A Simple Path to the Small Perturbation Method for Scattering from Slightly Rough Dielectric Surfaces

Abstract

We propose a perturbative method to compute electromagnetic scattering from slightly rough dielectric surfaces, which leads to the same result as the usual Small Perturbation Method (SPM) in a surprisingly simple way. The proposed method is based on three pillars: the volumetric perturbative approach, the reciprocity theorem, and a proper approximation of the electric field within the perturbation volume, that we name Internal Field Approximation (IFA). The proposed new mathematical derivation of the SPM turns out to be much simpler and more concise than the classical one. In addition, being based on a volumetric perturbation approach, it has the potential of dealing in future with surface and volume scattering within a unitary framework, which is useful in modelling scattering from, e.g., vegetated soil, snow-covered terrain, and inhomogeneous soils. Therefore, although the presented result is mainly theoretical, it can have important applications in remote sensing.

1. Introduction

The problem of electromagnetic scattering from a slightly rough interface between two homogeneous media was first solved over seventy years ago by Rice [1], based on a perturbative method previously proposed by Rayleigh to evaluate acoustic scattering from a rough wall [2]. This approach is now widely known as the Small Perturbation Method (SPM) [3,4], and its first-order solution leads to a closed-form expression of the scattered field intensity, and, thus, of the surface normalised radar cross section (NRCS). SPM has a rather limited range of validity, since it requires that the interface height deviation is much smaller than the electromagnetic wavelength and that its gradient is much smaller than unity. In spite of that, the SPM is widely used in several fields of physics and engineering, particularly in radar remote sensing. In fact, it is not only used as it is, but it is also employed in conjunction with the Geometrical Optics (GO) approach [3,4] to obtain the so-called Two-Scale Model (TSM) [5,6,7], with a much wider validity range. In addition, extensions of the method to the case of a layered medium with rough interfaces between different layers have been also proposed [8,9,10,11,12]. Finally, it is worth noting that the Bragg coefficients that appear in the SPM solution also play a fundamental role in the first-order Small Slope Approximation (SSA-1) [13], which has a much wider range of validity.

In spite of its widespread recognition, the SPM solution is commonly obtained via involved procedures that require tedious manipulations even in the first-order approximation. In the usual formulation [1,2,3,4], the actual scattering problem is obtained from the ideal (unperturbed, flat-surface) one by perturbing the boundary, and a series expansion of the fields for vanishing interface height deviation and vanishing height deviation gradient is performed. The original Rice approach [1,2,3] is based on the Rayleigh hypothesis [1,2,3], which can be avoided by using the extended boundary conditions along with the spectral representation of the Green’s function [4]. In both cases, lengthy calculations are needed to derive the scattered field expression (see, e.g., pages from 949 to 961 of [3], or 18 to 36 of [4]).

An alternative approach exists, in which the actual scattering problem is obtained from the ideal one by introducing a volumetric perturbation of the medium permittivity [14,15,16,17]. In [14,15,16], the Green’s function formalism is used, and this still leads to the need for rather cumbersome algebraic manipulations to obtain the final scattered field expression. In [17], the use of the reciprocity theorem allows for a much simpler derivation that leads to the usual SPM solution. However, in all cases (see [14,15,16,17]), distribution theory, and, in particular, Dirac delta functions, is used to describe the medium permittivity perturbation. This partly masks the assumptions on which the approach is based: seemingly, it is only needed that roughness height deviation must be small with respect to the wavelength λ, which is, in fact, the only assumption explicitly mentioned in [14,15,16,17]. However, as shown in [18], a deeper analysis reveals that the volumetric perturbative approach also implies the assumption of small height gradient, which is consistent with the assumptions made by the usual boundary perturbation approach of [1,2,3,4].

In this work, we propose a volumetric perturbative method for scattering from slightly rough surfaces that, similar to [17], employs the reciprocity theorem, but, at variance with [17], does not make use of distribution theory and delta functions to describe the medium permittivity perturbation. Instead, as we first suggested, in a different context, in [18], we use a proper approximation of the electric field within the perturbation volume, which we name Internal Field Approximation (IFA). The final result is coincident with the SPM solution, but the proposed procedure turns out to be much more concise than the classical one. In addition, at variance with available volumetric perturbative approaches, in our approach, it is clear why the assumption of small height gradient is necessary. Finally, the proposed method, being based on a volumetric perturbation approach, has the potential of dealing with surface and volume scattering within a unitary framework in the future, which may be important in remote sensing applications (for instance, in model-based polarimetric decomposition techniques [19,20] or in modelling scattering from vegetated soil, snow-covered terrain, and inhomogeneous soils).

We stress that, although the main concept of our approach was briefly presented in a short section of [18], here, for the first time, all details of the derivation are provided, and it is shown that the obtained result coincides with the usual SPM one.

The paper is organised as follows.

Section 2 recalls the general theory of volumetric perturbation and reciprocity for the evaluation of propagation and scattering in inhomogeneous media. In Section 3, this theory is applied to the problem of scattering from a slightly rough interface between two homogeneous media, and the IFA is introduced. In Section 4, the obtained result is briefly discussed, showing that it coincides with the SPM one. Conclusions are finally drawn in Section 5.

2. Volumetric Perturbative Reciprocal Approach

In this section, we introduce the volumetric perturbative reciprocal formulation for a general scattering problem. Throughout the paper, in the field expressions, a time factor exp(jωt) is understood and suppressed. In addition, we use bold characters to indicate vectors.

