### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. New Analytical Results: Isorefractive Media
- 3. Mathematical Motivation for Regularization
- 4. Spherical Cavities
- 5. Spheroidal Geometry
- 6. More General Scatterers
- 7. Bodies of Revolution
- 8. Propagation Studies
- 9. Riemann-Hilbert Approach
- 10. Conclusion
- Acknowledgments
- References

[1] Numerical techniques for Maxwell's equations or the Helmholtz equation often encounter difficulties of accuracy and convergence with scattering structures that feature cavities and sharp edges and are of moderate or large size (in wavelengths). Conversion to an equivalent integral equation formulation usually does not ameliorate the difficulty. This paper surveys some recent progress in techniques that address these difficulties for a variety of canonical and noncanonical structures. While some canonical cavity problems admit a purely analytical approach, another promising approach employs processes of analytical regularization that transform the basic integral equations to a second kind Fredholm matrix equation. The main tool is the Abel integral transform applied to trigonometric and other appropriate functions of hypergeometric type. The transformed equations are well conditioned (in contrast to the original formulation derived from Maxwell's equations); standard numerical techniques for their solution are easily applicable; near- and far-field scattered field results may be computed reliably and accurately. The process in spherical and spheroidal geometry is illustrated for the simplest canonical problem of a cavity excited by an axially located electric dipole. Extension of the technique to arbitrarily shaped (nonsymmetric) cavities has great practical implications, and progress on two-dimensional structures is described. Finally, recent developments from another class of noncanonical scatterers, arbitrarily shaped bodies of revolution, are reviewed.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. New Analytical Results: Isorefractive Media
- 3. Mathematical Motivation for Regularization
- 4. Spherical Cavities
- 5. Spheroidal Geometry
- 6. More General Scatterers
- 7. Bodies of Revolution
- 8. Propagation Studies
- 9. Riemann-Hilbert Approach
- 10. Conclusion
- Acknowledgments
- References

[2] The demand for reliable and effective methods for predicting the scattering and propagation of electromagnetic waves arises both at the fundamental scientific level and in technological application. Accurate predictions of the wave interaction with its environment, its transmitting antenna, a sensor, or a remotely sensed target, is essential in assessing the feasibility or success of the potential applications in areas such as communications, remote sensing and imaging, and so on. Robust theoretical and numerical methods are not merely cost-effective alternatives or supplements to experimental methods, but are primarily design tools for the ever expanding array of devices and sensors of modern electrical engineering.

[3] The growth of widely available and powerful computer resources has had a profound effect on the development and applications of numerical methods in this area. If an integral equation with a Green's function kernel (such as the electric field integral equation) is used to formulate the scattering problem, the method of moments, first popularized by *Harrington* [1968], produces a finite system of linear algebraic equations for the coefficients of the basis functions that have been selected to represent the desired surface current. A variety of applications is collected by *Miller et al.* [1992]. Alternatively, direct discretization of Maxwell's equations leads to methods such as the finite difference time domain (FDTD) method [*Yee*, 1966] or the finite element method (FEM) [*Silvester and Ferrari*, 1990]. The greatest success of both integral and differential approaches has been in the low- to intermediate- (or resonance) wavelength regime (where the scatterer dimensions are one to several wavelengths in size); the computational cost limits the size of scattering problem that may be effectively solved. The text [*Peterson et al.*, 1998] surveys these methods and their effective implementation.

[4] However, there is a more profound question about the accuracy of numerical methods as the wavelength dimensions increase: A dense system of linear equations of hundreds of thousands of variables cannot avoid the possibility or indeed likelihood of ill-conditioning that substantially degrades the accuracy of any numerical solution computed by direct or iterative means (even with preconditioning).

[5] The mathematical foundations for error estimation in numerical solutions of the integral equations that arise in electromagnetics has been carefully discussed by *Hsaio and Kleinman* [1997] and *Colton and Kress* [1992], and while progress has been made in obtaining reliable bounds for smooth closed scatterers (such as the sphere), open surfaces (with edges) and nonsmooth surfaces are rather more problematic.

[6] This recognition provides one of the enduring imperatives for analytical methods and solutions in electromagnetics: reliable benchmark solutions enable direct assessment of the accuracy of numerical methods in every aspect of choices of surface discretization, basis functions, and numerical algorithm to solve the discretized system (direct or iterative).

[7] Another enduring reason for the study of analytical solutions to Maxwell's equations is the identification of dominant scattering mechanisms: solutions of canonical problems chosen to highlight features, such as edges or cavity-backed apertures, provide reliable quantitative predictions and identify those target features that must be carefully addressed by purely numerical general purpose codes for other noncanonical objects, of arbitrary shape. The classic text by *Bowman et al.* [1987] provides an excellent survey of known scattering results for a variety of mainly closed canonical surfaces (such as the sphere).

[8] Related to analytic methods is the development of more accurate “semianalytical” or “analytical numerical” methods that are applicable to a wider class of scatterers than the idealized canonical shapes. The method of regularization (MoR) described in this article falls into this class: it addresses the key difficulty of error estimation encountered with the usual integral equations of electromagnetics by analytically transforming them to second-kind matrix systems that are intrinsically well conditioned and have a firm basis for error estimation in Fredholm theory. This rigorous approach to scattering was extensively developed by V. P. Shestopalov and coworkers in Kharkov (Ukraine) starting in the mid-1970s, in both electromagnetic and acoustic contexts [*Radin and Shestopalov*, 1973; *Vinogradov and Shestopalov*, 1977; *Vinogradov*, 1978].

