Analytical characterization and validation of creeping waves on dielectric coated and perfectly conducting cylinders



[1] Creeping waves play an important role in the analysis of electromagnetic scattering by large objects with curved boundaries. In this paper, an asymptotic model is presented for the back-scattered field on a dielectric-coated cylinder at aspect angles near broadside incidence, a region where the creeping wave has maximum intensity. The monostatic radar cross section is analytically derived from creeping wave poles and their residues, and is validated with data extracted from rigorous method of moments computation of the scattered field by utilizing a novel state space spectral estimation algorithm. Detailed computations reveal a stark contrast between dielectric-coated objects and uncoated metallic ones with regard to creeping wave propagation. Because of smaller curvature-induced leakage and coherent interaction between incident wave and the dielectric coating, creeping waves, strongly attenuated by metallic objects, become quite pronounced for coated objects. As the frequency increases, the creeping waves are partially trapped inside the dielectric layer and the scattered field becomes quite small. Therefore, in contrast to leakage on a metallic cylinder, which displays smooth monotonic reduction in amplitude with increasing frequency, creeping waves on a coated cylinder exhibit a strong cutoff akin to a guided wave.

1. Introduction

[2] Analysis of electromagnetic (EM) scattering by dielectric-coated curved objects is of significant practical interest, such as the radar cross section (RCS) of a missile structure or the installed radiation pattern of patch antennas conformal to rocket bodies. The underlying canonical configuration is a perfect electrically conducting (PEC) cylinder coated with one or more thin dielectric layers, the cylinder radius and length being much larger than the operating wavelength. In this paper, an asymptotic model is developed for the back-scattered field from a dielectric-coated cylinder, focusing on rigorous analytical characterization and validation of creeping waves. Detailed computations on a coated cylinder reveal stark contrast in propagation characteristics between the surface-guided field on a dielectric-coated cylinder and that on an uncoated metallic cylinder. The asymptotic approximations are facilitated by well-established representation of the source-excited field for a PEC cylinder in terms of an integral over the angular wave number spectrum using the theory of characteristic Green's functions [Felsen and Marcuvitz, 1973, pp. 685–697].

[3] The theoretical formulation describing the scattering by dielectric or dielectric-coated cylinders in terms of azimuthally propagating waves has existed for many years [Elliott, 1955; Helstrom, 1963; Streifer, 1964; Tang, 1957; Wait, 1960]. More recently, the advancement of numerical methods has enabled the computation of surface wave propagation constants and spectral integrals in the complex plane, resulting in the possibility of extending the asymptotic solution for a cylinder to more complex geometries. Several researchers have addressed the problem of asymptotic approximation of the EM field radiated by a line source or an infinitesimal dipole in the presence of an infinitely long coated cylinder, primarily focusing on numerical evaluation of azimuthally propagating waves derived from so-called creeping wave poles and their residues [Albertsen, 1989; Erturk and Rojas, 2000; Felsen and Naishadham, 1991; Logan and Yee, 1962; Kim and Wang, 1989; Krasnojen, 1998; Naishadham and Felsen, 1993; Paknys and Wang, 1986; Paknys and Jackson, 2005; Pearson, 1986, 1987; Wang, 1985]. Computation of the EM field itself, its verification with other methods, and its validation with either measured or computed data have received considerably less attention.

[4] In this paper, we evaluate asymptotically the monostatic RCS for a coated cylinder at normal incidence in terms of its geometrical optics (GO) field in the illuminated region and the creeping waves. Although the GO formulation is based on a standard approach [Felsen and Marcuvitz, 1973, pp. 693–697], the focus of this paper is on analytical treatment of creeping waves and evaluation of their contribution to the back-scattered field. We do not dwell on creeping wave propagation constants, as these are covered elsewhere [cf. Paknys and Wang, 1986]. Kim and Wang [1989] obtained the uniform theory of diffraction solution of the field exterior to a 2-D cylinder with a thin lossy coating by utilizing line source excitation at a fixed frequency. As the frequency increases, the coating becomes thicker relative to a wavelength in the dielectric, and the properties of wave propagation and the supported modes change significantly. To our best knowledge, the frequency behavior of the creeping wave modal field and its transition to leaky and trapped modes have not been investigated. We present new results on wideband characterization of creeping waves utilizing practical plane wave excitation, which show that creeping waves cannot be neglected for the coated cylinder. As the frequency is increased, the creeping wave exhibits a sharp modal cutoff, transitioning from an azimuthally propagating mode to a trapped mode. This should be contrasted with the relatively large monotonic decay of creeping waves with respect to frequency on a conducting cylinder.

[5] A second contribution of this paper is the rigorous validation of creeping wave scattering using the state space method (SSM) [Piou, 2005; Naishadham and Piou, 2005; Naishadham and Piou, 2008]. SSM is applied to extract spectral content and synthesize the wideband frequency response of creeping waves from a method of moments (MoM) solution for the scattering by the coated cylinder. It is shown that the extracted creeping wave contribution compares quite favorably with the analytical result for the creeping waves, on the basis of a 2-D formulation over a wide frequency band for a thick lossy dielectric coating.

