Scale transformation of Maxwell’s equationsand scattering by an elliptic cylinder

Lawrence A. Ferrari

Department of Physics and Astronomy, Ursinus College, 601 East Main Street, Collegeville, Pennsylvania 19426-1000,USA (lferrari500@comcast.net)

Received March 11, 2011; revised April 18, 2011; accepted April 18, 2011;posted April 19, 2011 (Doc. ID 144071); published May 27, 2011

A scale transformation that converts an ellipse into a circle has been suggested in the literature as a method foreliminating the need to evaluate the conventional Mathieu function solution for scattering by an elliptic cylinder.This suggestion is tested by examining the wave equation in the scaled coordinate system and by evaluating thescattering from a thin ellipse for conditions where it is expected that an approximate solution can be obtainedusing the scalar theory single-slit approximation. It is found that, for a plane electromagnetic wave normallyincident on a thin perfectly conducting ellipse, the position of the first minimum in the diffraction pattern, relativeto the central axis, differs by approximately a factor of 7 between the single-slit and the scaled theory approach tothe problem. The examination of the scaled wave equation and the scattering calculation suggests that, becausethe scale transformation generates an anisotropic medium, the use of a uniform medium solution in the scaledcoordinate system is not appropriate. © 2011 Optical Society of America

OCIS codes: 290.0290, 290.5825, 260.1960.

1. INTRODUCTIONOne of the relatively few problems dealing with the scatteringof electromagnetic waves that can be solved exactly is the onewhere a plane wave is normally incident on either an infinitelylong perfectly conducting or dielectric cylinder. The problemof the uniform dielectric cylinder was solved in 1881 by LordRayleigh [1]. The solution for the conducting cylinder can beobtained from the dielectric cylinder solution by letting theindex of refraction N → ∞ or without much difficulty bysolving the vector wave equation directly. At the present timethe solutions of both problems are discussed in many text-books on electromagnetic theory. The more general case ofoblique incidence has also been published [2]. The solutionsare usually written in terms of infinite series of integral-orderBessel functions of various types.

The problem of scattering of electromagnetic waves by aperfectly conducting elliptic cylinder was studied as earlyas 1908 by Sieger [3], but few numerical solutions were eval-uated. For elliptic cylinders the solution is in terms of aninfinite series of both the angular Mathieu and the modifiedor radial Mathieu functions. A numerical evaluation of thescattering by a perfectly conducting thin strip, which is anellipse with the minor diameter equal to zero, was publishedin 1938 by Morse and Rubenstein [4] when newly computedtables of Mathieu functions became available. Yeh [5] ob-tained the exact solution for scattering by a dielectric ellipticcylinder in 1963 for normal incidence and in 1964 [6] foroblique incidence. The solution for the perfectly conductingellipse is also contained in a basic text on Mathieufunctions [7].

The evaluation of the Mathieu function solution [4] dependson the value of a size parameter C ¼ πd=λ, where λ is thewavelength of the incident wave and d is a dimension thatcharacterizes the scattering element; for example, d couldbe the width of a thin strip, or the width of a slot in an opaque

screen. The series solution in this early study converged veryrapidly for small values of C; only three or four terms areneeded as long as C is less than about 4 [4].

Burke and Pao [8] investigate the propagation of wavesthrough obstacles such as buildings in an urban environmentwhere an entrance aperture such as a door or window couldbe approximated by a slot, which in turn is modeled by a thinellipse. They point out that, with the Mathieu function solu-tion, computation times become very long for elliptic cylin-ders whose major diameters are greater than about 20λ or30λ with C ¼ 20π and 30π, respectively. The solution wasevaluated for slot widths up to 40λ with good agreement witha two-dimensional moment method solution. It was foundthat, for accuracy sufficient to avoid a perceptible differenceof curves on a plot, the number of Mathieu functions summedshould be at least four times the slot width in wavelengths.They were unable to get accurate results for a slot widthof 60λ.