[9] The roots of the method may be traced to closely related regularization methods for potential theory problems [*Sneddon*, 1966; *Vinogradov et al.*, 2001]. The underlying guiding principle is to transform from first kind integral equations to second kind equations, by a process of analytical regularization. If we represent the integral equation in operator form as **AJ** = **E**, where the unknowns **J** (typically surface currents) are to be determined from a known excitation **E**, we seek a splitting of the operator **A** as the sum **A** = **A**_{0} + **A**_{1}, where the operator **A**_{0} is explicitly invertible (by some means). Then the equation can be transformed to (**I** + **H**)**J** = **A**_{0}^{−1}**E**, where **H** = **A**_{0}^{−1}**A**_{1}. Provided **H** has the correct property that makes this a second kind equation (**H** must be a compact operator in an appropriate function space), this transformed equation is the desired second kind regularized form (a compact perturbation of the identity **I**). Second kind functional equations have some intrinsic advantages over those of first kind, and there is a well developed literature and standard numerical techniques [*Atkinson*, 1997] for their solution. Moreover, Fredholm theory can be applied, the numerical convergence properties are excellent and the numerical error can be reliably estimated [*Kantorovich and Akilov*, 1974]. However reliable, if somewhat less easily applied, methods have been developed based upon coercive bilinear forms by E. P. Stephan, W. L. Wendland, and M. Costabel [see, e.g., *Carstensen and Stephan*, 1996]. The generic idea, of transformation to second kind equations, has been widely used in many areas of applied mathematics; for example it was used by *Jones* [1981] to establish the existence and uniqueness of the solution to the integral equation for the thin wire antenna (with exact kernel, rather than reduced kernel). An exposition of the method for scattering problems is given by *Vinogradov et al.* [2002].

[10] The validation of putative methods for calculating scattering from complex objects incorporating edges and cavity-backed apertures, which themselves may enclose a variety of other scatterers, depends entirely upon comparison with the results of other proven approaches, whether analytical, computational or experimental.

[11] This paper surveys some recent developments in analytical or analytical: numerical methods that illustrate their continuing importance in electromagnetic theory and its applications. Some recently developed analytical solutions for elliptically shaped cavities are described in section 2. The mathematical background that motivates the development of rigorous analytical-numerical methods such as MoR is described in section 3. Sections 4 and 5 explain the application of the MoR to canonical problems for spherical structures and spheroidal structures, by choosing the simplest representative example: the dipole excited spherical or spheroidal cavity. Sections 6 and 7 treat noncanonical problems for two classes of arbitrarily shaped structures: two-dimensional structures and bodies of revolution. Sections 8 and 9 briefly describe other applications and approaches to regularization methods.

### 3. Mathematical Motivation for Regularization

- Top of page
- Abstract
- 1. Introduction
- 2. New Analytical Results: Isorefractive Media
- 3. Mathematical Motivation for Regularization
- 4. Spherical Cavities
- 5. Spheroidal Geometry
- 6. More General Scatterers
- 7. Bodies of Revolution
- 8. Propagation Studies
- 9. Riemann-Hilbert Approach
- 10. Conclusion
- Acknowledgments
- References

[17] In this section we discuss the mathematical background that motivates the development of rigorous analytical-numerical methods such as MoR. Throughout, it will be convenient to employ Maxwell's equations for the electromagnetic field (**E**, **H**) in symmetrized form

where a time harmonic dependence of form *e*^{−iωt} (*ω* = *ck*) is assumed and suppressed (effectively the free space impedance *Z*_{0} is normalized to unity). To fix ideas, consider two-dimensional transverse magnetic (TM) scattering from an infinitely long cylindrical surface with cross section that is independent of one axial direction, fixed to be the *z* axis. The arbitrarily shaped surface or screen is assumed to be infinitely thin, perfectly conducting, and is open, i.e., has an aperture.

[18] The screen is illuminated by the *z*-independent plane electromagnetic wave

where = *x* + *y* is the position vector, is the wave vector with magnitude *k* related to wavelength *λ* by ∣∣ = *k* = 2*π*/*λ*. The electromagnetic field is of transverse magnetic type (*E* polarization); the only nonzero components are *E*_{z}, *H*_{x}, *H*_{y}. The total field

and the field

resulting from the scattering of the incident wave by the screen are to be found. The boundary value problem corresponding to this physical problem may be formulated in terms of the Helmholtz equation, with Dirichlet boundary conditions.

[19] Let *L* denote the two-dimensional cross section of the open screen (enclosing the cavity) in the *xy* plane. Let *p* = *p*(*x*, *y*) denote a point in the plane **R**^{2}. The scattered field *U*_{s}(*p*) satisfies the Helmholtz equation

at each off-body point *p*. In addition it must obey the Sommerfeld radiation conditions ensuring that it represents an outgoing wave at infinity. The boundary condition on *L* is the vanishing of the total field, so that

at each *q* ∈ *L*. For smooth closed scatterers, these conditions are adequate to ensure that a unique solution exists. However, if the scattering surface has singular points, or is open, the energy of the scattered field (**E**^{s}, **H**^{s}) in any finite volume *V* of space must be finite,

[20] This implies that for every finite region *S* of the cross-sectional plane,

[21] This provides the correct order of singularity in the field near the edge of the scatterer, of form *r*^{−1/2} where *r* is the distance of observation point to the nearest point on the edge. With this additional condition, the boundary value problem has a unique solution [*Jones*, 1986]. On the other hand, when the fields are represented by their Fourier expansions (or Fourier-Legendre expansions), this condition determines the Hilbert space of numerical sequences to which the unknown amplitude coefficients belong.

[22] A well-known form of solution uses the single layer potential representation of the scattered field in terms of the line current *J*_{z},

at each point *q *∈ **R**^{2}, where *dl*_{p} is the differential of arc length at the point *p *∈ *L*, *G*_{2}(*p*, *q*) = −1/4*iH*_{0}^{(1)}(*k*∣*p *− *q*∣) is the free space Green's function, *H*_{0}^{(1)} being the Hankel function of first kind and order zero. Defining the unknown normalized line current density *Z*(*p*) as

the scattered field representation becomes,

[23] Applying the boundary condition (6) to equation (11) yields the two-dimensional electric field integral equation

for the unknown current density *Z*(*p*). Once this is found, the scattered field at any point *q* can be found from (11).

[24] This is succinctly represented in operator form

where *A* denotes the integral operator in (12) with kernel *G*_{2}(*p*, *q*). Most numerical solution procedures for this equation proceed by choosing a finite basis of linearly independent functions *Z*_{1},…,*Z*_{n} and seeking the solution *Z* in the form

so that

[25] The constants *x*_{1}, …, *x*_{n} are determined by some testing procedure using the inner product defined for functions *W*_{1}, *W*_{2} by

[26] For example, a Galerkin procedure leads to the system of *n *linear equations

with an associated matrix *M* of elements *M*_{ij} = (*Z*_{i}, *AZ*_{j}). Inversion of the system produces the coefficients *x*_{1}, …, *x*_{n}. In this process the condition number of *M* is critical, especially as the number *n* increases (for local basis functions *n* can usually be related to the discretization of the cross section *L*). If the condition number increases unboundedly as *n * ∞, the solution *Z* constructed from (14) almost certainly will not converge because of the increasingly unstable inversion process (small numerical errors in the values (*Z*_{i}, −*U*_{0}) will generate large errors in the coefficients *x*_{i}.)

