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Keywords:

  • scattering and diffraction;
  • analytical regularization techniques;
  • integral equations

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. New Analytical Results: Isorefractive Media
  5. 3. Mathematical Motivation for Regularization
  6. 4. Spherical Cavities
  7. 5. Spheroidal Geometry
  8. 6. More General Scatterers
  9. 7. Bodies of Revolution
  10. 8. Propagation Studies
  11. 9. Riemann-Hilbert Approach
  12. 10. Conclusion
  13. Acknowledgments
  14. 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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. New Analytical Results: Isorefractive Media
  5. 3. Mathematical Motivation for Regularization
  6. 4. Spherical Cavities
  7. 5. Spheroidal Geometry
  8. 6. More General Scatterers
  9. 7. Bodies of Revolution
  10. 8. Propagation Studies
  11. 9. Riemann-Hilbert Approach
  12. 10. Conclusion
  13. Acknowledgments
  14. 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 = A0 + A1, where the operator A0 is explicitly invertible (by some means). Then the equation can be transformed to (I + H)J = A0−1E, where H = A0−1A1. 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.

2. New Analytical Results: Isorefractive Media

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

[12] While the text [Bowman et al., 1987] contains the solution to many important canonical scattering problems, until fairly recently there were few examples that incorporated the higher level of scattering complexity presented by apertures and cavities. Uslenghi and coworkers have developed new analytical solutions for scattering from structures incorporating isorefractive materials (of relative permittivity εr and relative permeability μr with product εrμr equal to that of free space). For structures that lie on coordinate surfaces in coordinate systems that admit solutions by the classic separation of variables technique, wave harmonic expansion in the isorefractive region may be matched to the expansion in free space and an analytic solution obtained in a closed form, as a series expansion in wave harmonics of the coordinate system.

[13] Thus a remarkable class of structures has been investigated, for example, an elliptically shaped cavity filled with isorefractive material [Uslenghi, 2004], specifically an elliptical “trench” which is cut into a perfectly conducting half-space, filled with isorefractive material and partially covered with a thin metal plate. This purely analytical model solution is expressed in terms of a two-dimensional elliptical harmonic expansion and can be obtained only for certain positions of the plate related to the focus of the ellipse (that allow the boundary conditions to be matched).

[14] The importance of this purely analytic solution is in providing a demanding benchmark test of purely numerically scattering codes that are designed to model scattering phenomena associated with complex material, edge effects and cavity backed apertures.

[15] Another interesting structure is the diaphanous (i.e., isorefractive) wedge, for which the analytic solution with plane wave or line source illumination was obtained in terms of Kantorovich-Lebedev transforms [Knockaert et al., 1997].

[16] Finally diffraction by a rectangular hole in an infinite conducting plate was solved [Hongo and Seriwaza, 1999] by using Weber-Schafheitlin integrals to obtain an infinite matrix system the solution of which is rapidly convergent under truncation methods.

3. Mathematical Motivation for Regularization

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. New Analytical Results: Isorefractive Media
  5. 3. Mathematical Motivation for Regularization
  6. 4. Spherical Cavities
  7. 5. Spheroidal Geometry
  8. 6. More General Scatterers
  9. 7. Bodies of Revolution
  10. 8. Propagation Studies
  11. 9. Riemann-Hilbert Approach
  12. 10. Conclusion
  13. Acknowledgments
  14. 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

  • equation image

where a time harmonic dependence of form eiωt (ω = ck) is assumed and suppressed (effectively the free space impedance Z0 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

  • equation image

where equation image = xequation image + yequation image is the position vector, equation image is the wave vector with magnitude k related to wavelength λ by ∣equation image∣ = k = 2π/λ. The electromagnetic field is of transverse magnetic type (E polarization); the only nonzero components are Ez, Hx, Hy. The total field

  • equation image

and the field

  • equation image

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 R2. The scattered field Us(p) satisfies the Helmholtz equation

  • equation image

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

  • equation image

at each qL. 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 (Es, Hs) in any finite volume V of space must be finite,

