Diffraction by an imperfect half plane in a bianisotropic medium

Authors


Abstract

[1] A general theory to study the electromagnetic diffraction by imperfect half planes immersed in linear homogeneous bianisotropic media is presented. The problem is formulated in terms of Wiener-Hopf equations by deriving explicit spectral domain expressions for the characteristic impedances of bianisotropic media, which allow one to exploit their analytical properties. In the simpler case of perfect electric conducting and perfect magnetic conducting half planes, the Wiener-Hopf equations involve matrices of order 2, which can be factorized in closed form if the constitutive tensors of the bianisotropic material are of special form. Four of these special cases are discussed in detail. In order to deal with the more general problem, a technique to numerically factorize the Wiener-Hopf matrix kernels is presented. Our numerical approach is discussed on one example, by considering the previously unsolved problem of a perfect electric conducting half plane in a gyrotropic medium. The reported numerical results show that the diffracted field contribution is obtained by use of the saddle point integration method.

1. Introduction

[2] The first rigorous studies of the diffraction by a perfect electric conducting (PEC) half plane immersed in a homogeneous isotropic medium are by Poincaré [1892] and Sommerfeld [1896]. Noticeable progress in the solution of these problems is attributed to the introduction and use of the Sommerfeld-Malyuzhinets (SM) technique and the Wiener-Hopf (WH) technique [Senior, 1978; Hurd and Luneburg, 1985; Budaev, 1995; Senior and Volakis, 1995; Lüneburg and Serbest, 2000; Daniele, 2003; Antipov and Silvestrov, 2006; Lyalinov and Zhu, 2006; Daniele and Lombardi, 2006]. In particular, we observe that the WH technique can deal with the most general problem of a half plane immersed in an anisotropic or bianisotropic medium, whereas the SM method cannot deal with it, as yet.

[3] The purpose of this paper is to establish a general WH theory to solve the electromagnetic problem of an imperfect half plane immersed in an arbitrary linear medium. Our theory yields to the factorization of matrices of order 2 and 4 in case of perfect and imperfect half plane, respectively. The solution of this kind of problems is known in closed form only for PEC or perfectly magnetic conducting (PMC) half planes surrounded by rather simple media; the perfectly conducting case is simpler because it requires the factorization of scalar kernels [Seshadri and Rajagopal, 1963; Jull, 1964; Przezdziecki, 2000] or of kernel matrices of order 2 [Hurd and Przezdziecki, 1981, 1985].

[4] To deal with the most general case, it is therefore convenient to introduce and use approximate techniques. For example, a general approximate factorization method has been introduced by Daniele [2004a, 2004b] and Daniele and Lombardi [2007], and the impenetrable half plane and wedge problems have been solved very efficiently by approximate factorization by Daniele and Lombardi [2006]. In this paper, this factorization method is applied to effectively solve a previously unsolved problem, thereby showing new results for a PEC half plane in a gyrotropic medium.

[5] In this connection we observe that problems involving very complex algebraic manipulations, such as those required by the problems of this paper as well as by those of Graglia et al. [1991], can nowadays be solved because powerful algebraic manipulator codes are readily available. The results of this paper were obtained by intensive use of the computing software Mathematica®. In fact, although the matrix formulas of this paper may look rather simple, the general explicit expression of each matrix coefficient in terms of the electromagnetic parameters usually occupies several pages.

[6] To facilitate the reading and the comprehension of the material presented in this paper, section 2 considers in detail the simpler cases of a PEC and of a PMC half plane; the most general case of an imperfect half plane is reported in Appendix A. The characteristic impedances and admittances of the bianisotropic medium that surrounds the half plane are defined in section 3, where we also provide two different methods for their evaluation. Special cases of PEC and PMC half planes amenable to closed-form solution are then discussed in section 4, whereas the general method to numerically factorize the WH kernels is given in section 5. One example of application of our numerical factorization method is also discussed in section 5, thereby showing with numerical results that the diffracted field contribution can be obtained by the saddle point integration method.

2. Wiener-Hopf Formulation of the Half Plane Problem

[7] We consider the frequency domain diffraction problem of a plane wave impinging on an imperfect half plane immersed in a homogeneous bianisotropic medium. With reference to Figure 1, the y axis of the Cartesian reference frame {z, x, y} is normal to the half plane surface {x < 0, y = 0}, whereas the z axis lies along the half plane edge. The space-time dependence factor of the incident electromagnetic wave is

equation image

where ω is the angular frequency, t the time, and ko the free-space wave number. Notice that the unnormalized components of the vector wave number of the incident wave are

equation image

and that the time dependence factor exp(t) is assumed and suppressed throughout the paper, whereas for phase continuity, the z dependence factor exp(−0z) is common to all incident and diffracted field components.