Let us consider a source current density J(r) that radiates an electromagnetic field E(r), H(r) in an inhomogeneous medium with relative dielectric permittivity distribution $\epsilon \left(\mathbf{r}\right)$. The electromagnetic field then satisfies the Maxwell’s equations:

$$\left\{\begin{array}{c}\nabla \times \mathbf{E}\left(\mathbf{r}\right)=-j\omega {\mu }_0\mathbf{H}\left(\mathbf{r}\right)\hfill \\ \nabla \times \mathbf{H}\left(\mathbf{r}\right)=j\omega {\epsilon }_0\epsilon \left(\mathbf{r}\right)\mathbf{E}\left(\mathbf{r}\right)+\mathbf{J}\left(\mathbf{r}\right)\hfill \end{array}\right.$$

(1)

where ${\mu }_0$ and ${\epsilon }_0$ are the permeability and the permittivity of the vacuum, respectively. Now, let us assume that the considered medium can be seen as an unperturbed medium with relative permittivity ${\epsilon }^{\left(0\right)}\left(\mathbf{r}\right)$ to which a perturbation $\delta \epsilon \left(\mathbf{r}\right)$ is applied, so that $\epsilon \left(\mathbf{r}\right)={\epsilon }^{\left(0\right)}\left(\mathbf{r}\right)+\delta \epsilon \left(\mathbf{r}\right)$, and let us define the unperturbed field ${\mathbf{E}}^{\left(0\right)}\left(\mathbf{r}\right)$, ${\mathbf{H}}^{\left(0\right)}\left(\mathbf{r}\right)$ as the field radiated by $\mathbf{J}\left(\mathbf{r}\right)$ in the unperturbed medium:

$$\left\{\begin{array}{l}\nabla \times {\mathbf{E}}^{\left(0\right)}\left(\mathbf{r}\right)=-j\omega {\mu }_0{\mathbf{H}}^{\left(0\right)}\left(\mathbf{r}\right)\\ \nabla \times {\mathbf{H}}^{\left(0\right)}\left(\mathbf{r}\right)=j\omega {\epsilon }_0{\epsilon }^{\left(0\right)}\left(\mathbf{r}\right){\mathbf{E}}^{\left(0\right)}\left(\mathbf{r}\right)+\mathbf{J}\left(\mathbf{r}\right)\end{array}\right.$$

(2)

By subtracting (2) from (1), we obtain

$$\left\{\begin{array}{c}\nabla \times \delta \mathbf{E}\left(\mathbf{r}\right)=-j\omega {\mu }_0\delta \mathbf{H}\left(\mathbf{r}\right)\hfill \\ \nabla \times \delta \mathbf{H}(\mathbf{r})=j\omega {\epsilon }_0{\epsilon }^{(0)}(\mathbf{r})\delta \mathbf{E}(\mathbf{r})+{\mathbf{J}}_{eq}(\mathbf{r})\hfill \end{array}\right.$$

(3)

where $\delta \mathbf{E}\left(\mathbf{r}\right)=\mathbf{E}\left(\mathbf{r}\right)-{\mathbf{E}}^{\left(0\right)}(\mathbf{r})$, $\delta \mathbf{H}\left(\mathbf{r}\right)=\mathbf{H}\left(\mathbf{r}\right)-{\mathbf{H}}^{\left(0\right)}(\mathbf{r})$ is the electromagnetic field perturbation, and

$${\mathbf{J}}_{eq}\left(\mathbf{r}\right)=j\omega {\epsilon }_0\delta \epsilon \left(\mathbf{r}\right)\mathbf{E}\left(\mathbf{r}\right)$$

(4)

is the equivalent current density, so that the field perturbation can be regarded as radiated by this equivalent source.

In order to compute the perturbed field in a generic point r₀, we define a test source

$$\overline{\mathbf{J}}\left(\mathbf{r}\right)=\widehat{\mathbf{t}}J\delta \left(\mathbf{r}-{\mathbf{r}}_0\right)$$

(5)

where $\widehat{\mathbf{t}}$ is an arbitrarily oriented unit vector, $\delta \left(\mathbf{r}\right)$ is the Dirac delta function, and J = 1 A·m is a unitary constant introduced to ensure dimensional consistency. This test source radiates a field ${\overline{\mathbf{E}}}^{\left(0\right)}(\mathbf{r})$ in the unperturbed medium. Accordingly, ${\mathbf{E}}^{\left(0\right)}$ and ${\overline{\mathbf{E}}}^{\left(0\right)}$ are the fields radiated in the unperturbed medium by the real and test sources, respectively.

By applying the reciprocity theorem, we obtain:

$${\iiint }_V\left\{\delta \mathbf{E}\left(\mathbf{r}\right)·\overline{\mathbf{J}}\left(\mathbf{r}\right)-{\overline{\mathbf{E}}}^{\left(0\right)}\left(\mathbf{r}\right)·{\mathbf{J}}_{eq}\left(\mathbf{r}\right)\right\}d\mathbf{r}=0$$

(6)

where V is a volume that encloses all sources. The use of (4) and (5) in (6) leads to

$$\delta \mathbf{E}\left({\mathbf{r}}_0\right)·\widehat{\mathbf{t}}={\displaystyle \frac{j\omega {\epsilon }_0}{J}}{\iiint }_{V_p}{\overline{\mathbf{E}}}^{\left(0\right)}\left(\mathbf{r}\right)·\delta \epsilon \left(\mathbf{r}\right)\mathbf{E}\left(\mathbf{r}\right)d\mathbf{r}\mathrm{}$$

(7)

where $V_p$ is the perturbation volume, i.e., the volume in which $\delta \epsilon \left(\mathbf{r}\right)$ is different from zero.

