Radio Science

Electromagnetic scattering by semi-infinite circular and elliptic cones

Authors


Abstract

[1] The scattering of a plane electromagnetic wave by a perfectly conducting semi-infinite cone with an elliptic or a circular cross section is treated by means of a multipole analysis in sphero-conal coordinates, in which the elliptic (including the circular) cone is one of the coordinate surfaces. Starting from the dyadic Green's function of the elliptic cone given in terms of periodic and nonperiodic Lamé functions, first the exact surface current on the cone is obtained. This surface current is the source of the scattered far field which is completely analytically found by means of the dyadic Green's function of the free space in sphero-conal coordinates. The finally obtained infinite series do not converge by a straight-forward summing-up procedure; however, a simple linear (consistent) sequence transformation technique is sufficient to come to stable and physically correct asymptotic results.

1. Introduction

[2] Electromagnetic and acoustic scattering by semi-infinite cones with a circular cross section have often been treated in the literature [Siegel et al., 1955; Felsen, 1955, 1957a, 1957b] (for an overview, see Bowman et al. [1987]). The interest in this canonical structure is mainly motivated by the fact that it possesses a tip which is associated to a field- and current singularity, and the related tip diffraction coefficient could be a further element to complete asymptotic methods such as the Geometrical Theory of Diffraction [Keller, 1962]. In this context, the (not that often investigated) elliptic cone (that is, a cone with an elliptic cross section) is of particular interest as it provides a large variety of different geometries and corresponding tips, particularly including the circular cone and the plane angular sector.

[3] The diffraction of a plane electromagnetic wave by an electrically perfectly conducting (PEC) semi-infinite elliptic cone can be investigated by evaluating the corresponding dyadic Green's function, which is given by sums of dyads each consisting of vector multipole functions in sphero-conal coordinates. In work by Blume et al. [1993] [see also Blume and Klinkenbusch, 1999], methods have been represented how the outwardly traveling spherical waves of the total field can be used to calculate radar cross sections and diffraction coefficients of the elliptic cone. The main advantage of this method is that it automatically preserves the correct boundary conditions as it is purely based on the eigenfunction expansion of the elliptic cone. However, the outwardly traveling spherical waves do also include a part of the incident plane wave, in particular, they contain the transmitted field. These singular parts influence the behavior of the resulting series at all other directions as well, and nonlinear series summation techniques had to be applied to come to stable solutions with the disadvantage that the consistency of such nonlinear methods have not be proven for the cases in question [Blume and Krebs, 1998].

[4] The present paper describes a different approach to solve this scattered-field problem. First, from the dyadic Green's function of the elliptic cone we derive the exact surface current on the cone's surface for an incident plane wave by using a dipole source in the far-field and by employing an appropriate transformation [Klinkenbusch, 1993]. Next, we apply the equivalence principle, replace the PEC cone by this electric surface current, and obtain by means of the bilinear form of the free-space dyadic Green's function in sphero-conal coordinates the isolated scattered far-field. The corresponding limiting-value process which is necessary to obtain the scattered far-field has been successfully validated by the identically treatable half-plane problem [Klinkenbusch, 2006], which numerically yields the well-known Keller [1962] GTD results.

[5] The mathematical key to that solution is given by integrals which represent the coupling between the two (cone and free space) field expansions. In particular, all of these integrals are solved completely analytically, moreover, they explicitly provide quantitative rules about the “relative convergence” between the two expansions. Nevertheless, the attempt to simply summing up the final expansion (given in terms of free-space multipole functions) fails. That is expected since we are trying to represent the scattered far field which contains singularities by means of a multipole expansion which does not explicitly contain such singularities. However, useful numerical results can be obtained by employing suitable sequence transformation techniques, where, unlike the method mentioned above, a very simple linear (consistent) scheme, namely the Cesàro transform, is sufficient to come to physically meaningful asymptotic results. A comparison with available Physical Optics data for the bistatic scattering cross section of the nose-on illuminated circular cone yields a good agreement. Moreover, the numerical evaluation includes the scattering coefficients of semi-infinite circular and elliptic cones for non–nose-on-incident plane waves.

