Diffraction corrections to GO backscattering from a perfectly conducting 3-D rough surface

Authors


Abstract

[1] High-frequency radio wave diffraction by a perfectly conducting arbitrarily shaped smooth surface is considered analytically. The results of solving the integral equation for surface currents at a gentle surface, when multiple scattering and surface shadowing can be neglected, are presented. The diffraction corrections to GO values of the surface currents were obtained (up to the second order of the small parameter ∼1/k, where k is the wave number of the incident EM wave) depending on the local angle of incidence, surface curvatures, and their space derivatives. It is shown that diffraction caused by the local surface curvature results not only in small corrections to the GO currents, but also gives rise to the surface current components orthogonal to those induced by the incident wave in the local GO limit. These general results are applied to the specific problem of backscattering from a Gaussian statistically rough surface at normal incidence. The diffraction corrections to the GO statistically averaged backscattering cross sections, HH/VV polarization ratio (which is equal to unity in the GO limit) and cross-polarization coefficient are obtained as functions of the 2nd (surface slope variances) and 4th (surface curvature variances) moments of the surface roughness power spectra. It is shown that diffraction results in the appearance of the cross-polarized component in the backscattered field, which is equal to zero in the GO limit.

1. Introduction

[2] In this paper, we extend the results obtained in the previous paper Fuks [2007b] for scattering of electromagnetic (EM) waves by a 2-D perfectly conducting rough surface to the 3-D case. To avoid repetition, readers are referred to the Introduction in Fuks [2007b], where the motivation, the short history of the problem, and all necessary bibliographic references can be found.

[3] The plan of the paper is as follows: in section 2 we find the high-frequency asymptotic expansions for currents induced by the plane incident EM wave on a smooth perfectly conducting surface of a general shape. These expansions are obtained by consecutive iterations of the integral equation for surface currents up to the terms of order of ∼1/k2, where k is the wave number of the incident wave. The first term of this expansion corresponds to the geometric optics (GO) approximation, when the induced currents at the surface are the same as they would be at the plane tangent to the surface at every point. The subsequent terms correspond to the diffraction corrections caused by the surface curvatures and their spatial derivatives. It is supposed that these corrections are related only to the local surface curvature parameters in the vicinity of every surface point: this is equivalent to the assumption that multiple scattering and shadowing can be neglected. In section 3, we employ the explicit expressions obtained for surface currents in order to calculate diffraction corrections to the backscattering cross sections of linearly and circularly polarized incident EM waves scattered by a statistically rough surface in the particular case of normal incidence. Along with these corrections, which should be small, the explicit equations for scattering cross sections into the orthogonal polarizations are derived. In the GO approximation, these cross-polarized components are equal to zero in a backscattered field. They appear only due to diffraction by surface curvature. In section 4, these general results are applied to the specific case of a surface with a Gaussian power spectrum of roughness, which allows us to analyze the dependence of cross-polarization coefficients and diffraction corrections to backscattering cross sections on surface roughness parameters: height variance, correlation radii, and surface anisotropy. The main results are summarized and discussed in the last section.

2. Integral Equation for the Surface Current

[4] To calculate an electromagnetic (EM) field scattered by an arbitrarily shaped surface S, it is necessary to have the explicit expression for the current J(r), requation imageS, induced at the surface by the incident EM wave. In particular, for a monochromatic incident wave with circular frequency ω (below, the dependence of time ∼exp(−iωt) is suppressed), the scattered magnetic field Hsc(R) at every space point R(x, y, z) can be obtained from the Huygens-Kirchhoff principle:

equation image

where G(R, r) is the Green function

equation image

of the Helmholtz equation

equation image

Here, k = ω/c, and c is a light speed. In the case of the perfectly conducting surface considered here, the current J(r) is a solution of the integral equation (e.g., Holliday [1987]),

equation image

which is the direct consequence of the Stratton-Chu equation Stratton [1941] for tangential components of surface EM fields. In (4), J0(r) = 2[n × H0] is the current induced at the surface S by the incident EM wave H0(r) in the local GO approximation, neglecting the multiple scattering and shadowing, and n is the unit normal vector to the surface S directed to the half-space where the EM field sources are located. The difference in signs before the integral term in (4) and in the corresponding equation (6) in Holliday [1987] is the result of the difference in our Green function definition (2) and definition (3) in Holliday [1987].