If the perturbation is small, i.e., if the perturbation volume and/or $\delta \epsilon \left(\mathbf{r}\right)$ are small, it is reasonable to approximate the actual electric field $\mathbf{E}\left(\mathbf{r}\right)$ within the perturbation volume $V_p$ (i.e., the internal field) with the unperturbed field ${\mathbf{E}}^{\left(0\right)}\left(\mathbf{r}\right)$. This is the distorted Born approximation. Accordingly, it is possible to compute the field perturbation, in the first-order limit, from knowledge of the medium perturbation and of the unperturbed fields produced by the real and test sources. However, as shown in the next section, the distorted Born approximation is not the only possible choice.

3. Scattering from a Slightly Rough Surface

3.1. Medium Perturbation

Let us now specialise the approach presented in Section 2 to the case of scattering from a slightly rough interface between two homogeneous media. In particular, let us consider a rough surface, whose deviation with respect to the plane z = 0 is $\zeta \left(x,y\right)=\zeta \left({\mathbf{r}}_{\perp }\right)$, that separates vacuum (or air) from a possibly lossy homogeneous dielectric medium with relative permittivity ${\epsilon }_r$ (see Figure 1). Accordingly, we can let

$$\epsilon \left(\mathbf{r}\right)=\left\{\begin{array}{c}1 \mathrm{f}\mathrm{o}\mathrm{r} z>\zeta \left({\mathbf{r}}_{\perp }\right)\\ {\epsilon }_r \mathrm{f}\mathrm{o}\mathrm{r} z<\zeta \left({\mathbf{r}}_{\perp }\right)\end{array}\right., {\epsilon }^{\left(0\right)}(\mathbf{r})=\left\{\begin{array}{c}1 \mathrm{f}\mathrm{o}\mathrm{r} z>0\\ {\epsilon }_r \mathrm{f}\mathrm{o}\mathrm{r} z<0\end{array} \right.$$

(8)

so that

$$\delta \epsilon \left(\mathbf{r}\right)=\left\{\begin{array}{l}{\epsilon }_r-1 \mathrm{f}\mathrm{o}\mathrm{r}\mathrm{} \mathbf{r}\in V^{+}\\ 1-{\epsilon }_r \mathrm{f}\mathrm{o}\mathrm{r}\mathrm{} \mathbf{r}\in V^{-}\\ 0 \mathrm{f}\mathrm{o}\mathrm{r} \mathbf{r}\notin V^{+} \mathrm{a}\mathrm{n}\mathrm{d} \mathbf{r}\notin V^{-}\end{array}\right.$$

(9)

where (see Figure 2) $V^{+}=\left\{\left({\mathbf{r}}_{\perp },z\right) \mathrm{such} \mathrm{that} 0<z<{\zeta }^{+}\left({\mathbf{r}}_{\perp }\right)\right\}$ and $V^{-}=\left\{\left({\mathbf{r}}_{\perp },z\right) \mathrm{such} \mathrm{that} {\zeta }^{-}\left({\mathbf{r}}_{\perp }\right)<z<0\right\}$, with

$${\zeta }^{+}\left({\mathbf{r}}_{\perp }\right)=\max \left\{\zeta \left({\mathbf{r}}_{\perp }\right),0\right\}, {\zeta }^{-}\left({\mathbf{r}}_{\perp }\right)=\min \left\{\zeta \left({\mathbf{r}}_{\perp }\right),0\right\}$$

(10)

so that $V_p=V^{+}\cup V^{-}$, i.e., the perturbation volume, is divided into its two parts above ($V^{+}$) and below ($V^{-}$) the plane z = 0.

Figure 1. Geometry of the problem.

Figure 2. Decomposition scheme for the perturbation volume.

3.2. Internal Field Approximation

If surface deviations are small compared to wavelength, then the unperturbed electric field dependence on z in the perturbation volume can be ignored, and in (7), we can let

$${\overline{\mathbf{E}}}^{\left(0\right)}\left(\mathbf{r}\right)\cong \left\{\begin{array}{c}{\overline{\mathbf{E}}}^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{+}\right) \mathrm{f}\mathrm{o}\mathrm{r} \mathbf{r}\in V^{+}\\ {\overline{\mathbf{E}}}^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{-}\right) \mathrm{f}\mathrm{o}\mathrm{r} \mathbf{r}\in V^{-}\end{array}\right.$$

(11)

At this point, we aim to approximate the actual field $\mathbf{E}\left(\mathbf{r}\right)$ in the perturbation volume with the unperturbed field ${\mathbf{E}}^{\left(0\right)}\left(\mathbf{r}\right)$. First of all, we note that this approximation can be accurate only if the surface slope is small, so that the normal to the surface is almost parallel to the z axis and the boundary conditions for the actual and unperturbed fields are approximately the same. In addition, we note that the use of the distorted Born approximation in conjunction with the small surface height deviation assumption would imply to let

$$\mathbf{E}\left(\mathbf{r}\right)\cong \left\{\begin{array}{c}\mathbf{E}\left({\mathbf{r}}_{\perp },0\right)\cong {\mathbf{E}}^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{+}\right) \mathrm{f}\mathrm{o}\mathrm{r} \mathbf{r}\in V^{+}\\ \mathbf{E}\left({\mathbf{r}}_{\perp },0\right)\cong {\mathbf{E}}^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{-}\right) \mathrm{f}\mathrm{o}\mathrm{r} \mathbf{r}\in V^{-}\end{array}\right.$$

(12)

However, $\mathrm{f}\mathrm{o}\mathrm{r} \mathbf{r}\in V^{+}$, $\mathbf{E}\left(\mathbf{r}\right)$ is the actual field within the medium of permittivity ${\epsilon }_r$, and $\mathrm{f}\mathrm{o}\mathrm{r} \mathbf{r}\in V^{-}$, it is the actual field in vacuum, so that it is intuitive that a better approximation is

$$\mathbf{E}\left(\mathbf{r}\right)\cong \left\{\begin{array}{c}\mathbf{E}\left({\mathbf{r}}_{\perp },0\right)\cong {\mathbf{E}}^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{-}\right) \mathrm{f}\mathrm{o}\mathrm{r} \mathbf{r}\in V^{+}\\ \mathbf{E}\left({\mathbf{r}}_{\perp },0\right)\cong {\mathbf{E}}^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{+}\right) \mathrm{f}\mathrm{o}\mathrm{r} \mathbf{r}\in V^{-}\end{array}\right.$$

(13)

which we call IFA. According to (13), in $V^{+}$, the actual electric field, which is the actual field just beneath the actual interface, is replaced by the unperturbed field evaluated just beneath the unperturbed interface. Similarly, in $V^{-}$, the actual electric field, which is the actual field just above the actual interface, is replaced by the unperturbed field evaluated just above the unperturbed interface.