2. Sphero-Conal Coordinates

[6] The sphero-conal coordinate system r, ϑ, ϕ is related to Cartesian coordinates by

equation image

with ranges of variability 0 ≤ r < ∞, 0 ≤ ϑ ≤ π, 0 ≤ ϕ ≤ 2π and real nonnegative parameters k, k′ which satisfy k2 + k2 = 1. The normalized metric scaling coefficients will be denoted by

equation image
equation image

The coordinate surfaces are shown in Figure 1. In particular, ϑ = ϑ0 represents a semi-infinite elliptic cone around the positive (ϑ0 < π/2) or negative (ϑ0 > π/2) z axis. k2 is directly related to the ellipticity of that cone. As for k = 1 the sphero-conal turn into usual spherical coordinates with the z axis as the polar axis, consequently the mentioned elliptic cone turns into a circular one. Moreover, ϑ0 = 0 and ϑ0 = π each describe a plane angular sector symmetric to the +z and −z axis, respectively.

Figure 1.

Sphero-conal coordinate surfaces.

3. Solution of Maxwell's Equations in Sphero-Conal Coordinates

[7] The Helmholtz equation

equation image

is completely separable in sphero-conal coordinates, that is, the corresponding elementary solutions can be written in the form

equation image

They consist of products of spherical Bessel functions zvr), periodic Lamé functions Φv(ϕ) and nonperiodic Lamé functions Θv(ϑ). The latter two satisfy the differential equations

equation image
equation image

which are coupled by the two separation constants v(v + 1) and λ.

[8] For all of the problems discussed in the rest of this paper the periodic Lamé functions have to be 2π periodic. It has been shown [Jansen, 1976] that for any such solution the values of v and λ have to be lying on so-called eigenvalue curves, as exemplarily shown in Figure 2. Note that the course of the eigenvalue curves only depends on k2. For the boundary-value problem posed by the PEC elliptic cone, the nonperiodic Lamé functions have to satisfy either the Dirichlet or the Neumann boundary condition on the cone's surface ϑ = ϑ0. The corresponding (v, λ) pairs are then discrete Dirichlet and Neumann eigenvalues on the eigenvalue curves mentioned above (see Figure 2). For solutions in the free space the eigenvalues v are given by all natural numbers (v = n = 1, 2, 3, …). The products of nonperiodic and periodic Lamé functions will be referred to as the Lamé products. They can be represented by a linear combination of four appropriately defined function types,

equation image

The four types of periodic Lamé functions are described by each two alternative Fourier series,

equation image
equation image
equation image
equation image

while the nonperiodic Lamé functions are expanded in terms of associated Legendre functions of the first kind [Jansen, 1976],

equation image
equation image
equation image
equation image

Here the algebraic factors T(i) are recursively defined as

equation image

While the expansions (9)–(12) uniformly converge in the entire domain 0 ≤ ϕ ≤ 2π, the expansions (13)–(16) generally do converge only in the interval 0 ≤ ϑ ≤ arccos(−k). For ϑ > arccos(−k) they are calculated by an appropriate numerical routine with a starting point at ϑ = π/2. Note that only products of periodic and nonperiodic Lamé functions belonging to the same function type l form a valid Lamé product. Moreover, the expansion coefficients of corresponding Lamé functions in (9)–(16) are really identical. This can be shown by inserting the expansions into the differential equations (6) and (7). For each set of coefficients we then obtain equivalent three-term recurrence formulae which each can be written as an infinite algebraic eigenvalue equation. Since the range of valid eigenvalues is limited in λ according to 0 ≤ λv(v + 1) (see Figure 2), this infinite system can (to any desired degree of accuracy) be approximated by a finite one. It is worth noting that the two Fourier series for each periodic Lamé function type belong to identical eigenvalues but lead to different coefficients (eigenvectors) in general. For integral eigenvalues n = 1, 2, 3, …, it has been shown [Jansen, 1976] that for each n always one expansion per function type in (9)–(12) becomes finite while the expansions (13)–(16) all are finite. To each integral eigenvalue n there exist 2n + 1 eigenfunctions in total, distributed among the function types. Hence in that case (which will be relevant for free-space expansions) we shall speak about Lamé polynomials Φn,m(l) and Θn,m(l) (instead of Lamé functions), and the integral numbers m stand for the different eigenfunctions per n. For sake of clarity, in the following derivations we shall write Lamé products, functions, and polynomials without indicating the function type. For the evaluation, however, generally all function types have to be considered except of cases where owing to symmetry reasons only certain function types might be relevant.

Figure 2.

Eigenvalue curves of the Lamé functions (function type 1) for k2 = 0.5, with discrete Dirichlet eigenvalues (crosses) and Neumann eigenvalues (circles).

[9] As in usual spherical coordinates [Stratton, 1941], any solenoidal solution of the vector Helmholtz equation (equivalently, any solution of Maxwell's equations in a homogeneous source-free domain) can be expressed in the form of a vector spherical-multipole expansion,

equation image
equation image

where Z = equation image denotes the intrinsic impedance of the medium. Here av and bv will be referred to as the multipole amplitudes. The vector spherical-multipole functions (“vector wave functions”) are derived from the scalar solutions of the Helmholtz equation in the usual way and can be written in the form

equation image
equation image

where the prime denotes the differentiation with respect to r, equation image = ωequation image is the wave number of the medium, and equation image = equation image/r. The transverse multipole functions equation imagev and equation imagev are related to the Lamé products by

equation image
equation image

Since the tangential part of the electric field (18) must vanish on the cone's surface, we have to choose Dirichlet eigenvalues v = σ and Neumann eigenvalues v′ = τ, implicitly defined by

equation image
equation image

respectively. As discussed above, within a domain unbounded with respect to ϑ and ϕ (as in the free space) we have integral eigenvalues v = v′ = n = 1, 2, 3, … with each 2n + 1 Lamé products Yn,m (ϑ, ϕ).

4. Derivation of the Scattered Field

[10] Outside the source region, the diffracted (total) electric field in the homogeneous domain bounded by a PEC elliptic cone can be represented by the volume integral

equation image

where equation imagee is an arbitrary current density distribution and equation imageC is the dyadic Green's function of the PEC elliptic cone. The latter can be derived in the usual way in terms of dyadic products each consisting of vector multipole functions as

equation image

provided that r < r′. The choice of Dirichlet eigenvalues σ and of Neumann eigenvalues τ enforces the tangential electric field components to vanish on the cone's surface. The upper indices I and II stand for the use of spherical Bessel functions of the first kind (jn(equation imager)) to ensure regularity of the field everywhere, and of spherical Hankel functions of the second kind (hn(2)(equation imager)) to satisfy the radiation condition, respectively. It is worth noting that the correct field singularity at the tip is analytically included through the eigenvalues 0 < σ < 1 and 0 < τ < 1 [Blume and Kahl, 1985]. Now suppose that the current distribution in (26) is given by a Hertzian dipole located in the far field at equation imageinc and perpendicularly polarized to equation imageinc. It has been shown [Klinkenbusch, 1993] that the resulting total field for a Hertzian dipole source can exactly be transformed into the total field caused by an incident plane wave. We then obtain for the total field outside the PEC elliptic cone (i.e., in the domain equation image0equation imageπ) the multipole expansion

equation image
equation image

with the multipole amplitudes:

equation image
equation image

Note that the plane wave of amplitude E0 is incident from (ϑinc, ϕinc) and electrically polarized in the direction equation image. The exact surface current on the cone's surface is then found as

equation image

Applying equivalence, we replace the PEC elliptic cone by this electric surface current and calculate the pure scattered field in the domain ϑ0 < ϑ ≤ π by Green's theorem as

equation image

where equation image0 (equation image, equation image′) is the dyadic Green's function of the free space in sphero-conal coordinates:

equation image

Note that this step is valid only in the limiting case r → ∞, such that r is always chosen to be larger than r′. We insert (32) and (34) into (33) and find for the scattered electric far field the expansion

equation image

with the multipole amplitudes

equation image
equation image

Depending on the desired angles of incidence and observation, for the numerical evaluation actually not all function types of the Lamé products may have to be considered. Note that all integrals in (36) and (37) exist and can be evaluated purely analytically, as shown in Appendix A. Moreover, from the behavior of the integrals involving spherical Bessel functions it follows that for each n, m in (36) and (37) the corresponding series in σ and τ do converge with a maximum contribution around nσ and nτ, respectively.

[11] For the sake of a better interpretability the final result will be represented in usual spherical coordinates θ, ϕ (with the polar axis z) and corresponding components. It can be written in the usual matrix notation as

equation image

where the dimensionless scattering coefficients Dθequation image, Dθequation image, Dϕequation image, and Dϕequation image still depend on both, the angles of incidence θinc, ϕinc and the angles of observation θ, ϕ.

5. Numerical Evaluation

[12] To practically come to numerical results for the case of an elliptic cone, first the eigenvalues and eigenfunctions of the two-parametric problem with two coupled Lamé equations have been found numerically by using the series expansions of the periodic and nonperiodic Lamé functions. The resulting eigenvalues and eigenvectors are then stored, and the FORTRAN procedures which evaluate the multipole amplitudes in (36) and (37) as well as the first (nontransformed) partial-sum series of the electric far field are then accessing these stored data. The series transformations due to Cesàro are finally performed in a postprocessing step using appropriate MATLAB© routines.

[13] To start with a near-field evaluation of (29), Figure 3 shows the real part of the total magnetic field in the xz plane of an elliptic cone. We observe strong interference between incident and reflected fields for x > 0, the less strongly disturbed incident field in that part of the domain x < 0, which is “visible” for the incident plane wave, and a wave with an almost spherical wavefront in the shadow region. The evaluation of the near field by means of (28)/(29) is always exactly possible because of the convergence-enforcing parts in the series given by the spherical Bessel functions of the first kind, jnr) for finite values of r. However, the multipole series of the far field (with κr → ∞) given by (35) does not converge by simply adding the series elements. This is due to the fact that this scattering problem involves a constellation, where the scattering geometry is of semi-infinite extend, both the source and observation points are at infinity, and we do not have a convergence-enforcing element such as the spherical Bessel functions of the first kind. Alternately, one may argue that we are trying to describe a two-dimensional (ϑ and ϕ depending) field which contains singularities (at the angles of reflection) by means of a spherical-multipole expansion, which of course does not explicitly have such singularities. In principle, the situation is comparable with the attempt to evaluate the completeness relation (with respect to odd functions on −π ≤ ϕ ≤ π) of the δ-distribution

equation image

at ϕ ≠ ϕ′ by means of a simple summing-up procedure. As can be immediately seen, the corresponding partial-sum series alternates between ±1, independently of the upper limit of the sum, though it should yield zero. Similarly, an alternating behavior can be observed for the results obtained with (35). For such series it is appropriate to apply sequence transformation techniques to find out the assigned limiting value. Among these techniques the linear ones (that is, the relation between the original partial-sum sequence and the transformed partial-sum sequence can be described by a matrix with constant elements) are easy to use and have proven to be efficient and effective especially for alternating series. Moreover and most important, the linear methods are always consistent, that is, if the transformed series converges then the obtained limiting value is the correct one [Hardy, 1963]. This behavior has not been generally proven for nonlinear methods. Linear methods (as well as nonlinear ones) have been used in previous works for the evaluation of multipole expansions related to semi-infinite cones. For instance [Siegel et al., 1955] used linear sequence transformations due to Euler for the evaluation of the radar cross section of a symmetrically (nose-on) illuminated circular PEC cone.

Figure 3.

Real part of the magnetic field near the tip of an elliptic cone with half opening angles αx = 45°, αy = 60° for a plane wave (wave length Λ) incident as indicated and (electrically) polarized in the xz plane.

[14] In this paper, for the numerical evaluation of the multipole expansions we have applied a most simple linear sequence transform due to the Italian mathematician Cesàro. The original partial-sum sequence sn (n = 0, 1, 2, 3, …) is transformed into a new one sn through the instruction

