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.
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 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. ). 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.
 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.  [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].
 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  GTD results.
 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
 The sphero-conal coordinate system r, ϑ, ϕ is related to Cartesian coordinates by
with ranges of variability 0 ≤ r < ∞, 0 ≤ ϑ ≤ π, 0 ≤ ϕ ≤ 2π and real nonnegative parameters k, k′ which satisfy k2 + k′2 = 1. The normalized metric scaling coefficients will be denoted by
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.
3. Solution of Maxwell's Equations in Sphero-Conal Coordinates
 The Helmholtz equation
is completely separable in sphero-conal coordinates, that is, the corresponding elementary solutions can be written in the form
They consist of products of spherical Bessel functions zv(κr), periodic Lamé functions Φv(ϕ) and nonperiodic Lamé functions Θv(ϑ). The latter two satisfy the differential equations
which are coupled by the two separation constants v(v + 1) and λ.
 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,
The four types of periodic Lamé functions are described by each two alternative Fourier series,
while the nonperiodic Lamé functions are expanded in terms of associated Legendre functions of the first kind [Jansen, 1976],
Here the algebraic factors T(i) are recursively defined as
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 ≤ λ ≤ v(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.
 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,
where Z = 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
where the prime denotes the differentiation with respect to r, = ω is the wave number of the medium, and = /r. The transverse multipole functions v and v are related to the Lamé products by
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
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
 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
where e is an arbitrary current density distribution and C 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
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(r)) to ensure regularity of the field everywhere, and of spherical Hankel functions of the second kind (hn(2)(r)) 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 inc and perpendicularly polarized to inc. 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 0 ≤ ≤ π) the multipole expansion
with the multipole amplitudes:
Note that the plane wave of amplitude E0 is incident from (ϑinc, ϕinc) and electrically polarized in the direction . The exact surface current on the cone's surface is then found as
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
where 0 (, ′) is the dyadic Green's function of the free space in sphero-conal coordinates:
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
with the multipole amplitudes
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 n ≈ σ and n ≈ τ, respectively.
 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
where the dimensionless scattering coefficients Dθ, Dθ, Dϕ, and Dϕ still depend on both, the angles of incidence θinc, ϕinc and the angles of observation θ, ϕ.
 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, jn(κr) 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
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.
 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
To demonstrate the function of the method, Figure 4 shows the partial-sum series for the real part of the scattering coefficient Dθ 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  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 Dθ 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 Dϕ 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°.
 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.
 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
where Γ(x) denotes the Gamma function. For λ = 1 we have [Watson, 1995, section 13.41(7)]
On the basis of the relation of the spherical Bessel function to (ordinary) Bessel functions
which can be written in the form
with sinc (x) = sin(x)/x. It is seen that
With the addition theorem
A2. Integrals Involving Periodic Lamé Functions
 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
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