[5] In this section, we consider the plane linear polarized incident wave H0 = h0eikr, where k is the wave vector and h0 is the unit vector orthogonal to k. After separating the phase factor eikr in surface currents J(r) = j(r)eikr and J0(r) = j0(r) eikr, (4) takes the form:

equation image

where j0(r) = 2[n(r) × h0] and ρ = r′ − r.

[6] Introduce the local Cartesian frame of reference (ξ, η, ζ), which origin O is located at the surface point r, axis is directed along the normal n, and the axis belongs to the plane of incidence, which contains vectors k and n (Figure 1). The equation of surface S in this frame of reference can be written in the form ζ = s(ρ), where ρ = (ξ, η), the wave vector k has the components k = (k sin ϑ, 0, −k cos ϑ), where ϑ is the local incidence angle, and n(r) = (0, 0, 1). The double-cross vector product in (5) acquires the appearance:

equation image

where jζ(r′) = j(r′) · n(r).

Figure 1.

Geometry of scattering in the local frame of reference.

[7] In the high-frequency limit (k → ∞), for a smooth and gentle (mildly sloping) surface S, if it is possible to neglect the multiple scattering and shadowing, only the small vicinity of every point r affects the current j(r) at this point. This means that in the local GO limit, the surface S at every point r can be replaced by the tangent plane ζ = 0. In this case, expression (6) turns to zero, and from (5) it follows that j(r) = j0(r). To obtain the diffraction corrections to this GO value of the surface current, we expand the integrand in (5) in a series of a small deviation s(ρ) of the real surface S from its tangent plane approximation up to the second order of s(ρ), and will solve the integral equation (5) by iterations. The first iteration j1(r) can be found by substituting in the integral term in (5), instead of j(r′), its GO value j0(r′) = 2[n′ × h0], where n′ ≡ n(r′) is the normal vector to surface S in the point r′ (Figure 1):

equation image

Here, eξ, eη, and eζ are the unit vectors in the , , and directions correspondingly. To obtain the diffraction corrections (which should be small) to the GO surface current j0(r), the integral over dρ in (5) can be replaced by the integral in the infinite limits over the projection dρ = dξdη of dρ at the plane ζ = 0, using the equation dρ = dρ/nζ. Taking into account the decomposition ρ = ρ + eζs(ρ) of vector ρ in the local frame of reference, from (5) we obtain the explicit equation for the first iteration j1(r) for the surface current:

equation image

Substituting this expression into the integral term in (5) instead of j(r′), we obtain the equation for the second iteration j2(r), and so on.

[8] Below, we consider two cases of the incident wave polarization separately: (a) “V” - vertical polarization (TM), or “p”-polarization in optics, when h0 = (0, 1, 0) is perpendicular to the plane of incidence ξOζ, and (b) “H” -horizontal polarization (TE), or “s” -polarization in optics, when vector h0 = (cos ϑ, 0, sin ϑ) is located in the plane of incidence.

2.1. Vertically Polarized Incident Wave

[9] For a vertically polarized incident wave, when h0 = eη, we have j0(r) = −2eξ, and the expression in braces in (8) takes the form:

equation image

Bearing in mind that in the high-frequency limit, only a small vicinity of the point r is significant in the integral (8), we can expand function s(ρ) and slopes γξ(ρ) and γη(ρ) into the Taylor's series near ρ = 0:

equation image
equation image
equation image

where common notations are used for the partial derivatives at ρ = 0: sξξ ≡ ∂2s/∂ξ2, sξη ≡ ∂2s/∂ξη, sξξξ ≡ ∂3s/∂ξ3 etc. Substituting (10) and (11) into (9), and then in (8), and shifting in (8) from the Cartesian variables of integration ξ and η to the dimensionless polar variables t and ϕ ( = t cos ϕ, = t sin ϕ), for the components j1ξ and j1η of the surface current j1 = j1ξeξ + j1ηeη we obtain with accuracy of O(1/k2):

equation image
equation image

[10] Integration in (13) and (14) can be carried out in the explicit form, using the integrals given in Appendix A:

equation image
equation image

[11] Substituting these results for the first iteration j1(r) of the surface current into the integral in the right-hand part of the integral equation (5) instead of j(r′), we can perform calculations similar to those above, and for the components j2ξ and j2η of the second iteration j2(r) = j2ξeξ + j2ηeη we obtain:

equation image
equation image

From (15)(18), it follows that the total current j(r) induced at the point requation imageS by a vertically polarized incident wave of unit amplitude H0 = eηeikr can be written with the accuracy O(1/k2) in the form j(r) = jV(r) = jξVeξ + jηVeη, where

equation image

and jηV = j1η, where j1η is given by (16).

2.2. Horizontally Polarized Incident Wave

[12] For a horizontally polarized incident wave, when the electric field has only one component along the axis , i.e., E0 = eηeikr, the polarization vector h0 of the magnetic field is located in the plane of incidence: h0 = eξ cos ϑ + eζ sin ϑ. In the GO limit, the electric current j0 induced by this wave at the surface point requation imageS is directed along the axis :

equation image

For the first iteration of the surface current j1, we can use (8), where the expression in braces takes the form:

equation image

Substituting (21), (10), (11) and (12) into (8), and switching to the dimensionless variables of integration t and ϕ, as in the previous subsection, for components j1ξ and j1η of the first iteration of the surface current j1 = j1ξeξ + j1ηeη we obtain equations similar to (13) and (14):

equation image
equation image

Only terms of O(1/k2) are retained in these equations. Using the integrals given in Appendix A, from (22) and (23) we obtain:

equation image
equation image

For the second iteration we have:

equation image
equation image

From (20) and (24)(27), it follows that the total current j(r) induced at the point requation imageS by a horizontally polarized incident wave of unit amplitude can be written with the accuracy O(1/k2) in the form j(r) = jH(r) = jξHeξ + jηHeη, where

equation image

and jξH = j1ξ, where j1ξ is given by (24).

[13] Note that diffraction by the local surface curvature results in the appearance of surface current components orthogonal to those induced by the incident wave in the local GO limit: jη for the “V” polarized wave, and jξ for the “H” polarized wave. The phase of the first-order (∼1/k) diffraction corrections to surface currents is shifted by ±π/2 relative to the GO currents, which results in the absence of terms ∼1/k in the high-frequency asymptotic expansions of current amplitudes. The sign of this phase shift depends on the incident wave polarization and on the sign of the expression (sξξsηη cos2ϑ), where sξξ is the curvature of the surface cross section by the plane of incidence, and sηη is the curvature of the surface cross section by the plane orthogonal to the plane of incidence. Whereas the first-order (∼1/k) diffraction corrections to surface currents depend only on the second derivatives sξξ and sηη of the surface in the local frame of reference, the second-order (∼1/k2) corrections depend on the third derivatives as well. At normal incidence (ϑ = 0), the reciprocity equations for induced surface currents hold: jηHequation imagejξV, jξHequation imagejηV when ξequation imageη.

3. Backscattering Cross Sections at Normal Incidence

[14] Substituting into (1) the explicit expressions for the surface currents J(r) obtained above and asymptotically evaluating the integral (1) in the high-frequency limit (k → ∞), we can obtain the diffraction corrections to the GO results for many scattering problems. In particular, it is possible to obtain the high-frequency asymptotic expansions of EM fields scattered by a solitary specular point, as it was done in Fuks [2005a, 2005b, 2006] for 2-D (i.e., cylindrical) surfaces. It was shown in the papers cited that even in this simplest 2-D case, the final expressions for the diffraction corrections (∼1/k2) to GO results are very bulky, and they contain the derivatives of the surface directrix up to the sixth order. In the 3-D case, we can expect that the number of terms in this asymptotic expansion (even only up to ∼1/k2) will be at least an order of magnitude more than in the 2-D case, because it will depend on all derivatives over two variables of the surface at a specular point up to the sixth order.