[27] Equation (12) is an integral equation of first kind and unfortunately such equations exhibit this increasingly unstable behavior. The smoother the kernel, the more rapidly is the onset of numerical instability observed as *n* increases; in this case the singularity of the Green's function (the kernel) means that the effect becomes noticeable only at finer discretizations; however, when the structure is able to trap energy, it will be found that coarser levels of discretization are inadequate and the problem of achieving accuracy at finer discretizations becomes more pressing. This difficulty is commonly observed when the Method of Moments is applied to highly resonant (high-Q) structures.

[28] The key mathematical question is whether the operator equation is well posed, meaning that, in addition to the obvious requirements of solution existence and uniqueness, is it insensitive to small perturbations in the excitation function? More precisely, is there a constant *C*, depending only upon the operator *A*, so the difference of two solutions *Z*_{1} and *Z*_{2} to *AZ*_{1} = *F*_{1} and *AZ*_{2} = *F*_{2}, respectively, is bounded (in a suitable norm) according to

(This implies that as ∥*F*_{1} − *F*_{2}∥ 0, also ∥*Z*_{1} − *Z*_{2}∥ 0.)

[29] Unfortunately, the integral equation (12) is not well posed, and the difficulty can be traced to the fact that *A* is compact [see *Adams*, 1975]. It should be noted that the well posedness of the operator depends upon the precise image space and its associated norm. If we can find a norm for which the range of the operator is closed and restrict the choice of the forcing functions (*F*_{1}, *F*_{2}) to have finite norm, then the inverse operator *A*^{−1} is bounded and the operator equation is well posed. However, when the domain and range of the operator *A* in equation (13) are associated with the same normed function space, the operator is compact and has an unbounded inverse.

[30] This may be demonstrated by applying the adjoint operator *A** to the integral equation to obtain the equivalent equation

[31] It may be shown that the operator *A** *A* is self-adjoint and compact, and has an infinite sequence of real eigenvalues *λ*_{n} and an associated complete set of orthogonal eigenfunctions *e*_{n} satisfying

furthermore, *λ*_{n} 0 as *n* ∞. Thus if we express

for suitable constants *μ*_{n}, then the solution is

[32] No eigenvalue *λ*_{n} is zero, as we have assumed existence and uniqueness of the solution; however, it is now obvious that small perturbations, numerically generated or otherwise, in the excitation function may certainly cause large perturbations in the solution.

[33] By contrast second kind integral equations give rise to operator equations of the form

where *I* denote the identity operator and *A* is compact. The magnetic field integral equation is of this type. These are well-posed equations (as can be seen by a similar eigenvalue analysis), and as mentioned above there is a well developed theory for estimating the accuracy of solutions to such equations [*Kantorovich and Akilov*, 1974]. However, the magnetic field integral equation is not applicable to the open surfaces and screens that are mainly considered in this paper.

[34] Three common approaches to stabilizing first kind equations may be mentioned. The first determines the eigenfunctions *e*_{n} and uses a representation of the exciting field that ensures *μ*_{n}/*λ*_{n} 0 sufficiently rapidly as *n* ∞. This is computationally intensive. Practically, it requires the computation of the singular values and the singular value decomposition of the matrix *M* coupled with the choice of a threshold below which the singular values are ignored and the associated least squares solution determined. It is difficult to avoid some arbitrariness in the choice of the threshold and consequent uncertainty about the accuracy of the solution (as *n* increases).

[35] A second approach is Tikhonov regularization [*Tikhonov and Goncharsky*, 1987]. Instead of determining the solution which satisfies

we determine the solution of

where *B* is some bounded operator (the simplest choice is the identity) and ɛ is the so-called regularization parameter. The solution of this minimization problem is obtained by solving the linear system

[36] This problem is now well posed; however, the optimal choice of the parameter is difficult to specify in advance, and various methods for its determination have been proposed [e.g., *Hansen and O'Leary*, 1993]. In any case, the solution is not the exact solution of the original problem, but some solution hopefully not too removed from the actual physical solution.

[37] It is therefore desirable, wherever possible, to convert the operator equation to one of second kind with a compact operator for which the Fredholm alternative holds. This third approach is known as (analytical) regularization. Formally, the bounded linear operator *R* is called a (left) regularizer of *A* if

where *K* is a compact operator (on a suitable function space). Some general properties of regularizers are described by *Kress* [1995, chapter 5]. Application of the regularizer *R* to (13) produces an equation of the desired format:

[38] The boundedness of *R* ensures that perturbations to *U*_{0} do not lead to arbitrarily large perturbations of *RU*_{0} which would otherwise defeat the purpose of this regularization.

[39] In general, the construction of *R* may be difficult, if not impossible. However, the dual series equations arising from the potential problems and diffraction problems considered by *Vinogradov et al.* [2001, 2002] can indeed be regularized; the regularization process is explicitly described, although the regularizer appears only implicitly in the analytical treatment of the dual series equations. The regularized equations enjoy all the advantages of second-kind equations for which the Fredholm alternative holds, including precise estimates of the error, or difference of any solution computed to a truncated system, from the true solution (as a function of truncation number *N*_{tr}). The error decays to zero as *N*_{tr} ∞ (and in practice quite rapidly beyond a certain cutoff point, usually related to the electrical size of the body in diffraction problems).

[40] In concluding this section, it is worth remarking that another approach to solving the operator equation (13) is to determine appropriate function spaces *H*_{1} and *H*_{2} for the domain and range of the operator *A*, in which it has a bounded inverse. Effectively, the problem is regularized by this determination; much of the effort in this approach focusses on the definition of *H*_{1} and *H*_{2} and on the discretization process within the associated inner product and norm [*Carstensen and Stephan*, 1996].

