[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.

[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/k^{2}, 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), rS, 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 H_{sc}(R) at every space point R(x, y, z) can be obtained from the Huygens-Kirchhoff principle:

where G(R, r) is the Green function

of the Helmholtz equation

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]),

which is the direct consequence of the Stratton-Chu equation Stratton [1941] for tangential components of surface EM fields. In (4), J_{0}(r) = 2[n × H_{0}] is the current induced at the surface S by the incident EM wave H_{0}(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 H_{0} = h_{0}e^{ikr}, where k is the wave vector and h_{0} is the unit vector orthogonal to k. After separating the phase factor e^{ikr} in surface currents J(r) = j(r)e^{ikr} and J_{0}(r) = j_{0}(r) e^{ikr}, (4) takes the form:

where j_{0}(r) = 2[n(r) × h_{0}] 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 Oζ is directed along the normal n, and the axis Oξ 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:

where j_{ζ}(r′) = j(r′) · n(r).

[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) = j_{0}(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 j_{1}(r) can be found by substituting in the integral term in (5), instead of j(r′), its GO value j_{0}(r′) = 2[n′ × h_{0}], where n′ ≡ n(r′) is the normal vector to surface S in the point r′ (Figure 1):

Here, e_{ξ}, e_{η}, and e_{ζ} are the unit vectors in the Oξ, Oη, and Oζ directions correspondingly. To obtain the diffraction corrections (which should be small) to the GO surface current j_{0}(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_{⊥}/n′_{ζ}. Taking into account the decomposition ρ = ρ_{⊥} + e_{ζ}s(ρ_{⊥}) of vector ρ in the local frame of reference, from (5) we obtain the explicit equation for the first iteration j_{1}(r) for the surface current:

Substituting this expression into the integral term in (5) instead of j(r′), we obtain the equation for the second iteration j_{2}(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 h_{0} = (0, 1, 0) is perpendicular to the plane of incidence ξOζ, and (b) “H” -horizontal polarization (TE), or “s” -polarization in optics, when vector h_{0} = (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 h_{0} = e_{η}, we have j_{0}(r) = −2e_{ξ}, and the expression in braces in (8) takes the form:

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 γ_{ξ}(ρ_{⊥}) and γ_{η}(ρ_{⊥}) into the Taylor's series near ρ_{⊥} = 0:

where common notations are used for the partial derivatives at ρ_{⊥} = 0: s_{ξξ} ≡ ∂^{2}s/∂ξ^{2}, s_{ξη} ≡ ∂^{2}s/∂ξ∂η, s_{ξξξ} ≡ ∂^{3}s/∂ξ^{3} 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 (kξ = t cos ϕ, kη = t sin ϕ), for the components j_{1ξ} and j_{1η} of the surface current j_{1} = j_{1ξ}e_{ξ} + j_{1η}e_{η} we obtain with accuracy of O(1/k^{2}):

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

[11] Substituting these results for the first iteration j_{1}(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 j_{2ξ} and j_{2η} of the second iteration j_{2}(r) = j_{2ξ}e_{ξ} + j_{2η}e_{η} we obtain:

From (15)–(18), it follows that the total current j(r) induced at the point rS by a vertically polarized incident wave of unit amplitude H_{0} = e_{η}e^{ikr} can be written with the accuracy O(1/k^{2}) in the form j(r) = j^{V}(r) = j_{ξ}^{V}e_{ξ} + j_{η}^{V}e_{η}, where

and j_{η}^{V} = j_{1η}, where j_{1η} 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 Oη, i.e., E_{0} = e_{η}e^{ikr}, the polarization vector h_{0} of the magnetic field is located in the plane of incidence: h_{0} = e_{ξ} cos ϑ + e_{ζ} sin ϑ. In the GO limit, the electric current j_{0} induced by this wave at the surface point rS is directed along the axis Oη:

For the first iteration of the surface current j_{1}, we can use (8), where the expression in braces takes the form:

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 j_{1ξ} and j_{1η} of the first iteration of the surface current j_{1} = j_{1ξ}e_{ξ} + j_{1η}e_{η} we obtain equations similar to (13) and (14):

