Eigenmode analysis of the electromagnetic field scattered by an elliptic cone

The vector spherical-multipole analysis is applied to determine the scattering of a plane electromagnetic wave by a perfectly electrically conducting (PEC) semi-infinite elliptic cone. From the eigenfunction expansion of the total field in the space outside the elliptic cone, the scattered far field is obtained as a multipole expansion of the free-space type by a single integration over the induced surface currents. As for the evaluation of the free-space-type expansion it is necessary to apply suitable series transformation techniques, a sufficient number of eigenfunctions has to be considered. The eigenvalues of the underlying two-parametric eigenvalue problem with two coupled Lam é equations belong to the Dirichletor the Neumann condition and can be arranged as so-called eigenvalue curves. It has been observed that the eigenvalues are in two different domains: In the first one Dirichletand Neumann eigenvalues are either nearly coinciding, while in the second one they are strictly separated. The eigenfunctions of the first (coinciding) type look very similar to free-space modes and do not contribute to the scattered field. This observation allows to significantly improve the determination of diffraction coefficients.


Introduction
Electromagnetic scattering by a PEC elliptic cone is of practical importance for several reasons.First, the geometry includes a tip, and the related tip-diffraction coefficient could be used to improve asymptotically valid methods like GTD and UTD in describing scattering by complex electrically large systems.Second, as the geometry can be treated mostly analytically using sphero-conal coordinates and the vector spherical-multipole expansion, the obtained results (e.g., for Correspondence to: L. Klinkenbusch (lbk@tf.uni-kiel.de)a finite elliptic cone with the degenerations sector and circular cone) can serve as a reference for numerical computations.Scattering by circular and (less often) by elliptic cones have been treated by many authors in the literature.Summaries on the different approaches can be found in Bowman et al. (1987); Klinkenbusch (2007).This paper starts with a description of sphero-conal coordinates which are used to describe the elliptic cone geometry.For an incident plane electromagnetic wave the exact total field outside and the corresponding surface current on the elliptic cone are determined by a spherical-multipole (eigenfunction) expansion based on suitable solutions of the scalar homogeneous Helmholtz equation.The far field is then found as a freespace type spherical-multipole expansion from an integration over the surface current.In sphero-conal coordinates the solution of the scalar Helmholtz equation leads to a twoparametric eigenvalue problem with two coupled Lamé differential equations, that is, the differential equations of the periodic and of the non-periodic Lamé functions.The two separation parameters can be arranged on so-called eigenvalue curves on which the Dirichlet-and Neumann eigenvalues are discretely distributed.It turns out that the numerical determination of some of these eigenvalues is difficult, however, as will be shown exactly these eigenvalues do not significantly contribute to the scattered far field and can be automatically sorted out.Finally, this procedure leads to an improved accuracy in determining the scattering coefficient magnitudes and phases.
Published by Copernicus Publications on behalf of the URSI Landesausschuss in der Bundesrepublik Deutschland e.V. M. Kijowski and L. Klinkenbusch: Eigenmode analysis of the field scattered by a The ranges of the values are 0 ≤ r < ∞, 0 ≤ ϑ ≤ π, 0 ≤ ϕ ≤ 2π and the ellipticity parameters k and k' satisfy The coordinate surfaces are shown in Fig. 1.The elliptic cone is identical to the coordinate surface ϑ = ϑ 0 and can be characterized by the half opening angles ϑ x = ϑ 0 and ϑ y = arccos(kcosϑ 0 ).The normalized metric scaling coefficients are given by Note that the elliptic cone includes several interesting degenerations: For k = 1 the elliptic cone turns into a circular cone while ϑ 0 = π describes a plane angular sector with half-opening angle arccos(k).

Eigenfunction Expansion of the Total Field
In a linear, homogeneous, and isotropic domain outside the elliptic cone the total electromagnetic field can be derived as a multipole (eigenfunction) expansion in sphero-conal coordinates which is based on the corresponding solution of the scalar homogeneous Helmholtz equation where κ = ω √ µε denotes the wave number.A first separation ansatz For the given problem it is necessary to have so are (at least) 2π -periodic in ϕ and fulfill canon conditions at ϑ = ϑ 0 : A second separation ansatz of the form with a second separation constant λ yields two ferential equations: Equation ( 11) is referred to as the differential e periodic Lamé functions Φ ν (ϕ) while Eq.(1 the differential equation of the non-periodic L Θ ν (ϑ).The periodic Lamé functions can be de finite Fourier series and the non-periodic Lamé expanded into infinite series using associated L tions of the 1st kind (Boersma and Jansen, 1 of solutions in the free unbounded space the ei integers (ν = n = 1,2,3,..), the series become fi sequently the solutions turn into periodic and Lamé polynomials Φ nm (ϕ) and Θ nm (ϑ), resp For a given value of ν and of the paramet responding second separation constant λ can b determined.The resulting (ν,λ)-pairs lie on eigenvalue curves sorted by numbers m = 0 arbitrary pair of eigenvalues (ν,λ) lying on t curves leads to a valid solution of the eigenvalu the Lamé products (7).Additional Dirichletboundary conditions imposed upon the nonp functions Θ ν at ϑ 0 result in a discrete spectrum pairs (ν i ,λ i ) (i = 1,2,3,..) lying on the eigen Figure 2 exemplarily shows the eigenvalue cu

Sphero-conal coordinates
Sphero-conal coordinates r, ϑ, ϕ are related to Cartesian coordinates by (Boersma and Jansen, 1990) The ranges of the values are 0 ≤ r < ∞, 0 ≤ ϑ ≤ π, 0 ≤ ϕ ≤ 2π and the ellipticity parameters k and k satisfy The coordinate surfaces are shown in Fig. 1.The elliptic cone is identical to the coordinate surface ϑ = ϑ 0 and can be characterized by the half opening angles ϑ x = ϑ 0 and ϑ y = arccos(kcosϑ 0 ).The normalized metric scaling coefficients are given by Note that the elliptic cone includes several interesting degenerations: For k = 1 the elliptic cone turns into a circular cone while ϑ 0 = π describes a plane angular sector with half-opening angle arccos(k).

Eigenfunction expansion of the total field
In a linear, homogeneous, and isotropic domain outside the elliptic cone the total electromagnetic field can be derived as a multipole (eigenfunction) expansion in sphero-conal coordinates which is based on the corresponding solution of the scalar homogeneous Helmholtz equation where κ = ω √ µε denotes the wave number.A first separation ansatz with a first separation constant ν(ν + 1) leads to the differential equation of the spherical Bessel functions with solutions z ν (κr) and to the eigenvalue equation of surface spherical harmonics which are referred to as Lamé products in case of sphero-conal coordinates For the given problem it is necessary to have solutions which are (at least) 2π -periodic in ϕ and fulfill canonical boundary conditions at ϑ = ϑ 0 : A second separation ansatz of the form with a second separation constant λ yields two ordinary differential equations: Equation ( 11) is referred to as the differential equation of the periodic Lamé functions ν (ϕ) while Eq. ( 12 and Klinkenbusch, 1999) where the expansion functions which are referred to as the vector spherical-multipole functions can be derived from the elementary solutions of the scalar homogeneous Helmholtz equation Ψ ν (r) by with r = r/r denoting the unit vector and κ = ω √ ε 0 µ 0 being the wave number in the free space.The transverse vector functions are defined as and the electric and magnetic multipole amplitudes are given by a σ and b τ , respectively.Note that the indices σ and τ symbolize the Dirichlet-and Neumann conditions as defined in ( 8) and ( 9) to ensure the vanishing of the tangential elec-infinity and multiplying the resulting field by an appropria factor (Blume and Klinkenbusch, 1999).For a plane wa with amplitude E 0 incident from (θ inc ,φ inc ) and electrical polarized in the direction Ĉ, the multipole amplitudes of t total field are found as where Z = µ 0 /ε 0 is the intrinsic impedance of the fr space.

Spherical-Multipole Expansion of the Scattered Fie
The scattered field is determined from the surface curre J S = − θ × H tot | ϑ 0 on the cone's surface by where the dyadic Green's function of the free space in bili ear form is deduced as At a time-factor e jωt the upper indices I and II stand for t use of spherical Bessel functions of the first kind (z n = j and of spherical Hankel functions of the second kind (z n h (2) n ), respectively.It has been shown (Klinkenbusch,200 that the scattered electric far field can be written in form o multipole expansion leads to a valid solution of the eigenvalue equation of the Lamé products Eq. ( 7).Additional Dirichlet-and Neumann boundary conditions imposed upon the non-periodic Lamé functions ν at ϑ 0 result in a discrete spectrum of eigenvalue pairs (ν i ,λ i ) (i = 1,2,3,..) lying on the eigenvalue curves.(Boersma and Jansen, 1990).
Outside the PEC elliptic cone the total electromagnetic field can be expressed in the form of a sphericalmultipole (eigenfunction) expansion (Stratton, 1941;Blume and Klinkenbusch, 1999) where the expansion functions which are referred to as the vector spherical-multipole functions can be derived from the elementary solutions of the scalar homogeneous Helmholtz equation ν (r) by with r = r/r denoting the unit vector and κ = ω √ ε 0 µ 0 being the wave number in the free space.The transverse vector functions are defined as and the electric and magnetic multipole amplitudes are given by a σ and b τ , respectively.Note that the indices σ and τ symbolize the Dirichlet-and Neumann conditions as defined in Eqs. ( 8) and ( 9) to ensure the vanishing of the tangential electric field on the surface of the PEC elliptic cone.The incident plane wave is realized by locating a Hertzian dipole at infinity and multiplying the resulting field by an appropriate factor (Blume and Klinkenbusch, 1999).For a plane wave with amplitude E 0 incident from (θ inc ,φ inc ) and electrically polarized in the direction Ĉ, the multipole amplitudes of the total field are found as where Z = √ µ 0 /ε 0 is the intrinsic impedance of the free space.

Spherical-multipole expansion of the scattered field
The scattered field is determined from the surface current J S = − θ × H tot ϑ 0 on the cone's surface by where the dyadic Green's function of the free space in bilinear form is deduced as At a time-factor e j ωt the upper indices I and I I stand for the use of spherical Bessel functions of the first kind (z n = j n ) and of spherical Hankel functions of the second kind (z n = h (2) n ), respectively.It has been shown (Klinkenbusch, 2006) that the scattered electric far field can be written in form of a multipole expansion available eigenvalues and eigenfunctions it is necessary to investigate the relevance of eigenvalues and eigenmodes which will be sketched in the following section.

Eigenmode analysis
In Fig. 2 E sc θ (θ,φ) ) hile the series in ( 24) and ( 25) converge and yield stable ultipole amplitudes of the scattered field, the resulting sees in (23) do not converge.In order to obtain a meaningful iting value it is necessary to apply a suitable sequence ansformation.In contrast to nonlinear techniques (like the anks transform) linear sequence transformations always eld consistent results.For the linear Cesàro transform the ansformed partial sum sequence s ′ n is obtained from the iginal partial sum sequence s n by he sequence transformation can be repeatedly applied to force faster convergence of the resulting partial sum seence.Figure 3 shows the double transformed partial sum quence of the scattering coefficent D θθ as a function of Clearly, a higher order of the original series and hence a gher number of eigenvalues is desired to obtain more acrate results.In order to increase the maximum number of  Finally, the scattered far field can be written as a function of the incident field by means of a scattering matrix as While the series in Eqs. ( 24) and ( 25) converge and yield stable multipole amplitudes of the scattered field, the resulting series in Eq. ( 23) do not converge.In order to obtain a meaningful limiting value it is necessary to apply a suitable sequence transformation.In contrast to nonlinear techniques (like the Shanks transform) linear sequence transformations always yield consistent results.For the linear Cesàro transform the transformed partial sum sequence s n is obtained from the original partial sum sequence s n by The sequence transformation can be repeatedly applied to enforce faster convergence of the resulting partial sum sequence.Figure 3 shows the double transformed partial sum sequence of the scattering coefficent D θ θ as a function of n.Clearly, a higher order of the original series and hence a higher number of eigenvalues is desired to obtain more accurate results.In order to increase the maximum number of available eigenvalues and eigenfunctions it is necessary to investigate the relevance of eigenvalues and eigenmodes which will be sketched in the following section.

Eigenmode analysis
In Fig. 2  separated.Figure 4 shows a closer view of this phenomenon revealing that the coinciding eigenvalues all are very near to integral values (integers) of ν.Since the numerical computation of these coinciding eigenvalues turns out to imply some numerical difficulties limiting the maximum number of computable eigenvalues we will now investigate that case in more detail.
The left column in Fig. 5 shows a sequence of plots of nonperiodic Lamé functions each satisfying the Dirichlet boundary condition at ϑ 0 = 160 • as a function of the argument ϑ.In the right column we see plots of the non-periodic Lamé polynomials (with integral eigenvalues ν = n) at (n,λ)-pairs on the same eigenvalue curve nearest by those ones of the corresponding non-periodic Lamé functions.We observe that for nearly integral eigenvalues of the non-periodic Lamé func-tions not only their values at ϑ 0 are vanishing but also the derivatives.Moreover, these eigenfunctions look very sim lar to the corresponding non-periodic Lamé polynomials.A non-integral eigenvalues only the values of the non-period Lamé functions vanish but not their derivatives, and the curves are different from the corresponding non-period Lamé polynomials, at least in the vicinity of the boundarie ϑ = 0 and ϑ = π.This general behavior is typical and can b observed for any other eigenvalue as well.
Due to numerical reasons, the computation of the nearl integral eigenvalues and -functions turns out to be difficul However, as we can deduce from the representations of th multipole amplitudes ( 24) and ( 25) the modes belonging t these eigenvalues do not significantly contribute to the sca tered far field.Each part of ( 24) and ( 25) has a factor of on of the following forms Clearly, if both function and derivative of a non-period Lamé function are small at ϑ 0 , the corresponding scatterin mode is also small compared to the other scattering mode In other words, these eigenmodes of the PEC cone do not sig nificantly lead to a surface current on the cone, or, the con is nearly invisible for these eigenmodes.Consequently, the are very similar to free-space modes, which are characterize by integral eigenvalues.Following this observation, these nearly-integral eigenva ues and eigenfunctions don't need to be exactly calculated and the modified algorithm allows to calculate much mor relevant eigenvalues and eigenfunctions to come to more ac curate scattering coefficients.

Scattering coefficients
Figure 6 shows the amplitude and the phase of the electri far field scattered by a PEC semi-infinite elliptic cone illum nated by a plane wave electrically polarized in the xz plan and incident from θ inc = 105 • , φ inc = 0 • .The amplitude o the scattering coefficient D θθ is shown for the maximum o der n max = 40 including the integral-eigenvalue modes an n max = 60 excluding these non-contributing modes.Th comparison between the phases shows marginal difference however, the differences in amplitudes reveal the improve ment of the results by considering more relevant eigenmode Finally, Fig. 7 proves that the errors of amplitudes an phases of the scattering coefficient are actually margin when all of the near-integer eigenvalues are neglected.

Conclusions
It has been found that the accuracy of computed scatterin coefficients for a PEC elliptic cone can be greatly improved Neumann eigenvalues nearly coincide and into a lower region where Dirichlet-and Neumann eigenvalues are strictly separated.Figure 4 shows a closer view of this phenomenon revealing that the coinciding eigenvalues all are very near to integral values (integers) of ν.Since the numerical computation of these coinciding eigenvalues turns out to imply some numerical difficulties limiting the maximum number of computable eigenvalues we will now investigate that case in more detail.
The left column in Fig. 5 shows a sequence of plots of nonperiodic Lamé functions each satisfying the Dirichlet boundary condition at ϑ 0 = 160 • as a function of the argument ϑ.
In the right column we see plots of the non-periodic Lamé polynomials (with integral eigenvalues ν = n) at (n,λ)-pairs on the same eigenvalue curve nearest by those ones of the corresponding non-periodic Lamé functions.
We observe that for nearly integral eigenvalues of the nonperiodic Lamé functions not only their values at ϑ 0 are vanishing but also their derivatives.Moreover, these eigenfunctions look very similar to the corresponding non-periodic Lamé polynomials.At non-integral eigenvalues only the values of the non-periodic Lamé functions vanish but not their derivatives, and their curves are different from the corresponding non-periodic Lamé polynomials, at least in the vicinity of the boundaries ϑ = 0 and ϑ = π.This general behavior is typical and can be observed for any other eigenvalue as well.
Due to numerical reasons, the computation of the nearly integral eigenvalues and -functions turns out to be difficult.However, as we can deduce from the representations of the multipole amplitudes Eqs. ( 24) and ( 25) the modes belonging to these eigenvalues do not significantly contribute to the scattered far field.Each part of Eqs. ( 24) and ( 25) has a factor of one of the following forms Clearly, if both function and derivative of a non-periodic Lamé function are small at ϑ 0 , the corresponding scattering mode is also small compared to the other scattering modes.
In other words, these eigenmodes of the PEC cone do not significantly lead to a surface current on the cone, or, the cone is nearly invisible for these eigenmodes.Consequently, they are very similar to free-space modes, which are characterized by integral eigenvalues.Following this observation, these nearly-integral eigenvalues and eigenfunctions don't need to be exactly calculated, and the modified algorithm allows to calculate much more relevant eigenvalues and eigenfunctions to come to more accurate scattering coefficients.

Scattering coefficients
Figure 6 shows the amplitude and the phase of the electric far field scattered by a PEC semi-infinite elliptic cone illuminated by a plane wave electrically polarized in the xz plane and incident from θ inc = 105 • , φ inc = 0 • .The amplitude of the scattering coefficient D θ θ is shown for the maximum order n max = 40 including the integral-eigenvalue modes and n max = 60 excluding these non-contributing modes.The comparison between the phases shows marginal differences, however, the differences in amplitudes reveal the improvement of the results by considering more relevant eigenmodes.
Finally, Fig. 7 proves that the errors of amplitudes and phases of the scattering coefficient are actually marginal when all of the near-integer eigenvalues are neglected.It has been found that the accuracy of computed scattering coefficients for a PEC elliptic cone can be greatly improved if computationally difficult but non-relevant modes of the scattered field are neglected.These non-scattered modes have nearly integral eigenvalues and are very similar to the freespace modes of the incident field.Further work will include an investigation into the nature of these non-scattered modes.
Figure 2 exemplarily shows the eigenvalue curves with discrete Dirichlet-and Neumann eigenvalues.Due to the Sturm-Liouville properties of the Lamé differential equations the discrete Dirichlet-and Neumann eigenvalues strictly must alternate on the eigenvalue curves

Fig. 3 .
Fig. 3. Partial sum sequence of the real part of the scattering coefficient D θθ for the maximum order of n max = 40.Dotted curve is original partial sum sequence, dashed curve is single Cesàro transformed sequence and solid curve is double Cesàro transformed sequence.

Fig. 3 .
Fig. 3. Partial sum sequence of the real part of the scattering coefficient D θθ for the maximum order of n max = 40.Dotted curve is original partial sum sequence, dashed curve is single Cesàro transformed sequence and solid curve is double Cesàro transformed sequence.

Fig. 7 .
Fig. 7. Amplitude and phase of the scattering coefficient D θ θ (dashed line) in the xz plane of a PEC semi-infinite elliptic cone with the half opening angles α x = 45 • , α y = 60 • and n max = 40.The plane wave is incident from θ inc= 105 • , φ inc = 0 • .The solid line using the right scale is the difference between the first amplitude (phase) resulting from all eigenvalues and the second amplitude (phase) resulting from all eigenvalues except nearly integer eigenvalues.