By using (9), (11), and (13) in (7), we obtain

$$\begin{array}{c}\delta \mathbf{E}\left({\mathbf{r}}_0\right)·\widehat{\mathbf{t}}\cong {\displaystyle \frac{j\omega {\epsilon }_0}{J}}\left\{\left({\epsilon }_r-1\right){\iiint }_{V^{+}}{\overline{\mathbf{E}}}^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{+}\right)·{\mathbf{E}}^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{-}\right)d\mathbf{r}+\left(1-{\epsilon }_r\right){\iiint }_{V^{-}}{\overline{\mathbf{E}}}^{\left(0\right)}\left({{\mathbf{r}}_{\perp },0}^{-}\right)·{\mathbf{E}}^{\left(0\right)}\left({{\mathbf{r}}_{\perp },0}^{+}\right)d\mathbf{r}\right\}\hfill \\ ={\displaystyle \frac{j\omega {\epsilon }_0}{J}}\left\{\left({\epsilon }_r-1\right)\iint d{\mathbf{r}}_{\perp }\underset{0}{\overset{{\zeta }^{+}\left({\mathbf{r}}_{\perp }\right)}{\int }}{\overline{\mathbf{E}}}^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{+}\right)·{\mathbf{E}}^{\left(0\right)}\left({{\mathbf{r}}_{\perp },0}^{-}\right)dz+\left(1-{\epsilon }_r\right)\iint d{\mathbf{r}}_{\perp }\underset{{\zeta }^{-}\left({\mathbf{r}}_{\perp }\right)}{\overset{0}{\int }}{\overline{\mathbf{E}}}^{\left(0\right)}\left({{\mathbf{r}}_{\perp },0}^{-}\right)·{\mathbf{E}}^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{+}\right)dz\right\}\hfill \end{array}$$

(14)

The boundary conditions on the plane z = 0 for the unperturbed fields are

$$\left\{\begin{array}{l}{\overline{\mathbf{E}}}_{\perp }^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{-}\right)={\overline{\mathbf{E}}}_{\perp }^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{+}\right)\\ {\overline{E}}_z^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{-}\right)={\displaystyle \frac{1}{{\epsilon }_r}}{\overline{E}}_z^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{+}\right)\end{array}\right.$$

(15)

$$\left\{\begin{array}{l}{\mathbf{E}}_{\perp }^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{-}\right)={\mathbf{E}}_{\perp }^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{+}\right)\\ E_z^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{-}\right)={\displaystyle \frac{1}{{\epsilon }_r}}{E}_z^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{+}\right)\end{array}\mathrm{}\right.$$

(16)

where ${\mathbf{E}}_{\perp }^{\left(0\right)}=E_x^{\left(0\right)}\widehat{\mathbf{x}}+E_y^{\left(0\right)}\widehat{\mathbf{y}}$ and ${\overline{\mathbf{E}}}_{\perp }^{\left(0\right)}={\overline{E}}_x^{\left(0\right)}\widehat{\mathbf{x}}+{\overline{E}}_y^{\left(0\right)}\widehat{\mathbf{y}}$.

By using (15) and (16) in (14), we obtain

$$\begin{array}{c}\delta \mathbf{E}\left({\mathbf{r}}_0\right)·\widehat{\mathbf{t}}\cong {\displaystyle \frac{j\omega {\epsilon }_0}{J}}\left({\epsilon }_r-1\right)\left\{\iint {\int }_0^{{\zeta }^{+}\left({\mathbf{r}}_{\perp }\right)}\left[{\overline{\mathbf{E}}}_{\perp }^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{+}\right)·{\mathbf{E}}_{\perp }^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{+}\right)+{\displaystyle \frac{1}{{\epsilon }_r}}{\overline{E}}_z^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{+}\right)E_z^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{+}\right)\right]dz d{\mathbf{r}}_{\perp }-\iint {\int }_{{\zeta }^{-}\left({\mathbf{r}}_{\perp }\right)}^0\left[{\overline{\mathbf{E}}}_{\perp }^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{+}\right)·{\mathbf{E}}_{\perp }^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{+}\right)+{\displaystyle \frac{1}{{\epsilon }_r}}{\overline{E}}_z^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{+}\right)E_z^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{+}\right)\right]dz d{\mathbf{r}}_{\perp }\right\}\hfill \\ ={\displaystyle \frac{j\omega {\epsilon }_0}{J}}\left({\epsilon }_r-1\right)\iint {\int }_0^{\zeta \left({\mathbf{r}}_{\perp }\right)}\left[{\overline{\mathbf{E}}}_{\perp }^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{+}\right)·{\mathbf{E}}_{\perp }^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{+}\right)+{\displaystyle \frac{1}{{\epsilon }_r}}{\overline{E}}_z^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{+}\right)E_z^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{+}\right)\right]dz d{\mathbf{r}}_{\perp }\hfill \\ ={\displaystyle \frac{j\omega {\epsilon }_0}{J}}\left({\epsilon }_r-1\right)\iint \left[{\overline{\mathbf{E}}}_{\perp }^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{+}\right)·{\mathbf{E}}_{\perp }^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{+}\right)+{\displaystyle \frac{1}{{\epsilon }_r}}{\overline{E}}_z^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{+}\right)E_z^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{+}\right)\right]\zeta \left({\mathbf{r}}_{\perp }\right)d{\mathbf{r}}_{\perp }\hfill \end{array}$$

(17)

3.3. Unperturbed Fields

If we assume that the field source (i.e., the transmitting antenna) is placed in the upper half-space and it is in the far zone with respect to the surface area that it illuminates, then the unperturbed field E⁽⁰⁾(r) is the field obtained when an arbitrarily polarised (locally) plane wave ${\mathbf{E}}_i(\mathbf{r})$ impinges over the flat interface:

$${\mathbf{E}}_i\left(\mathbf{r}\right)=e^{-j{\mathbf{k}}_{i\perp }·{\mathbf{r}}_{\perp }}{e}^{-j{k}_{iz}z}\left(E_{ih}{\widehat{\mathbf{h}}}_i+E_{iv}{\widehat{\mathbf{v}}}_i\right)$$

(18)

where ${\mathbf{k}}_i={\mathbf{k}}_{i\perp }+k_{iz}\widehat{\mathbf{z}}$ is the incident wavenumber vector,

$$\begin{array}{c}{\mathbf{k}}_{i\perp }=k\sin {\vartheta }_i\cos {\phi }_i\widehat{\mathbf{x}}+k\sin {\vartheta }_i\sin {\phi }_i\widehat{\mathbf{y}}\hfill \\ k_{iz}=-k\cos {\vartheta }_i \hfill \end{array}$$

(19)

$k=\omega \sqrt{{\epsilon }_0{\mu }_0}=2\pi /\lambda$ is the propagation constant in vacuum, ${\vartheta }_i$ and ${\phi }_i$ are the incidence polar and azimuth angles (see Figure 1), and

$${{{\widehat{\mathbf{h}}}_i=\widehat{\mathbf{k}}}_{i\perp }\times \widehat{\mathbf{z}},\hspace{1em}\hspace{1em}\hspace{1em}\widehat{\mathbf{v}}}_i={\widehat{\mathbf{h}}}_i\times {\widehat{\mathbf{k}}}_i$$

(20)

Accordingly, E⁽⁰⁾(r) above the unperturbed illuminated interface, in its vicinity, is the sum of the incident and reflected plane waves. In particular, on the plane z = 0, we have

$$\begin{array}{c}{\mathbf{E}}_{\perp }^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{+}\right)=e^{-j{\mathbf{k}}_{i\perp }·{\mathbf{r}}_{\perp }}\left\{E_{ih}\left[1+R_h\left({\vartheta }_i\right)\right]{\widehat{\mathbf{h}}}_{i }+E_{iv}\cos {\vartheta }_i\left[1-R_v\left({\vartheta }_i\right)\right]{\widehat{\mathbf{k}}}_{i\perp }\right\},\hfill \\ E_z^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{+}\right)=e^{-j{\mathbf{k}}_{i\perp }·{\mathbf{r}}_{\perp }}{E}_{iv}\mathrm{}\sin {\vartheta }_i\left[1+R_v\left({\vartheta }_i\right)\right]\hfill \end{array}$$

(21)

where $R_h\left(\vartheta \right)$ and $R_v\left(\vartheta \right)$ are the usual Fresnel reflection coefficients for horizontal and vertical polarisation, respectively:

$$\begin{array}{l}{R}_h\left(\vartheta \right)={\displaystyle \frac{\cos \vartheta -\sqrt{{\epsilon }_r-{\sin }^2\vartheta }}{\cos \vartheta +\sqrt{{\epsilon }_r-{\sin }^2\vartheta }}}\\ R_v\left(\vartheta \right)={\displaystyle \frac{{\epsilon }_r\cos \vartheta -\sqrt{{\epsilon }_r-{\sin }^2\vartheta }}{{\epsilon }_r\cos \vartheta +\sqrt{{\epsilon }_r-{\sin }^2\vartheta }}}\end{array}$$

(22)

Similarly, if we assume that the test source defined in (5) is placed in the upper half-space and it is in the far zone with respect to the illuminated surface area, then the unperturbed field ${\overline{\mathbf{E}}}^{(0)}(\mathbf{r})$ is the field obtained when an arbitrarily polarised (locally) plane wave ${\overline{\mathbf{E}}}_i(\mathbf{r})$ with wavenumber vector ${-\mathbf{k}}_s$ impinges over the flat interface:

$${\overline{\mathbf{E}}}_i\left(\mathbf{r}\right)=-j\omega {\mu }_0{e}^{j{\mathbf{k}}_{s\perp }·{\mathbf{r}}_{\perp }}{e}^{j{k}_{sz}z}{\displaystyle \frac{e^{-jk{r}_s}}{4\pi r_s}}J\widehat{\mathbf{t}}$$

(23)

where ${\mathbf{k}}_s={\mathbf{k}}_{s\perp }+k_{sz}\widehat{\mathbf{z}}$ is the scattering wavenumber vector,

$$\begin{array}{c}{\mathbf{k}}_{s\perp }=k\sin {\vartheta }_s\cos {\phi }_s\widehat{\mathbf{x}}+k\sin {\vartheta }_s\sin {\phi }_s\widehat{\mathbf{y}}\hfill \\ k_{sz}=k\cos {\vartheta }_s \hfill \end{array},$$

(24)

${\vartheta }_s$ and ${\phi }_s$ are the scattering polar and azimuth angles (see Figure 1), $\widehat{\mathbf{t}}$ may coincide with either ${\widehat{\mathbf{h}}}_s$ or ${\widehat{\mathbf{v}}}_s$,

$${\widehat{\mathbf{h}}}_s={\widehat{\mathbf{k}}}_{s\perp }\times \widehat{\mathbf{z}},\hspace{1em}\hspace{1em}\hspace{1em}{\widehat{\mathbf{v}}}_s={\widehat{\mathbf{h}}}_s\times {\widehat{\mathbf{k}}}_s,$$

(25)

and $r_s=\left|{\mathbf{r}}_0\right|$ is the distance of the test source (i.e., of the point at which we are computing the field perturbation) from the origin, so that ${\mathbf{r}}_0\equiv \left\{r_s,{\vartheta }_s, {\phi }_s\right\}$ in a spherical coordinate system.