### 4. Spherical Cavities

- Top of page
- Abstract
- 1. Introduction
- 2. New Analytical Results: Isorefractive Media
- 3. Mathematical Motivation for Regularization
- 4. Spherical Cavities
- 5. Spheroidal Geometry
- 6. More General Scatterers
- 7. Bodies of Revolution
- 8. Propagation Studies
- 9. Riemann-Hilbert Approach
- 10. Conclusion
- Acknowledgments
- References

[41] The analytical regularization methods described by *Vinogradov et al.* [2001, 2002] may be simply illustrated in the context of spherically shaped cavities. We use standard spherical coordinates (*r*, θ, ϕ).

[42] Consider the perfectly conducting open spherical shell of radius *a* with a circular aperture subtending an angle θ_{0} at the origin. Locate an electric dipole on the *z* axis a distance *d* (<*a*) in the positive direction from the origin; the *z* axis is the axis of symmetry and the dipole moment of strength *p* is aligned with the axis. See Figure 1. In this transverse magnetic case, the only nonzero field components are (*E*_{r}, *E*_{θ}, *H*_{ϕ}) and *E*_{r}, *E*_{θ} may be expressed in terms of *H*_{ϕ} as

[43] The incident dipole field has magnetic field component that may be expanded in spherical harmonics,

here *ψ*_{n}(*x*) = *xj*_{n}(*x*), *ζ*_{n}(*x*) = *xh*_{n}^{(1)}(*x*) and *j*_{n}, *h*_{n}^{(1)} denote the spherical Bessel functions of order *n*.

[44] A rigorously correct formulation of the diffraction problem providing a unique solution to Maxwell's equations is as follows. First, the solution must be continuous across surface *r* = *a* and satisfy the mixed boundary conditions of continuity of the magnetic field across the aperture,

and vanishing of the electric field on the perfectly conducting surface,

[45] Second, the scattered field (**E**^{s}, **H**^{s}) must represent an outgoing spherical wave at infinity, and finally, the energy of the scattered field in any arbitrary finite region *V* of space, including the cavity edges, must be finite. Uniqueness of the solution is guaranteed by this set of boundary, edge and radiation conditions [*Jones*, 1986, section 9.1; *Tikhonov and Samarskii*, 1963, chapter 7, section 4].

[46] We thus seek the scattered field in the form

where

[47] By using the asymptotic behavior of the spherical Bessel functions and their derivatives, the finite energy condition (7) effectively constrains the as yet unknown coefficient sequence *x*_{n} to be square summable. This provides the correct mathematical setting of a Hilbert space in which all the operations described below are justifiable.

[48] Matching the internal cavity field and external region field across the aperture via the mixed boundary conditions produces the dual series equations

and

when θ_{0} < θ < *π*; here

and

is an asymptotically small parameter: as *n* ∞, ɛ_{n} = *O*((*ka*/*n*)^{2}).

[49] This system of dual series equations is equivalent to a first kind Fredholm integral equation (essentially the electric field integral equation): it exhibits ill-conditioning that becomes especially noticeable under the discretization process of numerical methods such as the Method of Moments, in which the cavity grid is increasingly refined.

[50] The method of regularization converts this system into a second-kind Fredholm matrix equation. The motivating idea may be explained as follows. Suppose that (34) were replaced by

so that the pair (38), (35) would be

[51] Then we could use the orthogonality of the family of functions {*P*_{n}^{1}(cos θ), *n* = 1, 2, …} to convert (39) to a system of linear equations for the coefficients *x*_{n}. With appropriate scaling this system would be well conditioned.

[52] However, the difficulty is the imbalance in the decay rate of the individual terms in (34), (35). Because the unknown coefficient sequence is square summable, it can be readily estimated that the general term of series (34) and (35) decays at rate *O*(*n*^{−3/2}) and *O*(*n*^{−1/2}), respectively, as *n * ∞. Thus the first series (34) converges uniformly, while the second (35) is nonuniformly convergent. It is necessary to transform these series to a form employing another family of orthogonal functions: effectively, we are changing basis.

[53] Recognizing that

and that term-by-term integration of the series (35) is permissible, we obtain following uniformly converging series:

where *C*_{1} is a constant of integration, for θ_{0} < θ < *π*.

[54] The basis conversion is effected by an Abel integral transform of a rather general type. It may be applied to harmonic expansions comprising trigonometric and other special functions of hypergeometric type [*Vinogradov et al.*, 2001, section 1.6]. In this particular case the transform reduces to the Mehler-Dirichlet formulae

valid for *n* ≥ 0, from which are deduced the integral representations

[55] Inserting these identities in the equations (34) and (41) produces

and

for θ_{0} < θ < *π*. An interchange of the order of integration and summation is justified, and produces the two Abel integral equations

where

and

[56] Both integral equations have a unique solution, namely the zero solution, and we deduce

[57] Exploiting the orthogonality of the family {sin (*n *+ (1/2))θ, *n* = 1, 2, …}, we may convert (50) to a system of linear equations for the unknowns *x*_{n} and the constant *C*_{1}. Elimination of *C*_{1} from this system yields

where *m* = 1, 2, …,

and the so-called “incomplete scalar product”

when *n* = *m*, the first term of (53) inside the braces is replaced by θ_{0}.

[58] The transformed equations (50) are well conditioned and standard numerical techniques based upon simple truncation methods are easily applicable and produce reliable results; moreover the computed solution of the system truncated to *N*_{tr} equations converges to the true solution as the truncation number *N*_{tr} increases [*Kantorovich and Akilov*, 1974].

[59] The surface current density has one component

and the radiation pattern *S*_{1}, defined by *E*_{θ}, *H*_{ϕ} ∼ *S*_{1}(θ) *e*^{ikr}/*r* as *r * ∞, equals

[60] It should be noted that the series (54) is slowly convergent and that a convergence acceleration technique [*Vinogradov et al.*, 2002, p. 190] facilitates more accurate representation of the computed surface current.