[6] Mathematical details on the formulation of the creeping wave field, starting from line source excitation of an infinite coated cylinder and developing plane wave synthesis as a subsequent limiting case, are presented in section 2. SSM is summarized in section 3 for completeness. Representative computed results parameterized in terms of dielectric constant, polarization, and frequency are presented in section 4, emphasizing corroboration with wave phenomena extracted using the MoM/SSM spectral estimation approach. Important conclusions are summarized in section 5.

2. Electromagnetic Field Formulation

[7] We begin with the line source-excited configuration in cylindrical coordinates for formulating the EM field in terms of transverse magnetic (TM) and transverse electric (TE) potentials [Harrington, 1961], and compute the scattered response for plane wave incidence in the limit the source point approaches infinity. In general, for oblique incidence on a dielectric cylinder, the fields are TE-TM coupled. However, for angles approaching broadside incidence, such coupling can be neglected and the scattered field can be constructed asymptotically from the line source potentials. For angles off broadside, one should employ the dipole Green's functions formulation [Pearson, 1986; Naishadham and Felsen, 1993] as the starting point for plane wave synthesis of creeping waves.

[8] Invoking the line source formulation of TE and TM potentials for a coated cylinder [Paknys and Wang, 1986] and using completeness relations of the characteristic Green's functions in cylindrical coordinates [Felsen and Marcuvitz, 1973, sections 3.3, 6.6], each potential is represented as a spectral integral in the angular wave number (ν) domain, with the azimuthal (ϕ) domain extended to span the infinite range −∞ < ϕ < ∞, thereby removing the 2π-periodicity constraint and allowing for propagation of creeping waves. The required physical periodicity of the field at equation imageϕ-ϕ′equation image = π is restored by summing over image sources located at ϕ′ ± 2, n = 1,2,⋯, in the infinite angular domain. The spectral amplitudes depend only on the radial coordinates owing to separation of variables in the Helmholtz equation for the potentials. This radial domain problem is solved by a superposition of inward and outward propagating waves, with their amplitudes determined by invoking the required continuity conditions across the layer boundaries. The (source-free) potentials for plane wave incidence are obtained by moving the source point to infinity and evaluating the corresponding spectral integrals using the stationary phase asymptotic method. Standard far-field approximations are then invoked to compute the scattered field.

[9] Consider a perfectly conducting cylinder of radius a and length h (the latter assumed to be much longer than the wavelength), which is depicted in the Cartesian coordinate system in Figure 1. The cylinder is illuminated by a plane wave traveling along the direction equation imagei = −(equation imagesinequation image′ + equation imagecosequation image′), where θ′ denotes the angle of incidence. Because of azimuthal symmetry, without loss of generality, the incident propagation vector is assumed to lie in the xz-plane (ϕ′ = 0) The incident wave may be arbitrarily polarized in a plane perpendicular to equation imagei at an angle γp measured from the xz-plane, with the electric field intensity, of unit amplitude, given by the following equation:

equation image
Figure 1.

Cylindrical scatterer geometry and plane wave excitation.

[10] A time dependence exp(jωt), ω = 2πf being the angular frequency, is assumed, and k0 is the wave number in free space. The polarization angle γp = 0 corresponds to horizontal (or TM) polarization and γp = π/2 denotes vertical (or TE) polarization. The observation point is located at (r, θ, ϕ) in spherical coordinates. For the coated cylinder, the PEC core is covered with a dielectric layer of relative permittivity ɛr and thickness d = ba where b is the outer radius. We are primarily interested in the field at normal incidence obtained by making θ′ = π/2.

2.1. Spectral Integral Representation

[11] For a line source positioned at r′ = (ρ′, ϕ′), the z-directed TM (magnetic) and TE (electric) vector potentials can be expressed as spectral integrals in the complex angular wave number:

equation image
equation image

The EM field can be readily derived from the potentials [cf. Harrington, 1961, equations (5–18), (5–19)]. The ν-integral above is recognized as the continuous angular spectrum that follows from the application of the Watson transformation to the conventional angular harmonic series representation [Watson, 1919]. However, we may obtain the integral representation in equation (2) directly by invoking a completeness relation for the Dirac δ function in the infinite angular domain [Felsen and Marcuvitz, 1973, pp. 306–313]. Henceforth, carets denote the spectral amplitudes in the transform domain. The vector r = (ρ, ϕ) locates the observation point, and i = 0, 1 specifies free space and dielectric, respectively. The radial variation of the fields is obtained by solving Bessel's differential equation as a superposition of cylindrical waves traveling in opposite directions. The spectral amplitudes equation imagei, equation imagei are determined by imposition of boundary conditions on the tangential field at the interfaces ρ = a and ρ = b.

[12] The source-excited field propagates on the surface of the coated cylinder along circumferential trajectories formed by varying ν. The integration path Cν in the complex ν-plane is parallel to and just below the real axis. For an observer shadowed by the cylinder, we close this path in the lower half plane around the poles of the spectral amplitudes in equation (2) and evaluate the ν-integral as a summation of the corresponding residues. Modal amplitudes of the creeping waves are proportional to the residue at a given pole location multiplied by the appropriate excitation coefficient. The index n accounts for multiple circumnavigations of the creeping waves, and the infinite n-summation can be expressed in closed form (see equation (12) below). The n = 0 term is evaluated asymptotically by integrating along the real axis in the ν-plane and accounts for the GO field in the illuminated region [Felsen and Marcuvitz, 1973, pp. 693–697].

[13] A few points are in order regarding the computational advantages of the spectral representation in equation (2). Although this representation is valid for any cylinder radius, it is especially useful for large-radii cylinders. In this case, higher-order creeping waves decay rapidly for both PEC and coated cylinders, and the residue series converges efficiently, so that only the dominant residue (usually p = 1 term) would suffice. However, as we quantify in section 4, even after traversing partially around the cylinder and launching into the lit region, the dominant creeping wave on a coated cylinder has significant amplitude. Therefore, besides evaluation of the field in the deep shadow region, the asymptotic spectral representation of the creeping waves provides an efficient method to characterize their contribution to the back-scattered field.

[14] Next, we present expressions for the spectral amplitudes equation imagei, equation imagei in a form suitable for direct numerical implementation. The radial wave functions involve Hankel functions Hν(1,2)(kiρ) of complex order ν, where ki = ωequation image. In particular, spectral amplitudes in the free space region are given by the sum of incident and scattered fields:

equation image

where the incident field is derived from the potentials of a line source with amplitude J0 or M0:

equation image

The amplitudes of the scattered field potentials are given by the following equation:

equation image

The TM and TE reflection coefficients, Ree and Rhh, respectively, follow from the tangential field boundary conditions at ρ = a and ρ = b. Only the relevant equations are summarized below, with the reader referred to Paknys and Wang [1986] for details. The PEC boundary condition at ρ = a leads to the following calculations:

equation image

for the TM and TE reflection coefficients, respectively, at the conductor interface. Reflection coefficients at the outermost (dielectric) interface are computed as follows [Paknys and Wang, 1986; Kim and Wang, 1989]:

equation image
equation image

[15] It follows from equations (8) and (9) that the dispersion equation, whose zeros yield the poles νp(β) of the reflection coefficient in the complex ν-plane, is given by the following:

equation image

The solution to equation (10) has been discussed in Naishadham and Felsen [1993], classifying the poles as creeping wave type, trapped wave type, or leaky wave type, depending on whether Re(νp) ≈ k0b, Re(νp) > k0b, or Re(νp) < k0b. When b = a, ɛr = 1 and μr = 1, we observe that Ce → ∞, Ch → 0, with the result that Ree → Γe and Rhh → Γh. In other words, when the coating vanishes, the reflection coefficients in equations (8) and (9) correctly reduce to the TM and TE reflection coefficients of equation (7) for a PEC cylinder.

2.2. Plane Wave Incidence

2.2.1. Incident Field

[16] The line source field is now specialized to plane wave incidence by asymptotic evaluation of the incident and scattered field potentials in equations (5) and (6) for ρ′ → ∞. Using equations (2), (3), and (5) with ρ < ρ′, and setting J0 = M0 = 1 henceforth, the incident TM and TE line source Green's function is given by the following equation:

equation image

where we have replaced the infinite summation in equation (2) with the closed-form result

equation image

Clearly, the integrand has simple poles at νm = m, m = 0, ±1, ±2, ⋯ in the ν-plane. Because the integrand is odd-symmetric about the origin, the integration contour Cν may be deformed around the positive real axis (see Figure 2a), and by residue evaluation, equation (11) results in the 2D Green's function

equation image

Equation (13) is valid for observation angles in the z = 0 plane. The 3-D Green's function, appropriate to oblique plane wave incidence depicted in Figure 1, follows from the inverse Fourier transformation

equation image

where equation imageρ = Gρ with k0 replaced by equation image0 = equation image, and β is the axial wave number. Substitution of equation (13) in equation (14) yields

equation image

The second equality in equation (15) follows from the addition theorem

equation image

Stationary phase evaluation of the second integral in equation (15) yields the 3-D Green's function

equation image

Using the standard far-field approximations in spherical coordinates for r′ → ∞ along the ray defined by (θ′,ϕ′), equation (17) yields the incident TM and TE potentials

equation image

valid for illumination angle in the lit region 0 ≤ ψiψsb, where ψsb denotes the shadow boundary. Appropriate polarization is effected by deriving the fields from these potentials using standard expressions [cf. Harrington, 1961]. With ϕ′ = 0, the propagation phase in equations (18) and (1) become identical. For the 2-D formulation relevant to this paper, we set θ = θ′ = π/2.

Figure 2.

(a) Deformation of integration contour Cν around the positive real axis. (b) Poles of the reflection coefficient in the complex ν-plane: • • • leaky wave poles, x x x creeping wave poles, o o o trapped wave poles.

2.2.2. Reflected Field

[17] Next, we approximate the scattered field in equation (6) for ρ′ → ∞. In the lit region, the scattered field consists of the reflected and diffracted fields. The reflected field is obtained from the n = 0 term in equation (2) by asymptotic evaluation of the ν-integral along the real axis for Re(ν) < k0b [Felsen and Marcuvitz, 1973, pp. 693–697; Kim and Wang, 1989]. For Re(ν) > k0b, this integral does not contribute to the reflected field. For the diffracted field, however, the contour Cν is closed around poles νp of the reflection coefficient in the lower half plane (see Figure 2b) or zeroes of the dispersion equation (10). Residue evaluation results in surface-guided modes such as creeping waves. Unlike the GO field, surface diffraction occurs in the shadow region behind the object.

[18] A general treatment of plane wave scattering requires variation of elevation angle θ′ in Figure 1, whereas the 2-D reflected field derived in Kim and Wang [1989] is valid only in the z = 0 plane. Therefore, instead of equation (6), we start from the 3-D dipole-excited reflected field for a coated cylinder [Pearson, 1986; Naishadham and Felsen, 1993] and let the source point r′ → ∞ for plane wave synthesis. However, since we are interested in aspect angles near broadside, we neglect TE-TM coupling in the dipole reflection coefficient expressions. The TM and TE plane wave potentials of the reflected field for large k0b are given by the following equation (see Appendix A for details):

equation image

where equation imageee, equation imagehh are locally planar approximations of the reflection coefficients in equations (8)(9) for large k0b (see Appendix B), νs is the saddle point in the angular spectral integral, and the cylindrical coordinates of the specular point of reflection are given by (bb,zb). With θ and θ′ set to π/2 and zb = 0, equation (19) reduces to the 2-D-reflected field for plane wave incidence. The term within the radical denotes the spreading factor.

2.2.3. Surface-Diffracted Field

[19] The surface-diffracted field for creeping waves launched by an incident plane wave is obtained from the ν−spectral integration in equation (2) using the scattered line source field in equation (6) with the reflection coefficients given in equations (8) and (9). Closing the ν-integration path around the zeroes of equation (10) (see Figure 2b) and evaluating the integral in terms of residues equation imageee and equation imagehh leads to the following equations:

equation image
equation image

where Nii and Dii are the numerator and denominator, respectively, of the corresponding reflection coefficient in equations (8) and (9). The n-summation may be accomplished in closed form as indicated in equation (12) and large-argument asymptotic approximation may be utilized for the source Hankel function, yielding the following equation:

equation image

Equation (22), derived as a limiting case of the line source Green's function, does not capture the TE-TM mode coupling that occurs for arbitrary oblique aspect angles. Hence, unlike the incident field in equation (18) and the reflected field in equation (19), the asymptotic formulation in equation (22) for the creeping waves is valid only at normal incidence. The creeping waves in equation (22) implicitly assume multiple clockwise and counterclockwise circumnavigations through the closed-form sum in equation (12). The scattered field potentials are now given by the sum of reflected and diffracted components in equations (19) and (22), respectively. The reader is referred to Naishadham and Felsen [1993] for implementation details of various quantities in equation (22), in particular the Hankel functions with complex argument and order. Derivatives with respect to order in the residues in equation (21) are computed numerically using finite differences, and Davidenko's method, a complex-zero search procedure with exponential convergence, efficiently computes the ν-plane poles. We have observed that only the leading order term in the summation would suffice for coated cylinders (see section 4).

3. State Space Method

3.1. Spectral Model

[20] The back-scattered EM field for the cylinder, computed using the MoM, is denoted by the data sequence y(k), which comprises N uniformly spaced frequency samples, each represented as a sum of M complex sinusoids (or point scatterers), corrupted by white Gaussian noise w(k). Note that noise is considered only as a random input to the system to derive its transfer function, but not in the analysis. Over a given bandwidth, the signal measurements at these N frequencies can be modeled as follows:

equation image

where ai refers to the amplitude of the ith point scatterer, and the poles pi are given by the following:

equation image

In equation (24), αi and Ri denote the decay/growth rate and range, respectively, associated with the ith scatterer. The parameter c refers to the speed of light, and Δf is the sampling frequency. Each scattering center, and thus each αi, has a unique frequency dependence imposed by the data. For a quasi-stationary process, as applicable to predominantly linear scattering mechanisms, the interactions between scattering centers render the pole amplitudes to have weak frequency dependence, so that one can still employ linear system concepts to derive the transfer function. Our first task is to estimate the parameters αi and Ri, which are embedded in the data sequence y(k). The state space method provides an efficient computation of these parameters from the eigenvalues of an open-loop matrix derived from the data. As demonstrated by Naishadham and Piou [2008], SSM owes novelty to the fact that it identifies modal frequency responses relevant to the EM wave phenomena of interest in terms of specific range-classified pole contributions.

3.2. Computation of the State Matrix

[21] The input-output description of the model in equation (23) is given by the (state-space) difference equations

equation image

where x(k) ∈ equation imageM×1 is the state, Aequation imageequation imageM×M is the system's open loop or state matrix, and Bequation imageequation imageM×1 and Cequation imageM are constant matrices. The latter two matrices are not used in this work. The system transfer function is obtained by taking the z-transform of equation (25) and evaluating the ratio T(z) = Y(z)/W(z). Our objective is to derive the state matrix A, from which the model parameters, αi and Ri, can be calculated. A finite-size Hankel or forward prediction matrix is formed from the data samples as described by equation (26) below, and its singular value decomposition (SVD) is computed to derive A. The parameter L that appears in H denotes the correlation window. It is heuristically chosen to be L = [N/2], where the brackets denote the smallest integer less than or equal to the inserted quantity:

equation image

By computing its SVD and the low-rank truncation, the following M-rank reduction of H is obtained:

equation image

In equation (27), Usn denotes the signal component of the left-unitary matrix [U], Σsn is a diagonal matrix containing the singular values of H arranged in decreasing order, Vsn is the signal component of the right-unitary matrix [V], and * denotes conjugate transpose. By using the balanced coordinate transformation method [Kung et al., 1983], equation (27) can be further factorized as follows:

equation image


equation image

denote the observability and controllability matrices, respectively. The matrix Aequation imageequation imageM×M can be computed from either Ω or Γ. For example, if the derivation of A is based on the controllability matrix, then the following equation holds:

equation image

The matrices Γc1 and Γcl in equation (30) are obtained by deleting the first and last columns, respectively, of Γ. If the eigenvalues of A are assumed to be distinct, one has the equation

equation image

[22] It is well known from spectral estimation theory that the eigenvalues λi above are simply poles pi of the autoregressive (AR) transfer function T(z) corresponding to the model in equation (23). Instead, if one seeks representation in terms of a more general autoregressive moving average (ARMA) transfer function, its zeros are computed using eigenvalues of the matrix (ABC), as described by Naishadham and Piou [2008]. An AR model suffices for the objective in this paper, namely, to extract modal responses in a high signal-to-noise environment. Explicit in equation (24), the poles carry decay/growth and range information about the scattering centers. Thus, equation (24) yields the relationship between magnitude and phase of the eigenvalues, and the parameters αi and Ri, respectively:

equation image

where λi = |λi| ∠ ϕi. Once the poles are estimated, the amplitudes ai (i = 1, ⋯, M) can be obtained from a least squares fit to the EM field data, using the model in equation (23) and the parameters estimated in equation (32) [Naishadham and Piou, 2008].

4. Results and Discussion

[23] We examine the monostatic RCS of a coated cylindrical target with core radius a = 12.13 cm, length h = 0.85 m, and a dielectric coating of thickness 4.57 mm and relative permittivity ɛr = 3.5–j0.55. The reference solution to validate the analytical model is computed over a frequency band of 2–12 GHz and angular space 0 ≤ θ ≤ π, using the body-of-revolution MoM [Mautz and Harrington, 1977]. This computation yields monostatic RCS as a function of incident angle and frequency. The MoM data for specific angles is then processed with the state space spectral estimation algorithm, summarized in section 3, to extract the specular and creeping wave components. We assume that edge diffraction, both single edge as well as multiple creeping wave/edge diffraction, can be neglected at aspect angles near broadside. Therefore, the coherent addition of the two main wave constituents, the specular and creeping waves, generates the composite signal.

[24] Broadside incidence perpendicular to the cylinder axis corresponds to the maximum of the creeping wave radiation. In this direction, the back-scattered creeping wave is essentially a cylindrical wave, with all the creeping rays adding coherently at the receiver. The frequency-domain vertically copolarized (VV or TE/TE) RCS for θ = 90° computed by MoM is shown by the solid line in Figure 3b. To locate the creeping wave peak, the frequency response is Fourier transformed with a Hamming window to suppress the sidelobes, obtaining the composite range response shown in Figure 3a. The first peak denotes the specularly reflected field, and the next one denotes the creeping wave. The phase reference (zero range) is on the cylinder axis. Since the excitation of the broadside creeping wave is essentially a 2-D problem, using ray tracing, one may verify that the creeping wave should occur at πb/2 down-range from the zero-phase reference, where b is the outer cylinder radius. Thus, the range to the creeping wave peak should be 0.25 m as confirmed in Figure 3a. The specular return, on the other hand, appears to emanate from a line source located on the front side of the cylinder (−0.12 m range), as physical optics would predict [Ufimtsev, 2007].

Figure 3.

(a) VV (TE) polarized range response of the coated cylinder at broadside incidence. (b) Comparison of analytically evaluated composite frequency response with method of moments (MoM) data.

[25] Figure 3b depicts comparison between the total response computed by the analytical solution using equations (19) and (22) with the corresponding MoM-derived RCS at broadside. The interference pattern between the specular components of the GO return and the creeping wave is clearly evident at frequencies below the creeping wave cutoff around 7.5 GHz. Favorable comparison with the MoM data reveals that the dominant wave phenomena at broadside are the specular reflection and the creeping wave. Other diffraction effects, such as edge diffraction and multiple interactions (e.g., edge-diffracted creeping waves), which are included in the MoM but not considered in the analytical model, seem to be secondary. These secondary effects contribute to some error in the analytical solution for the total field at frequencies beyond the creeping wave cutoff. However, since the work reported herein focuses on creeping waves, we have not addressed the augmentation of the analytical model to include edge diffraction effects.

[26] Next, a 20th-order state-space model is employed to process the MoM-simulated frequency response for the broadside aspect, and using only three range-classified poles for each wave species, the specular and creeping wave constituents are extracted. Figure 4 depicts excellent corroboration between the analytical solution from Section 2 for these two wave components and the SSM-extracted reference data. The creeping wave response is plotted only until 8 GHz because it cuts off at 7.5 GHz. It is emphasized again that the discrepancy between MoM and analytical data at high frequencies is caused by ignoring secondary diffraction effects in the analytical solution. To the best of our knowledge, this work represents the first wideband validation of creeping waves on a coated cylinder using a rigorous MoM reference solution. The state space method allowed us to accurately extract creeping wave and specular constituents from the MoM data and facilitated such validation. Later in this section, we establish such validation even for perfectly conducting cylinders.

Figure 4.

TE analytical response versus MoM comparison for coated cylinder. (a) Geometrical optics (GO) specular. (b) Creeping wave.

[27] It is seen that the creeping wave cuts off to −40 dB at ∼7.5 GHz. Above the cutoff, the creeping wave transitions into a trapped mode, which bounces back and forth internal to the layer, similarly to a surface wave on a planar slab, and leaks outside tangentially on the surface due to curvature [Naishadham and Felsen, 1993]. The transition of a creeping wave to trapped mode can be illustrated by examining its propagation constant as a function of frequency. Figure 5 displays the dominant TE mode creeping wave propagation constant, ν1/b, computed by solving equation (10) for each frequency. As the creeping wave propagates, it attenuates by shedding energy tangentially because of curvature. The attenuation remains relatively constant until ∼7.5 GHz and then rapidly increases in magnitude with frequency, resulting in the trapped mode. Beyond the creeping wave cutoff, there is very little curvature-induced leakage. From the phase constant plot, it is observed that dispersion exists at higher frequencies. It has been observed that higher-order creeping waves (p > 1) suffer significantly larger attenuation than the dominant mode and are therefore neglected.

Figure 5.

TE creeping wave propagation constant for the coated cylinder. (a) Attenuation constant. (b) Phase constant.

[28] Next, we examine the TM or HH polarized response at broadside incidence. In this case, analogous to the PEC cylinder, the creeping wave should be weaker than its VV polarized counterpart. The range response in Figure 6a shows the specular at the down-range of −0.12 m, the same as in the TE case (Figure 3a). However, the next peak, about −40 dB in amplitude, occurs at 0.12 m and does not appear to be the creeping wave specular (expected at 0.25 m). To examine this further, we plot in Figure 6b the total response extracted from the MoM solution for broadside incidence along with the sum of analytically evaluated GO specular and creeping wave returns. If the creeping wave were present, it would beat with the GO specular until it is cut off, as we have observed for VV polarization in Figure 3b. Absence of such interference alludes to a weak creeping wave for HH polarization. On the basis of the ray path analysis sketched in Figure 7, we infer that the peak at 0.12 m is due to an edge-diffracted wave, which becomes dominant at high frequencies. Since the edge diffraction is lumped in the MoM solution and ignored in the analytical solution, there exists a significant discrepancy beyond 9 GHz between these two sets of data in Figure 6b.

Figure 6.

(a) HH (TM) polarized range response of the coated cylinder at broadside incidence. (b) Comparison of analytically evaluated composite frequency response with MoM data.

Figure 7.

Ray path for the edge-diffracted wave. Horizontal polarization.

[29] Figure 7 shows the ray path associated with an edge-diffracted wave, which attaches broadside to the cylinder, travels diametrically along the base to the detachment point, and returns broadside to the receiver. Since the electric field is tangential to the cylinder at the two diffraction points for horizontal polarization, this edge wave is stronger for HH than for VV polarization [Ufimtsev, 2007]. In contrast, the creeping wave along the curved surface of the electrically long cylinder is not affected by edge diffraction.

[30] The MoM-extracted specular and edge-diffracted components are plotted in Figure 8 along with the analytical solution for the specular and creeping wave responses for HH polarization. Again, a 20th-order state-space model is employed to process the MoM-simulated frequency response, and using only three range-classified poles for each, the specular and edge wave constituents are extracted. This plot reiterates that the creeping wave is negligibly small in comparison with the edge-diffracted wave. The creeping wave could not be extracted from MoM data as its amplitude is within the range of numerical “noise” (−100 to −80 dB). Also, the creeping wave for HH polarization does not exhibit the frequency rolloff observed in the VV polarization. Good agreement is observed between the analytical and MoM constituents of the specular field for frequencies less than 9 GHz.

Figure 8.

HH analytical response versus MoM comparison for coated cylinder. (a) GO specular. (b) Diffracted wave.

[31] Next, we present salient results on broadside RCS for a PEC cylinder of the same dimensions as the conducting core of the coated cylinder. First, the vertical polarization is considered. The range plot in Figure 9a for VV polarization shows the occurrence of specular and creeping waves at the anticipated locations. The frequency response in Figure 9b displays good corroboration of the analytical (specular + creeping wave) solution with the composite MoM-extracted data. Although the ripple caused by the interference between specular and creeping wave components is much smaller than that of the coated cylinder (Figure 3), weak oscillations persist throughout the 10 GHz bandwidth. The absence of a standing wave pattern in Figure 9b indicates that the PEC cylinder creeping wave does not have a steep high-frequency cutoff like the coated cylinder, as we show next.

Figure 9.

(a) VV (TE) polarized range response of the perfect electrically conducting (PEC) cylinder at broadside incidence. (b) Comparison of analytically evaluated composite frequency response with MoM data.

[32] Figure 10 depicts the analytically evaluated specular response and the creeping wave components in comparison with MoM-extracted wave objects for the PEC cylinder. SSM with a model order of 10 has been used in the extraction process, with only two range-classified poles to represent each wave species. At high frequencies, the broadside specular RCS increases as square root of the frequency, as can be shown easily from equation (A2) [cf. Kim and Wang, 1989], and does not depict the resonant standing wave pattern of its dielectric counterpart in Figure 4. Creeping wave for the PEC cylinder decreases monotonically with frequency and does not have the high-frequency rolloff associated with trapped waves on a coated cylinder. Figure 10 quantitatively demonstrates the strong attenuation of the creeping wave on a large conducting curved surface.

Figure 10.

VV analytical response versus MoM comparison for PEC cylinder. (a) GO specular. (b) Creeping wave.

[33] For completeness, horizontal polarization for the PEC cylinder is considered next. The range response in Figure 11a shows the specular at the down range of −0.12 m, the same as in the VV case (Figure 9a). However, the next peak, about −36 dB in amplitude, does not correspond to the creeping wave specular, but a manifestation of the edge diffraction along the path depicted in Figure 7. The frequency response in Figure 11b displays good corroboration of the analytical (specular + creeping wave) solution with the composite MoM-extracted data. The absence of ripple in the composite frequency response indicates that the creeping wave amplitude is very small, as one would expect for horizontal polarization. Indeed, this is corroborated in Figure 12, which compares the analytically evaluated specular response and the creeping wave components with MoM-extracted wave objects for the HH case. SSM with a model order of 10 has been used in the extraction process, with only two range-classified poles to represent each wave species. It is noted that the specular response is identical between the VV and HH cases for the PEC cylinder. However, the creeping wave is 50 to 80 dB smaller in the HH case and practically negligible. In comparison, edge diffraction, extracted from the MoM solution, is significantly larger in amplitude. It is interesting to note that edge diffraction for the PEC cylinder is similar in amplitude to that for the HH-polarized coated cylinder (see Figure 8b), whereas the creeping wave for the PEC cylinder is smaller than that for the coated cylinder, especially at high frequencies. This demonstrates conclusively that the creeping waves are strongly dependent on polarization and enhanced by the dielectric constant of the coating. At the higher frequencies, although creeping waves can be neglected for both coated and PEC cylinders in horizontal polarization, edge diffracted waves, relatively insignificant for the PEC cylinder, cannot be neglected for the coated cylinder. As is evident from Figure 6, ignoring edge diffraction causes considerable discrepancy between the MoM-extracted composite response for the coated cylinder and an analytical solution comprising the coherent addition of only the creeping waves and the specular response. In contrast, there is no such discrepancy for the PEC cylinder (see Figure 11).

Figure 11.

(a) HH (TM) polarized range response of the PEC cylinder at broadside incidence. (b) Comparison of analytically evaluated composite frequency response with MoM data.

Figure 12.

HH analytical response versus MoM comparison for PEC cylinder. (a) GO specular. (b) diffracted wave.