With the mentioned assumptions, ${\overline{\mathbf{E}}}^{\left(0\right)}(\mathbf{r})$ above the unperturbed illuminated interface, in its vicinity, is the sum of the incident and reflected plane waves. In particular, on the plane z = 0, for $\widehat{\mathbf{t}}={\widehat{\mathbf{h}}}_s$, we have

$$\begin{array}{c}{\overline{\mathbf{E}}}_{\perp }^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{+}\right)=-j\omega {\mu }_0{\displaystyle \frac{e^{-jk{r}_s}}{4\pi r_s}}{e}^{j{\mathbf{k}}_{s\perp }·{\mathbf{r}}_{\perp }}J\left[1+R_h\left({\vartheta }_s\right)\right]{\widehat{\mathbf{h}}}_s\hfill \\ {\overline{E}}_z^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{+}\right)=0 \hfill \end{array}$$

(26)

and for $\widehat{\mathbf{t}}={\widehat{\mathbf{v}}}_s$, we have

$$\begin{array}{l}{\overline{\mathbf{E}}}_{\perp }^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{+}\right)=j\omega {\mu }_0{\displaystyle \frac{e^{-jk{r}_s}}{4\pi r_s}}{e}^{j{\mathbf{k}}_{s\perp }·{\mathbf{r}}_{\perp }}J\cos {\vartheta }_s\left[1-R_v\left({\vartheta }_s\right)\right]{\widehat{\mathbf{k}}}_{s\perp }\\ {\overline{E}}_z^{\left(0\right)}\left({\mathbf{r}}_{\perp },0^{+}\right)=-j\omega {\mu }_0{\displaystyle \frac{e^{-jk{r}_s}}{4\pi r_s}}{e}^{j{\mathbf{k}}_{s\perp }·{\mathbf{r}}_{\perp }}J\sin {\vartheta }_s\left[1+R_v\left({\vartheta }_s\right)\right] \end{array}$$

(27)

3.4. Scattered Fields

The total scattered field is the sum of the field reflected by the flat unperturbed surface in the specular direction $({\vartheta }_s={\vartheta }_i, {\phi }_s={\phi }_i)$ and the perturbation field, which is non-null for all scattering directions. The latter can be computed by substituting (21), (26) and (27) in (17):

$$\begin{gathered} E_{sh}\left(r_s,{\vartheta }_s, {\phi }_s\right)=\delta \mathbf{E}\left({\mathbf{r}}_0\right)·{\widehat{\mathbf{h}}}_{\mathit{s}}\cong {\displaystyle \frac{k^2{e}^{-jk{r}_s}}{4\pi r_s}}\left({\epsilon }_r-1\right)·\{E_{ih}\left[1+R_h\left({\vartheta }_i\right)\right]\left[1+R_h\left({\vartheta }_s\right)\right]\cos \left({\phi }_s-{\phi }_i\right) \\ +E_{iv}\cos {\vartheta }_i\left[1-R_v\left({\vartheta }_i\right)\right]\left[1+R_h\left({\vartheta }_s\right)\right]\sin \left({\phi }_s-{\phi }_i\right)\} \tilde{\zeta }\left({\mathbf{k}}_{i\perp }-{\mathbf{k}}_{s\perp }\right) \end{gathered}$$

(28)

$$\begin{gathered} E_{sv}\left(r_s,{\vartheta }_s, {\phi }_s\right)=\delta \mathbf{E}\left({\mathbf{r}}_0\right)·{\widehat{\mathbf{v}}}_{\mathit{s}}\cong {\displaystyle \frac{k^2{e}^{-jk{r}_s}}{4\pi r_s}}\left({\epsilon }_r-1\right)·\{E_{ih}\left[1+R_h\left({\vartheta }_i\right)\right]\cos {\vartheta }_s\left[1-R_v\left({\vartheta }_s\right)\right] \\ \sin \left({\phi }_s-{\phi }_i\right)-E_{iv}\cos {\vartheta }_i\left[1-R_v\left({\vartheta }_i\right)\right]\cos {\vartheta }_s\left[1-R_v\left({\vartheta }_s\right)\right]\cos \left({\phi }_s-{\phi }_i\right) \\ +{\displaystyle \frac{E_{iv}}{{\epsilon }_r}}\sin {\vartheta }_i\left[1+R_v\left({\vartheta }_i\right)\right]\sin {\vartheta }_s\left[1+R_v\left({\vartheta }_s\right)\right]\}\tilde{\zeta }\left({\mathbf{k}}_{i\perp }-{\mathbf{k}}_{s\perp }\right) \end{gathered}$$

(29)

where

$$\tilde{\zeta }\left({\mathbf{k}}_{i\perp }-{\mathbf{k}}_{s\perp }\right)=\iint e^{-j\left({\mathbf{k}}_{i\perp }-{\mathbf{k}}_{s\perp }\right)·{\mathbf{r}}_{\perp }}\zeta \left({\mathbf{r}}_{\perp }\right)d{\mathbf{r}}_{\perp }$$

(30)

is the Fourier transform (FT) of the height deviation $\zeta \left({\mathbf{r}}_{\perp }\right)$, and we have used:

$$\begin{array}{c}{\widehat{\mathbf{h}}}_i·{\widehat{\mathbf{h}}}_s={\widehat{\mathbf{k}}}_{i\perp }·{\widehat{\mathbf{k}}}_{s\perp }=\mathbf{}\cos \left({\phi }_s-{\phi }_i\right) \\ {\widehat{\mathbf{k}}}_{i\perp }·{\widehat{\mathbf{h}}}_s={\widehat{\mathbf{k}}}_{i\perp }·{\widehat{\mathbf{k}}}_{s\perp }\times \widehat{\mathbf{z}}=\widehat{\mathbf{z}}·{\widehat{\mathbf{k}}}_{i\perp }\times {\widehat{\mathbf{k}}}_{s\perp }=\sin \left({\phi }_s-{\phi }_i\right) \\ {\widehat{\mathbf{k}}}_{s\perp }·{\widehat{\mathbf{h}}}_i={\widehat{\mathbf{k}}}_{s\perp }·{\widehat{\mathbf{k}}}_{i\perp }\times \widehat{\mathbf{z}}=\widehat{\mathbf{z}}·{\widehat{\mathbf{k}}}_{s\perp }\times {\widehat{\mathbf{k}}}_{i\perp }=\sin \left({\phi }_i-{\phi }_s\right)\end{array}$$

(31)

Equations (28) and (29) can be expressed in a more compact form as

$$E_{sq}={\displaystyle \frac{k^2{e}^{-jk{r}_s}}{4\pi r_s}}\left[{\alpha }_{hq}\left({\vartheta }_i,{\phi }_i;{\vartheta }_s,{\phi }_s\right)E_{ih}+{\alpha }_{vq}\left({\vartheta }_i,{\phi }_i;{\vartheta }_s,{\phi }_s\right)E_{iv}\right]\tilde{\zeta }\left(u_x,u_y\right)$$

(32)

where the subscript q can stand for h or v,

$$\begin{array}{c}{u}_x{=k}_{ix}-k_{sx}=k(\sin {\vartheta }_i\cos {\phi }_i-\sin {\vartheta }_s\cos {\phi }_s)\\ u_y{=k}_{iy}-k_{sy}=k(\sin {\vartheta }_i\sin {\phi }_i-\sin {\vartheta }_s\sin {\phi }_s)\end{array}$$

(33)

and

$${\alpha }_{hh}\left({\vartheta }_i,{\phi }_i;{\vartheta }_s,{\phi }_s\right)=\left({\epsilon }_r-1\right)\left[1+R_h\left({\vartheta }_i\right)\right]\left[1+R_h\left({\vartheta }_s\right)\right]\cos \left({\phi }_s-{\phi }_i\right)$$

(34)

$${\alpha }_{hv}\left({\vartheta }_i,{\phi }_i;{\vartheta }_s,{\phi }_s\right)=\left({\epsilon }_r-1\right) \cos {\vartheta }_s\left[1+R_h\left({\vartheta }_i\right)\right]\left[1-R_v\left({\vartheta }_s\right)\right]\sin \left({\phi }_s-{\phi }_i\right)$$

(35)

$${\alpha }_{vh}\left({\vartheta }_i,{\phi }_i;{\vartheta }_s,{\phi }_s\right)=\left({\epsilon }_r-1\right)\cos {\vartheta }_i\left[1-R_v\left({\vartheta }_i\right)\right]\left[1+R_h\left({\vartheta }_s\right)\right]\sin \left({\phi }_s-{\phi }_i\right)$$

(36)

$$\begin{gathered} {\alpha }_{vv}\left({\vartheta }_i,{\phi }_i;{\vartheta }_s,{\phi }_s\right)=-\left({\epsilon }_r-1\right)\{\cos {\vartheta }_i\cos {\vartheta }_s\left[1-R_v\left({\vartheta }_i\right)\right]\left[1-R_v\left({\vartheta }_s\right)\right]\cos \left({\phi }_s-{\phi }_i\right)- \\ {\displaystyle \frac{1}{{\epsilon }_r}}\sin {\vartheta }_i\sin {\vartheta }_s\left[1+R_v\left({\vartheta }_i\right)\right]\left[1+R_v\left({\vartheta }_s\right)\right]\} \end{gathered}$$

(37)

3.5. Normalised Radar Cross-Section

Assuming that $\zeta \left({\mathbf{r}}_{\perp }\right)$ is a zero-mean random process, the surface NRCS can be computed from (32) as

$${\sigma }_{pq}^0={\displaystyle \frac{⟨{\left|E_{sq}\right|}^2⟩4\pi r^2}{A{\left|E_{ip}\right|}^2}}={\displaystyle \frac{k^4}{4\pi }}{\left|{\alpha }_{pq}\left({\vartheta }_i,{\phi }_i;{\vartheta }_s,{\phi }_s\right)\right|}^{2}W\left(u_x,u_y\right)$$

(38)

where $A$ is the illuminated area, the symbol $⟨·⟩$ stands for the statistical mean, and

$$W\left(u_x,u_y\right)={\displaystyle \frac{1}{A}}⟨{\left|\tilde{\zeta }\left(u_x,u_y\right)\right|}^2⟩$$

(39)

is the roughness power spectral density (PSD).

4. Discussion

The analysis in the previous section shows that the field scattered by a slightly rough surface is proportional to the FT of the surface height deviation, and the proportionality coefficient can be easily obtained as the scalar product of the fields produced by the real and test sources over the flat surface. Each of these two fields is simply the interference of one incident and one reflected plane wave.

It can be shown that the obtained result in (32)–(39) is coincident with the usual SPM solution. In fact, from (22), we obtain

$$\begin{array}{l}1+R_h\left(\vartheta \right)={\displaystyle \frac{2\cos \vartheta }{\cos \vartheta +\sqrt{{\epsilon }_r-{\sin }^2\vartheta }}}\\ 1+R_v\left(\vartheta \right)={\displaystyle \frac{2{\epsilon }_r\cos \vartheta }{{\epsilon }_r\cos \vartheta +\sqrt{{\epsilon }_r-{\sin }^2\vartheta }}}\\ 1-R_v\left(\vartheta \right)={\displaystyle \frac{2\sqrt{{\epsilon }_r-{\sin }^2\vartheta }}{{\epsilon }_r\cos \vartheta +\sqrt{{\epsilon }_r-{\sin }^2\vartheta }}}\end{array}$$

(40)

The use of (40) in (34)–(37) leads to

$${\alpha }_{pq}=4\cos {\vartheta }_i\cos {\vartheta }_s{B}_{pq}$$

(41)

where $B_{pq}$ are the usual Bragg coefficients [3,7]:

$$B_{hh}={\displaystyle \frac{\left({\epsilon }_r-1\right)\cos \left({\phi }_s-{\phi }_i\right)}{\left(\cos {\vartheta }_i+\sqrt{{\epsilon }_r-{\sin }^2{\vartheta }_i}\right)\left(\cos {\vartheta }_s+\sqrt{{\epsilon }_r-{\sin }^2{\vartheta }_s}\right)}}$$

(42)

$$B_{hv}={\displaystyle \frac{\left({\epsilon }_r-1\right)\sqrt{{\epsilon }_r-{\sin }^2{\vartheta }_s}\sin \left({\phi }_s-{\phi }_i\right)}{\left(\cos {\vartheta }_i+\sqrt{{\epsilon }_r-{\sin }^2{\vartheta }_i}\right)\left({\epsilon }_r\cos {\vartheta }_s+\sqrt{{\epsilon }_r-{\sin }^2{\vartheta }_s}\right)}}$$

(43)

$$B_{vh}={\displaystyle \frac{\left({\epsilon }_r-1\right)\sqrt{{\epsilon }_r-{\sin }^2{\vartheta }_i}\sin \left({\phi }_s-{\phi }_i\right)}{\left(\cos {\vartheta }_s+\sqrt{{\epsilon }_r-{\sin }^2{\vartheta }_s}\right)\left({\epsilon }_r\cos {\vartheta }_i+\sqrt{{\epsilon }_r-{\sin }^2{\vartheta }_i}\right)}}$$

(44)

$$B_{vv}=-{\displaystyle \frac{\left({\epsilon }_r-1\right)\left\{\sqrt{{\epsilon }_r-{\sin }^2{\vartheta }_i}\sqrt{{\epsilon }_r-{\sin }^2{\vartheta }_s}\cos \left({\phi }_s-{\phi }_i\right)-{\epsilon }_r\sin {\vartheta }_i\sin {\vartheta }_s\right\}}{\left({\epsilon }_r\cos {\vartheta }_i+\sqrt{{\epsilon }_r-{\sin }^2{\vartheta }_i}\right)\left({\epsilon }_r\cos {\vartheta }_s+\sqrt{{\epsilon }_r-{\sin }^2{\vartheta }_s}\right)}}$$

(45)

Accordingly, we obtain the well-known usual SPM expression of the NRCS [3,7]:

$${\sigma }_{pq}^0={\displaystyle \frac{4}{\pi }}{k}^4{\cos }^2{\vartheta }_i{\cos }^2{\vartheta }_s· {\left|B_{pq}\left({\vartheta }_i,{\phi }_i;{\vartheta }_s,{\phi }_s\right)\right|}^{2}W\left(u_x,u_y\right)\mathrm{}\mathrm{}$$

(46)

It is finally worth noting that, although the case of perfectly conducting scattering surface cannot be directly treated by our approach, the correct SPM result is achieved anyway in this case by first considering a finite value of ${\mathsf{\epsilon }}_{\mathrm{r}}$, and then taking the limit for ${\mathsf{\epsilon }}_{\mathrm{r}}\to \mathrm{\infty }$ of the obtained result.

5. Conclusions

We have presented a new perturbative method for the evaluation of scattering from a slightly rough surface, based on the volumetric perturbation approach, on the reciprocity theorem and on the IFA. The proposed method leads to the usual SPM result in a surprisingly easy way. In particular, we show that the Bragg coefficients are simply the scalar product of the electric fields produced by the real antenna and by the test source (i.e., an elementary antenna placed in the point where the field must be computed) in the presence of a flat surface, on the flat surface itself. Therefore, each of the two fields can be straightforwardly computed as the sum of one incident and one reflected plane wave.

The comparison of the proposed field calculation procedure with those of [3,4] clearly shows the advantage of our procedure in terms of brevity and simplicity. We also highlight that, according to our formulation, the solution for the scattering through the interface can be similarly derived by using the expression for the unperturbed field (${\overline{\mathbf{E}}}^{\left(0\right)}$) obtained when the test source is placed in the lower half-space.

We underline that, since the scattered field expression coincides with the SPM one, the validity limits of the proposed method are identical to those of SPM. However, the proposed method also clarifies why this happens (which is not clear by using the other volumetric perturbation approaches of [14,15,16,17]). In particular, the approximation based on the replacement of the actual field within the perturbation volume with a uniform one is likely to perform poorly for dielectric surfaces featuring steep slopes, sharp edges, or corners. Therefore, not only is a small height deviation approximation needed, but also a small slope one, coherently with the usual SPM validity limits.