Only terms of O(1/k^{2}) are retained in these equations. Using the integrals given in Appendix A, from (22) and (23) we obtain:

For the second iteration we have:

From (20) and (24)–(27), it follows that the total current j(r) induced at the point rS by a horizontally polarized incident wave of unit amplitude can be written with the accuracy O(1/k^{2}) in the form j(r) = j^{H}(r) = j_{ξ}^{H}e_{ξ} + j_{η}^{H}e_{η}, where

and j_{ξ}^{H} = j_{1ξ}, where j_{1ξ} 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: j_{η} for the “V” polarized wave, and j_{ξ} 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 (s_{ξξ} − s_{ηη} cos^{2}ϑ), where s_{ξξ} is the curvature of the surface cross section by the plane of incidence, and s_{ηη} 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 s_{ξξ} and s_{ηη} of the surface in the local frame of reference, the second-order (∼1/k^{2}) corrections depend on the third derivatives as well. At normal incidence (ϑ = 0), the reciprocity equations for induced surface currents hold: j_{η}^{H} −j_{ξ}^{V}, j_{ξ}^{H} −j_{η}^{V} when ξη.

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/k^{2}) 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/k^{2}) 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 H_{0} = h_{0}e^{−ikz}. 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 H_{sc} takes the form:

where e_{x}, e_{y} are the unit vectors in Ox and Oy directions, correspondingly, and j_{x}, j_{y} 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 j_{ξ}, j_{η} of the current j in the local frame of reference (ξ, η, ζ) at every point rS, 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 j_{ξ}, j_{η} and j_{x}, j_{y} 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/k^{2}. Hereinafter, we will keep only terms up to O(1/k^{2}) 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.

[16] For the linear polarized incident wave, h_{0} = e_{x} cos ϕ + e_{y} sin ϕ, the surface current projections j_{x}, j_{y} in the integrand in (29) have the form:

Here, j_{ξ}^{V} and j_{η}^{H} are given by (19) and (28), correspondingly, and j_{η}^{V} = j_{1η}, j_{ξ}^{H} = j_{1ξ} where j_{1η} and j_{1ξ} are given by (16) and (24) at ϑ = 0.

[17] For a copolarized backscattered field (with the same polarization h_{0} as the incident wave), from (29) we have:

where r_{⊥} = (x, y), integration over dr_{⊥} = dxdy is performed over the projection S_{0} of S onto the plane z = 0, and the second derivatives Z_{xx} = ∂^{2}Z/∂x^{2}, Z_{xy} = ∂^{2}Z/∂x∂y, Z_{xx} = ∂^{2}Z/∂x^{2} are assumed to be functions of r_{⊥}. The specific (from the unit area) copolarized backscattering cross section is defined by the equation:

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

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 ≈, 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):

where Z_{x} = ∂Z/∂x, Z_{y} = ∂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 Z_{αβ}, where α, β = x, y, also can be expanded in the Taylor series:

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:

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., 〈Z_{x}Z_{y}〉 = 0. In this frame of reference, the second derivatives Z_{αβ} (r_{⊥}) do not correlate with any odd derivative at the same point r_{⊥}, and, in particular, with slopes Z_{α} and the third derivatives Z_{αβγ}. For Gaussian statistics, this results in their independence, which allows us to perform the statistical averaging of (34) over the second derivatives Z_{αβ}. Take into account that 〈Z_{αβ}〉 = 0 and express the variances of Z_{αβ} and their binary correlators through the fourth derivatives of W(x, y) at x = y = 0:

[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 Z_{αβγ}. We can perform the conditional statistical averaging of Z_{αβγ} for specified slopes Z_{x} and Z_{y}:

where h_{x}^{2} ≡ 〈Z_{x}^{2}〉 and h_{y}^{2} ≡ 〈Z_{y}^{2}〉 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 Z_{αβ} and Z_{αβγ}, 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 (Z_{x} and Z_{y}), multiplied by the factor exp [2ik(Z_{x}x + Z_{y}y)]. 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)}:

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

and integrating over the random slopes Z_{x}, Z_{y}, 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 S_{0}. Substituting 〈∣H_{co}∣^{2}〉 from (34) into (33), we obtain the final result for σ_{co} in the form:

where σ_{0} is the backscattering cross section in the GO limit:

[20] Note that the term Δ_{0}, 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:

Here, _{0} = ∂^{2}/∂x^{2} + ∂^{2}/∂y^{2} is the Laplace operator, and is a second-order differential operator:

is the tensor-matrix of the second derivatives:

where the temporary notations x → x_{1}, y → x_{2} are introduced, ()^{−1} is the inverse matrix of and ∇ = e_{x}∂/∂x + e_{y}∂/∂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:

Note that det W_{ik}(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 ��(κ_{x}, κ_{y}) of the surface roughness:

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

[22] For small-slope roughness with h_{x}^{2}, h_{y}^{2} ≪ 1, from (46) it follows that:

[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 �� (κ_{x}, κ_{y}) = δ(κ_{y}) ��_{1}(κ_{x}). 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 h_{0} = e_{x} 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 Δ_{HH}:

Here, 1/a_{1}^{2} ≡ 〈Z_{xx}^{2}〉 = is half of the averaged square of the directrix curvature at the specular points, where Z_{x} = 0. For ϕ = π/2, when h_{0} = e_{y} 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:

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

[24] In the opposite limiting case of statistically isotropic surface roughness, the power spectrum �� (κ_{x}, κ_{y}) is a function of κ = , and it can be represented in the form:

It is easy to check that in this case Δ_{2} = Δ_{4} = 0, and the diffraction correction Δ to the copolarized backscattering cross section in (43) transforms to the simple form:

Here, 1/a_{0}^{2} = is one-fourth of the square of the total (Gaussian) surface curvature averaged over all specularly reflecting points (Z_{x} = Z_{y} = 0), and γ_{0}^{2} = 〈(∇Z)^{2}〉 = + = is the surface slope variance. Note that Δ = 0 for γ_{0}^{2} = 1/, when the r.m.s. tangent of slopes in any surface cross section is equal to h_{x} = γ_{0}/ ≃ 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/k^{2}; 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, σ_{co}, at two orthogonal polarizations, from (46) we obtain with the accuracy of 1/k^{2}:

Note that backscattering cross sections at orthogonal polarizations are equal to each other (K_{⊥}(ϕ) = 1) when the polarization vector h_{0} of the incident wave makes the angle ϕ = ±π/4 with the “main” directions in the surface (Ox and Oy).

At ϕ = 0, this equation coincides with equation (29) in Fuks [2007b] for the HH/VV polarization ratio R_{HV} = K_{⊥}(0). For the opposite limiting case of the statistically isotropic surface, Δ_{2} = 0, and K_{⊥} = 1.

[27] Introduce the polarization vector h_{⊥} orthogonal to the polarization vector h_{0} of the incident wave:

For the cross-polarized backscattered field H_{cr}, from (29), (30) and (67) it follows:

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

The cross-polarization coefficient K_{cr} is equal to the cross-polarized backscattering cross section

normalized by the copolarized backscattering cross section σ_{co}. From (69), we can obtain for 〈∣H_{cr}∣^{2}〉 an expression similar to (34), where instead of F_{co}(r_{⊥}, ϕ) we have iF_{cr}(r_{⊥}, ϕ)/k. Performing for 〈∣H_{cr}∣^{2}〉 the same derivations (35)–(44) as for 〈∣H_{co}∣^{2}〉 in the previous subsection, we obtain with an accuracy of 1/k^{2}:

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

[29] For the circular, right-polarized incident wave, the polarization vector h_{0} has the form h_{0} = (e_{x} + ie_{y})/, and the Cartesian projections of the induced surface current can be expressed through the currents induced by the linear polarized wave (compare with (30)):

Here, the same notations as in (30) are used for surface current projections j_{x}^{H}, j_{x}^{V}, j_{y}^{H}, and j_{y}^{V}. Substituting j_{x} and j_{y} from (74) to (29), we obtain the equation for the backscattered field H_{sc}. The copolarized and cross-polarized components of H_{sc} can be obtained by multiplying it by h*_{0} and h_{0}, correspondingly:

Here, r_{⊥} = (x, y), integration over dr_{⊥} = dxdy is performed over the projection S_{0} 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 〈∣H_{co}∣^{2}〉 and 〈∣H_{cr}∣^{2}〉 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 σ_{co} and the cross-polarization coefficient K_{cr}:

where σ_{0} is the GO backscattering cross section (50), moments of the roughness power spectrum are defined by (57), and _{0} = ∂^{2}/∂x^{2} + ∂^{2}/∂y^{2} 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 h_{x}^{2}, h_{y}^{2} ≪ 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 h_{x}^{2}, h_{y}^{2} ≪ 1) coincide with diffraction corrections to the backscattering cross sections of acoustic waves from the perfectly rigid and free surfaces, which, for h_{x}^{2}, h_{y}^{2} ≪ 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 h_{x}^{2}, h_{y}^{2} ≪ 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 (h_{x}^{2} ≪ 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 h_{y}^{2}, h_{x}^{2} ≪ 1.

[32] For validity of the results obtained above, the diffraction corrections to the GO backscattering cross sections σ_{co}, normalized on the σ_{0}, as well as the cross-polarization coefficient K_{cr}, 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:

where h^{2} = 〈Z^{2}〉 is the roughness height variance, and l_{x} and l_{y} 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:

Substituting these parameters into (47)–(49), and introducing the coefficient of roughness anisotropy α ≡ l_{x}/l_{y}, for the linear polarized incident wave we obtain:

Without restricting the generality, we can assume l_{y}l_{x}, i.e., α 1. Note that these equations cannot be applied to the 2-D case, when l_{y} = ∞ and α = 0 for the same reasons as mentioned in the previous section (see remark below (58)): the surface curvature radii and correlation lengths l_{x} and l_{y} must be much smaller than the linear size of the surface S in the xOy plane. The plots of F_{0} and F_{2} as functions of h_{x}^{2} are depicted in Figures 3 and 4, correspondingly, for the set of parameter α.

[34] The cross-polarization coefficient K_{cr}(71) takes the form:

The plots of F_{1} and F_{4} as functions of α^{2} are depicted in Figure 5. For anisotropic roughness, when α < 1 and F_{4}(α) > 0, the cross-polarization coefficient K_{cr} has a maximum at ϕ = ±π/4

and a minimum for ϕ = 0, π:

From (86), it follows that the difference ΔK_{cr} between maximal and minimal values of the cross-polarization coefficient K_{cr} is equal to

For small-slope roughness, when h_{y}h_{x} ≪ 1, the azimuth independent diffraction correction Δ_{0} = (2kh)^{−2} to the copolarized backscattering cross section σ_{co} significantly exceeds the azimuth varying corrections (Δ_{2}/Δ_{0} = O(h_{x}^{2}), Δ_{4}/Δ_{0} = O(h_{x}^{4})), as well as the cross-polarization coefficient K_{cr}/Δ_{0} = O(h_{x}^{4}). The diffraction correction to the polarization ratio K_{⊥}(65) is proportional to Δ_{2}, and it is also small in comparison to Δ_{0}.

[35] In the case of statistically isotropic surface roughness, when l_{x} = l_{y} = l_{0} and α = 1, the diffraction correction Δ to σ_{co} and the cross-polarization coefficient K_{cr} are given by (64) and (73), correspondingly, where

[36] For the circular-polarized wave,

[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 h^{2}. In this subsection, parameter h^{2} appears only from the ratios of the second and the fourth derivatives of the autocorrelation function W(x, y):

For multiscale roughness, with the wide spatial power spectrum ��(κ_{x}, κ_{y}), slowly decreasing at a small roughness scale, these ratios (of the second and fourth moments of ��(κ_{x}, κ_{y})) can be significantly different from the surface roughness height variance h^{2}.

5. Conclusions

[38] We have found the diffraction corrections up to terms ∼1/k^{2} 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 s_{ξξ} and s_{ηη} of the surface cross sections in the local frame of reference (i.e., on the surface curvatures), the second-order (∼1/k^{2}) 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 dϕ in (13), (14), (22), (23) and similar to them:

where J_{0}(p) is the Bessel function of the zero order:

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

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: