IEEE Transactions on Nuclear Science, Vol. NS-32, No. 6, December 1985

SHORT-PULSE MICROWAVE COUPLING TO APERTURES IN A CONDUCTING PLANE*

P. H. Levy, J. E. Faulkner, and D. L. Shaeffer

Physics International CompanySan Leandro, California 94577

Abstract

Computational studies are made of the diffractionof microwaves by circular apertures in a conductingplane. The fields in and behind the apertures are cal-culated by linking high and low frequency numericalapproximations through the resonance region, using a

solution of the electric field integral equation(EFIE). The transient responses of the apertures toshort microwave pulses with varied rise and decay timesare obtained by Fourier inversion of the frequencydomain responses. Transfer functions are calculated inand behind the apertures. Agreement is found withexact theory and experimental results.

Introduction

The coupling of microwaves to apertures plays an

important role in many problems of applied physics andengineering. The diffraction of incident fields byinadvertent apertures can dominate the electromagneticcoupling to system enclosures at microwave frequencies.Although microwave diffraction problems have beenstudied for many years, until recently[1] only mono-

chromatic plane waves, normally incident on apertureshave been treated. The coupling of short microwavepulses (pulse widths of tens of cycles or less) cannotbe treated adequately by a steady state model, becausethe bandwidth of such pulses is too broad. Except fora few exact solutions for apertures of simple geometricshape, that employ complicated functions of mathemat-ical physics, the resonance response of apertureshas been ignored. 3 Since resonances may generateelectric fields that cause air breakdown in the aper-ture,[4 it is desirable to construct a straightforwardsolution to the aperture problem which treatsresonance.

This paper presents methods of modeling thecoupling of short microwave pulses to circular andelliptical apertures in an infinite, perfectlyconducting plane of infinitesimal thickness. Themethods utilize existing large and small hole approx-imate theories (with important modifications), whichare linked through the resonance region (occurring atka 1.5 for circular apertures, where a is theaperture radius and k is the wave number) usingsolutions of the electric field integral equation(EFIE). This integral equation is solved using amodified version of the computer code EFIE, [5 which wecall EFIE2. The approximate solutions are comparedwith predictions of exact theory and with recent exper-imental data.

It is well known that the classical Kirchoff-FresnelE61 diffraction theory and the Stratton andChu [7] vector representation fail to predict the fieldsin and immediately behind apertures in large conductingscreens. A number of approaches have been developed totreat this problem for the simplified case of a linear-ly polarized plane wave of frequency v, incident on anaperture in an infinite, perfectly conducting plane ofnegligible thickness. Bekefi [8] and independently

Silver and Ehrlich[9] developed a simplified Hertzvector formulation which assumes that the component of

the magnetic field tangential to the screen vanishesbehind the screen. For the circular aperture, thisapproximation is valid only for D/\ ) 1. Bouwkamp 10]has proposed a modified version of Bethe's small-hole theory. However, this approximation also fails inthe resonance region due to the slow convergence of theseries involved. Flammer 121 and Meixner and Andre-jewski[131 solved the circular aperture diffractionproblem using oblate spheroidal wave functions. Levineand Schwinger[14,15] developed a general variationalformulation of the diffraction by an aperture in aninfinite plane, which is valid over a fairly wide rangeof frequencies. More recently, Butler 161 has obtainedsolutions for the transmitted field as a function ofdistance from the aperture using dipole moments, andGraves[17 has obtained solutions using integralequations and a dipole field approximation. Cathey[18]and Ari[191 have developed a set of simplifiedequations and curves, based on physical optics andexperimental data, for estimating the coupling ?f anormally incident plane wave thirough a circularaperture. Except for Flammer s work, which isquite complex computationally, none of these approachestreats the resonance response. Furthermore, whileCathey's[181 approximate formulae may be used todetermine an upper bound on the aperture response,these expressions neglect the near-field oscillationsaltogether. These oscillations may not reach a peakvalue until a distance of 10 X or greater behindapertures with D/\ > 1.

Description of Coupling Problem

The problem considered is that of a microwavepulse, incident on a circular or elliptical aperture inan infinite, conducting plane of infinitesimal thick-ness. The plane waves comprising the pulse arelinearly polarized. The aperture geometry and orien-tation of the incident wave is shown in Figure 1. Thepulse is incident at an angle e to the plane of theaperture. The rise and decay time of the pulse arevariable. The aperture dimension and wavelength arevaried to achieve a broad range of D/X. For smallapertures the fields behind the aperture decay exponen-tially with distance near the aperture and inverselywith distance far from the aperture. For apertureswhich are large compared to a wavelength, the fieldamplitude, on an axis perpendicular to the aperture,oscillates with distance from the aperture as themaxima and minima of the Fresnel pattern[20,21] aretraversed. A

Hj

y

x

Y, Z)

z

Figure 1. Microwave coupling geometry.

*This work was supported in part by the Defense Nuclear Agency, under DNA001-84-C-0007.

0018-9499/85/1200-4333$01.00 ( 1986 IEEE

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The treatment of this problem is applicable onlyto unloaded apertures in thin shields (shield thicknessd << X), with edges that are many wavelengths from therim of the aperture, and with dimensions large comparedto the local radius of curvature of the surface (i.e.,the shield). The effect of finite aperture thicknesson the microwave coupling to slots in a conducting1plane has been addressed bW Bacon and Vittitoe,[2 andAuckland and Harrington. ] The coupling to circularapertures in Werical shields has been treated recent-

ly by Casey.[2 I

The computational methods developed here can beused to predict the fields in or behind apertures ofany size, as a function of frequency, but are invalidat the aperture rim where the solution is singular.

allows one to assume that the normal derivative of theHertz vector in the aperture is equal to the normalderivative of the incident Hertz vector, and thatoutside the aperture the normal derivative is zero.

It is possible to evaluate the above area integralby a line integral, and solve the equtions for thescattered fields in this fashion (see also Neuge-bauer).[271 We have also obtained an analytic form of

the axial transfer function, and a numerical transferfunction with high order accuracy off-axis. The axialelectric field transfer function for D/X > 1, is givenby

-ikz -ikR a2 -ikR ia2 -ikR¢ = e - e + e - e

2 2kR3(7)

Discussion of Large and Small Aperture Coupling ModelsThe models presented here for large and small

apertures (i.e., non-resonant) are based in part on thelarge-hole approximation of Bekefi and Buchsbaum[251and the small-hole theory of Bouwkamp.[10,261 Therange where neither approximation is valid is given by

1/i < D/X < 1 . (1)

This range is important, since resonances are expectedto occur in the neighborhood of D/X 1/i. Thisresonance region is treated by first solving the elec-tric field integral equation (EFIE) for the compli-mentary disk. The fictitious magnetic currents for theaperture are then determined from Babinet's principle,and the fields are calculated by integration of thecurrents over the aperture area. In all cases theaperture response is computed in the frequency domain.The transient behavior is obtained by inverse Fouriertransform of the product of the incident pulse fre-quency spectrum and the aperture transfer function.The magnitude and direction of the transmitted E and Hfields can be calculated at any point inside or behindthe aperture.

The diffracted fields for large holes are

evaluated at any location in the positive half-space(Z > 0) by an integration of the normal derivative ofthe incident Hertz vector over the aperture.

If we are given an infinite, perfectly conductingscreen at z = 0, and a plane wave traveling in thepositive z direction of Figure 1, we can write

^E1 -iwt-ikz(2E =xE e wiz(2)- 0

Biwt-ikz

B =yB0e 3

where

E cB (4)

The Hertz vector for the incident plane wave may bewritten

2u = E/k . (5)

The Hertz vector for z greater than zero is givenby

(6)J1 bul e-ikI - r |~~~u(r)= s- l 0 dA r'

where r is the position vector of the observation pointand r' is the position vector on the plane. In the

above equation, the value of u in the right hand side

is calculated from Equation (5). In the regime D > X,the situation is much like geometric optics, which

where a is the aperture radius, and R = a2 + z2. Forthe off-axis transfer function, it is convenient tocompute the Hertz vector on a grid and then compute thefields by numerical differentiation. This givesaccuracy comparable to that obtained from integrationof the derivatives of the Green's function. Thisapproach permits the fields to be computed anywhere inspace. The field calculations are not limited to oneaperture radius of the z axis as in Buchsbaum andBekefi's[251 treatment.

The singularity of the axial transfer function atk = 0 produces a spurious pole late in the diffractedpulse, when the spectrum of the incident pulse has anon-zero dc component. When the above transfer func-tion was modified by use of a simple high-pass filter,to yield qualitatively correct results as k + 0, thespurious dc component was eliminated since any dc com-ponent of the transfer function then had magnitude zero(as it should have for an infinite, equipotentialplane). Use of a high-pass filter is justifiedphysically so long as the spectrum of the incidentpulse is dominated by high-frequency components.

The diffracted fields for small holes are obtainedby integration of the fictitious magnetic currents(calculated for the complimentary disk) over theaperture. We have developed numerical transfer func-tions on and off axis behind the aperture. Bouw-kamp[263 has developed an analytic transfer function inthe plane of the aperture, which we have used in ourcalculations. The transfer functions for locationsbehind the aperture (z > 0) are evaluated numerically,using recursive techniques. Their generation andnumerical solution are discussed below.

The approach is to integrate the fictitiouscurrents over the aperture to get the fields. It isanalogous to calculating fields from real currents. Ifa is the aperture radius, xs and ys are coordinates onthe plane, and

p2 = x 2 + ys2 < a2I (8)

the fictitious currents are then-x y

k =x I

3n wa _p (9)

k= J-F2 a/_P77y 3it' 2 -]

The numbers are dimensionless (e.g.,'xs is really kxswhere k is the wave number).

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Next definek sinr

Fi = -i dEk sinr

F = Y dEly JJ r

k cosrF -dE2x r

k cosr

F = --r dE2y rJJ

The integral is taken over the aperture. The symbcdenotes the (dimensionless) distance from a point cthe aperture to the observer point. The symbols 12 denote real and imaginary parts respectively.

The results are

2 22x a 2y

B =F + +lx 2x b2 bxby

2 2B -F +yz 2ya xaay by

2F 2F

B = 2x+

ylz bxbz oyoz

62F 62FB = - F - 1x ly2x lx ax2 xby

a2 F 2FB = - F - lx 1y2y ly bxby by2

b2F 62FB

lx ly2z bxbz by6z

E = - 6Fl / z

Ely = F lx/az

Elz= OFly/x- F lx/by

E2x = -2y/az

E2y = aF2x/az

E2z = 6F2y/Ix 2

The fields are calculated by first constructinthe derivative of eikr/r and then integrating overaperture.

Resonant Aperture Model

The response of resonant apertures has beentreated by solving the scattering problem for thecomplimentary disk, and then integrating the fictitmagnetic currents to arrive at the aperture fields.have solved numerically the electric field integralequation for the disk using the modified scatteringcode EFIE2.J283 For a detailed discussion of theelectric field integral equation (EFIE) and itsapplication, see Rao and Wilton.J29]

A plane wave is incident on a circular aperturan infinite conducting plane, as shown in Figure 1.The plane is chosen as z = 0. The center of theaperture is at the origin. We now relate the surfacurrent induced on the disk to the aperture solutioThe case treated here is that of normal incidence,the principle can be applied to an arbitrary angleincidence.

( 10

1l ranand

The plane wave is given by

iwt-ikzE = x E e_

iwt-ikzB = y B e

whereE = cB -0 0

(13)

(14)

The calculation uses Babinet's principle in whichthe conducting plane with an aperture is represented bya disk. The plane wave is first rotated 900. Theplane wave then becomes

.1 iwt-ikzE = y E e_

A iwt-ikzB= -xB e_

(15)

The calculated electric currents are regarded asfictitious magnetic currents in the aperture. Thedetermination of the fields will be described later inthe paper.

Extended EFIE Model

The physical situation modeled is that of a per-fect conductor on which is incident a plane electro-magnetic wave of a given frequency. Using a cartesiancoordinate system, the orientation of the plane wave isdescribed in the input. The perfect conductor ismodeled by plane triangular patches. The user assignsindices and coordinates to the vertices of these

(11) triangles. Indices are assigned to the sides of thetriangles. This is done by assigning a side index andpair of vertex indices. The model then computes thesurface currents and charges by the method of moments.