Figure 1.

Geometry of the problem.

[8] In the frequency domain, the constitutive relations of the linear homogenous medium in which the half plane is immersed are [Graglia et al., 1991]

equation image

where E is the electric field, H the magnetic field, D and B the electric and magnetic flux densities, respectively; equation imageo is the electric permittivity and μo the magnetic permeability of free space; Zo and Yo = 1/Zo is the free space impedance and admittance, respectively; equation image and equation image are cross tensors that relate the magnetic and electric field to the electric and magnetic flux densities, respectively. The dimensionless overlined tensors appearing in (3) define the dimensioned constitutive tensors

equation image

[9] In the rectangular Cartesian coordinates {z, x, y} all vector quantities may be written as three-element column vectors; then, the constitutive tensors are written as (3 × 3) matrices. To compact the notation, in the following, the (n × n) identity matrix is indicated by equation imagen.

[10] For lossless media, the following conditions hold [Kong, 1975]:

equation image

where the superscript plus is used to denote the transpose and complex conjugate operation.

[11] The boundary conditions for the imperfect half plane (equation imagez, x < 0, y = 0) in terms of the tangential components Et = equation imageEz + equation imageEx and Ht = equation imageHz + equation imageHx of the electric and magnetic field on the upper (y = 0+) and lower (y = 0) half plane face are

equation image

where the (2 × 2) impedance matrices Za, Zb, Zab, and Zba depend on the material of the half plane; this is impenetrable if its faces are decoupled, that is for Zab = Zba = 0. Notice that the boundary condition valid for a PMC half plane (∀z, x < 0, y = 0)

equation image

usually poses simpler problems; the solutions of PMC problems are normally obtained by duality from those of the PEC problems. The boundary condition valid for a PEC half plane (∀z, x < 0, y = 0) is

equation image

[12] To formulate the WH problem, we introduce the one-dimensional Fourier transforms

equation image

and define

equation image

According to the uniqueness theorem, the knowledge of the tangential field Ht on a closed surface permits one to obtain the tangential electric field Et, and viceversa. By considering the whole y = 0 plane as a closed surface that bounds the homogeneous upper half-space, and the lower half-space, one can prove that the Fourier transforms are related by the following algebraic equations:

equation image
equation image

where equation image, equation image (and equation image, equation image) are (2 × 2) matrices that represent the impedance (admittance) of the upper (y > 0) and of the lower (y < 0) half-space, respectively. The coefficients of these matrices are functions of the spectral variable η, and depend on the electromagnetic properties of the linear medium that surrounds the imperfect half plane. Equations (11) and (12) extend the characteristic impedance concept to an arbitrary indefinite linear medium. The procedure to evaluate the characteristic impedance matrices is explained in section 3. To facilitate the reader, in sections 2.1 and 2.2 we consider in detail only the simpler case of a perfectly conducting half plane; the WH equations for the imperfect half plane are reported in Appendix A.

2.1. WH Equations for Perfectly Conducting Half Planes

[13] In the case of a PEC half plane, the boundary condition (8) yields Va = Vb = V+(η). By summing Ia and Ib (i.e., equations (11) and (12)) one gets

equation image
equation image

where A(η) is the Fourier transform of the total electric current induced on the half plane.

[14] In the case of a PMC half plane, the boundary condition (7) yields Ia = −Ib = I+(η). By subtracting Va and Vb (i.e., equations (11) and (12)) one gets

equation image
equation image

where M(η) is the Fourier transform of the total magnetic current induced on the half plane.

[15] For sake of brevity we omit, for a moment, a detailed discussion of the incident wave and simply assume an incident plane wave that, at y = 0, yields (see (1))

equation image

where e0 and h0 are incident polarization vectors, and where the factor exp(−ox) introduces, in the Fourier domain, a pole at η = ηo. The residues of this pole for V+(η) and I+(η) are known since they represent, in the Fourier domain, the contribution at y = 0 of the incident plane wave. From equations (9) and (17) one gets

equation image
equation image

which, together with equations (13) and (15), yield

equation image
equation image

with As(η) and Ms(η) regular at η = ηo. By substituting these latter equations into (13) and (15) one obtains the nonhomogeneous WH equations for the PEC and the PMC problem. These equations, as well as those reported in Appendix A, always take the following form:

equation image

2.2. Solution of the Nonhomogeneous Equations

[16] The factorization of the matrix kernel G(η) = G(η)G+(η) leads to [see Daniele, 2004a]

equation image
equation image

[17] In section 4 we illustrate four cases where closed-form factorization is possible. The technique to obtain the factorization in the general case is given in section 5, where we also apply this technique to obtain the diffraction coefficients for a PEC half plane surrounded by a gyrotropic medium, that is, an important, previously unsolved problem.

3. Half-Space Characteristic Impedances and Admittances

[18] The transverse field equations obtained by using the Bresler-Marcuvitz formalism [Bresler and Marcuvitz, 1956; Daniele, 1971, 2006] yield

equation image

with V and I defined in (9) and where P is a (4 × 4) matrix partitioned into four (2 × 2) submatrices

equation image

P(η), Te(η), Z(η), Y(η), and Th(η) are second-degree polynomial matrices, since all their coefficients are second degree polynomials of the variable η. The general expression of P(η) can be obtained as reported in Appendix B.

[19] We now discuss two methods to evaluate the characteristic impedances equation image and equation image introduced in (11) and (12) by first observing that the coefficients of the matrix

equation image
equation image

are dimensionless. Notice that P and equation image have the same eigenvectors, and that the eigenvalues of P are ko times those of equation image.

3.1. First Evaluation Method

[20] The general solution of (25) in terms of the four eigenvectors γi and eigenvalues [Vz(i), Vx(i), Iz(i), Ix(i)]t of the matrix P reads

equation image

where the scalar coefficients Ci, for i = 1, 4, are y-independent. If the medium surrounding the half plane is passive, one can suppose that two eigenvalues (γ1 and γ2) have a nonnegative real part, whereas γ3 and γ4 have a nonpositive real part. A proof of this conjecture, that is verified by all the passive media we have considered in our studies, is discussed by Daniele [2006]. In the lower half-space, the solution is obtained by taking C1 = C2 = 0, whereas one sets C3 = C4 = 0 in the upper y > 0 half-space. Thus, for y > 0, one has

equation image
equation image

which, together with (10) and (11), yield

equation image

Similarly, in the indefinite lower medium (y < 0) one gets (see (10) and (12))

equation image

3.2. Second Evaluation Method

[21] The expressions of the characteristic impedances (32) and (33) are in general, too complex to permit one to study their properties, in the complex η plane. More convenient expressions are obtained by eliminating I or V from (25) by use of (26):

equation image

and by assuming solutions of the form

equation image

to obtain

equation image

with

equation image
equation image

In general, the matrix equations (36) admit several solutions; for a passive medium we choose those with real positive (equation imageV, equation imageI) and real negative (equation imageV, equation imageI) eigenvalues for y > 0 and y < 0, respectively. From equations (25) and (26), one gets

equation image

which immediately yields

equation image

Although (32) and (33) differ from (40), these expressions are equivalent and yield the same numerical results in all cases we have considered.

3.3. Evaluation of the Eigenvalues

[22] The eigenvalues of γV and γI are the same; these eigenvalues are γ1 and γ2 (for equation imageV and equation imageI), and γ3 and γ4 (for equation imageV and equation imageI). In the most general case of a linear bianisotropic medium, one has to solve the matrix equation (36) which, by omitting the V and I subscripts, reads

equation image

Since this equation involves only (2 × 2) matrices, it is associated with the characteristic equation

equation image

where the scalar coefficients tγ and Δγ are the trace and the determinant of γ, respectively; these coefficients are

equation image

for the upper half-space y > 0, and

equation image

for the lower half-space y < 0. The problem posed by (41) is difficult whenever A and B are of arbitrary order and not commutative. Fortunately, in our case, equations (41) and (42) allows one to eliminate γ2 and yield

equation image

Now, according to Cayley's theorem, one has (A + tγ1)−1 = x1 + yA and from (45),

equation image

with

equation image

where tA and ΔA are the trace and the determinant of A, respectively.

[23] The characteristic impedances and admittances are obtained directly from (40) after evaluating and ordering the four eigenvalues of the matrix equation image given in (27)

equation image

The results provided by (48) always coincide with those of Graglia et al. [1991]; the compact expression of the four solutions of (48), for i = 1, 4, is

equation image

with

equation image
equation image
equation image
equation image
equation image

In this connection, it is of importance to point out that the quantities Te(η)/ko, Th(η)/ko, Z(η)/(koZo), and Y(η)/(koYo) appearing into (27) are independent from ko, Zo and Yo. Furthermore, these matrices result to be independent from the angular frequency ω if the dimensionless tensors equation image, equation image, equation image, and equation image do not depend on ω. Thus, by considering equations (40) and the normalized version of (46), one immediately recognizes that the normalized propagation matrices γV/ko and γI/ko, and the normalized characteristic impedance and admittance matrices equation image/Zo, equation image/Zo, equation image/Yo, and equation image/Yo are independent from ko, Zo and Yo. The main consequence of this behavior is that the eigenvectors of the normalized propagation matrices are equal to those of the unnormalized ones (γV and γI), and that the eigenvalues of equation imageV = γV/ko and equation imageI = γI/ko are independent from ko, Zo and Yo.

[24] By setting, as suggested by equation (2),

equation image

into equation (48), one obtains an algebraic equation of the form f(equation image, equation image) = 0 whose coefficient do not depend on ko, Zo, and Yo. The true physical eigenvalues γi, for i = 1, 4, of the P matrix are

equation image

where equation imagei are suitable solutions of the equation f(equation image, equation image) = 0. It can be proven that f(equation image, equation image) is an algebraic function of order 4 both in equation image and equation image.

[25] In order to appreciate the physical meaning of the function f(equation image, equation image), let us consider a linear lossless material and its two dispersion surfaces in the {α, η, equation image} space. In the present paper, α is constant, since we consider a problem invariant in the z direction. In fact, the value α = αo appears in the exponential factor exp(−oz) that expresses the z dependence of the electromagnetic fields. In the (equation image, equation image) plane, the solutions of the equation f(equation image, equation image) = 0 lie on two curves obtained by cutting the dispersion surfaces with the α = αo plane. Typical dispersion curves for a lossless medium are shown in Figure 2, where four real values of equation image (and consequently of γ = jkoequation image) are associated with a given value of equation image. The two solutions that hold in the y < 0 half-space are those with equation image < 0, the remaining two with equation image > 0 are valid in the y > 0 half-space.

Figure 2.

Dispersion curves for a lossless gyrotropic medium with equation image = 1, and equation image as in (117). The curves show the (equation image, equation image) values for equation image = equation imageo = 1/2; the corresponding vector wave numbers are k = ko (equation imageoequation image + equation imageequation image + equation imageequation image). The four branch points given in (120) and (121) are circled.

[26] To apply the WH technique, one has to know the properties of the multivalued functions γi(equation image) = jkoequation imagei(equation image). By noticing that the coefficient of equation image4 in (48) is equal to one and by considering f(equation image, equation image) a function of the variable equation image, the branch points of the functions γi in the equation image plane belong to the set of points defined by all the zeroes of the discriminant D(equation image) of the polynomial function f(equation image, equation image) [Bliss, 2004]. In all the cases we have studied, the zeroes of the discriminant D(equation image) coincide with the zeroes of the function s(equation image) given in (54). In general, one has 12 branch points. For lossless media, four of these are real; the equation image values of the real branch points are the abscissas of the four vertical lines which are tangent to the two dispersion curves. In Figure 2, the branch points are marked by circles.

[27] As previously noticed, the propagation matrices equation image (for y > 0) and equation image (for y < 0) depend on (γ1 + γ2), γ1γ2 and on (γ3 + γ4), γ3γ4, respectively. It is remarkable that with the exception of the four real branch points, all the other branch points of γ1, γ2, γ3, and γ4 do not appear in the quantities (γ1 + γ2), (γ3 + γ4), γ1γ2, and γ3γ4. This means that the characteristic impedance (and admittance) has only four branch points, that are real for lossless media. Although it is rather simple to prove this property for an f(equation image, equation image) biquadratic in equation image, as it happens for media with vanishing Te(η) and Th(η) matrices [Hurd and Przezdziecki, 1981], in the most general case this property can only be proved after algebraic computer manipulations, for example by expanding in the neighborhood of the complex branch point equation imagebr the quantities (γ1 + γ2), γ1γ2, (γ3 + γ4), and γ3γ4 in terms of Puiseux series. The point equation imagebr is an ordinary point if, in the Puiseux series, the term containing the factor (equation imageequation imagebr)1/r, with r integer, is associated to a coefficient numerically equal to zero.

4. Special Cases With Closed-Form Solutions

[28] The general expressions of the characteristic impedances and admittances (32), (33), or (40) simplify in special cases. This section reports four important special cases for which the half plane diffraction problem is solvable in closed form by using the WH approach.

4.1. Isotropic and Bi-isotropic Media

[29] In case of a bi-isotropic (chiral) material defined by the parameters equation image = equation imageequation image3, μ = μequation image3, ξ = −jϑequation image3, ζ = +jϑequation image3 one gets

equation image
equation image

with

equation image
equation image
equation image
equation image

and where

equation image
equation image
equation image
equation image
equation image

[30] When the chirality factor equation image vanishes, the characteristic impedances simplify into the expected free-space result:

equation image

In order to ascertain the possibility to obtain closed form factorizations of the matrix kernels appearing into (13) and (15), or into (A4), it is of importance to know the expression of the characteristic impedance matrices. Notice that the known closed form solutions available in the literature are relative only to the case of a PEC or a PMC half plane; these solutions involve the factorization of (2 × 2) matrices. At skew incidence, that is for α0 ≠ 0, the factorization of these matrices is often simplified by the transformation [Senior, 1978; Lüneburg and Serbest, 2000]

equation image

For example, in case of a PMC half plane immersed in a chiral medium, one has to factorize the matrix (equation image + equation image) (see (15)), with

equation image

Thus factorization is accomplished by factorizing the scalars χ1χ2, and (k1χ2 + k2χ1). Similar considerations apply to the PEC case, exhaustively studied by Przezdziecki [2000].

4.2. Bianisotropic Media With Diagonal Te and Th Matrices

[31] Explicit solutions can also be obtained when the matrix coefficients of (36) commute, as it happens, for example, in case of diagonal Te(η) = Teequation image2 and Th(η) = Thequation image2. In this case one gets

equation image
equation image
equation image

with

equation image
equation image

In spite of the fact that the cases of diagonal Te(η) and Th(η) can hardly be classified, we notice that this certainly happens in the special case

equation image
equation image

In this case, the kernel (equation image + equation image) and (equation image + equation image) commute with polynomial matrices and can be factorized in closed form [Daniele, 2004a]; that is to say that the closed form solution for the problem of a PEC or a PMC half plane immersed in a bianisotropic medium defined by (76) and (77) can always be obtained.

4.3. Bianisotropic Media With Vanishing Te and Th Matrices

[32] For appropriate values of the electromagnetic constitutive parameters, the matrices Te(η) and Th(η) vanish. This happens, for example, for ξ = 0, ζ = 0 and equation image and μ given as in (76). This particular case has been addressed by Hurd and Przezdziecki [1981] after reducing the factorization problem to a Hilbert problem. At any rate, in this case, equations (36) simplify into

equation image

so that one is actually faced with the classical transmission line problem, extensively studied in the literature. In this case, the characteristic impedances are [Paul, 1975]

equation image

with

equation image

By taking into account that γV commutes with the polynomial matrix ZY, or that γI commutes with the polynomial matrix YZ, one can express the kernel (equation image + equation image) and (equation image + equation image) in terms of a polynomial matrix multiplied by a matrix that commutes with a polynomial matrix. In this connection, once again, we remark that matrices that commute with a polynomial matrix are factorized in closed form [Daniele, 2004a].

4.4. Anisotropic Media of the Hurd-Przezdziecki Problem

[33] We conclude this section by briefly considering the case of the PEC half plane for ξ = 0, ζ = 0, μ = μequation image3 and

equation image

which has been solved by Hurd and Przezdziecki [1985] with a different approach. In this case the Te and Th matrices do not vanish, and our formulation yields

equation image

Although, for the sake of brevity, the rational matrices Qo,1(η), So,1(η) and the not-rational functions fo,1(η), go,1(η) are not reported here, we notice that the matrices appearing in (82) commute with polynomial matrices and can certainly be factorized in closed form.

5. Numerical Factorization and Far-Field Evaluation

5.1. Normalization of the WH Equations

[34] The method to numerically factorize the WH kernel and to evaluate the far-field quantities is illustrated, for the sake of clarity, only for the simpler case of a PEC half plane. The extension of this method to deal with imperfect half planes is rather straightforward.

[35] In terms of the dimensionless normalized admittance

equation image

the PEC half plane problem yields (see (13) and (20))

equation image

where Ro = jE0yt(η0)e0t is a known term whereas As(η) is the Fourier transform of the total scattered current induced on the half plane, that is the total minus the Physical Optic (PO) current.

[36] Factorization in closed form is possible whenever yt(η) commute with a polynomial matrix; if this does not happen, one has to use the approximate technique described by Daniele [2004a], which is based on the numerical solution of a Fredholm integral equation of the second kind. This technique requires that the WH matrix kernel and its inverse exist and are finite for η → ±∞ on the real axis. Therefore we need to modify the kernel yt(η) because of the following asymptotic behaviors for η → ∞

equation image
equation image

where aij and bij are not vanishing constants. By introducing the matrices

equation image
equation image

equation (84) yields

equation image

with

equation image
equation image
equation image

and where Ge(η) and Ge(η)−1 exist and are finite for η → ±∞.

[37] By introducing the normalized quantities equation imageo = ηo/ko (as per equation (2)), and

equation image
equation image
equation image

equation (89) reduces to

equation image

where all the involved quantities do not depend on ko, Zo and Yo. That is to say that the normalized WH equation (96) does not change for a fictitious medium that has the same normalized matrices equation image, equation image, equation image, and equation image, but different values of ko, Zo and Yo. As a matter of fact, values of ko with a vanishing or a very small imaginary part make the evaluation of the eigenvalues and the numerical factorization of Ge difficult; for this reason it is convenient to introduce a fictitious lossy medium, and in our numerical simulations we have used

equation image

In the following, to avoid any confusion, the quantities computed for the fictitious medium are indicated by a tilde. The eigenvalue equation imagei = jequation imageoequation imagei computed for the fictitious medium permits one to evaluate the corresponding physical eigenvalue γi = koequation imagei/equation imageo of the real medium.

[38] In the fictitious medium, equation (89) now reads

equation image

where numerical evaluation of equation image+(η) by the Fredholm method given by Daniele [2004a] is now possible. The physical solution X+(η) is obtained by noticing that

equation image

which yields

equation image

This permits one to solve the WH problem (89) and to obtain the functions V+(η), Ia(η) = equation imageV+, and Ib(η) = equation imageV+.

5.2. Field Evaluation by Inverse Fourier Transformation

[39] In the following, without loss of generality, we consider only the evaluation of the transverse component of the electric field for the PEC half plane problem where, to simplify the notation, we set equation imageV = equation image, equation imageV = equation image. The transverse component of the magnetic field can be similarly obtained. The longitudinal field components Ey(z, x, y) and Hy(z, x, y) are obtained from the transverse ones by Maxwell's equations [see Daniele, 2006].

[40] The plus function V+(η) is obtained by solving equation (84). The transverse field Et(z, y, x) is then obtained by inverse Fourier transformation

equation image

with equation image = equation image for y > 0, and equation image = equation image in the region y < 0, and where the integration path B+ is a horizontal straight line located above all the singularities of V+(η). The primary field Etp(z, y, x) = Eti(z, y, x) + Etr(z, y, x) represents the contribution of the incident plus the reflected field; the primary field is evaluated by assuming an entire PEC plane at y = 0. Notice that Etp is equal to zero for y < 0 if the incident wave impinging of the half plane propagates in the negative y direction. Vice versa, the primary field is zero for y > 0 whenever the incident wave propagates in the positive y direction.

[41] Let us now assume that the incident plane wave propagates in the negative y direction. By using the Cayley representation for the exponential factor exp(−equation imagey) in (101) one gets

equation image

for y > 0, and

equation image

for y < 0. By further assuming an incident plane wave with a propagation factor equal to exp[−ozoxγ3(ηo)y] (without loss of generality, the case of an incident plane wave with propagation factor equal to exp[−ozoxγ4(ηo)y] can be omitted) one gets

equation image
equation image

where e0t is the eigenvector of equation image(ηo) associated with the eigenvalue γ3(ηo), whereas the amplitude vectors Et1r and Et2r can be obtained by geometrical optics considerations. For example, while dealing with a PEC plane, one has Et1r = c1et1 and Et2r = c2et2, where et1 and et2 are the eigenvectors of equation image(ηo) associated with the eigenvalue γ1(ηo) and γ2(ηo); the scalar coefficients c1 and c2 are simply obtained by enforcing Etp = 0 on the PEC plane.

[42] The far-field contributions are evaluated by applying the saddle point method [Felsen and Marcuvitz, 1973] to each integral of (102) and (103). For each given observation point with azimuthal angle equation image, this method requires the determination of the saddle points equation images of the function

equation image

with

equation image

and the determination of the steepest descent paths (SDP) that cross the saddle points. In fact, the integration path B+ of (102) and (103) is warped into a SDP, and each saddle point has is own SDP. Occasionally, more saddle points may occur for i = 1, 2, 3, or 4; in these cases, application of the saddle point method is more difficult. For the sake of simplicity we do not discuss these cases, and we assume that each integral has only one significant saddle point. We also observe that the equation images are real for a lossless medium. Since the observation angle equation image appearing in (106) depends on the location of the observation point, it may happen that the SDP captures the pole η = ηo of the function V+. A detailed study, not reported here, shows that the pole is captured only if equation images > equation imageo; when this happens, the pole η = ηo is always captured clockwise. A careful study of the residues of the integrands and use of the boundary condition for the PEC half plane permit one to write, for y > 0

equation image

whereas for y < 0, one gets

equation image

The previous results show that the total field does not present any reflected wave contribution in the region y > 0 if the saddle points equation imagesi satisfy the condition equation imagesi > equation imageo, for i = 1, 2. Furthermore, for y < 0, the total field does not present any geometrical optics contribution (in particular, the incident wave contribution) if the saddle point equation images3 satisfies the condition equation images3 < equation imageo.

[43] As far as the evaluation of the SDP integrals is concerned, after some algebraic manipulations one may prove that all the transverse field components, including the magnetic field ones, can be expressed in terms of scalar integrals of the form

equation image

where ρ is the radial distance from the observation point to the edge of the half plane, and q is given in (106). The saddle points equation images are obtained by evaluating the zeroes of the derivative of the phase term

equation image

with

equation image

The saddle points are therefore given by the solutions, with respect to equation image, of the following system of algebraic equations

equation image

which does not change by changing ϕ into (ϕ − π). As a matter of fact, for lossless media, the saddle points can be obtained graphically [Felsen and Marcuvitz, 1973, p. 110] by considering the dispersion curves f(equation image, equation image) and the vector distance ρ = ρ(equation image cos ϕ + equation image sin ϕ) from the observation point to the edge of the half plane, as shown in Figure 3. In Figure 3, at the saddle points equation images1 and equation images2, the straight lines tangent to the dispersion curve equation image(equation image) are orthogonal to ρ, and this happens only and for all the saddle points.

Figure 3.

Geometrical evaluation of the saddle points for lossless media.

[44] More generally, since equations (113) are algebraic, the saddle points equation images are defined to be the zeroes of the polynomial in equation image obtained from the resultant

equation image

of the two equations (113) with respect to equation image [Bliss, 2004].

[45] For lossless media, the contribution of each real saddle point to the integral (110) is

equation image

where the plus or minus sign in the exponential factor is chosen according to the sign of qequation images) [Felsen and Marcuvitz, 1973, p. 387].

[46] In this connection we recall that the function V+(η) has one pole in η = ηo, with residue T = jEoe0t. The function T/(ηηo) is the Fourier transform of the geometrical optic field over the aperture (x > 0, y = 0). The Physical Optics (PO) contribution to the diffracted field is calculated by approximating A(koequation image) with the value obtained by using V+PO = T/(ηηo), IaPO = equation imageT/(ηηo), or IbPO = equation imageT/(ηηo) in (108) and (109). For this reason, the quantity

equation image

evaluated at the saddle points is able to represent the difference between the true diffracted field and the PO contribution to the diffracted field. Notice that the superscript d in (116) stays for difference. This difference is considered in the numerical case study that follows.

5.3. Numerical Results for a PEC Half Plane in a Gyrotropic Medium

[47] Let us consider a lossless gyrotropic medium with equation image = 1, equation image = equation image = 0 and

equation image

By choosing equation imageo = 1/2, from equations (48) and (56), we obtain

equation image

with

equation image

These data yield 12 normalized branch points in the normalized equation image plane; four of them are real

equation image

and eight are complex

equation image

[48] The equation image values at the four real branch points are

equation image

Recall that the real branch points individuate the four branch points of the characteristic impedance and admittance. The dispersion curves for this case are shown in Figure 2, where it is evident that two different kinds of waves can propagate in this gyrotropic medium.

[49] We now consider an incident plane wave of the second kind (the wave 2 in the dispersion diagram of Figure 2), with Eo = 1, equation imagei = −3π/4, and

equation image

that is for equation imageo = equation imageo cos βi, equation imageo = −equation imageo cos ϕi, and βi = π/3. In the free space, and not in the gyrotropic medium, this wave would correspond to a plane wave incident with an azimuthal angle ϕi, and a zenithal angle βi. The polarization vector in the gyrotropic medium

equation image

does not depend on the value of equation imageo.

[50] The incident wave originates two diffracted waves, one of the first and one of the second kind, which are computed very accurately by numerically solving, with the technique explained by Daniele [2004a], the Fredholm equation associated with the factorization of the WH kernel. The results of Figure 4 show the real and the imaginary part of the two normalized components of the quantity equation image+d(equation image) = koV+d(equation image) given in (116). As previously explained, in order to obtain the far diffracted field according to (115) it is enough to know the saddle points and the value of V+d(η) at the saddle points. The location ηs of the saddle points is obtained from (114), and the values of ηs in the observation angular region {−180° < ϕ ≤ 180°} are reported in Figure 5. Tables 1 and 2 report the values of the diffracted field components

equation image

relative to the first and second wave for different observation angles. The values for the first kind wave are given in Table 1, whereas Table 2 shows the values relative to the wave of the second kind. Notice from equation (124) that in general, the phase of the cylindrical waves 1 and 2, with respect to the phase one has for an isotropic medium, is corrected by a θc factor which depends on the azimuthal observation angle ϕ as well as on the kind (1 or 2) of the diffracted wave.

Figure 4.

Real and imaginary part of the function (left) V1+d(equation image) and (right) V2+d(equation image), given in (116), for equation imageo = equation imageo = 1/2.

Figure 5.

Saddle points ηs in the observation angular region {−180° < equation image ≤ 180°} for the first and second wave of the dispersion diagram of Figure 2, in case of equation imageo = 1/2. The two values of ϕ where ηs is the same for the first and second wave are reported. Notice that the first wave has two or three saddle points in the region around ϕ = −140°.

Table 1. Field Components for the First Wave Diffracted Field at Four Different Observation Angles equation image, in Case of equation imageo = equation imageo = 1/2a
ϕExPOExdEzPOEzdθc
45°0.1232 − j0.04480.0029 + j0.02180.0365 − j0.2505−0.0038 − j0.01231.50575
135°−0.0182 + j0.0163−0.0633 − j0.0239−0.0172 + j0.02590.0298 − j0.11991.60743
−45°−0.4736 − j3.3692−0.0230 − j0.0271−5.1562 + j0.03840.0504 − j0.21971.46782
−135°0.1488 + j1.44651.4096 − j2.99302.2875 − j0.07571.0800 − j2.34241.34645
Table 2. Field Components for the Second Wave Diffracted Field at Four Different Observation Angles equation image, in Case of equation imageo = equation imageo = 1/2a
ϕExPOExdEzPOEzdθc
45°−0.8191 + j0.1432−0.0109 − j0.0201−0.0302 + j1.3237−0.0567 + j0.00191.09735
135°0.2144 − j0.0396−0.0848 − j0.07540.0087 − j0.3219−0.1371 + j0.04971.00051
−45°0.3280 + j2.39360.0332 + j0.03373.6851 − j0.0347−0.0640 + j0.26991.12522
−135°−0.2353 − j1.7434−0.4665 + j1.6206−2.7566 + j0.0255−0.3959 + j1.36211.19875

6. Conclusion

[51] A very general Wiener-Hopf approach to study the electromagnetic diffraction by an imperfect half plane immersed in a linear homogeneous bianisotropic medium is presented. The effects of the material are summarized by introducing characteristic impedance and admittance matrices, which indeed allow for a straightforward formulation of the Wiener-Hopf problem.

[52] In the simpler case of perfect electric conducting and perfect magnetic conducting half planes, the Wiener-Hopf equations involve matrices of order 2, which are factorized in closed form for special form of the material constitutive tensors. Four of these special cases are discussed in detail.

[53] The superiority of the Wiener-Hopf technique with respect to other existing techniques is rather evident when one deals with the most general problem, where the Wiener-Hopf matrix kernels must be factorized numerically by using a technique presented in this paper. Our numerical approach is discussed in detail on one example, by considering the previously unsolved problem of a perfect electric conducting half plane immersed in a gyrotropic medium. Numerical results are reported to show that the diffracted field contribution can be obtained by the saddle point integration method.

Appendix A

[54] For the imperfect half plane problem, by taking into account the boundary conditions (6), the arrays

equation image

are plus functions of the complex variable η, since they are Fourier transforms of functions that vanish for x < 0. Furthermore, by taking into account the continuity of the electromagnetic field on the aperture half plane region {x > 0, y = 0}, it is straightforward to prove that the arrays

equation image

are minus functions of the complex variable η, since they are obtained by Fourier transforming functions that vanish for x > 0. By eliminating Va, Vb, Ia, and Ib from (A1) and (A2) and from the first equation of (11) and (12), one gets

equation image

with

equation image
equation image
equation image

The voltage and current functions in terms of the minus WH unknowns are

equation image
equation image

Appendix B

[55] The (4 × 4) matrix P given in (26) is obtained by erasing the vanishing third and sixth rows and third and sixth columns of the (6 × 6) matrix

equation image

where Γ0t is the transpose of Γ0, and where we have introduced the following (6 × 6) matrices

equation image
equation image
equation image
equation image
equation image
equation image
equation image
equation image
equation image
equation image
equation image

and the following (3 × 3) matrices

equation image
equation image

Acknowledgments

[56] This work was supported by NATO in the framework of the Science for Peace Programme under the grant CBP.MD.SFPP 982376.

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