[61] A near-field characteristic of some interest is the stored or accumulated energy. The energy *W*_{i} accumulated inside (*r* ≤ *a*) the open spherical shell may be distinguished from that energy accumulated in the exterior (*r *> *a*), or more exactly, that part of scattered energy that adheres to the open shell, i.e., the reactive part of the energy. (Physically, the difference between the total energy of the scattered field and the energy carried by the outgoing travelling wave is exactly the stored external energy *W*_{e}.) Explicit expressions for *W*_{i} and *W*_{e} are readily derived in terms of the coefficients *x*_{n}. A typical calculation is shown in Figure 2. A number of strongly resonant features in this spherical cavity (θ_{0} = 30°) excited by a dipole located with *d*/*a* = 0.9. The condition number remains well controlled even at resonance frequencies to guarantee accurate calculation of the field. A typical value of the truncation number is *N*_{tr} = [*ka*] + 12.

[62] This technique has been extended to a wide variety of cavity structures possessing some spherical symmetry [*Vinogradov et al.*, 2002, chapters 5 and 6]: this includes spherical cavities with one or several apertures; the cavity may have metal or dielectric inclusions, and may have dipole (electric or magnetic) or plane wave excitation. Varying the geometric parameters of such canonical structures provides a large number of physically interesting benchmark solutions, against which the accuracy of solutions computed by another general purpose scattering code may be examined. Among these structures are various types of spherical lens reflectors including the Luneberg Lens reflector that is accurately examined by *Vinogradov et al.* [2007].

### 5. Spheroidal Geometry

- Top of page
- Abstract
- 1. Introduction
- 2. New Analytical Results: Isorefractive Media
- 3. Mathematical Motivation for Regularization
- 4. Spherical Cavities
- 5. Spheroidal Geometry
- 6. More General Scatterers
- 7. Bodies of Revolution
- 8. Propagation Studies
- 9. Riemann-Hilbert Approach
- 10. Conclusion
- Acknowledgments
- References

[63] After spherical geometry, spheroidal geometry provides the simplest setting for three-dimensional scattering theory. Depending on its aspect ratio, a closed spheroidal surface takes widely differing shapes ranging from the highly oblate spheroid (the disc at one extreme) to the sphere to the prolate spheroid (and in the limit, a thin cylinder of finite length). In this setting a significant extension of the spherical shell studies to spheroidal cavity structures is possible.

[64] The prolate spheroidal coordinate system (*ξ*, *η*, ϕ) is related to rectangular coordinates by [*Flammer*, 1957]

where *d* is the interfocal distance and −1 ≤ *η* ≤ 1, 1 ≤ *ξ* < ∞, 0 ≤ ϕ ≤ 2*π*; *ξ* and *η* = cos θ play the role of radial and angular parameters, respectively. The quantity 1/2*kd*, known as the wave parameter, is conventionally denoted by *c*.

[65] The simplest situation corresponding to the spherical cavity study of section 4 is the excitation of a metallic spheroidal cavity by an axially located vertical electric dipole. In this axially symmetric situation, only the components *H*_{ϕ}, *E*_{ξ}and *E*_{η}do not vanish. Consider then an infinitely thin perfectly conducting spheroidal shell having one circular hole located so that the *z* axis is normal to the aperture plane and the structure is axisymmetrical. It is excited by a vertical (axially aligned) electric dipole placed along the *z* axis at *ξ* = *ξ*_{1}, *η* = 1 (see Figure 3). The screen occupies the surface

and the aperture is given by

[66] The magnetic field of the electric dipole of moment *p* has spheroidal harmonic expansion [see *Dyshko and Konyukhova*, 1995]

where *R*_{1l}^{(1)}(*c*, *ξ*) and *R*_{1l}^{(3)}(*c*, *ξ*) are the prolate radial spheroidal functions (p.r.s.f.) of first and third kind, respectively, *S*_{1l}(*c*, *η*) is the prolate angular spheroidal function (PASF), and the coefficients *N*_{1l}(*c*) and *χ*_{1l}(*c*) are normalization constants defined by *Flammer* [1957] (or see *Vinogradov et al.* [2002, section 1.2]). The total field *H*_{ϕ} is a sum of the incident and the scattered field *H*_{ϕ}^{sc}

and satisfies Maxwell's equations subject to the following conditions that are analogous to those imposed on the spherical cavity: (1) the tangential component of the total electric field must be continuous across the spheroidal surface *ξ* = *ξ*_{0}; (2) the mixed boundary conditions

are imposed on the total electric field on the screen and on the total magnetic field across the aperture, respectively; (3) the scattered field satisfies radiation conditions, and (4) the scattered energy in any finite volume *V* of space must be finite. These conditions guarantee uniqueness of the solution [*Jones*, 1986, section 9.1; *Tikhonov and Samarskii*, 1963, chapter 7, section 4].

[67] Representing the scattered field in terms of spheroidal harmonics and enforcing both the continuity condition on the shell surface *ξ* = *ξ*_{0}, and the radiation condition, produces

where the unknown Fourier coefficients *A*_{l} are to be found, and

[68] Imposing the mixed boundary conditions (64) on the expansion (65) produces the following dual series equations defined on subintervals corresponding to the screen surface and aperture,

where *η* ∈ (−1, *η*_{0}), and

the coefficients *B*_{l} determined by the incident field (61) equal

[69] It is convenient to define new coefficients *x*_{l} by *A*_{l} = *Z*_{l}^{(3)} (*c*, *ξ*_{0})*x*_{l}(2*l *+ 1)/*l*(*l *+ 1) so that

[70] This set of dual series equations (70)–(71), containing the prolate angle spheroidal functions with the azimuthal index *m* = 1, is the analogue of the spherical cavity set and is to be solved for the coefficients *x*_{l}.

[71] The asymptotic expansions established by *Farafonov* [1983] for the prolate radial spheroidal functions show that

is *O*(*l*^{−2}) as *l * ∞, so that equations (70)–(71) may be written

[72] The next step is to transform the dual series equations (73)–(74) to equations with associated Legendre function kernels that are amenable to solution by the Abel integral transform technique.

[73] At this point, the regularization approach employed for spherical geometry must be significantly adapted because the angular spheroidal functions do not possess Abel type integral representations. Successful treatment of this and more general problems for open spheroidal shells depends upon the conversion of these dual series equations with PASF kernels to dual series equations with associated Legendre function kernels that are amenable to regularization techniques employing the Abel integral transform. This key idea relies on the absolutely convergent expansion of the PASF as a series in associated Legendre functions [*Flammer*, 1957; *Wait*, 1969]

where the coefficients *d*_{r}^{ml}(*c*) depend only on the wave parameter *c* = *kd*/2; in this summation the prime indicates that it is taken over only even or odd values of *r* according as *l *− *m* is even or odd, respectively. The coefficients rapidly decrease, provided *r *> *l*, and the ratio *d*_{r}^{ml}(*c*)/*d*_{r−2}^{ml}(*c*) = *O*(*c*^{2}/4*r*^{2}) as *r* ∞. For given *l* and *m*, and provided *l *> *c*, the dominant coefficient is *d*_{l−m}^{ml}(*c*). Thus

is a sum of a main term and a remainder that is asymptotically small (when *l* ∞). It is important to observe that the angle and wave dependences in the PASF kernel are now effectively separated.

[74] Insert the representation (76) with index *l* = 1 in (73)–(74), set *η* = cos , *η*_{0} = cos θ_{0} and rearrange to obtain

when θ ∈ (0, θ_{0}), and

when θ ∈ (θ_{0}, *π*); here *y*_{l} = *x*_{l}*d*_{l−1}^{1l}(*c*) and *g*_{l} = −*ic**d*_{l−1}^{1l}(*c*)*B*_{l}.

[75] The edge condition implies that the coefficient sequence *y*_{l} is square summable, (∣*y*_{l}∣^{2} < ∞); this is the correct mathematical setting for the solution to the dual series equations (77)–(78), and ensures the mathematical validity of all operations for constructing the regularized solution, which is carried out in the manner similar to that employed for the spherical cavity. It leads to the regularized system of form

where *s* = 1, 2, ….

[76] It may be established that the matrix operator of the system (79) is a compact perturbation of the identity operator, a second kind operator. Thus this system of equations can be effectively solved by the truncation method, with a desired accuracy depending on the truncation number *N*_{tr}. Solutions of engineering accuracy are obtained if *N*_{tr} exceeds the maximal electrical size of the spheroid: *N*_{tr }> *kb* = *cξ*_{0}, where *b* is the semimajor axis.

[77] The accuracy of this estimation is illustrated by the dependence of normalized error versus truncation number *N*_{tr} presented in Figure 4. The error was computed in the maximum-norm sense,

where {*y*_{l}^{N}}_{l=1}^{N}denotes the solution to (79), truncated to *N* equations. The Fredholm nature of this system guarantees that *e*(*N*_{tr}) 0 as *N*_{tr } ∞. The condition number of the system is well controlled, even near values of quasi-eigenfrequencies corresponding to internal cavity resonances, as illustrated in Figure 5. The sharp spikes in the plot occur at frequencies very near those of the internal cavity resonances of the corresponding closed spheroid.

[78] This technique has been extended to a variety of spheroidal cavity structures [*Vinogradov et al.*, 2002, chapter 7] including cavities with one or several apertures. As with spherical cavities, varying the geometric parameters of these canonical structures provides a large number of physically interesting benchmark solutions for the validation of general purpose scattering codes. One interesting extension includes an impedance lining of the cavity [*Smith and Vinogradova*, 1999].

### 6. More General Scatterers

- Top of page
- Abstract
- 1. Introduction
- 2. New Analytical Results: Isorefractive Media
- 3. Mathematical Motivation for Regularization
- 4. Spherical Cavities
- 5. Spheroidal Geometry
- 6. More General Scatterers
- 7. Bodies of Revolution
- 8. Propagation Studies
- 9. Riemann-Hilbert Approach
- 10. Conclusion
- Acknowledgments
- References

[79] The method of regularization has been extended to treat scattering from certain classes of noncanonical objects of more general and arbitrary shape. Although presently restricted to two-dimensional scattering or three-dimensional bodies of revolution, the extension has practical implications for calculations, such as radar cross sections of aircraft engine inlets, or aspects of reflector antenna design.

[80] Consider the cylindrical structure, with cross section *L* as described in section 3. The surface current is determined by the EFIE (12). The fundamental idea here is to split the kernel *G*_{2}(*p*, *q*) as a sum of two terms, *G*_{0}(*p*, *q*) + *G*_{1}(*p*, *q*), choosing *G*_{0}(*p*, *q*) so that the integral equation obtained from (12) upon replacing the kernel *G*_{2} by *G*_{0}, is soluble by analytic means. Upon rewriting (12) as

explicit inversion of the left-hand side transforms this equation to one overtly of second kind. Some constraint on the choice of *G*_{0} and *G*_{1} is necessary to ensure that the transformed equation is genuinely of second kind. As will be seen below, *G*_{0} should contain the dominant singularity of *G*_{2} and the remainder term *G*_{1} should be continuous (actually *G*_{1} will be smooth).

[81] In this analysis it is necessary to regard the open screen cross section *L* as part of a larger closed structure *S*, that is parameterized by two functions *x*(), *y*(θ) on the interval θ ∈ [−*π*, *π*] that are 2*π* periodic so that *x*(−*π*) = *x*(*π*), *y*(−*π*) = *y*(*π*). Thus

the cross section *L* is parameterized by −θ_{0} ≤ θ ≤ θ_{0}, while the aperture *L*′ is created by the removal from *S* of the segment parameterized by −*π* ≤ θ ≤ −θ_{0} or θ_{0} ≤ θ ≤ *π*. The choice of *L*′ is not unique, and may be chosen as conveniently as possible for the problem at hand; however, it must be chosen so that the surface *S* is smooth, particularly at the joining points of *L* and *L*′. With this parameterization, the differential of arc length is *l*(*τ*) = .

[82] Setting *z*_{0}(*τ*) = *Z*(*x*(*τ*), *y*(*τ*)) *l*(*τ*), the functional equation (12) takes the form

where *R*(, *τ*) = is the distance between two points of the cavity parameterized by θ and *τ*, and *u*(θ) = −*U*_{0}(*x*(θ), *y*(θ)). Introduce the new unknown function

so that (8) becomes

[83] Equation (85) together with the requirement that *z* vanishes outside −θ_{0} ≤ θ ≤ θ_{0} is completely equivalent to equation (83).

[84] Recognizing that the Hankel function *H*_{0}^{(1)}(*s*) has a logarithmic singularity at *s* = 0, split the kernel of (85) into a singular part and a remainder,

[85] It will be supposed that the surface *S* is such that *H*(θ, *τ*) is smooth and continuously differentiable with respect to θ and *τ*. Thus

[87] The splitting and subsequent expansion of the kernel that we employ is well known [see, e.g., *Kress*, 1995, section 13.2]. It leads to a dual series equation with exponential functions *e*^{inθ}. The singular part of the Green's function has the Fourier series expansion

where the prime over the summation sign means that *n* ≠ 0. Expand the incident function *u*(θ) and the solution *z*(*τ*) in Fourier series

and expand *H* in a double Fourier series,

[88] The smoothness of *H* ensures that the Fourier coefficients *h*_{np} decay sufficiently rapidly, so that

[89] Insertion of these Fourier expansions (88)–(91) into equation (85) and coupled with the vanishing of *z*(*τ*) outside the interval −θ_{0} ≤ θ ≤ θ_{0}, leads to the dual series equations:

[90] Thus the integral equation (85) is converted to an equivalent dual series equation defined on two subintervals of [−*π*, *π*] with the unknowns {*ζ*_{n}}_{n=−∞}^{∞} to be found. If the right-hand side of (93) is temporarily presumed to be known, this dual series is soluble by the means described in *Vinogradov et al.* [2001, section 2.2]. To transform the system (93)–(94) with exponential kernels to a system with trigonometric functions (cos (*n*θ) and sin (*n*θ)), introduce the new unknowns

where *n* = 1, 2, …, and set *g*_{n}^{+} = *g*_{n} + *g*_{−n}, *g*_{n}^{−} = *g*_{n} − *g*_{−n}. The system may be manipulated to the trigonometric kernel form

and

where

and the matrix coefficients are defined by

if *n*, *p *≥ 0,

if *n *≥ 0, *p *≥ 1,

if *n *≥ 1, *p *≥ 0, and

if *n*, *p *≥ 1.

[91] Equations (97)–(99) are coupled systems of dual series equations with trigonometric function kernels and may be regularized in the standard way described by *Vinogradov et al.* [2001, section 2.2]. The details are omitted. It produces a Fredholm matrix equation of second kind with rescaled unknowns

[92] The matrix operator *H* = (*H*_{mn}) of this equation is compact in the functional space of square summable sequences *l*_{2} and the unknowns *X*_{n}, *Y*_{n} are asymptotically *O*(*n*^{−1}) as *n * ∞. This Hilbert space is the correct setting in which all the mathematical operations are justified.

[93] A truncation method is effective for solving the final matrix equation. As the number *N*_{tr} of equations to be solved increases, its solution rapidly converges to the exact solution of infinite system. The associated matrix is well conditioned and the algorithm is numerically stable. The matrix elements are in closed form; they are simple functions of the Fourier coefficients *h*_{np} that may be computed practically and efficiently using the fast Fourier transform.

[94] As an illustration, consider the thin metallic open cavity specified with two parameters *a* and *q* by

[95] Fix *a* = 1 and take *q* in the range 0 < *q *< 1. See Figure 6. The cavity surface (or metal) and aperture corresponds to the intervals ∣∣ ≤ θ_{0} and θ_{0} ≤ ∣θ∣ ≤ *π*, respectively; the point θ_{0} = 0 always lies on the metal. The parameter *q* determines the width of the structure; its electrical length, fully closed, is approximately 2*ka*.

[96] The regularized system is sufficiently well conditioned to provide accurate solutions, even near those frequencies corresponding to quasi-eigenvalues of the cavity. An example of the dependence of condition number on wave number is shown in Figure 7. The sharp spikes on the plot correspond to quasi-eigenvalues of the cavity. The solution allows the calculation of line current density and radar cross section (RCS). The normalized line current density

may be calculated from (90). It is easy to estimate that *ζ*_{n} = *O*(*n*^{−1/2}) as *n * ∞ so the series (90) is slowly convergent; however, there is a standard method for accelerating its convergence using the technique described by *Vinogradov et al.* [2001, section 4.2].

[97] The radar cross section

when normalized by the factor *πa*, is given by

where the coefficients *β*_{n} are defined via the Fourier expansion of the incident plane wave,

here *τ* = *π* + *α*, and *α* is the angle of incidence of the plane wave to the *x* axis.

[98] The calculation of the scattered far field (and associated RCS (108)) presents few difficulties because the term *β*_{nm} in (109) is exponentially decreasing when ∣*n*∣ or ∣*m*∣ > *kL*_{0}, where *kL*_{0} is electrical size of the screen. Accurate results for the RCS calculation are obtained with truncation number *N*_{tr }> 2*kL*_{0} + 20.

[99] The RCS was computed for the open cavity with the wave directly incident on the aperture (*α* = 0, Figure 8), over the range 1 < *ka *< 40. The spikes in the condition number plot (Figure 7) clearly define quasi-eigenvalues of the cavity. The condition numbers at these frequencies lie in the range 10^{4} to 10^{5}, thus establishing that even at these nearly resonant frequencies, the matrix is sufficiently well conditioned to provide stable and accurate solutions.

### 7. Bodies of Revolution

- Top of page
- Abstract
- 1. Introduction
- 2. New Analytical Results: Isorefractive Media
- 3. Mathematical Motivation for Regularization
- 4. Spherical Cavities
- 5. Spheroidal Geometry
- 6. More General Scatterers
- 7. Bodies of Revolution
- 8. Propagation Studies
- 9. Riemann-Hilbert Approach
- 10. Conclusion
- Acknowledgments
- References

[101] As indicated previously, the Method of regularization has recently been significantly extended to treat scattering by three dimensional bodies of revolution [*Smith et al.*, 2007]. At present the method provides a satisfactory treatment for two situations, where the source of excitation is an axisymmetrically located vertical electric dipole, or a scalar (acoustic) plane wave. To indicate the slightly broader class of possibilities, we formulate and solve the latter case.

[102] Consider a closed body of revolution *S* which is generated by the rotation of a plane curve *L* about the *z* axis, the ends of which lie on the axis. An aperture is opened in an axisymmetric way, creating bowl-shaped screen of revolution *S*_{0} (see Figure 9). Employ standard cylindrical coordinates (*ρ*, ϕ, *z*) and assume that the curve *L* is parameterized, in the half plane ϕ = 0, by smooth functions *ρ*(*τ*), *z*(*τ*), where *τ* ∈ [0, *π*]. As before, the differential of arc length is

and must be positive. The surface *S* is smooth at all points, especially on the *z* axis provided *z*′(0) = *z*′(*π*) = 0, and *ρ*(0) = *ρ*(*π*) = 0. Let the parameter value *τ* = *τ*_{0} (0 < *τ*_{0} < *π*) define the shell edge.

[103] Suppose the scalar wave *u*_{i}(*q*), defined for all points *q* ∈ **R**^{3}, is incident on the screen. The diffracted field *u*_{s}(*q*) for the Dirichlet problem is to be found. A unique solution satisfying the Helmholtz equation is guaranteed by enforcing (1) the boundary conditions *u*_{i}(*q*) + *u*_{s}(*q*) = 0 at each point *q* on *S*_{0}, (2) the Sommerfeld radiation conditions, and (3) a condition on the finiteness of field energy within any confined volume of space.

[104] As is standardly deduced, this problem reduces to solving a Fredholm integral equation of first kind

for each point *q* on *S*_{0}, for *J*_{D}(*p*) is the unknown surface current density; here

is the free space Green's function, and *R*(*q*, *p*) = ∣*q *− *p*∣ is the distance function. Once the surface density is determined, all the associated near and far-field quantities are readily determined.

[105] First we rescale the surface density, analogously to the two dimensional case. Define the scale factor

for a sphere this factor is independent of *τ*. Introduce the unknown function *X*, related to *J*_{D}, but defined for the full interval of the parameter *τ*, by

with 0 ≤ ϕ ≤ 2*π*. Because of axial symmetry of the screen, (111) can be separated into a set of integral equations over *L*:

where the superscript *m* = 0, ±1, ±2, … denotes the *m*th azimuthal Fourier coefficient of the corresponding function.

[106] A careful analysis of the singularities of the azimuthal Green's function *G*^{m}(*τ*_{q}, *τ*_{p}) shows it can be decomposed as

where *g*^{m}(*τ*_{q}, *τ*_{p}) denotes the *m*th azimuthal Fourier coefficient of the Green's function of the unit sphere, and the remainder *D*^{m}(*τ*_{q}, *τ*_{p}) is a sufficiently smooth function, in the sense that all its first derivatives are continuous and all its second-order partial derivatives are square integrable.

[107] Now (115) can be rewritten into the following form:

where the integral equation kernel is separated into the “spherical canonical part” *g*^{m}(*τ*_{q}, *τ*_{p}) involving all singularities and a continuous remainder *D*^{m}(*τ*_{q}, *τ*_{p}). If the remainder term is deleted from (117), the modified equation has an exact analytical solution that provides a basis to regularize (117).

[108] Now fix *m*, and expand all quantities in equation (117) in Fourier-Legendre series, using the normalized associated Legendre function of order *m* and degree *n*:

[109] Insertion of these series transforms (117) to a set of dual series equations, to be solved for the unknowns **x**^{m} = {*x*_{n}^{m}, *n* = ∣*m*∣, *n* = ∣*m*∣ + 1, …}:

[110] The edge condition imposes the condition on the coefficients **x**^{m} that

providing the correct Hilbert space setting for this problem. The dual series are equivalent to an operator equation of first kind in this Hilbert space [*Poyedinchuk et al.*, 2000], so that direct truncation techniques are generally inappropriate.

[111] Dual series with associated Legendre function kernels can be regularized. The detailed solution is given by *Vinogradov et al.* [2001, section 2.3]. By transferring the term involving the coefficients *D*_{nl}^{m} to the right-hand side of (122), this method of analytical regularization, again based on Abel's integral transformation of the associated Legendre function, removes this ill-conditioning and produces an algebraic system, of form

where the matrix **D**^{m} and vector **u**^{m} are the array of coefficients *D*_{nl}^{m} and components *u*_{n}^{m}, respectively; **Q** is a matrix of “incomplete scalar products” [*Vinogradov et al.*, 2001, p. 346], which is equivalent to an orthoprojection operator. The compactness of the matrix operator **D**^{m} follows from the decay rate of its coefficients:

[112] Thus (124) is equivalent to a Fredholm operator equation of the second kind, which solution can be efficiently calculated with any prescribed accuracy by a truncation technique.

[113] In accordance with the Meixner condition, the Fourier-Legendre series describing the surface density has to exhibit a singularity when *τ* *τ*_{0}, yet vanish when *τ* ∈ [*τ*_{0}, *π*]. The accurate representation of such behavior requires quite a large number of terms because the series is slowly convergent. However, the dominant part of the singularity at the screen edge can be extracted and analytically summed, so that the convergence of the modified series representing the remainder of the surface density is vastly accelerated [*Smith et al.*, 2007].

[114] Two examples may be used to illustrate results. In Figure 10 the prolate spheroid with major and minor semiaxes *a* and *b* is illuminated by a plane wave normal to the aperture plane. The aspect ratio is *b*/*a* = 0.15. In the indicated parameterization *r*(*τ*) = is the polar radius of the generating curve. The monostatic cross section *σ*_{0} computed from the backscattered field as a function of *k*, is displayed over the range 0 < *k *< 20. As the truncation number *N* = *N*_{tr} of the regularized system is increased the condition number *ν*_{N} of the system approaches a limit, dependent upon *k*. This limiting value of the condition number *ν*_{N} is displayed over the same range. A strongly resonant feature is visible near *k* = 16.6; however, the condition number remains well controlled so that the accuracy of the solution is guaranteed. Away from resonance the condition number and solution accuracy are also well controlled. These results were validated against those obtained by a different method based upon the approach described in section 4.

[115] An example of a noncanonical open shell is provided by rotation of the “Pascal Limaçon” shown in Figure 11 about an axis. Again the monostatic cross section *σ*_{0} and the limiting value of the condition number *ν*_{N} are displayed over the range 0 < *k *< 10. Also the condition number is well controlled even in the vicinity of highly resonant feature near *k* = 9.2.

[116] This approach for solving the scalar Dirichlet diffraction problem for arbitrarily shaped open screen of revolution is seen to be rigorous and effective. The analytical regularization procedure which transforms the integral equation to an algebraic system of second kind permits the problem to be efficiently solved with any prescribed accuracy.