An analytical-numerical technique for the solution of the two-dimensional electromagnetic plane wave scattering by a finite set of dielectric circular cylinders buried in a dielectric half-space is presented. The problem is solved for both the near- and far-field regions, for transverse magnetic and transverse electric polarizations. The scattered field is represented in terms of a superposition of cylindrical waves, and use is made of the plane wave spectrum to take into account the reflection and transmission of such waves by the interface. The validity of the approach is confirmed by comparisons with results available in the literature, with very good agreement, and by self-consistency tests. Applications of the method to objects of arbitrary cross section simulated by suitable configurations of circular cylinders are shown.
If you can't find a tool you're looking for, please click the link at the top of the page to "Go to old article view". Alternatively, view our Knowledge Base articles for additional help. Your feedback is important to us, so please let us know if you have comments or ideas for improvement.
 The two-dimensional scattering problem of electromagnetic waves by buried cylindrical objects has wide application in remote sensing of buried pipes, conduits and cables, in the understanding of mutual interactions between buried objects and surrounding media, in the communication through the earth, in the backscattering from randomly distributed scatterers. For this reason, such problem has been discussed by many authors in the past, both from a theoretical and a numerical point of view, and several different techniques have been proposed to solve it.
Howard  solved a two-dimensional Fredholm integral equation for the scattered field from a subterranean cylindrical inhomogeneity excited by a line source employing an eigenfunction expansion. In the pioneering analytical work [D'Yakonov, 1959], an interesting solution is obtained as a limiting case of a boundary value problem involving two nonconcentric cylinders, but the results are not suitable for numerical evaluation. D'Yakonov's work is extended by Ogunade  in a form appropriate for obtaining numerical data, using conventional eigenfunction expansions; the author considered a conducting circular cylinder inside a dielectric cylinder and took the limit that the radius of the dielectric cylinder goes to infinity; therefore his results cannot be extended to arbitrary configurations and cross sections. Budko and van den Berg  modeled a finite-sized object embedded in a lossy half-space in terms of an effective homogeneous circular scatterer, and a Green's function approximate approach is employed. Bertoncini et al.  applied the uniform geometrical theory of diffraction to calculate the scattering from a polygonal perfectly conducting object buried in a lossy half-space.
Ellis and Peden  computed the scattering from buried inhomogeneous dielectric objects, in the presence of an air-earth interface, employing a two-dimensional method of moment formulation and utilizing cylindrical pulse basis functions and point matching. Surface integral equation methods have been used to treat a buried dielectric cylinder by Butler and Xu  and Diamandi and Sahalos . Volume integral equation methods have been used for determining the scattering from a two-dimensional buried lossy dielectric object [Izadian et al., 1984], obtaining both the frequency domain solution and the time domain response. Xu and Ao  employed a hybrid finite element and moment method to analyze the scattering by an inhomogeneous lossy dielectric/ferrite cylinder located below a planar interface between two half-spaces of different electromagnetic properties.
 The concept of plane wave spectrum of a cylindrical wave [Cincotti et al., 1993] has been used by Di Vico et al.  for the case of a finite set of buried conducting cylinders. The reflection properties of such waves in the presence of plane interfaces have been discussed by Borghi et al. , by introducing suitable reflected cylindrical functions, while the transmission of the cylindrical waves outside the dielectric half-space has been studied through transmitted cylindrical functions and the relevant spectral integrals by Di Vico et al. . Such integrals have been numerically solved employing suitable adaptive integration techniques of Gaussian type, together with convergence acceleration algorithms [Borghi et al., 1999, 2000].
Lawrence and Sarabandi  considered the scattering problem from a dielectric cylinder embedded in a dielectric half-space, in the presence of a slightly rough interface. The solution utilizes the plane wave representation of the fields, taking into account all the multiple interactions between the surface and the cylinder. Asymptotic evaluation of integrals is required for far-field results, and when the cylinder is deeply buried. For the particular case of a flat surface, their method is equivalent to the solution proposed by Borghi et al.  for cylinders placed above the interface.
 In this paper we extend the work of Di Vico et al.  by considering the two-dimensional plane wave scattering problem by a finite set of dielectric circular cylinders, with possibly different radii and arbitrarily placed, buried in a dielectric half-space. We take into account all the cylinder-cylinder interactions and the multiple reflections between the cylinders and the interface.
 The present method may be applied for any value of the cylinder size and of the distance between the obstacles and the surface. It deals with both transverse magnetic (TM) and transverse electric (TE) polarization cases and yields results in both the near- and far-field zones. Moreover, it is able to characterize two-dimensional obstacles of arbitrary shape simulated by a suitable array of circular cylinders, as it was proposed for the free-space case by Elsherbeni and Kishk .
 In section 2 the analytical characteristics of the approach are described. In section 3 details about the numerical solution are given and in section 4 relevant numerical results are reported. Convergence checks are presented. Comparisons are made in terms of scattered field. Examples of scattering by two-dimensional objects of generic shape are shown.
2. Theoretical Analysis
 The geometry of the scattering problem is shown in Figure 1a: N dielectric circular cylinders, with possibly different radii and permittivities, are buried in a linear, isotropic, homogeneous, dielectric, lossless half-space (medium 1, with permittivity ɛ1 = ɛ0ɛr1 = ɛ0n12, where n1 is the refractive index). Each cylinder is parallel to the y axis: the structure is assumed to be infinite along the y direction; therefore the problem is reduced to a two-dimensional form. A monochromatic plane wave, with wave vector ki lying in the xz plane, impinges at an angle ϕi from medium 0 (a vacuum) on the planar interface with medium 1.
 We introduce a main reference frame MRF (O, ξ, ζ), with normalized coordinates ξ = k0x and ζ = k0z, k0 = 2π/λ0 being the vacuum wave number. Moreover, a reference frame RFq is centered on the qth cylinder axis (q = 1,.,N): both rectangular (ξq, ζq) and polar (ρq, θq) coordinates are considered, with ξq = k0xq = ξ − χq, ζq = k0zq = ζ − ηq, ρq = k0rq. The qth cylinder, with dimensionless radius αq = k0aq and permittivity ɛcq = ɛ0ncq2, where ncq is the refractive index, has axis located in (χq, ηq) in MRF.
 The presence of the planar interface is taken into account by means of its complex plane wave reflection and transmission coefficients (for the assumed polarization), Γfg(n∥) and Tfg(n∥), respectively: the unit vector n = n⊥ + n∥ is parallel to the wave vector k of the generic plane wave impinging from medium f on the interface with medium g (f,g = 0,1). Here and in the following, the symbols ⊥ and ∥ are used to indicate the orthogonal and parallel (to the interface) components, respectively, of a generic vector. The time dependence of the field is assumed to be e−iωt, where ω is the angular frequency, and will be omitted throughout the paper.
 The solution to the scattering problem is carried out in terms of V(ξ, ζ), which represents the y component of the electric/magnetic field: V = Ey(ξ, ζ) for TM(y) polarization and V = Hy(ξ, ζ) for TE(y) polarization. Once V(ξ, ζ) is known, it is possible to derive all the other components of the electromagnetic field by using Maxwell's equations.
 In order to obtain a rigorous solution for V(ξ, ζ), the total field is expressed as the superposition of the following seven terms, produced by the interaction between incident field, interface and cylinders (see Figure 1b).
incident plane wave field;
reflected field, due to the reflection in medium 0 of Vi by the interface;
transmitted field, due to the transmission in medium 1 of Vi by the interface;
field scattered by the cylinders in medium 1;
scattered-reflected field, due to the reflection in medium 1 of Vs;
scattered-transmitted field, due to the transmission in medium 0 of Vs;
field inside the qth dielectric cylinder.
 The explicit expressions of Vi, Vr, Vt, Vs, Vsr, and Vst have been given by Di Vico et al. . In particular, we have for Vi, Vr, and Vt
 In equations (1)–(3), V0 is the incident plane wave amplitude and n∥i, n⊥i are the components of ni, the unit vector parallel to the incident wave vector ki; ϕt is the angle of the transmitted plane wave, according to Snell's law; in equation (3), use has been made of the plane wave expansion in terms of first-kind integer-order Bessel functions Jℓ [Abramowitz and Stegun, 1972]. The scattered field Vs has the following expression:
 In equation (4), CWℓ(n1ξh, n1ζh) = , where Hℓ(1) is the first-kind Hankel function of integer order ℓ; note that equation (4) gives the field associated to the point having coordinates (ξ, ζ) in MRF as a function of the coordinates (ξh, ζh) in RFh. The scattered-reflected field Vsr is given by the sum of the fields scattered by each buried cylinder and reflected by the interface:
 The field Vcq inside the qth dielectric cylinder is given by an expansion in terms of first-kind Bessel functions, with unknown coefficients dqm [Balanis, 1989]:
 Once the expressions of all the fields have been given, the boundary conditions on the cylinder surfaces have to be imposed:
where q = 1,.,N and
 By using the orthogonality property of the exponential functions, after some manipulations, it is possible to obtain a linear system for the unknown coefficients chℓ and dqm, for both polarization cases:
with m = 0, ±1, ±2, …, q = 1, ., N, and
where δ is the Kronecker symbol, Tm(1)(x) = Jm(x)/Hm(1)(x), Tm(2)(x) = Jm′(x)/Hm(1)′(x), and pq = ncq/n1 or n1/ncq for TM or TE polarization, respectively.
 A way to solve system (14) is to eliminate the coefficients dqm, thus obtaining a linear system for the sole chℓ coefficients. From equation (14) it follows that
where Dmℓhq = Gmq(2)Amℓhs(1) − Gmq(1)Amℓhq(2) and Mmq = Bmq(1)Lmq(2) − Bmq(2)Lmq(1). In particular, equation (20) shows that the computational effort is similar to the buried perfectly conducting cylinders case [Di Vico et al., 2005].
 Once the chℓ coefficients are known, it is possible to evaluate the field Vcq, inside the qth dielectric cylinder, by means of dqm coefficients in a straightforward way. Indeed, from equation (19), and by recalling equations (15)–(18), after some algebra we obtain the equality
 From this expression it is easily seen how the effect of the plane surface on the internal field is contained in the last two terms in curly braces, which take into account the scattered-reflected Vsr and transmitted Vt fields. In absence of the surface, the first of these terms vanishes while in the second one T01 = 1 (Vt coincides with the incident field Vi).
 By knowledge of the chℓ and dqm coefficients, the total electromagnetic field is completely determined in any point of space and for both polarizations.
3. Numerical Solution
 Although no approximations have been introduced in the theoretical analysis, to obtain numerical results it is necessary to approximate the series in equations (14) with a finite number of terms. The choice of a truncation index M ≅ 3n1α, where α is the maximum dimensionless radius αq (q = 1,.,N), usually reveals an efficient criterion, showing a good compromise between accuracy and computational heaviness [Elsherbeni, 1994]. A check of such criterion is shown further. However, in all the performed computations we required a convergence on the third significant figure: to this aim, in some cases we use larger values of M.
 The evaluation of the reflected and transmitted cylindrical functions RWℓ(u,v) (equation (6)) and TWℓ(u, v, χ) (equation (10)) has to be numerically carried out. In fact, in all practical cases, the expressions of the reflection and transmission coefficients Γ and T do not allow an analytical evaluation of the integrals in equations (6) and (10). In order to develop an efficient integration algorithm, one has to take into account the infinite extension of the integration domain, since the solution cannot neglect the evanescent components of the spectrum. Moreover, the spectrum of the CWℓ functions shows a highly oscillating behavior as the expansion order ℓ increases. A suitable algorithm for the integration of the RWℓ functions has been developed by Borghi et al. [1999, 2000], where adaptive techniques of Gaussian type have been employed, together with convergence acceleration procedures. Afterward, the algorithm has been generalized to integrate also the transmitted wave functions [Di Vico et al., 2005]: in the following, we give some details about the numerical evaluation of TWℓ.
the expression of the TWℓ functions becomes, after some manipulations,
 Here, the first term represents the contribution to TWℓ of the homogeneous spectral plane waves of the spatial Fourier spectrum, and the second term is the contribution of the evanescent spectral plane waves.
 For what concerns the integration of the evanescent spectrum, the integrand of equation (22) presents a singularity in ∣n∥| = 1, which can be removed with the position n∥ = ; moreover, assuming the parity of the transmission coefficient, after some algebra we obtain
 To solve integral (26) we adopted a generalized Gaussian integration method, consisting of a decomposition of the integration interval in subintervals of suitable length on which a fixed low-order Gaussian rule (for example Gauss-Legendre rule) gives good accuracy.
 As far as the integration of the homogeneous spectrum is regarded, studying how the sign of the square root arguments in equation (23) varies as a function of n∥, and keeping in mind that ∣n∥∣ < 1, n1 > 1 and χ > 0, it is possible to decompose equation (23) into two terms, as follows:
 Note that decomposition (27) is linked to a physical phenomenon: in fact, all the components of the homogeneous spectrum are totally reflected when ∣n∥| > 1/n1.
 In order to point out the oscillating behavior of the integrand in equation (28), let us make the positions n∥ = cos t, t1 = arccos (1/n1) and t2 = arccos (−1/n1), thus obtaining
where γ(t) = − (u + χ) + n1v cos t + n1χ sin t + ℓt. In practical cases, the term T10(cos t) is a fairly regular function and does not greatly affect the integration technique. The kernel eiγ(t), instead, shows a highly oscillating behavior as the parameters u, v, χ, and ℓ, increase; moreover, this kernel presents an oscillation rate which is a nonlinear function of the integration variable.
 To integrate equation (30), we take as a starting point the adaptive generalized Gaussian quadrature rule proposed by Borghi et al. , which is based on a decomposition of the integration interval in a suitable number of subdomains; the number of subdomains and their amplitudes depend on the oscillatory behavior of the integrand. The aforementioned adaptive integration algorithm, however, is based on the assumption that the local oscillation rate is monotonic; on the other hand, in the present case the oscillation rate
is not a monotonic function, as one can see, when the parameters n1, u, v, χ, and ℓ vary. Therefore in order to calculate the integral in equation (30), it is necessary to make a previous decomposition of the whole interval in a suitable number of subintervals in which fi(t) behaves monotonically.
 By using the adaptive algorithm, we obtained the decomposition of the integration domain in subintervals shown in Figure 2a, as an example. The imaginary part of the oscillating integrand of equation (30) is shown together with its subinterval endpoints, when T10 = 1, u = −50, v = 10, χ = 3, ℓ = −50 and n1 = 2. It can be noted that, even if the number of oscillations of the integrand is high, the adaptive algorithm is able to track them.
 The integral of equation (29) is the contribution of the totally reflected components of the homogeneous spectrum. In order to remove the singularity of the integrand function in n∥ = ±1, we put n∥ = cos t; moreover, we make use of the parity property of the transmission coefficient T10[cos(π − t)] = T10(cos t), and rewrite equation (29) as
 The integrand in equation (33) is the product of a purely oscillating function, showing an oscillation rate which is a nonlinear function of the integration variable, times an exponentially decreasing function (in fact, in medium 0 it is always u + χ < 0). As the parameters v, χ and ℓ increase, the oscillating rate increases too, causing an irregular behavior of the integrand function. As u increases, the exponential becomes predominant and the integrand function rapidly approaches 0, with negligible oscillations. Because of the just discussed peculiarities, the integration requirements are similar to the case of the homogeneous contribution Oℓ (u, v, χ) and a similar adaptive generalized Gaussian quadrature rule has been developed.
 In Figure 2b the real part of the oscillating integrand of equation (33) is shown together with its subinterval endpoints chosen by the adaptive decomposition algorithm, when T10 = 1, u = −4, v = 190, χ = 3, ℓ = −10 and n1 = 2. It can be noted that the adaptive algorithm chooses the integrand extrema as subinterval endpoints.
4. Numerical Results
 To validate the employed approach, a comparison is performed with results given in the literature for the case of one buried cylinder. Lawrence and Sarabandi  proposed an analytical-numerical solution for the electromagnetic scattering from a dielectric circular cylinder embedded in a dielectric half-space, in the presence of a flat or rough interface. In Figure 3a, the far-field radar cross section σ is shown, as a function of the scattering angle θ′ = θ − 90° (see Figure 1a), and for different values of the relative permittivity of medium 1 (ɛr1 = 1,1.2,4); the other parameters are: nc = 1.5, α = 0.32π, χ = 2.6π (η = 0), ϕi = 30°, for TE polarization. Our results are superimposed with those in Figure 2 of Lawrence and Sarabandi : the agreement is very good (the curves are practically coincident).
 For the same case as in Figure 3a, when ɛr1 = 4, we show in Figure 3b the behavior of the magnitude of the expansion coefficients c1ℓ ≡ cℓ for different values of the truncation index M. In particular, we set M as the nearest integer greater than μ n1α, with μ = 1, 2, 3. It can be noted that the convergence is stable. The coefficients are negligible for suitable values of ℓ when μ exceeds 2, so that the rule suggesting the choice μ = 3 is well satisfied [Borghi et al., 1996]. The matrix size of the system in equation (20) is N2(2M + 1)2. A typical estimation of the computer time for our FORTRAN code is about 60 ms to obtain one point in Figure 3a, when ɛr1 = 4, on a Pentium IV, CPU 2.8 GHz, RAM 2 GB.
 Again for the same case as in Figure 3b, we report in Figure 3c a two-dimensional plot of the magnitude of the total magnetic field, as a function of ξ and ζ. In the upper part of the map, above the interface, that is, in medium 0, ∣Vtot∣ = ∣Vi + Vr + Vst∣; in medium 1 ∣Vtot∣ = ∣Vt + Vs + Vsr∣. The field values are codified through a gray scale ranging from black (lowest ∣Vtot∣) to white (highest ∣Vtot∣). We can observe the small variations of the field pattern in medium 0 with respect to a standing wave configuration, while the effects of the presence of the scatterer are very pronounced in medium 1, with a shadow region corresponding to the transmission angle ϕt ≅ 14°.
 As is already known for the free space case [Elsherbeni and Kishk, 1992], the solution for the scattering problem by N buried circular cylinders may allow us to study the scattering by a two-dimensional object of arbitrary cross section. To show this possibility, and at the same time to perform a self-consistency check on our method, we consider here the simulation of one buried circular cylinder of normalized radius R by a circular cylinder of normalized radius rint, and a suitable set of N′ = N − 1 circular cylinders of same normalized radius r (see the inset of Figure 4).
 In the literature, it is stated that the best simulation is obtained when the modeling cylinders satisfy the same-volume rule [Elsherbeni and Kishk, 1992]: in particular, the total volume (for unit length along y) of the N modeling cylinders has to be equal to the volume of the simulated cylinder. Therefore it has to be πR2 = πrint2 + N′πr2.
 We have checked the scattered-transmitted field. In particular, Figure 4a shows the behavior of ∣Vst∣ as a function of ζ in the near field (ξ = −0.1), for the following values of the parameters relevant to the simulated cylinder: n1 = 2, R = 1, χ = 2.57 (η = 0), ϕi = 0, TM polarization. The curve obtained with rint = 0.8 and N′ = 55 (limit value over which the small modeling cylinders overlap one another) is displayed and compared to our reference solution for one buried cylinder (solid line). In Figure 4b the relative error ∣εrel∣ is shown in magnitude for the same case as in Figure 3a. It can be seen that ∣εrel∣ is less than 10−1 in the shown range.
 Finally, we show in Figure 5 some examples of simulation of arbitrary-shaped buried dielectric obstacles by means of a suitable set of N circular cylinders satisfying the same-volume rule; ∣Vst∣ is reported, versus ζ and for different values of N, when n1 = 2, nc = 5, χ = 2.5, ξ = −1, ϕi = 0, TM polarization. In Figure 5a we consider a rectangular section scatterer of normalized sides w = 2 and h = 1 along ζ and ξ directions, respectively. In Figure 5b a circular shell with normalized radii Rint = 0.8 (internal) and Rext = 1 (external) is simulated. The computer time for our FORTRAN code to obtain one point in Figure 5b is about 445 ms, 4 s 60 ms, and 63 s 70 ms, when N = 16, 64, and 256, respectively, on a Pentium IV, CPU 2.8 GHz, RAM 2 GB. Figure 5c refers to a slab of normalized length L = 1.5, thickness t = 0.1, and tilt angle ψ = 45°. It is noted that the shape of the field is strongly related to the scatterer geometry. Moreover, in Figure 5c the slope of the obstacle clearly reflects on the asymmetry of ∣Vst∣.
 We have considered the two-dimensional plane wave scattering problem by a set of dielectric circular cylinders buried in a dielectric half-space. The proposed method deals with both TM and TE polarization cases and yields results in both the near- and far-field zones. Moreover, it may be applied for any value of the cylinder size and of the distance between the obstacles and the surface.
 The presented technique could be used for the characterization of suitable scenarios in the context of surface-penetrating radar applications (SPR) [Daniels, 1996], which usually employ purely numerical finite difference time domain techniques [Gurel and Oguz, 2000]. Furthermore, our method may be employed in iterative algorithms for the solution of inverse problems, where fast, efficient and accurate forward solvers are needed, since the forward problem has to be solved repeatedly [Wang and Chew, 1989].
 Convergence checks and self-consistency tests have been performed. Comparisons have been made with results available in the literature, with good agreement. It has been shown that with our approach it is possible to simulate and characterize two-dimensional buried obstacles of arbitrary shape.