  • equation image

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

  • equation image

[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 Jz,

  • equation image

at each point q R2, where dlp is the differential of arc length at the point p L, G2(p, q) = −1/4iH0(1)(kp q∣) is the free space Green's function, H0(1) being the Hankel function of first kind and order zero. Defining the unknown normalized line current density Z(p) as

  • equation image

the scattered field representation becomes,

  • equation image

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

  • equation image

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

  • equation image

where A denotes the integral operator in (12) with kernel G2(p, q). Most numerical solution procedures for this equation proceed by choosing a finite basis of linearly independent functions Z1,…,Zn and seeking the solution Z in the form

  • equation image

so that

  • equation image

[25] The constants x1, …, xn are determined by some testing procedure using the inner product defined for functions W1, W2 by

  • equation image

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

  • equation image

with an associated matrix M of elements Mij = (Zi, AZj). Inversion of the system produces the coefficients x1, …, xn. 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 [RIGHTWARDS ARROW] ∞, the solution Z constructed from (14) almost certainly will not converge because of the increasingly unstable inversion process (small numerical errors in the values (Zi, −U0) will generate large errors in the coefficients xi.)

[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 Z1 and Z2 to AZ1 = F1 and AZ2 = F2, respectively, is bounded (in a suitable norm) according to

  • equation image

(This implies that as ∥F1F2[RIGHTWARDS ARROW] 0, also ∥Z1Z2[RIGHTWARDS ARROW] 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 (F1, F2) 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

  • equation image

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

  • equation image

furthermore, λn [RIGHTWARDS ARROW] 0 as n [RIGHTWARDS ARROW] ∞. Thus if we express

  • equation image

for suitable constants μn, then the solution is

  • equation image

[32] No eigenvalue λn 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

  • equation image

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 en and uses a representation of the exciting field that ensures μn/λn [RIGHTWARDS ARROW] 0 sufficiently rapidly as n [RIGHTWARDS ARROW] ∞. 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

  • equation image

we determine the solution of

  • equation image

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

  • equation image

[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

  • equation image

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:

  • equation image

[38] The boundedness of R ensures that perturbations to U0 do not lead to arbitrarily large perturbations of RU0 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 Ntr). The error decays to zero as Ntr [RIGHTWARDS ARROW] ∞ (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 H1 and H2 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 H1 and H2 and on the discretization process within the associated inner product and norm [Carstensen and Stephan, 1996].

4. Spherical Cavities

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. New Analytical Results: Isorefractive Media
  5. 3. Mathematical Motivation for Regularization
  6. 4. Spherical Cavities
  7. 5. Spheroidal Geometry
  8. 6. More General Scatterers
  9. 7. Bodies of Revolution
  10. 8. Propagation Studies
  11. 9. Riemann-Hilbert Approach
  12. 10. Conclusion
  13. Acknowledgments
  14. 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 θ0 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 (Er, Eθ, Hϕ) and Er, Eθ may be expressed in terms of Hϕ as

  • equation image
image

Figure 1. Spherical cavity excited by an axially located vertical electric (or magnetic) dipole.

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

  • equation image

here ψn(x) = xjn(x), ζn(x) = xhn(1)(x) and jn, hn(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,

  • equation image

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

  • equation image

[45] Second, the scattered field (Es, Hs) 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

  • equation image

where

  • equation image

[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 xn 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

  • equation image

and

  • equation image

when θ0 < θ < π; here

  • equation image

and

  • equation image

is an asymptotically small parameter: as n [RIGHTWARDS ARROW] ∞, ɛn = 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

  • equation image

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

  • equation image

[51] Then we could use the orthogonality of the family of functions {Pn1(cos θ), n = 1, 2, …} to convert (39) to a system of linear equations for the coefficients xn. 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 [RIGHTWARDS ARROW] ∞. 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

  • equation image

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

  • equation image

where C1 is a constant of integration, for θ0 < 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

  • equation image
  • equation image

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

  • equation image

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

  • equation image

and

  • equation image

for θ0 < θ < π. An interchange of the order of integration and summation is justified, and produces the two Abel integral equations

  • equation image
  • equation image

where

  • equation image

and

  • equation image

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

  • equation image

[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 xn and the constant C1. Elimination of C1 from this system yields

  • equation image

where m = 1, 2, …,

  • equation image

and the so-called “incomplete scalar product”

  • equation image

when n = m, the first term of (53) inside the braces is replaced by θ0.

