* --* TRIPLE SUM HYBRID (SELF+1IM)"

*^^*\ —

-10 >•—

d

-20 —

-30 — \

/<w^yv^ ^*^>^^f ^=*~^*^

-40 l

1 1 ! I 1 1 1 1 l7T 2?r 3v An Sir Sir ?w Bw Sir 1 Btt

k R

Figure 7: Comparison of the mode sum and the hybrid sum with

G(r,r') for off-centered source point with l.Ow £ k R t < 10w

33
OFF-CENTERED SOURCE POINT

Id

D

_J

CE

>

_J

CE

U

M

u

z

D

20

10

-10

-20

-30

-40

T T

# # MODE SUM (S @ HYBRID (SELF)

* TRIPLE SUM O HYBRID (SELF+1IM)

1 _L 0.1ir 0.27T 0.3?r 0.4-tt . 5tt 0.Gn 0.7tt . 8tt 0.97T lir k Rj

Figure 8: Comparison of the mode sum and the hybrid sum with

G(r.r') for off -centered source point with 0.3* < k Rn < l.Oir, o 1

34
FIRST

IMAGE

x-z PLANE

Figure 9: A cross -sectional view of the first image distance R. and

the distance R between the source and the obersvation point

o

r

35
Ld

D

_l

CE>

_J

CEU

M

a:u

E

D

Z

-G

-12

-18 -

-24 -

•30

* TRIPLE SUM — HYBRID, WR=0.6 —a HYBRID, HR=0.4 —b HYBRID, WR=0.

1

—c HYBRID, WR=0.0B

WR=NIDTH RRTIO

.2tt 0. 3;r 0.4tt 0.577- 0-Btt

ko^o

Figure 10: Comparison of the hybrid sum with G(r.r') for different

width ratios.

36
Figure A-l: A typical range of integration for the finite sum.

37
Figure A- 2: Arrangement of a finite set of summation intervals

38
flBS-114A (REV. 2-80) U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBS/TN-1312

2. Performing Organ. Report No.' 3. Publication Date September 1987

4. TITLE AND SUBTITLE

An Investigation of a Ray-Mode Representation of the Green's Function

in a Rectangular Cavity

5. AUTHOR(S)

D. I. Wu and D. C. Chang

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

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

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

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)

It is well known that a point-source excited field in a rectangular cavity can be

represented either in terms of summation of modes or in terms of rays produced by

the equivalent image sources. Both representations involve series that are slowly

convergent, so computation of fields inside the cavity is difficult. To obtain

a numerically efficient scheme, a hybrid ray-mode representation is developed here

using the finite Poisson summation formula. The modal representation is modified

in such a way that all the modes near resonance are retained while the truncated

remainder of the mode series is expressed in terms of a weighted contribution of

rays. For a large cavity, the contribution of rays from far away images becomes

small, therefore the ray sum can be approximated by one or two dominant terms

without a loss of numerical accuracy. To illustrate the accuracy and the computational simplification of this ray-mode representation, numerical examples are included with the conventional mode series (summed at the expense of

long computation time) serving as a reference.

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

Green's function; hybrid representation; modal representation; Poisson

summation formula; ray-mode representation; rectangular cavity.

13. AVAILABILITY

[X] Unlimited

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|X | Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

]] Order From National Technical Information Service (NTIS), Springfield, VA. 2216 I

14. NO. OF

PRINTED PAGES

48

15. Price

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