equation image

To demonstrate the function of the method, Figure 4 shows the partial-sum series for the real part of the scattering coefficient Dθequation image as a function of the order n of the multipole expansion (35), evaluated for a certain angle of incidence and for a certain observation point. Note that the summation of all the contributions belonging to the same value n (indicated by m in (35)) is performed without any sequence transform, because in this summation “direction” the number of terms is limited anyway. Obviously, for this configuration we achieve a good approximation of the limiting value already within a few n after a double-time Cesàro transform. By applying this technique for each angle of observation, we obtained the electric far field scattered by different circular and elliptic PEC cones. The relevant geometry is sketched in Figure 5. In order to initially validate the method we perform a comparison with a result available in the literature. Among the (rather few) works which contain computed scattering coefficients and scattering cross sections (where all of them are based on a ‘nose-on’ incidence of the plane wave), Blume and Kahl [1987] derived a Physical Optics result for the bistatic scattering cross section of an elliptic (including a circular) cone. Figure 6 shows the normalized bistatic scattering cross section of a nose-on illuminated circular cone (with half opening angle 60°) in the two symmetry planes, as compared to the cited PO result which is valid in the range 0 ≤ θ < 60°. It should be noted that the Physical Optics result of the nose-on radar cross section has been supposed to be a very good approximation [Blume, 1996], and the result in Figure 6 shows that this is true for the bistatic scattering cross section as well, at least for nose-on incidence. Next, we have numerically evaluated the scattering coefficients for an elliptic cone with half opening angles αx = 45°, αy = 60 for a plane wave incident from θinc = 105°, ϕinc = 0°. Figures 7 and 8 show amplitude and phase of the scattering coefficient Dθequation image in the xz plane, each for different upper limits of the considered orders of the spherical-multipole expansion, nmax = 40 and nmax = 45. Apart from the angles nearby the reflection and forward scattering directions, both the amplitudes and phases of the approximate evaluations compare well. Note that of course the singularities are approximated only by high amplitudes at the corresponding angles, and increasing nmax leads to an increase of these amplitudes. Figures 9–12 show amplitudes and phases of the scattering coefficients in the yz and xy plane, respectively. Owing to symmetry reasons there is no Dϕequation image coefficient in the xz plane (plane of incidence). Finally, Figures 13–18 show the amplitudes and phases of the electric far field scattered by a semi-infinite PEC circular cone (half opening angle αx = αy = 45°) illuminated by a plane wave electrically polarized in the xz plane and incident from θinc = 45°, ϕinc = 0°.

Figure 4.

Evaluation of the real part of the scattering coefficient Dθθ(θ, ϕ; θinc, ϕinc) as a function of the order n of the multipole expansion (35). Here θinc = 105°, ϕinc = 0°, θ = 68°, ϕ = 0°. Dotted line is original partial-sum sequence, dashed line is single Cesàro transformed, and solid line is double Cesàro transformed.

Figure 5.

Geometry of the PEC cone illuminated by a plane electromagnetic wave.

Figure 6.

Normalized (to the squared wavelength Λ2) bistatic scattering cross section for a circular cone (half opening angles αx = αy = 60°) in the xz plane (solid line) and in the yz plane (dashed line), illuminated by a plane wave polarized in the y direction from θinc = 0° (nose-on), compared to the Physical Optics result given by Blume and Kahl [1987]. The circular cone has the half opening angle αx = αy = 60°. The PO result is valid only in the range 0 ≤ θ < 60° and identical in both planes.

Figure 7.

Amplitude of the scattering coefficient Dθθ in the xz plane of a semi-infinite elliptic cone with half opening angles αx = 45°, αy = 60° as a function of θ for two different upper limits nmax of the order of the multipole expansion (35). Plane wave is polarized in the xz plane and is incident from θinc = 105°, ϕinc = 0°.

Figure 8.

Phase of the scattering coefficient Dθθ in the xz plane of a semi-infinite elliptic cone with half opening angles αx = 45°, αy = 60° as a function of θ for two different upper limits nmax of the order of the multipole expansion (35). Plane wave is polarized in the xz plane and is incident from θinc = 105°, ϕinc = 0°.

Figure 9.

Amplitudes of the scattering coefficients Dθθ and Dϕθ in the yz plane of a semi-infinite elliptic cone with half opening angles αx = 45°, αy = 60° as a function of θ. Plane wave is polarized in the xz plane and is incident from θinc = 105°, ϕinc = 0°, nmax = 40.

Figure 10.

Phases of the scattering coefficients Dθθ and Dϕθ in the yz plane of a semi-infinite elliptic cone with half opening angles αx = 45°, αy = 60° as a function of θ. Plane wave is polarized in the xz plane and is incident from θinc = 105°, ϕinc = 0°, nmax = 40.

Figure 11.