In the original version of EFIE,[5] the wavelengthof the incident field was arbitrarily set at one meter.This has been modified in EFIE2 to give the capabilityof specifying multiple wavelengths or frequencies. Aprogram has been written to construct the input for thecomplementary disk; this avoids tedious hand calcula-tions. This program (EFDAT) represents the disk as apolygon. For reasons of symmetry, the number of sidesof the polygon is divisible by four. The input con-sists of the sides of the polygon (divided by four) and

(12) the distance from the center of the polygon to avertex. The EFDAT program then constructs the zoning

!X/ay of the problem. For the problem of scattering from adisk with normal incidence, it is clear that if theplane of the disk is the xy plane, then there is a

the symmetry with respect to reflection in either the x orthe y axis. EFDAT constructs the zoning to maintain this

symmetry.

Since the rest of the EFIE2 input is relativelysimple compared to the zoning, it is included in theEFDAT input. Specifically, this consists of the

.iousfrequency (or wavelength) of the incident field and itsorientation.

We

Aplcto toRsnt Apertures

e in

,cen.butof

While the resonance problem may be solved throughthe use of oblate spheroidal wave functions,[12'13] thecomplexity of this approach is a motivation for seekingsimpler methods. The techniques for apertures smalland large compared to a wavelength, discussed in thepreceding section, break down for ka = 1. The EFIEmethod provides an efficient tool for teating thiscase.

The surface currents are calculated, using theextended EFIE model (EFIE2), for the case of the planewave of Equations (13) incident on a perfectly

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conducting disk of radius a. The fictitious magnetic

currents are denoted by the vector J. The disk is

modeled as an octagon constructed from triangular

patches. The current is calculated at the center of

each patch, and then the fields are calculated by

integration over the aperture (disk) area. This method

cannot accurately predict the fields near the edge due

to the geometrical approximation used. However, the

experimental results of King, et al.[301 show that,

near the center, the response of circular and square

apertures is nearly identical.

The electric field at any point in or behind the

aperture, normalized to the magnitude of the incident

field, can be obtained by integrating the product of

the J and the derivative of the appropriate Green'sfunction (a tensor quantity) over the aperture area,

E(x,y,z,w)

E / G(x,y,z,x',ry',-)

J(x',y',w)dx'dy' = A(x,y,z,w) (16)

The time domain response, E(x,y,z,t), is then obtained

by taking the Fourier inverse of A(x,y,z,w) F(w),

where F(u) is the Fourier spectrum of the incidentpulse.

Discussion of Theoretical Results and Comparison With

Experiment

Much work has been devoted to establishing the

field distributions in and immediately behind apertureslarge compared to the incident wavelength, where inter-

ference is pronounced. Andrews'[31] work stimulated

much interest in the subject after World War II, and

Ehrlich and Silver[321 investigated the near-zone field

structure experimentally for circular apertures up to

D = 36 X, confirming the complexity of the field

structure in the aperture first pointed out by Andrews.

Buchsbaum and Bekefi[25] showed reasonable agreementbetween experimental and theoretical field distribu-

tions for circular and elliptical agertures above and

below resonance. Recently Cathey 1 ] has attempted to

develop some simple empirical formulae, based upon this

data, for estimating the coupling of a normally inci-

dent plane wave to circular apertures in thin shields.

The results presented here include circular

aperture transfer functions, calculated through the

resonance region up to aperture diameters large com-

pared to a wavelength. The transient response is also

presented for a double-exponentially modulated sine

wave with various rise and decay times. Like other

workers on this problem, we have concentrated on the

case of a plane wave incident normally on the diffract-

ing structure (see Figure 1), although the models

discussed above can treat oblique incidence and other

types of phase differences over the aperture. For

example, a pulse incident on the aperture after

traversing random media.