[15] Here, as an application of the results obtained above for the diffraction corrections to the GO surface currents J(r), we consider the problem of EM backscattering from a statistically rough Gaussian surface at normal incidence when the incident wave has the form H0 = h0eikz. In the “laboratory” Cartesian frame of reference (x, y, z), the surface S is given by the equation z = Z(x, y) (Figure 2), where Z(x, y) is a random Gaussian statistically homogeneous function of two variables. In a far zone, at distance R from the surface, according to (1), the backscattered field Hsc takes the form:

equation image

where ex, ey are the unit vectors in Ox and Oy directions, correspondingly, and jx, jy are the projections of the surface current j in the Cartesian frame of reference (x, y, z). In the previous section we have obtained the projections jξ, jη of the current j in the local frame of reference (ξ, η, ζ) at every point requation imageS, where the incident wave vector k makes the angle ϑ with the local normal n (Figure 2). In the high-frequency limit (k → ∞), only a small vicinity of the specularly reflecting points, where ϑ = 0, makes the main contribution to the integral (29). At these points, the local (ξ, η, ζ) and the laboratory (x, y, z) frames of reference coincide. It is possible to make sure that the difference between current projections jξ, jη and jx, jy at the vicinity of the specular points, where ϑ ≠ 0, makes contribution to the integral (29) only in terms of higher order of magnitude than 1/k2. Hereinafter, we will keep only terms up to O(1/k2) in the high-frequency asymptotic expansions of scattered field and backscattering cross sections, which allows us to put ϑ = 0 in the explicit expressions for surface current diffraction corrections obtained above.

Figure 2.

Geometry of scattering in the “laboratory” frame of reference.

[16] For the linear polarized incident wave, h0 = ex cos ϕ + ey sin ϕ, the surface current projections jx, jy in the integrand in (29) have the form:

equation image

Here, jξV and jηH are given by (19) and (28), correspondingly, and jηV = j1η, jξH = j1ξ where j1η and j1ξ are given by (16) and (24) at ϑ = 0.

3.1. Copolarized Backscattering (Linear Polarization)

[17] For a copolarized backscattered field (with the same polarization h0 as the incident wave), from (29) we have:

equation image
equation image

where r = (x, y), integration over dr = dxdy is performed over the projection S0 of S onto the plane z = 0, and the second derivatives Zxx = ∂2Z/∂x2, Zxy = ∂2Z/∂xy, Zxx = ∂2Z/∂x2 are assumed to be functions of r. The specific (from the unit area) copolarized backscattering cross section is defined by the equation:

equation image

Here, 〈…〉 stands for statistical averaging over realizations of the random function Z(x, y). From (31) it follows that:

equation image

where (*) means a complex conjugation. In the high-frequency limit (k → ∞), the exponential factor in (34) oscillates very fast when separation between points r and r increases. So only the small vicinity of point r gives the contribution to the integral over dr. The linear size of this area in every direction has the order of magnitude of the Fresnel zone ≈equation image, where a is the curvature radius of the normal surface cross section in that direction, which is supposed to be much smaller than the roughness horizontal scale (correlation length) and the linear size of the surface S in the xOy plane. Introduce the vector ρ = r′ − r (Figure 2) and expand the integrand in (34) in series of ρ ≡ (x, y), for simplicity using the same notations (x, y) for projection of ρ onto plane z = 0, as for r (this will not result in any confusion in further derivations):

equation image

where Zx = ∂Z/∂x, Zy = ∂Z/∂y are the surface slopes at the point r. In the imaginary terms of the order of ∼1/k in the expression F(r,ϕ) F* (r, ϕ), the differences of the second derivatives Zαβ, where α, β = x, y, also can be expanded in the Taylor series:

equation image

As a result of these expansions, the integrand in (34) transforms into the function of derivatives of Z(r) up to the first three orders. For statistically homogeneous random function Z(r) with 〈Z(r)〉 = 0, we can introduce the surface roughness autocorrelation function:

equation image

and choose the axes Ox and Oy of the frame of reference along the “main” directions of the surface roughness, where surface slopes do not correlate, i.e., 〈ZxZy〉 = 0. In this frame of reference, the second derivatives Zαβ (r) do not correlate with any odd derivative at the same point r, and, in particular, with slopes Zα and the third derivatives Zαβγ. For Gaussian statistics, this results in their independence, which allows us to perform the statistical averaging of (34) over the second derivatives Zαβ. Take into account that 〈Zαβ〉 = 0 and express the variances of Zαβ and their binary correlators through the fourth derivatives of W(x, y) at x = y = 0:

equation image
equation image

[18] Note that W(n,m) = 0 for n + m = 2N + 1, where N is the integer number. The integrand in (34) linearly depends on the third derivatives Zαβγ. We can perform the conditional statistical averaging of Zαβγ for specified slopes Zx and Zy:

equation image
equation image
equation image
equation image

where hx2 ≡ 〈Zx2〉 and hy2 ≡ 〈Zy2〉 are the surface slope variances. These equations are the particular cases of the general equation (6.10) in Feller [1966].

[19] After averaging over random derivatives Zαβ and Zαβγ, the integrand in (34) remains to be the product of the fourth-order polynomial of two variables (x and y), the coefficients of which are the linear functions of slopes (Zx and Zy), multiplied by the factor exp [2ik(Zxx + Zyy)]. Now, in (34), we can perform the integration over dρ = dxdy in the infinite limits (if the linear size of the surface S in the xOy plane significantly exceeds the characteristic surface curvature radii), which results in the appearance of δ-functions and their derivatives δ(n):

equation image

The final statistical averaging can be performed by multiplying (34) by the roughness slope PDF:

equation image

and integrating over the random slopes Zx, Zy, which can be performed easily because of the presence of δ-functions and their derivatives δ(n) in the integrand. Because of the spatial statistical inhomogeneity, the result of this integration does not depend on variable r, and the last integration over dr gives the total illuminating area S0. Substituting 〈∣Hco2〉 from (34) into (33), we obtain the final result for σco in the form:

equation image
equation image
equation image
equation image

where σ0 is the backscattering cross section in the GO limit:

equation image

[20] Note that the term Δ0, which corresponds to the diffraction correction independent of the azimuth angle ϕ, can be written in the operator invariant form, independent of the choice of the frame of reference:

equation image

Here, equation image0 = ∂2/∂x2 + ∂2/∂y2 is the Laplace operator, and equation image is a second-order differential operator:

equation image

equation image is the tensor-matrix of the second derivatives:

equation image

where the temporary notations xx1, yx2 are introduced, (equation image)−1 is the inverse matrix of equation image and ∇ = ex∂/∂x + ey∂/∂y is the gradient operator, dot (·) means the scalar (inner) product. The GO backscattering cross section (50) also can be written in the invariant form:

equation image

Note that det Wik(ρ) > 0 at ρ = 0, because it is the Gaussian (total) curvature of the surface z = W(x, y) at its global maximum ρ = 0.

[21] In many cases, instead of the correlation function W(x, y), it is more convenient to deal with the spatial power spectrum ��(κx, κy) of the surface roughness:

equation image

where integration is performed in the infinite limits. The statistical parameters introduced above can be expressed as the moments of the power spectrum:

equation image
equation image

[22] For small-slope roughness with hx2, hy2 ≪ 1, from (46) it follows that:

equation image

[23] The equations obtained above cannot be applied directly to the specific case of the 2-D (i.e., cylindrical) surface S with the directrix z = Z(x), when �� (κx, κy) = δ(κy) ��1(κx). As a matter of fact for a 2-D surface, the integrand in (34) does not oscillate in the direction of surface generatrix Oy, where Z(x) = const., and we cannot expand to infinity the limits of integration over dy in (44) for this case. Nevertheless, if we formally put W(0,4) = W(2,2) = 0, only the first terms remain in the square brackets in (47)(49), and (46) transforms to the results obtained in Fuks [2007b] for a 2-D surface. For ϕ = 0, it corresponds to “HH”- polarization backscattering, when h0 = ex and the electric field of the incident wave is parallel to the surface generatrix Oy. In this case, the diffraction correction Δ to the GO backscattering cross section coincides with equation (27) in Fuks [2007b] for ΔHH:

equation image

Here, 1/a12 ≡ 〈Zxx2〉 = equation image is half of the averaged square of the directrix curvature at the specular points, where Zx = 0. For ϕ = π/2, when h0 = ey and the electric field of the incident wave is orthogonal to the surface generatrix Oy, from (47)(49) it follows that the result for “VV” - polarization backscattering cross section is:

equation image

which coincides with equation (28) in Fuks [2007b].

[24] In the opposite limiting case of statistically isotropic surface roughness, the power spectrum �� (κx, κy) is a function of κ = equation image, and it can be represented in the form:

equation image

and instead of (56), (57) we have:

equation image
equation image

It is easy to check that in this case Δ2 = Δ4 = 0, and the diffraction correction Δ to the copolarized backscattering cross section in (43) transforms to the simple form:

equation image

Here, 1/a02 = equation image is one-fourth of the square of the total (Gaussian) surface curvature averaged over all specularly reflecting points (Zx = Zy = 0), and γ02 = 〈(∇Z)2〉 = equation image + equation image = equation image is the surface slope variance. Note that Δ = 0 for γ02 = 1/equation image, when the r.m.s. tangent of slopes in any surface cross section is equal to hx = γ0/equation image ≃ 0.537, which corresponds to the characteristic slope angle ≃28.2°. Vanishing of Δ means that the diffraction corrections to the GO backscattering cross section in this case are proportional to the higher order of 1/k than 1/k2; i.e., the GO approach is valid for the lower frequencies than in a general case.

[25] For the polarization ratio, determined as the ratio of the backscattering cross sections, σco, at two orthogonal polarizations, from (46) we obtain with the accuracy of 1/k2:

equation image

Note that backscattering cross sections at orthogonal polarizations are equal to each other (K(ϕ) = 1) when the polarization vector h0 of the incident wave makes the angle ϕ = ±π/4 with the “main” directions in the surface (Ox and Oy).

[26] In the 2-D case, (65) has the form:

equation image

At ϕ = 0, this equation coincides with equation (29) in Fuks [2007b] for the HH/VV polarization ratio RHV = K(0). For the opposite limiting case of the statistically isotropic surface, Δ2 = 0, and K = 1.

3.2. Cross-Polarized Backscattering (Linear Polarization)

[27] Introduce the polarization vector h orthogonal to the polarization vector h0 of the incident wave:

equation image

For the cross-polarized backscattered field Hcr, from (29), (30) and (67) it follows:

equation image

Using the explicit expressions for the surface currents obtained in the previous section, from (68) we obtain:

equation image

The cross-polarization coefficient Kcr is equal to the cross-polarized backscattering cross section

equation image

normalized by the copolarized backscattering cross section σco. From (69), we can obtain for 〈∣Hcr2〉 an expression similar to (34), where instead of Fco(r, ϕ) we have iFcr(r, ϕ)/k. Performing for 〈∣Hcr2〉 the same derivations (35)(44) as for 〈∣Hco2〉 in the previous subsection, we obtain with an accuracy of 1/k2:

equation image

For a 2-D (cylindrical) rough surface, z = Z(x), when W(0,4) = W(2,2) = 0, from (71) it follows that:

equation image

where a1 is the same as in (59).

[28] In the opposite limiting case of 3-D statistically isotropic roughness, from (71) we have:

equation image

where a0 is the same as in (64).

3.3. Circular Polarization

[29] For the circular, right-polarized incident wave, the polarization vector h0 has the form h0 = (ex + iey)/equation image, and the Cartesian projections of the induced surface current can be expressed through the currents induced by the linear polarized wave (compare with (30)):

equation image

Here, the same notations as in (30) are used for surface current projections jxH, jxV, jyH, and jyV. Substituting jx and jy from (74) to (29), we obtain the equation for the backscattered field Hsc. The copolarized and cross-polarized components of Hsc can be obtained by multiplying it by h*0 and h0, correspondingly:

equation image
equation image
equation image

Here, r = (x, y), integration over dr = dxdy is performed over the projection S0 of S onto the plane z = 0, and all second derivatives of the surface z = Z(r) relate to the point r. The calculation of 〈∣Hco2〉 and 〈∣Hcr2〉 can be performed in the same way, as was done above for the linear polarized waves, which results in the following equations for the backscattering cross section σco and the cross-polarization coefficient Kcr:

equation image
equation image

where σ0 is the GO backscattering cross section (50), moments of the roughness power spectrum are defined by (57), and equation image0 = ∂2/∂x2 + ∂2/∂y2 is the Laplace operator. The invariant form of (78) has almost the same appearance as the azimuth independent diffraction correction (51) to the copolarized backscattering cross section of the linear polarized probing signal: the only difference is in the factor “16” in (78) instead of “12” in (51).

[30] For small-slope roughness with hx2, hy2 ≪ 1, (78) coincides with the backscattering cross section for the linear polarized wave (58), i.e., in this case, diffraction corrections do not depend on the polarization of the incident wave. Moreover, (58) and (78) (in the case hx2, hy2 ≪ 1) coincide with diffraction corrections to the backscattering cross sections of acoustic waves from the perfectly rigid and free surfaces, which, for hx2, hy2 ≪ 1, are given by the first term in the square brackets in equation (58) in Fuks [2007a]. This universal form of diffraction corrections is caused by the fact that all of them originate from the expansion of the “propagation exponent” (35), which, according to the Huygens-Kirchhoff principle (1), is the same for all kinds of propagating waves, but not from the diffraction corrections to the surface currents. This means that these corrections are generated by interference in the space of waves propagating from a rough surface, where they were specularly reflected at every surface point according to the GO laws, rather than by diffraction at surface roughness.

[31] Note that for hx2, hy2 ≪ 1, these corrections to the backscattering cross sections, caused by the interference of the specularly reflected waves, are always positive. It can be interpreted as an effective “smoothing” of the surface with an increase of the wavelength of the incident field: the GO backscattering cross section at normal incidence is in inverse proportion to the roughness slope variance (e.g., (50) for Gaussian statistics), and increasing the backscattering cross section is equivalent to the roughness slope decreasing, i.e., roughness “smoothing”. This effect of surface “smoothing” was first obtained in Lynch [1970] for the specific case of scalar (acoustic) waves scattering by a perfectly “free” (with zero-pressure boundary conditions) 2-D surface with small-slope roughness (hx2 ≪ 1). Later, the same effect of surface “smoothing” was obtained in a 3-D case for the acoustic Rodriguez [1989] and EM waves Rodriguez [1991] in the “small momentum transfer limit”: i.e., for extremely gentle surfaces with hy2, hx2 ≪ 1.

[32] For validity of the results obtained above, the diffraction corrections to the GO backscattering cross sections σco, normalized on the σ0, as well as the cross-polarization coefficient Kcr, must be small in comparison with unity.

4. Numerical Results for the Gaussian Correlation Function of Surface Roughness

[33] Here, we apply the general equations obtained above to the specific case of the Gaussian autocorrelation function of surface roughness:

equation image

where h2 = 〈Z2〉 is the roughness height variance, and lx and ly are the correlation lengths in the Ox and Oy directions, correspondingly. The surface statistical parameters, upon which the diffraction corrections to the backscattering cross sections depend, have the following appearance in this case:

equation image
equation image

Substituting these parameters into (47)(49), and introducing the coefficient of roughness anisotropy αlx/ly, for the linear polarized incident wave we obtain:

equation image
equation image
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Without restricting the generality, we can assume lyequation imagelx, i.e., αequation image 1. Note that these equations cannot be applied to the 2-D case, when ly = ∞ and α = 0 for the same reasons as mentioned in the previous section (see remark below (58)): the surface curvature radii and correlation lengths lx and ly must be much smaller than the linear size of the surface S in the xOy plane. The plots of F0 and F2 as functions of hx2 are depicted in Figures 3 and 4, correspondingly, for the set of parameter α.

Figure 3.

Dependence of factor F0 in equation (83) on the slope variance hx2 for the set of the anisotropy coefficient α.

Figure 4.

Dependence of factor F2 in equation (84) on the slope variance hx2 for the set of the anisotropy coefficient α.

[34] The cross-polarization coefficient Kcr(71) takes the form:

equation image
equation image

The plots of F1 and F4 as functions of α2 are depicted in Figure 5. For anisotropic roughness, when α < 1 and F4(α) > 0, the cross-polarization coefficient Kcr has a maximum at ϕ = ±π/4

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and a minimum for ϕ = 0, π:

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From (86), it follows that the difference ΔKcr between maximal and minimal values of the cross-polarization coefficient Kcr is equal to

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For small-slope roughness, when hyequation imagehx ≪ 1, the azimuth independent diffraction correction Δ0 = (2kh)−2 to the copolarized backscattering cross section σco significantly exceeds the azimuth varying corrections (Δ20 = O(hx2), Δ40 = O(hx4)), as well as the cross-polarization coefficient Kcr0 = O(hx4). The diffraction correction to the polarization ratio K(65) is proportional to Δ2, and it is also small in comparison to Δ0.

Figure 5.

Dependence of factors F1 and F4 in equation (86) on the anisotropy coefficient α.

[35] In the case of statistically isotropic surface roughness, when lx = ly = l0 and α = 1, the diffraction correction Δ to σco and the cross-polarization coefficient Kcr are given by (64) and (73), correspondingly, where

equation image

[36] For the circular-polarized wave,

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[37] Note that the general equations for diffraction corrections to the GO copolarized backscattering cross section (46)(49) and cross-polarization coefficient (71) depend only on the variances of the first and second derivatives of the surface profile (the second and the fourth derivatives of the autocorrelation function W(x, y), correspondingly), and they do not depend directly on the surface height variance h2. In this subsection, parameter h2 appears only from the ratios of the second and the fourth derivatives of the autocorrelation function W(x, y):

equation image

For multiscale roughness, with the wide spatial power spectrum &#55349;&#56494;(κx, κy), slowly decreasing at a small roughness scale, these ratios (of the second and fourth moments of &#55349;&#56494;(κx, κy)) can be significantly different from the surface roughness height variance h2.

5. Conclusions

[38] We have found the diffraction corrections up to terms ∼1/k2 to surface currents induced by a plane incident wave at a perfectly conducting smooth surface, neglecting multiple scattering and surface shadowing. It was shown that the phases of the first-order corrections (∼1/k) are shifted by ±π/2 relative to the GO currents, which results in the absence of terms ∼1/k in the high-frequency asymptotic expansions of current amplitudes and, ultimately, in backscattering cross sections. Whereas the first-order (∼1/k) diffraction corrections to surface currents depend only on the second derivatives sξξ and sηη of the surface cross sections in the local frame of reference (i.e., on the surface curvatures), the second-order (∼1/k2) corrections depend on the third derivatives as well. Note that diffraction by local surface curvatures results not only in corrections to GO currents (which are assumed to be small), but also gives rise to the surface current components orthogonal to those induced by the incident wave in the local GO limit. These current components are responsible for backscattering into the cross-polarized EM waves, which is absent in the GO limit.

[39] For gentle surface roughness, with small slope variances, diffraction corrections to backscattering cross sections at normal incidence are always positive, and they do not depend on the polarization of the incident wave. These corrections coincide with those obtained previously for scattering of sound waves, and they originate not from the diffraction corrections to the surface currents, but from the expansion of the propagation exponent in the scalar Green function of the Helmholtz equation, i.e., from the interference in the space of upward propagating waves. The positivity of these corrections can be interpreted as effective smoothing of surface roughness as the wavelength of the incident wave increases. In contrast, cross-polarized backscattering is generated by the orthogonal to GO components of surface currents, which are caused entirely by diffraction by surface roughness curvatures, but not by wave interference in the space.

Appendix A

[40] In section 3, to calculate the diffraction corrections to the surface currents, we used the obvious expressions for the integrals over in (13), (14), (22), (23) and similar to them:

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where J0(p) is the Bessel function of the zero order:

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In particular, from (A1) it follows:

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The integrals over dt in the equations mentioned above have the appearance:

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With the help of the software package Mathematica, these integrals can be evaluated in an explicit form. To perform the final calculations in section 2, we need only five of them:

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Ancillary