Amplitudes of the scattering coefficients Dθθ and Dϕθ in the xy plane of a semi-infinite elliptic cone with half opening angles αx = 45°, αy = 60° as a function of θ. Plane wave is polarized in the xz plane and is incident from θinc = 105°, ϕinc = 0°, nmax = 40.

Figure 12.

Phases of the scattering coefficients Dθθ and Dϕθ in the xy plane of a semi-infinite elliptic cone with half opening angles αx = 45°, αy = 60° as a function of θ. Plane wave is polarized in the xz plane and is incident from θinc = 105°, ϕinc = 0°, nmax = 40.

Figure 13.

Amplitude of the scattering coefficient Dθθ in the xz plane of a semi-infinite circular cone with half opening angle αx = αy = 45° as a function of θ. Plane wave is polarized in the xz plane and is incident from θinc = 45°, ϕinc = 0°, nmax = 40.

Figure 14.

Phase of the scattering coefficient Dθθ in the xz plane of a semi-infinite circular cone with half opening angle αx = αy = 45° as a function of θ. Plane wave is polarized in the xz plane and is incident from θinc = 45°, ϕinc = 0°, nmax = 40.

Figure 15.

Amplitudes of the scattering coefficients Dθθ and Dϕθ in the yz plane of a semi-infinite circular cone with half opening angle αx = αy = 45° as a function of θ. Plane wave is polarized in the xz plane and is incident from θinc = 45°, ϕinc = 0°, nmax = 40.

Figure 16.

Phases of the scattering coefficients Dθθ and Dϕθ in the yz plane of a semi-infinite circular cone with half opening angle αx = αy = 45° as a function of θ. Plane wave is polarized in the xz plane and is incident from θinc = 45°, ϕinc = 0°, nmax = 40.

Figure 17.

Amplitudes of the scattering coefficients Dθθ and Dϕθ in the xy plane of a semi-infinite circular cone with half opening angle αx = αy = 45° as a function of θ. Plane wave is polarized in the xz-plane and is incident from θinc = 45°, ϕinc = 0°, nmax = 40.

Figure 18.

Phases of the scattering coefficients Dθθ and Dϕθ in the xy plane of a semi-infinite circular cone with half opening angle αx = αy = 45° as a function of θ. Plane wave is polarized in the xz plane and is incident from θinc = 45°, ϕinc = 0°, nmax = 40.

6. Conclusions

[15] The scattering of a plane electromagnetic wave by a perfectly conducting semi-infinite cone with elliptic or circular cross section has been investigated by means of mixed cone and free-space eigenfunction expansions which consist of Lamé functions and polynomials. The method analytically yields the pure scattered far field at arbitrary directions in the form of a free-space spherical-multipole expansion. This series does not converge in the usual sense, but a simple Cesàro sequence transformation suffices to consistently transform the series into a convergent one and to obtain asymptotic results. Future work will include systematic studies in more sophisticated sequence transformation techniques, as well as investigations of suitable combinations of the exact spherical-multipole based results with asymptotic (e.g., Physical Optics) ones.

Appendix A

A1. Integrals Involving Spherical Bessel Functions

[16] The two types of integrals involving spherical Bessel functions which have to be solved in (36) and (37) will be denoted by I1 and I2 in the following. The “critical case” of the Weber-Schafheitlein integral [Watson, 1995, section 13.41(2)] is given by

equation image
equation image

where Γ(x) denotes the Gamma function. For λ = 1 we have [Watson, 1995, section 13.41(7)]

equation image

On the basis of the relation of the spherical Bessel function to (ordinary) Bessel functions

equation image

it holds

equation image

which can be written in the form

equation image

with sinc (x) = sin(x)/x. It is seen that

equation image

With the addition theorem

equation image

we have

equation image

A2. Integrals Involving Periodic Lamé Functions

[17] The two types of integrals involving periodic Lamé functions which have to be solved in (36) and (37) will exemplarily be treated and denoted by I3 and I4. As mentioned in section 3, for each n always one of the two alternative series representations of (9)–(12) becomes finite. Exemplarily, we have the finite expansions

equation image

for the function type 2, where the upper limit M is found to be the nearest integral number smaller (n − 1)/2. With (10) we have

equation image

Similarly, we derive for

equation image

Ancillary