While software packages such as Maple and Mathematicacan evaluate the Mathieu functions, there is also the possibi-lity that a software package might contain errors in evaluatingthe radial functions [9] as the argument gets larger than aboutthree or four. In addition, it has been pointed out [10] thatexpressions traditionally used to evaluate the radial Mathieufunctions result in inaccurate function values over someparameter ranges due to unavoidable subtraction errors thatoccur in the series evaluation. For many of the expressions,the errors for small orders increase without bound as the sizeparameter increases.

A possible solution to the difficulties associated with theevaluation of the Mathieu function solution has been sug-gested in the literature [11–13]. The suggestion is to use a scaletransformation to transform an ellipse into a circle and thenfind a coordinate system that corresponds to one where aknown solution to the cylinder scattering problem exists. This

L. A. Ferrari Vol. 28, No. 6 / June 2011 / J. Opt. Soc. Am. A 1285

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solution is then transformed back to the original coordinatesystem as the solution of the scattering by an elliptic cylinder.This assertion, if correct, turns a difficult computation pro-blem into one that is relatively easy. The scale transformationconcept was also used to investigate the scattering character-istics of an ellipsoid by transforming the ellipsoid into asphere [12].

In the first part of this work, the concept of the scaletransformation and its application to Maxwell’s equations isreviewed. The scale transformation leaves the form ofMaxwell’s equations unchanged [14–16], but in this process,a uniform medium is transformed to an anisotropic medium.In order to test the assertion made in [13], the solution for thescattering by a perfectly conducting cylinder, obtained in auniform medium, and applied to an ellipse transformed intoa circle, called the scaled theory in what follows, is examinedfor conditions where it is expected that an approximatesolution can be obtained using the scalar theory single-slitapproximation. The scattering of electromagnetic waves nor-mally incident on a thin perfectly conducting elliptic cylinderwhere the ratio of the major axis to the minor is 7 and the ratioof the maximum diameter to the wavelength is approximately60 and 120 is investigated. With these aspect ratios, it isexpected that the scalar theory single-slit approximation fordiffraction by a thin conducting strip should give an approx-imate value for the position of the first minimum in the diffrac-tion pattern. It is found that the position of the first minimumfrom the scalar theory differs by a factor of approximately6.75 from the scaled theory. Because of this large discrepancy,it is concluded that the application of a scattering formulaobtained for uniform media cannot be applied directly toan anisotropic medium.

The wave equation for the case where the electric field ofthe incident wave is parallel to the ellipse axis is obtained inthe scale-transformed coordinate system. This turns out to bea very complicated equation. Upon examination, it is foundthat the uniform media solution in the scaled coordinatesystem is correct only when the original elliptic cylinder isactually a circular cylinder, that is, when the scaled mediumis no longer anisotropic.

2. SCALE TRANSFORMATION OFMAXWELL’S EQUATIONSIn uniform media with permittivity εo, permeability μo, and nosources, Maxwell’s equations are given by

∇ × E ¼ iωμ ·H; ∇ ×H ¼ −iωε · E; ð1Þ

where the time dependence expð−iωtÞ has been used, andthe permittivity ε and permeability μ have been written asdiagonal tensors whose elements are all equal:

ε ¼" εo 0 00 εo 00 0 εo

#; ð2Þ

μ ¼" μo 0 00 μo 00 0 μo

#: ð3Þ

In the following discussion the basic notation defining thescale transformation as given in [13] is used.

Consider the scale transformation [13] defined by x ¼ sxx0,y ¼ syy0, and z ¼ szz0, where sx, sy, and sz are constants. It iseasy to show that, under this transformation,