5. Conclusions

[34] An asymptotic model has been developed for the specular and creeping wave components of the field scattered by a dielectric-coated cylinder, starting from the solution for an infinitely long cylinder subject to line source excitation and moving the source point to infinity to derive the plane wave scattered field. Numerical computations using this model reveal that the dominant creeping wave is essentially along the specular direction and can be adequately characterized by the 2-D line source characteristic Green's function emphasizing azimuthal propagation. The state space method, recently developed by the authors to extract the scatterer's physical features from target signatures, has been applied for the first time to extract broadside creeping waves from a rigorously computed full-wave (moment method) solution to the monostatic RCS of a long coated cylinder. Excellent corroboration has been established between the analytical model of the creeping waves and the extracted full-wave reference solution over a wide bandwidth, for both PEC and coated cylinders, thus validating the simpler analytical model. The creeping wave is shown to be much stronger for the coated cylinder than for the PEC cylinder and is significantly influenced by polarization. An asymptotic solution to the canonical problem of the coated cylinder forms the basis for its generalization to more practical structures, such as coated cones and ogives. The numerical computations reveal interesting physical behavior of creeping waves on a coated cylinder. In contrast to leakage on a metallic cylinder, which displays smooth monotonic reduction in amplitude with increasing frequency, creeping waves on a coated cylinder exhibit a strong cutoff akin to a guided wave caused by trapping within the dielectric layer.

Appendix A: Geometrical Optics Reflected Field

[35] Let the Green's function for either an electric or a magnetic line source in the presence of a coated cylinder be denoted as equation image(ρ, ρ′). The corresponding dipole-excited field can be derived from the line source-excited field by performing the inverse Fourier transformation

equation image

with ki replaced by equation imagei = equation image, i = 0,1, in the line source result. The line source reflected field, asymptotically derived in Felsen and Marcuvitz [1973, section 6.7], using first-order saddle point integration of the ν-integral in (2) with n = 0, is given by the following equation:

equation image

In equation (A2), equation image is the asymptotic spectral reduction of the reflection coefficient in equation (8) or (9) for large k0b, ′ and are distances to the specular reflection point from source and observer locations, respectively, and θi is the incidence angle at that point relative to the surface normal (see Figure A1b). The real saddle point νs in the asymptotic integration leading to (A2) is the solution of the following equation (see Figure A1b):

equation image

The last equality in equation (A3) ensures that Re(ν) < k0b. It follows from geometrical interpretation of the GO reflected field (Figure A1b) that the saddle point specifies the specular point location through the relation

equation image

Substituting equation (A2) in equation (A1), we obtain the reflected field for the dipole excitation given by the following equation:

equation image

where equation image is the asymptotic approximation of either Ree or Rhh, discussed in Appendix B.

Figure A1.

GO ray path description for the reflected field. (a) Dipole source at P. (b) Line source at P [Felsen et al., 1990].

[36] The integral in equation (A5) has been asymptotically evaluated by Felsen et al. [1990] for an acoustic point source. By means of the transformation β = k0 sin w, the integration path is mapped into the Sommerfeld contour Cs in the complex w-plane, thus removing the branch points at β = ±k0. Furthermore, the space coordinates are mapped according to the following equation (see Figure A1a):

equation image

so that equation (A5) yields

equation image

The real saddle point in the w-plane is given by

equation image

We note from Figure A1 that L′ and L denote the incident and specularly reflected ray paths for the dipole field, while ′ and are their respective projections on the z = z′ source plane. Thus, γ is simply the elevation angle between either L′ and ′ or L and . Likewise, ψi specifies the angle of incidence (and reflection) in the 3-D space and θi is its projection. Carrying out the integration in equation (A7) using the method of stationary phase results in the following equation:

equation image

As the source point approaches infinity for plane wave incidence, we observe from Figure A1 that

equation image

where rb denotes position vector to the specular point on the cylinder. Then, invoking the standard far-field approximations in spherical coordinates for r′ → ∞, we approximate the TM and TE reflected field from equation (A9) for plane wave incidence as follows:

equation image

where νs = k0bsinθi. We note that equation image is simply the Fresnel reflection coefficient for the TM or TE modes on a grounded dielectric slab (see Appendix B), and the square root term is the spreading factor for ray divergence upon reflection by the cylinder.

Appendix B: Asymptotic Approximations of the Reflection Coefficients

[37] For −∞ < Re(ν) < k0b, we use the Debye asymptotic approximations of the Hankel functions [Felsen and Marcuvitz, 1973, pp. 710–712] given by the following equation:

equation image

where cos α = ν/(k0b), to simplify the reflection coefficients in equations (8) and (9) for large k0b. Similar asymptotic approximations for large argument k1a are applied to equation imagee, equation imageh. The reflection coefficients thus approximated are indicated by equation image in Appendix A. To make Ree and Rhh applicable to plane wave reflection, we replace ki with equation imagei in equations (8) and (9) prior to the asymptotic integration in equation (A7). Since the phase of equation image changes slowly, it does not enter the stationary phase calculations. Therefore, substituting the saddle points νs = k0b cos αb (or α = αb from equation (A3)) and κi = ki cos wski sin θ′ in the Debye approximations for the Hankel functions, the reflection coefficients may be written as follows:

equation image
equation image

We note that at normal incidence (θ′ = π/2 = αb), equations (B2) and (B3) correctly reduce to the Fresnel reflection coefficients for a grounded dielectric slab of the same thickness as the coating on the cylinder. For brevity, the asymptotic approximations of Ce and Ch are not given above. The interested reader is referred to Kim and Wang [1989].