[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 Ntr equations converges to the true solution as the truncation number Ntr increases [Kantorovich and Akilov, 1974].

[59] The surface current density has one component

  • equation image

and the radiation pattern S1, defined by Eθ, HϕS1(θ) eikr/r as r [RIGHTWARDS ARROW] ∞, equals

  • equation image

[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 Wi accumulated inside (ra) 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 We.) Explicit expressions for Wi and We are readily derived in terms of the coefficients xn. A typical calculation is shown in Figure 2. A number of strongly resonant features in this spherical cavity (θ0 = 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 Ntr = [ka] + 12.

image

Figure 2. Internal (solid) and external (dashed) stored energy for cavity (θ0 = 30°) excited by electric dipole with d/a = 0.9.

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[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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. New Analytical Results: Isorefractive Media
  5. 3. Mathematical Motivation for Regularization
  6. 4. Spherical Cavities
  7. 5. Spheroidal Geometry
  8. 6. More General Scatterers
  9. 7. Bodies of Revolution
  10. 8. Propagation Studies
  11. 9. Riemann-Hilbert Approach
  12. 10. Conclusion
  13. Acknowledgments
  14. 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]

  • equation image
  • equation image
  • equation image

where d is the interfocal distance and −1 ≤ η ≤ 1, 1 ≤ ξ < ∞, 0 ≤ ϕ ≤ 2π; ξ and η = cos θ play the role of radial and angular parameters, respectively. The quantity 1/2kd, 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 Hϕ, Eξand Eηdo 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 = ξ1, η = 1 (see Figure 3). The screen occupies the surface

  • equation image

and the aperture is given by

  • equation image
image

Figure 3. Spheroidal cavity (semiaxes a, b) excited by an axially located vertical electric (or magnetic) dipole.

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

  • equation image

where R1l(1)(c, ξ) and R1l(3)(c, ξ) are the prolate radial spheroidal functions (p.r.s.f.) of first and third kind, respectively, S1l(c, η) is the prolate angular spheroidal function (PASF), and the coefficients N1l(c) and χ1l(c) are normalization constants defined by Flammer [1957] (or see Vinogradov et al. [2002, section 1.2]). The total field Hϕ is a sum of the incident and the scattered field Hϕsc

  • equation image

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 = ξ0; (2) the mixed boundary conditions

  • equation image
  • equation image

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 = ξ0, and the radiation condition, produces

  • equation image

where the unknown Fourier coefficients Al are to be found, and

  • equation image

[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,

  • equation image

where η ∈ (−1, η0), and

  • equation image

the coefficients Bl determined by the incident field (61) equal

  • equation image

[69] It is convenient to define new coefficients xl by Al = Zl(3) (c, ξ0)xl(2l + 1)/l(l + 1) so that

  • equation image
  • equation image

[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 xl.

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

  • equation image

is O(l−2) as l [RIGHTWARDS ARROW] ∞, so that equations (70)(71) may be written

  • equation image
  • equation image

[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]

  • equation image

where the coefficients drml(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 drml(c)/dr−2ml(c) = O(c2/4r2) as r [RIGHTWARDS ARROW] ∞. For given l and m, and provided l > c, the dominant coefficient is dlmml(c). Thus

  • equation image

is a sum of a main term and a remainder that is asymptotically small (when l [RIGHTWARDS ARROW] ∞). 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 equation image, η0 = cos θ0 and rearrange to obtain

  • equation image

when θ ∈ (0, θ0), and

  • equation image

when θ ∈ (θ0, π); here yl = xldl−11l(c) and gl = −icequation imagedl−11l(c)Bl.

[75] The edge condition implies that the coefficient sequence yl is square summable, (equation imageyl2 < ∞); 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

  • equation image

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 Ntr. Solutions of engineering accuracy are obtained if Ntr exceeds the maximal electrical size of the spheroid: Ntr > kb = 0, where b is the semimajor axis.

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

  • equation image

where {ylN}l=1Ndenotes the solution to (79), truncated to N equations. The Fredholm nature of this system guarantees that e(Ntr) [RIGHTWARDS ARROW] 0 as Ntr [RIGHTWARDS ARROW] ∞. 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.

image

Figure 4. Dependence of normalized error e(Ntr) upon truncation number Ntr (a/b = 0.5, ξ0 = 1.155, θ0 = 30°).

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image

Figure 5. Dependence of condition number upon wave number kb (a/b = 0.5, ξ0 = 1.155, θ0 = 30°).

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[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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. New Analytical Results: Isorefractive Media
  5. 3. Mathematical Motivation for Regularization
  6. 4. Spherical Cavities
  7. 5. Spheroidal Geometry
  8. 6. More General Scatterers
  9. 7. Bodies of Revolution
  10. 8. Propagation Studies
  11. 9. Riemann-Hilbert Approach
  12. 10. Conclusion
  13. Acknowledgments
  14. 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 G2(p, q) as a sum of two terms, G0(p, q) + G1(p, q), choosing G0(p, q) so that the integral equation obtained from (12) upon replacing the kernel G2 by G0, is soluble by analytic means. Upon rewriting (12) as

  • equation image

explicit inversion of the left-hand side transforms this equation to one overtly of second kind. Some constraint on the choice of G0 and G1 is necessary to ensure that the transformed equation is genuinely of second kind. As will be seen below, G0 should contain the dominant singularity of G2 and the remainder term G1 should be continuous (actually G1 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(equation image), y(θ) on the interval θ ∈ [−π, π] that are 2π periodic so that x(−π) = x(π), y(−π) = y(π). Thus

  • equation image

the cross section L is parameterized by −θ0 ≤ θ ≤ θ0, while the aperture L′ is created by the removal from S of the segment parameterized by −π ≤ θ ≤ −θ0 or θ0 ≤ 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) = equation image.

[82] Setting z0(τ) = Z(x(τ), y(τ)) l(τ), the functional equation (12) takes the form

  • equation image

where R(equation image, τ) = equation image is the distance between two points of the cavity parameterized by θ and τ, and u(θ) = −U0(x(θ), y(θ)). Introduce the new unknown function

  • equation image

so that (8) becomes

  • equation image

[83] Equation (85) together with the requirement that z vanishes outside −θ0 ≤ θ ≤ θ0 is completely equivalent to equation (83).

[84] Recognizing that the Hankel function H0(1)(s) has a logarithmic singularity at s = 0, split the kernel of (85) into a singular part and a remainder,

  • equation image

[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

  • equation image

[86] Integral equations with logarithmic kernels are attractive because various means for their solution appear in the literature [see Porter and Stirling, 1990, pp. 170 and 320]; Jones [1981] exploited this in his proof of the existence and uniqueness of the solution to the integral equation modeling the thin wire antenna [see also van Beurden and Tijhuis, 2007].

[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 einθ. The singular part of the Green's function has the Fourier series expansion

  • equation image

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

  • equation image
  • equation image

and expand H in a double Fourier series,

  • equation image

[88] The smoothness of H ensures that the Fourier coefficients hnp decay sufficiently rapidly, so that

  • equation image

[89] Insertion of these Fourier expansions (88)(91) into equation (85) and coupled with the vanishing of z(τ) outside the interval −θ0 ≤ θ ≤ θ0, leads to the dual series equations:

  • equation image
  • equation image

[90] Thus the integral equation (85) is converted to an equivalent dual series equation defined on two subintervals of [−π, π] with the unknowns {ζn}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 (nθ) and sin (nθ)), introduce the new unknowns

  • equation image

where n = 1, 2, …, and set gn+ = gn + gn, gn = gngn. The system may be manipulated to the trigonometric kernel form

  • equation image
  • equation image

and

  • equation image
  • equation image

where

  • equation image
  • equation image
  • equation image

and the matrix coefficients are defined by

  • equation image

if n, p ≥ 0,

  • equation image

if n ≥ 0, p ≥ 1,

  • equation image

if n ≥ 1, p ≥ 0, and

  • equation image

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

  • equation image

[92] The matrix operator H = (Hmn) of this equation is compact in the functional space of square summable sequences l2 and the unknowns Xn, Yn are asymptotically O(n−1) as n [RIGHTWARDS ARROW] ∞. 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 Ntr 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 hnp 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

  • equation image

[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 ∣equation image∣ ≤ θ0 and θ0 ≤ ∣θ∣ ≤ π, respectively; the point θ0 = 0 always lies on the metal. The parameter q determines the width of the structure; its electrical length, fully closed, is approximately 2equation imageka.

image

Figure 6. Cross section of the axially slotted cylinder.

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[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

  • equation image

may be calculated from (90). It is easy to estimate that ζn = O(n−1/2) as n [RIGHTWARDS ARROW] ∞ 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].

image

Figure 7. Condition number of the cavity with θ0 = 120°, q = 0.5.

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[97] The radar cross section

  • equation image

when normalized by the factor πa, is given by

  • equation image

where the coefficients βn are defined via the Fourier expansion of the incident plane wave,

  • equation image

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 βnm in (109) is exponentially decreasing when ∣n∣ or ∣m∣ > kL0, where kL0 is electrical size of the screen. Accurate results for the RCS calculation are obtained with truncation number Ntr > 2kL0 + 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 104 to 105, thus establishing that even at these nearly resonant frequencies, the matrix is sufficiently well conditioned to provide stable and accurate solutions.

image

Figure 8. Radar cross section of the cavity with θ0 = 120°, q = 0.5.

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[100] Another example with applications to reflector antenna design appears in the work by Vinogradova and Smith [2001].

7. Bodies of Revolution

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. New Analytical Results: Isorefractive Media
  5. 3. Mathematical Motivation for Regularization
  6. 4. Spherical Cavities
  7. 5. Spheroidal Geometry
  8. 6. More General Scatterers
  9. 7. Bodies of Revolution
  10. 8. Propagation Studies
  11. 9. Riemann-Hilbert Approach
  12. 10. Conclusion
  13. Acknowledgments
  14. 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 S0 (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

  • equation image

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 = τ0 (0 < τ0 < π) define the shell edge.

image

Figure 9. Body of revolution with surface S0 and “aperture surface” S1.

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[103] Suppose the scalar wave ui(q), defined for all points qR3, is incident on the screen. The diffracted field us(q) for the Dirichlet problem is to be found. A unique solution satisfying the Helmholtz equation is guaranteed by enforcing (1) the boundary conditions ui(q) + us(q) = 0 at each point q on S0, (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

  • equation image

for each point q on S0, for JD(p) is the unknown surface current density; here

  • equation image

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

  • equation image

for a sphere this factor is independent of τ. Introduce the unknown function X, related to JD, but defined for the full interval of the parameter τ, by

  • equation image

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

  • equation image

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

[106] A careful analysis of the singularities of the azimuthal Green's function Gm(τq, τp) shows it can be decomposed as

  • equation image

where gm(τq, τp) denotes the mth azimuthal Fourier coefficient of the Green's function of the unit sphere, and the remainder Dm(τq, τp) 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:

  • equation image

where the integral equation kernel is separated into the “spherical canonical part” gm(τq, τp) involving all singularities and a continuous remainder Dm(τq, τp). 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 equation image of order m and degree n:

  • equation image
  • equation image
  • equation image

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

  • equation image
  • equation image

[110] The edge condition imposes the condition on the coefficients xm that

  • equation image

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 Dnlm 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

  • equation image

where the matrix Dm and vector um are the array of coefficients Dnlm and components unm, 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 Dm follows from the decay rate of its coefficients:

  • equation image

[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 [RIGHTWARDS ARROW] τ0, yet vanish when τ ∈ [τ0, 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) = equation image is the polar radius of the generating curve. The monostatic cross section σ0 computed from the backscattered field as a function of k, is displayed over the range 0 < k < 20. As the truncation number N = Ntr of the regularized system is increased the condition number νN of the system approaches a limit, dependent upon k. This limiting value of the condition number νN 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.

image

Figure 10. Monostatic cross section of the prolate spheroid versus wave number k, with corresponding condition number.

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[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 σ0 and the limiting value of the condition number νN 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.

image

Figure 11. Monostatic cross section of the body of revolution generated by the “Pascal Limaçon” versus wave number k, with corresponding condition number.

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[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.

8. Propagation Studies

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

[117] The Method of regularization may be adapted to study structures such as shielded microstrip structures, particularly to determine the propagation constant of higher-order modes. See the cross-sectional Figure 11, in which perfectly conducting walls enclose the microstrip line lying on a dielectric substrate of thickness t. The method is particularly effective for some limiting cases that are notoriously difficult to calculate with any accuracy [Smith et al., 2003].

[118] Using standard modal expansions in the regions above and below the strip line and matching across the boundaries of these regions leads to some coupled dual series equations for the unknown modal coefficients; these are enforced on the line y = t in the metallized region (∣x∣ < w/2) and outside (w/2 < ∣x∣ < Lx/2). In this form the system is quite ill-conditioned for standard numerical solution based on discretizing the intervals ∣x∣ < w/2, w/2 < ∣x∣ < Lx/2. The dual series are of trigonometric type, having much in common with the pair (97), (99). Although it is more complex, the MoR may be applied to transform this system to a well-conditioned system that is suitable for the accurate determination of propagation constants.

9. Riemann-Hilbert Approach

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

[119] The regularization methods in this paper have employed the Abel integral representations of associated Legendre functions (and related functions) as the principal tool to effect conversion to a second kind system. However, there are other available tools, based on the Riemann-Hilbert problem. A large class of two-dimensional scattering problems are amenable to treatment by this method. The approach has been substantially developed by V. P. Shestopalov and coworkers; a survey of some recent work is given by Poyedinchuk et al. [2000].

[120] An interesting recent exampl0e is the study of electromagnetic wave diffraction by a grating with a chiral layer by Panin and Poyedinchuk [2002]. Chiral composites exhibit some unusual characteristics such as optical activity and circular dichroism. This structure has the capability of transforming elliptically polarized waves to linear polarization.

[121] A modal approach to the problem results in some coupled dual series equations for the unknown modal coefficients; and the application of the Riemann-Hilbert method transforms it to a well-conditioned system that is suitable for the accurate determination of desired field quantities. Some further details are given by Panin et al. [2007].

10. Conclusion

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

[122] By their nature, analytical and analytical-numerical methods are rather more difficult to apply than the conventional numerical techniques of electromagnetics. However, they possess the unique advantage of providing reliable and robust solutions to scattering problems that are of guaranteed accuracy. This is of particular importance for those structures where conventional tools fail. Moreover solutions to scattering problems so obtained provide benchmarks for the validation of general purpose codes. The key feature of this approach is the analytical conversion of the more conventional formulations of scattering (differential or integral) to a second-kind matrix system for which there is a sound theoretical understanding of the reliability of numerical solution methods.

[123] At the heart of the regularization method principally employed in this paper are the Abel integral representations of associated Legendre functions and related hypergeometric functions. They provide the tools to effect the transformation to second kind equations. This is most clearly seen in the canonical problems of the dipole excited spherical or spheroidal cavity, though the class of problems soluble by regularization methods in these geometries is rather broader in dealing with wave phenomena associated with complex media, edge and cavity effects. More general classes of scattering problems are amenable to these methods. Noncanonical problems are treatable, at least for two classes of arbitrarily shaped structures: two-dimensional structures and bodies of revolution.

[124] One notable feature of the MoR is its incorporation of the edge condition constraining the class of admissible solutions to the scattering problem (as well as radiation and boundary conditions). Since many conventional numerical methods do not take account of this edge condition, it may represent a source of error, particularly in highly resonant structures.

[125] There are many challenging scattering and diffraction problems to stimulate further development of rigorous analytical-numerical methods, such as periodic structures and related structures that present some difficulties to present-day methods.

Acknowledgments

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

[126] I wish to acknowledge helpful discussions with my colleagues Sergey Vinogradov, Elena Vinogradova and Sergey Panin in the preparation of this article. Also I wish to thank the referees for their helpful comments on the manuscript.

References

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