The magnitude of transfer functions (ratio of the

tangential component of the electric field to the

incident field, |EX/EoIi as a function of frequency)for circular apertures has been calculated in and at

various distances behind the aperture on axis. These

calculations are plotted in ka space to emphasize the

dependence on the ratio of aperture dimension to wave-

length. Figure 2 shows the calculated in-aperture

2.0

ExIE.

1.0

0.010.05.0

ka

Figure 2. Magnitude of electric field transferfunction in center of circular aperture

(Z = 0).

transfer function (Z = 0) compared with an exact

solution using oblate spheroidal wavefunctions.The oscillatory behavior, starting at about ka = 6, is

due to the diffraction terms in Equation (7). For odd

integer values of D/X there is a maxima in the center,

while for even values a minima occurs in the center.

The solid line in Figure 2 for ka greater than 3

was constructed from the large hole theory. The solid

line for ka less than one was constructed from the

small hole theory. At the time Figure 2 was construc-

ted, we were only able to generate currents (rather

than fields) from the EFIE program. An estimate of the

transfer function at resonance was obtained by examin-ing the EFIE currents. The low frequency and high

frequency portions of the transfer function were joined

to the resonance point by linear interpolation. It is

expected that a proper calculation of the transferfunction from the EFIE currents would more nearlycoincide with the exact theory (dotted line) of

Figure 2.

Figures 3 and 4 show our calculated transferfunctions at Z/D = 0.08 and Z/D = 0.g[ compared withexperimental results of King, et al. °] of Lawrence

Livermore National Laboratory. The measured transferfunctions were obtained by Fourier transform of D probemeasurements behind a circular aperture in a thin

(< 1/16-inch) aluminum plate, illuminated by a video

pulse. The radiating monocone in the LLNL facility

launches a spherical wave; however, over the area of

the plate there is excellent plane wave fidelity. The

aperture radius was 6.37 cm. Although measurementswere made over a broader frequecy range, reliable datawere obtained between 100 MHz and 3 GHz (ka 4 for

this case). The large spikes in the experimental dataat about 200 MHz (ka 0.25) are due to late-time

reflections from the radiating cone back to the screen.

The approximate theory results agree very well

with the exact theory calculations of Bekefi and

Buchsbaumt25] in the aperture, and with experimentbehind the aperture. The small resonance predicted at

Z/D = 0.08 (Figure 3) is not evident in the data due to

the presence of noise. However, the data are consis-

tent with the predicted resonance. No in-aperturemeasurements were available for comparison with the

theoretical predictions shown in Figure 2.

i- APPROXIMATE THEORY

LINKED THROUGHRESONANCE

--_ EXACT THEORY(Buchsbaum and Bekefi)

I- I-

i~~~~

I~N---

0.0

behind the aperture are larger than the fields in thecenter of the aperture due to the contribution of thediffracted wave, as expected. These field enhancementsare comparable to those of resonant circular aperturesin and very close to the aperture. The peak in theFresnel pattern for the case D/X = 10 is not shown. Itoccurs at about Z = 30 X. If D = nX, where n is aninteger, the number of maxima in the diffractionpattern behind the aperture is given by D/2X for neven, and by 1/2 (D/X - 1) for n odd. The total numberof maxima and minima is equal to D/X - 1.

2.0

1.5

1.0

0.5Figure 3. Magnitude of electric field transferfunction at Z/D = 0.08.

0.0

D/X = 10

IES.0 10.0 20.0

Z/I

1.0

E.Eo

0.5

0.00.0 2.0

ka

E0 1.0

0.5

0.0

2.0

1.5

Ex 1.0

.Eo

0.5

4.0

Figure 4. Magnitude of electric field transferfunction at Z/D = 0.8.

The calculations indicate that the resonantfrequency occurs at about ka = 1.5, or X = 1.33 na,rather than at X exactly equal to the aperturecircumference as observed by Hoffman, et al.[331 andKing, et al.J30] for high aspect ratio apertures. Thedifference in resonant frequency is due to the factthat circular apertures have a small aspect ratio, and,therefore, the resonance does not occur at X equal tothe circumference as it does for high aspect ratioapertures. In fact, the experimental results forcircular apertures presented by King, et al.[333 showresonance at ka 1.5, in good agreement with ourtheoretical predictions. Both the predicted andmeasured magnitude of the transfer function increasesat the resonant frequency as one moves closer to theaperture, as expected (at Z/D = 0.8 the resonancedissappears completely). However, the predictedmagnitude of the transfer function decreases rapidlyover a small distance (from Z = 0, Figure 2, toZ/D = 0.08, Figure 3) behind the aperture. The magni-tude of the transfer function increases slowly abovethe resonant frequency at Z/D = 0.8. This is due tothe fact that the resonance has decayed significantly;so the response is dominated by the higher frequencyfield penetration (at D/X = 1, or ka = i, a maxima inthe field distribution occurs on-axis).

The electric field as a function of distancebehind the aperture on axis is shown for several largeapertures in Figure 5. The near-zone response exhibitsthe classic Fresnel patterns. The fields at many x

0.00.0 10.0

ZIX

Figure 5. Electric field asdistance behind afor various D/X.

a function ofcircular aperture

The transient response of resonant and above-resonance apertures excited by a double-exponentiallymodulated sinusoid of the form

(e a -e Pt) sin v t0

has been studied as a function of pulse rise and decaytimes. The ratio 0/a was varied from 5 to 100 for aresonant circular aperture, while a/v0 was fixed at 0.2(where v0 is the pulse carrier frequency). As theratio of 0/a is increased from 5 to 100 (i.e., as therisetime is reduced), the magnitude of the peak elec-tric field in the center of the aperture increasesroughly 60%. It is very close to the predicted steadystate value for P/a = 100. Figures 6 to 8 show theincreasing amplitude of the electric field with in-creasing 0/a, plotted as a function of the dimension-less variable vot. Of course, as p goes to infinityand a goes to zero the transient response approachesthe steady state value. Compared with the response of

2.0

1.5

ExE.

1.0

0.5

0.0

0.0

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2.0

ka

4.0

(17)

1.0

0.5

E0

-0.5

1.0

0.0

0.4

E.

E.

5.0 10.0

'. t

Figure 6. Resonant aperture response for :/a = 5,

a/vO = 0.2.

2.0

1.0

E.

Eo

-1.0

-20

0.0 5.0 10.0

". t

Figure 7. Resonant aperture response for:/a = 20, a/vO = 0.2.

1.0

E1

- 1.0

0.0 5.0 10.0

'. t

Figure 8. Resonant aperture response for

:/a = 100, a/vO = 0.2.

a large (D/Xo = 2) aperture for :/a = 5, shown in

Figure 9, the peak electric field is larger and appearsto occur later due to the time required for oscilla-

tions of the electric field to reach a maximum.

0.0 10.0 20.0

Figure 9. Large aperture (D/Xo = 2)response for 0/a = 5, a/vo = 0.2.

Conclusions

The coupling of microwave pulses to circularapertures has been studied throughout frequency space,

including resonance (ka = 1.5). New approximatesolutions have been developed which agree well withexact theory predictions and experimental results inand behind circular apertures. The results may be used

to predict the diffracted fields as a function of

aperture physical size versus frequency, for unloadedapertures in thin shields. The predicted resonance

behind the aperture is not evident in the experimentaldata due to noise. However, the resonance isconsistent with the data. Further experiments withwell-controlled monochromatic sources and aperturesphysically large, to avoid screen-probe interactions,are necessary to determine whether or not significantresonance occurs for circular apertures.

The diffraction patterns observed behind largeapertures show that peak enhancements of the incidentelectric field can occur up to greater than 10 wave-

lengths behind the aperture. These enhancements are

comparable to the enhancements observed in andimmediately behind the aperture at steady state

resonance. However, the amplitude of the transientresonance response is lower than the steady stateamplitude for incident pulses with small ratios of

decay to rise time. Also, the peak of the in-apertureresonance response occurs later in time than the above

resonance peak. Given an adequate treatment of air

breakdown, the models presented here can be used to

predict at what time breakdown occurs.

The helpful suggestions of Dr. F. M. Tesche,LuTech, and Dr. D. V. Giri, University of California,Berkeley, are gratefully acknowledged. We wish to

thank Drs. R. J. King and H. G. Hudson of Lawrence

Livermore National Laboratory for generously providingresults of experiments performed in their laboratory.This work was supported in part by the Defense Nuclear

Agency, under DNA001-84-C-0007.

References and Notes

[1] Considerable attention has been devoted to

understanding the transient response of various

canonical objects to a nuclear electromagneticpulse (EMP). However, until the recent resurgenceof interest in microwaves as a defensive weapon,

most work at microwave frequencies (where irradi-

ated objects can be quite large electrically)addressed the steady state diffraction problem.

4338

4339

[2] Circular apertures have been treated using oblatespheroidal functions, and the infinite slit hasbeen treated using Mathieu functions.

[31 Recently Harrington has developed a general solu-tion using characteristic modes, which treatsapertures of arbitrary size and shape. See R. F.Harrington and J. R. Mautz, IEEE Trans. MicrowaveTheory and Techniques, MTT-33, 6, 500, June 1985.

[4] Electric field enhancements as large as 100 orgreater have been reported for large aspect ratioapertures.

[5] D. R. Wilton, A. W. Glisson, and S. M. Rao,EFIE - A Computer Code for ElectromagneticScattering Problems Involving Arbitrarily ShapedConducting Surfaces, Dept. of Electrical Engineer-ing, University of Mississippi.

[6] See for instance: B. B. Baker and E. T. Copson,The Mathematical Theory of Huygen's Principle,Oxford University Press, 1950, second edition,chap. II.

[7] J. A. Straton and L. J. Chu, Phys. Rev. 56, 99(1949).

[8] G. Bekefi, J. Appl. Phys. 23, 1405 (1952).

[9] S. Silver, M. J. Ehrlich, Antenna LaboratoryReport No. 181 (1951), Dept. of Engineering,University of California, Berkeley.

[10] C. J. Bouwkamp, Phillips Research Repts. 5, 401(1950).

[11] H. A. Bethe, Phys. Rev. 66, 163 (1944).

[12] C. Flammer, J. Appl. Phys., 24, 1224 (1953).

[13] J. Meixner and W. Andrejewski, Annalen der Physick(6), 7, 157 (1950).

[14] H. Levine, J. Schwinger, Phys. Rev. 74, 958(1948).

[15] H. Levine, J. Schwinger, Phys. Rev. 75, 1423(1949).

[16] C. M. Butler, Y. Rahmat-Samii, and R. Mittra, IEEETrans. Antennas and Propagation, AP-26, 82,Jan. 1978, and C. M. Butler, A. 0. Howard, and R.D. Nevels, J. Appl. Phys. 48, 4886 (1977).

[17] B. D. Graves, T. T. Crow, and C. D. Taylor, IEEETrans. Electromagnetic Compatibility, EMC-18, 154,Nov. 1976.

[18] W. T. Cathey, IEEE Trans. ElectromagneticCompatibility, EMC-25, 3, 339, August 1983.

[19] N. Ari, D. Hansen, and H. Schar, Proceedings ofthe IEEE, 73, 2, 368, Feb. 1985.

[20] M. Born and E. Wolf, Principles of Optics.Macnillan & Co., New York, 1964, Chap. 8.

[21] F. A. Jenkins and H. E. White, Fundamentals ofOptics. McGraw-Hill, New York, 1957, Chap. 18.

[22] High Power Microwave Research Quarterly ProgressReport, July-September 1984, Sandia National

[23] D. T. Auckland and R. F. Harrington, IEEE Trans.on Microwave Theory and Techniques, MTT-26, 7,499, July 1978.

[24] K. F. Casey, IEEE Trans. ElectromagneticCompatibility, EMC-27, 1, 13, Feb. 1985.

[251 G. Bekefi and S. J. Buchsbaum, J. Appl. Phys. 24,1123 (1953).

[26] C. J. Bouwkamp, Repts. Prog. in Phys. 17, 35(1954).

[27] H. E. J. Neugebauer, J. Appl. Phys. 23, 1406(1952).

[28] P. H. Levy and J. E. Faulkner, PhysicsInternational High Power Microwave Research NoteNo. 9, Nov. 1984.

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