∇ ¼ T∇0; ð4Þwhere

T ¼" 1=sx 0 0

0 1=sy 00 0 1=sz

#: ð5Þ

If we let

E ¼ TE0; ð6Þ

H ¼ TH0; ð7Þ

Maxwell’s equations become

∇0 × E0 ¼ iωμ0 ·H0; ∇0 ×H0 ¼ −iωε0 · E0; ð8Þ

where the new permittivity ε0 and permeability μ0 tensors aregiven by

ε0 ¼" εx 0 00 εy 00 0 εz

#¼ εo

" sysz=sx 0 00 szsx=sy 00 0 sxsy=sz

#; ð9Þ

μ0 ¼" μx 0 00 μy 00 0 μz

#¼ μo

" sysz=sx 0 00 szsx=sy 00 0 sxsy=sz

#: ð10Þ

By comparing Eqs. (1) and (8) it can be seen that Maxwell’sequations have the same form under the scale transformationas has been demonstrated previously [14–16]. It is also seenthat, while the new permittivity and permeability tensors arestill diagonal, the elements are no longer equal; therefore, theuniform medium has been transformed into one that isanisotropic.

A. Scale Transformation of an EllipseThe geometry of interest in this analysis is the elliptic cylinderas shown in Fig. 1(a). This coordinate system is called the Σsystem. A perfectly conducting elliptic cylinder with majordiameter 2a and minor diameter 2b is oriented with its axisalong the z axis. The equation for the ellipse is given by

x2

a2þ y2

b2¼ 1: ð11Þ

If we let

sx ¼ 2aaþ b

; sy ¼ 2baþ b

; sz ¼ 1; ð12Þ

under the scale transformation the equation of the ellipsebecomes

x02 þ y02 ¼ R02; with R0 ¼ aþ b2

; ð13Þ

which is the equation of a circle with radius R0. The ellipsetransformed into a circle is shown in Fig. 1(b). This coordinatesystem is called the Σ0 system.

1286 J. Opt. Soc. Am. A / Vol. 28, No. 6 / June 2011 L. A. Ferrari

B. Scattering of a Plane Wave with the Wave ElectricField in the Direction of the Ellipse AxisA plane wave with wave vector k inclined at an angle ϕo withrespect to the x axis as shown in Fig. 1(a) is incident on theperfectly conducting infinite elliptic cylinder. The incidentwave is polarized with its electric vector along the z axis.Thus, we write for the incident wave electric field:

E ¼ ð0; 0; Ezðx; yÞÞ; ð14Þ

E0 ¼ ð0; 0; E0zðx0; y0ÞÞ; ð15Þ

for the wave electric field in the Σ0, system.The wave equation, obtained from Maxwell’s equations, for

E0z is

1μy

∂2E0z

∂x02þ 1μx

∂2E0z

∂y02þ ω2εzE0

z ¼ 0; ð16Þ

where μx ¼ μoðsy=sxÞ, and μy ¼ μoðsx=syÞ.The phase velocity of a plane wave propagating in the x0

direction is v0x ¼ c=sx, and for a plane wave propagating inthe y0 direction it is v0y ¼ c=sy, where c is the speed of lightin the vacuum. Because the primed medium is anisotropic,the phase velocities in the x0 and y0 directions are different.

In the Σ0 system the incident plane wave is given by

E0z ¼ E0

oeiðk0xx

0þk0yy0Þ; ð17Þ

and the wavenumber k0 is

k0 ¼ ω ffiffiffiffiffiffiffiffiffiffiffiffiffiεosxsyp �

cos2 ϕ0o

μyþ sin2 ϕ0

o

μx

�−1=2

: ð18Þ

With some algebraic manipulation, obtain

k0 ¼ k

��cosϕ0

o

sx

�2þ�sinϕ0

o

sy

�2�

−1=2; ð19Þ

with the relations between the angles in the Σ and Σ0 systemsgiven by

ϕ0o ¼ tan−1

�sxsy

tanϕo

�; ϕ0 ¼ tan−1

�sxsy

tanϕ�: ð20Þ

The wavenumber k0 referred to the angle ϕo is

k0 ¼ kgogoo

; ð21Þ

where

go ¼�cos2 ϕo

s2xþ sin2 ϕo

s2y

�1=2

; ð22Þ

goo ¼�cos2 ϕo

s4xþ sin2 ϕo

s4y

�1=2

; ð23Þ

and the relations between the wavenumber components are

k0x ¼ ksxgoo

cosϕo; k0y ¼ ksygoo

sinϕo: ð24Þ

Now choose a new coordinate system, the ðx00; y00Þ or Σ00

system, that is rotated an angle ϕ0o with respect to the

ðx0; y0Þ system as shown in Fig. 1(b). In this system the inci-dent wave vector is along the x00 axis.

Transform Eq. (16) to the Σ00 system. The transformationequations are

x00 ¼ cosϕ0ox0 þ sinϕ0

oy0; y00 ¼ − sinϕ0ox0 þ cosϕ0

oy0:ð25Þ

Equation (16) in the Σ00 system is

�cos2 ϕ0

o

μyþ sin2 ϕ0

o

μx

�∂2E0

z

∂x002þ 2 sinϕ0

o cosϕ0o

�1μx

−1μy

�∂2E0

z

∂x00∂y00

þ�cos2 ϕ0

o

μxþ sin2 ϕ0

o

μy

�∂2E0

z

∂y002þ ω2εzE0

z ¼ 0: ð26Þ

This is a complicated equation due to the fact that themedium is anisotropic.

In polar coordinates ðρ00;ϕ00Þ in the Σ00 system, this equationbecomes even more complicated, i.e.,

fK1 cos2 ϕ00 þ K2 sinϕ00 cosϕ00 þ K3 sin2 ϕ00g ∂2E0

z

∂ρ002

þ 1

ρ002 f2ðK1 − K3Þ sinϕ00 cosϕ00

þ K2ðsin2 ϕ00 − cos2 ϕ00Þg ∂E0z

∂ϕ00

þ 2ρ00 fðK3 − K1Þ sinϕ00 cosϕ00

þ K2ðcos2 ϕ00 − sin2 ϕ00Þg ∂2E0z

∂ρ00∂ϕ00

þ 1ρ00 fK1 sin2 ϕ00 − 2K2 sinϕ00 cosϕ00 þ K3 cos2 ϕ00g ∂E

0z

∂ρ00

þ 1

ρ002 fK1 sin2 ϕ00 − K2 sinϕ00 cosϕ00 þ K3 cos2 ϕ00g ∂2E0

z

∂ϕ002

þ ω2εzE0z ¼ 0; ð27Þ

where

K1 ¼cos2 ϕ0

o

μyþ sin2 ϕ0

o

μx; ð28Þ

K2 ¼ 2 sinϕ0o cosϕ0

o

�1μx

−1μy

�; ð29Þ

K3 ¼cos2 ϕ0

o

μxþ sin2 ϕ0

o

μy: ð30Þ

C. Scattering by a Perfectly Conducting Cylinder andApplication to an Elliptic CylinderWhen a plane wave, polarized with the electric field in thez direction and propagating in the positive x direction, isincident on an infinitely long perfectly conducting cylinder ofradius a, located with its axis along the z axis, the waveequation for Ez in polar coordinates ðρ;ϕÞ is

1ρ

�ρ ∂Ez

∂ρ

�þ 1

ρ2∂2Ez

∂ϕ2 þ ω2

c2Ez ¼ 0; ð31Þ

L. A. Ferrari Vol. 28, No. 6 / June 2011 / J. Opt. Soc. Am. A 1287

and the solution is

Ezðρ;ϕ; tÞ ¼ Eo

Xþ∞

n¼−∞

in�JnðkρÞ −

JnðkaÞHð1Þ

n ðkaÞHð1Þ

n ðkρÞ�einϕe−iωt;

ð32Þwhere JmðkρÞ is the Bessel function of the first kind,Hð1Þ

m ðkρÞ ¼ JmðkρÞ þ iYmðkρÞ is the cylindrical Hankel func-tion of the first kind, and YmðkρÞ is the Bessel function ofthe second kind or Neumann function. The first term onthe right-hand side of Eq. (32) is the incident plane-wave fieldrepresented in terms of cylindrical waves, and the second isthe scattered field. Examination of Eq. (32) shows that theboundary condition on the electric field Ezða;ϕ; tÞ ¼ 0 issatisfied. This problem is discussed in a number of bookson electromagnetic theory [17,18].

It is asserted [11–13] that, since the scale transformationconverts an ellipse into a circle, previously obtained resultsfor the scattering by a perfectly conducting cylinder can beapplied in the transformed coordinate system provided the in-cident wave is in the proper direction. In the Σ00 system theincident wave propagates in the x00 direction, satisfying thatrequirement, and therefore it is asserted that the formulafor the scattered field contained within Eq. (32) can be applieddirectly in the Σ00 system. Thus, it is expected that the scat-tered field, without the time factor, should be given by

E00zsðρ00;ϕ00Þ ¼ −E00

o

Xþ∞

n¼−∞

inJnðk0RÞHð1Þ

n ðk0RÞHð1Þ

n ðk0ρ00Þeinϕ00: ð33Þ

Since ϕ0 ¼ ϕ0o þ ϕ00, and if we define the angle of interest in

the Σ system to be the angle ϕ1 relative to the incident wavevector, ϕ1 is given by

ϕ1 ¼ ϕ − ϕo ¼ tan−1�sysx

tanðϕ0o þ ϕ00Þ

�: ð34Þ

Translating Eq. (33) back into the original system, oneobtains [13]

Ezsðρ;ϕÞ¼−Eo

Xþ∞

n¼−∞

inJnðk0RÞHð1Þ

n ðk0RÞHð1Þ

n ðkρÞgngno

×�cosnϕsx

þ isinnϕsy

�

×�cosnϕo

sx− i

sinnϕo

sy

�; ð35Þ

where

gn ¼��

cosnϕsx

�2þ�sinnϕsy

�2�

1=2; ð36Þ

gno

��cosnϕo

sx

�2þ�sinnϕo

sy

�2�

1=2: ð37Þ

Equation (16) is the wave equation for E0z in the Σ0 system,

and the plane-wave solution, Eq. (17), leads to k0 given byEq. (19), where k0 is related to ϕ0

o, the incident wave vectorangle in the Σ0 system. However, the k0, or k00, used in [13] ap-pears to relate this quantity to ϕo, i.e., in [13] k0 is given by

k0 ¼ k��

cosϕo

sx

�2þ�sinϕo

sy

�2�

−1=2; ð38Þ

which is not the same as Eq. (21).The solution to the scattering by an elliptic cylinder via the

scaled theory is, therefore, only slightly more complicatedthan the solution to the scattering by a circular cylinder; thereis no need to evaluate Mathieu functions.

The assertion [11–13] that leads to Eq. (35) assumes thatEq. (33) is the solution of Eq. (27), the wave equation inthe Σ00 system. This is clearly not correct. The suggestion hereis that this occurs because the uniform medium of the Σ sys-tem has been converted via the scale transformation to an an-isotropic medium, and Eq. (33) is the solution of the waveequation in a uniform medium. This can be seen by examiningEq. (27). With μx ¼ μy, it is found that K1 ¼ K3, and K2 ¼ 0.Further, sx ¼ sy ¼ 1, which leads to E0

z ¼ Ez, a ¼ b, and theellipse is now a circle; Eq. (27) becomes Eq. (31). Ifμx ≠ μy, a transformation on E0

z must be found that transformsEq. (27) into the form of Eq. (31); it is not clear that such atransformation can be found.

D. Plane Wave Normally Incident on a Thin PerfectlyConducting EllipseIn order to test the assertion given in [13] without consideringthe form of the wave equation in Σ00, we consider the scatter-ing of a wave normally incident on a thin perfectly conductingelliptic cylinder oriented with its major axis perpendicular tothe incident wave. The incident wave propagates in the posi-tive x direction. The major diameter is now 2b, and the minordiameter is 2a. This geometry is chosen in order to simulate a

Fig. 1. (a) Elliptic cylinder with major diameter 2a andminor diameter 2b in theΣ system. The wave vector k of the incident plane wave is inclinedan angle ϕo with respect to the x axis. (b) Scaled, or Σ0, coordinate system with the ellipse of (a) transformed into a circle of radius R0. The wavevector k0 of the incident plane wave is inclined at an angle ϕ0

o with respect to the x0 axis. The Σ00 system is rotated an angle ϕ0o with respect to the x0

axis. In this system k0 is parallel to the x00 axis.

1288 J. Opt. Soc. Am. A / Vol. 28, No. 6 / June 2011 L. A. Ferrari

thin conducting strip of width 2b. We use values of parametersthat are typical of a laser scattering experiment. For a laserwavelength λ ¼ 632:8 nm, ϕo ¼ 0, two different values of b,and the aspect ratio b=a ¼ 1:75=0:25 ¼ 7:00, the numerical va-lues of the various relevant parameters used in the numericalevaluation are listed in Table 1.

From the Babinet principle [19], which applies for both sca-lar and electromagnetic waves when 2b=λ ≫ 1 and 2b=ρ ≪ 1,the scattering of a plane wave by a thin conducting strip ofwidth 2b will be, apart from a phase factor, the same asthe scattering from an opening of width 2b in an opaquescreen. In the same limit, the scattering by a perfectly con-ducting cylinder of radius b is approximated very well bythe scalar theory single-slit approximation for scattering ofa plane wave by an opening of width 2b in an opaque screen[20]. The single-slit approximation, therefore, will be used as areference with which to compare the scaled theory.

Figure 2 is a plot of the absolute value of Ezs=Eo fromEq. (35) for the two different values of 2b=λ and the aspectratio of b=a ¼ 7:00 for the angle of incidence ϕo ¼ 0°. Alsoshown in the graph is the scattered wave amplitude usingthe single-slit approximation normalized to Ezs=Eo at ϕ ¼ 0.It is clear that there is a significant difference between theprediction of the scaled theory, Eq. (35), and the single-slitapproximation. While the shapes of the graphs are different,the most significant difference is the position of the first mini-mum since this data point is frequently used in the analysis ofexperimental data. Equation (35) predicts that the first mini-mum for 2b=λ ¼ 120 is at ϕ1 ¼ 55:3 × 10−3, while, since kb ¼378, the single-slit approximation predicts ϕ1ss ¼ 8:31 × 10−3,ϕ1 > ϕ1ss; the difference is not quite a factor of 7. Similarresults are obtained for 2b=λ ¼ 60.

If the ellipse is rotated 90° with the major axis along thex axis with ϕo ¼ 0°, a ¼ 3:81 × 10−5 m, the position of the firstminimum in the diffraction pattern from Eq. (35) for 2b=λ ¼ 17is at ϕ1 ¼ 8:26 × 10−3, while, since kb ¼ 54, the single-slit ap-proximation gives ϕ1ss ¼ 5:82 × 10−2. The difference betweenthe two positions is still about 7, but now the numerical valuesare reversed, ϕ1ss > ϕ1. Similar results are obtained fora ¼ 1:90 × 10−5 m; the difference between the first minimumsis still about 7.

For another approach to finding the position of the firstminimum, we evaluate Eq. (33) in the Σ00 coordinate systemand then transform that angle to the Σ system. We have

k0ρ00 ¼ kρ gogoo

�cos2 ϕs2x

þ sin2 ϕs2y

�1=2

: ð39Þ

Figure 3 is a graph of jE00zs=E00

o j versus ϕ00. The solid curve re-sults from Eq. (33), and the dashed curve is obtained from thesingle-slit approximation. The position of the first minimum isϕ001 ¼ 5:62 × 10−2, and with k0R ¼ 54, ϕ00

1ss ¼ 5:82 × 10−2. Theshape of jE00

zs=E00o j and the positions of the first minimums

are what is expected from the scattering by a perfectly con-ducting cylinder with this value of k0R. The angle ϕ00

1 trans-formed to the Σ system is equal to ϕ1 ¼ 0:375, and sincekb ¼ 378, ϕ1ss in the Σ system is equal to ϕ1ss ¼ 8:31 × 10−3;the ratio is equal to ϕ1=ϕ1ss ¼ 45:2.

Equation (33) transformed to the Σ system with a differentset of transformation equations than in [13] is

Ezsðρ;ϕÞ ¼ −Eo

Xþ∞

n¼−∞

inJnðk0RÞHð1Þ

n ðk0RÞHð1Þ

n ðkρÞeinðϕ0−ϕ0oÞ; ð40Þ

with ϕ0 and ϕ0o given by Eq. (20).

The data plotted in Fig. 3 transformed to the Σ system to-gether with Ezsðρ;ϕÞ from Eq. (40) are plotted together inFig. 4. The solid curve is obtained from Eq. (33), and the solidpoints is obtained from Eq. (40). It is clear that there is nodifference between the two evaluations. It is also clear thatthe field distribution shown in Fig. 4 is significantly differentfrom the field distribution obtained from Eq. (35) that isshown in Fig. 2.

Fig. 2. Solid curves show the amplitude of the scattered field from athin elliptic cylinder obtained by evaluating the scaled theory formula,Eq. (35), as a function of the angle ϕ in the Σ system for two differentvalues of 2b=λ with ρ ¼ 5:2m. The dashed curves indicate the ampli-tude of the scattered wave obtained from the scalar theory single-slitapproximation for each value of 2b=λ. The single-slit data are normal-ized to the scaled theory value at ϕ ¼ 0.

Table 1. Values of the Parameters Used for the

Evaluation of the Scattering of a Plane Wave

Normally Incident on a Thin Elliptic Cylinder

b 2b=λ kb 2a=λ ka R k0R

1:90 × 10−5 m 60 188 8.5 27 1:085 × 10−5 m 273:81 × 10−5 m 120 377 17 54 2:177 × 10−5 m 54

Fig. 3. Amplitude of the scattered field from the thin elliptic cylinderas a function of the angle ϕ00 in the Σ00 system with ρ ¼ 5:2m and2b=λ ¼ 120. The continuous curve is obtained from Eq. (33) andthe dashed curve from the scalar theory single-slit approximation.The single-slit evaluation is normalized to jE00

zs=E00o j at ϕ00 ¼ 0 and

2b=λ ¼ 120.

L. A. Ferrari Vol. 28, No. 6 / June 2011 / J. Opt. Soc. Am. A 1289

If the ellipse is rotated 90° so that the major axis is along thex axis and all other dimensions maintained, the position of thefirst minimum is ϕ1 ¼ 1:176 × 10−3, ϕ1ss ¼ 5:813 × 10−2, andthe ratio is equal to ϕ1=ϕ1ss ¼ 1=49:4, which is approximatelythe reciprocal of the ratio of the opposite orientation. It is alsoimportant to observe that the behavior of Ezs with ϕ shown inFigs. 3 and 4 resembles the typical behavior of the scatteringfrom a circular cylinder, while the distribution shown in Fig. 2does not.

3. CONCLUSIONSBased on the analysis presented here, we conclude that, inorder to apply a scattering formula in the scaled Σ00 system,one must first obtain the wave equation in this system andthen find a solution; it seems clear that, because the scaletransformation generates an anisotropic medium, the use ofthe uniform medium solution, Eq. (35), in the scaled systemis not appropriate. The scale transformation suggested forscattering by a perfectly conducting elliptic cylinder doesnot produce a relatively simple alternative to the Mathieufunction solution.

ACKNOWLEDGMENTSI would like to thank Professor D. Nagy for his hospitality andsupport and Professors R. Coleman and T. Goebeler for avaluable discussion. I would also like to thank an anony-mous referee for numerous suggestions that improved themanuscript.

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Fig. 4. Amplitude of the scattered field from the thin elliptic cylinderin the Σ system. The solid curve is the data plotted in Fig. 3 trans-formed to the Σ system, and the solid points are obtained fromEq. (40) with ρ ¼ 5:2m and 2b=λ ¼ 120.

1290 J. Opt. Soc. Am. A / Vol. 28, No. 6 / June 2011 L. A. Ferrari