# Inversion of computer generated and experimental in-plane, co-polarized
light scattering data in p and s polarization to obtain the normalized
surface-height autocorrelation function and the rms-roughness of a two-
dimensional randomly rough metal surface

I. Simonsen<EMAIL_ADDRESS>Surface du Verre et Interfaces, UMR 125
CNRS/Saint-Gobain, F-93303 Aubervilliers, France  Department of Physics, NTNU
– Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
J.B. Kryvi  Department of Physics, NTNU – Norwegian University of Science and
Technology, NO-7491 Trondheim, Norway  A.A. Maradudin<EMAIL_ADDRESS>Department of Physics and Astronomy, University of California, Irvine CA
92697, U.S.A.

###### Abstract

An approach is presented for the inversion of experimental in-plane, co-
polarized light scattering data in p and s polarization to obtain the
normalized surface-height autocorrelation function and the rms-roughness of a
two-dimensional randomly rough metal surface. The approach is based on an
expression, obtained on the basis of second-order phase perturbation theory,
for the contribution to the mean differential reflection coefficient from the
light scattered diffusely by the rough surface. The inversion scheme is
validated by using several sets of computer generated scattering data for
rough silver surfaces defined by Gaussian surface height correlation
functions. The reconstructions obtained by this approach are found to be
rather accurate for weakly rough surfaces illuminated by p- and s-polarized
incident light; this is also true in cases where the contributions to the
input data from multiple scattering of surface plasmon polaritons is not
insignificant. Finally, the inversion scheme is applied to experimental
scattering data obtained for characterized two-dimensional randomly rough gold
surfaces, and the results obtained in this way, compare favorably to what is
obtained by directly analyzing the surface morphology. Such results testify to
the attractiveness of the computationally efficient inversion scheme that we
propose.

randomly rough surface, rough surface scattering, inverse scattering problem,
surface-height autocorrelation function, phase perturbation theory

###### pacs:

##  I Introduction

Randomly rough surfaces are abundant in natural and man-made systems alike
Ogilvy (1987). There is therefore a strong desire to characterize such
surfaces and, in particular, to determine their statistical properties.
Statistical information on a randomly rough surface is contained in its
distribution of heights and in its normalized surface-height autocorrelation
function Ogilvy (1987); Stout K.J. (2000); Simonsen (2010). These two
functions are often parametrized by the root-mean-square (rms) height and the
transverse correlation length of the surface roughness. In principle, the most
direct and straightforward approach to the statistical properties of a
randomly rough surface, is first to obtain a map of its surface morphology
over a grid of points in the mean plane and then to calculate the statistical
properties of interest on the basis of these data. For instance, a map of the
surface height can be obtained by atomic force microscopy and/or by contact
profilometry Kaupp (2006); Thomas (1999) two of many possible scanning probe
microscopy techniques. What is measured by such techniques is the surface
morphology convoluted by the tip of the probe. Moreover, such techniques are
typically contact methods which make them time-consuming to perform and
typically put constraints on the size of the spatial region in the mean plane
of the surface that can be measured with sufficient spatial resolution.

Indirect methods represent alternative approaches to the determination of the
statistical properties of a randomly rough surface for which maps of the
surface morphologies are not required. Such methods are typically non-contact
methods. A wave approach of this kind is based on inverse scattering theory.
Here one first measures the angular dependence of the intensity of the light
that has been scattered diffusely into the far field by the rough surface when
it is illuminated by a plane incident wave and next use these measured data to
determine the statistical properties of the rough surface. There are several
advantages of this approach. The technique is non-invasive, fast, relatively
cheap to implement, and typically can cover a large region of the mean plane
of the surface which is important when aiming to determine the statistical
properties of the rough surface. For two dimensional randomly rough surfaces,
this problem was initially studied by Chandley Chandley (1976), and later by
Marx and Vorburger Marx and Vorburger (1990). Chandley assumed that the
intensity of the light scattered by the randomly rough surface could be
modeled by scalar diffraction theory in combination with the use of a thin
phase screen model. In this way and due to the form of the expressions for the
scattered intensity, the angular dependence of the mean scattered intensity
could be readily inverted for the statistical properties of the surface by the
use of a two-dimensional Fourier transform. Furthermore, Chandley also made
the additional assumption that the autocorrelation function of the wave front
is the same as the surface-heigh autocorrelation of the surface. This is only
a good approximation for sufficiently small polar angles of incidence and
scattering and for surfaces which are not too rough. An additional
disadvantage of this approach is that one needs to measure the full angular
distribution of the scattered light, something that is both time-consuming and
requires rather specialized optical equipment (photo goniometers).

Almost one and a half decades after the initial work of Chandley, Marx and
Vorburger Marx and Vorburger (1990) assumed the Kirchhoff approximation and
used it to calculate the mean scattered intensity resulting from a plane
incident scalar wave that is scattered from a perfectly conducting surface. By
assuming a particular form of the normalized surface-height autocorrelation
function, such a model was used in a least-square procedure to determine the
transverse correlation length and the rms-roughness of the surface from
experimental scattering data. This approach had the advantage over the
approach of Chandley that the full angular intensity distribution of the
scattered light was not required; for instance, it was sufficient to only
consider the intensity distribution in the plane of incidence (in-plane
scattering).

Two decades after the initial publication by Chandley Chandley (1976), a
generalization of part of his work was conducted by Zhao and colleagues Zhao
_et al._ (1996). Like Chandley, these authors used a Fourier technique for the
inversion of the scattering data, while the scattering model that they used
was based on the Kirchhoff approximation, unlike Chandley, but similar to what
was assumed by Marx and Vorburger Marx and Vorburger (1990). The advantage of
the approach of Zhao et al. is that it is computationally efficient, due to
the use of the Fourier transform, and that it does not require the full
angular intensity distribution of the scattered light as an input. Moreover,
this approach is not limited to small polar angles of incidence and scattering
and it is not restricted to surfaces for which the rms-roughness over the
wavelength of the incident light is much less than unity (as was the case for
the approach of Chandley).

Assuming scalar wave theory and the Kirchhoff approximation, Zamani et al.
Zamani  _et al._ (2016) recently derived within certain additional
approximations an analytic relation between the surface-height autocorrelation
function and the intensity of the scalar wave that has been scattered
diffusely by the rough surface; also see Refs. Zamani  _et al._ , 2012a, b.
This expression these authors were able to invert analytically for the
surface-height autocorrelation function of rough surface. It should be
remarked that previously a similar analytic inversion approach had been
introduced by Zhao et al. Zhao  _et al._ (1996). However, the expressions that
the analytic inversions were based on in Refs. Zhao  _et al._ , 1996 and
Zamani  _et al._ , 2016 are different, and so are the results for the
correlation functions obtained by inversion. These differences are due to the
two group of authors using different assumptions and mathematical
approximations in deriving these expressions.

Only recently have inversion schemes been applied to experimental scattering
data. For instance, Zamani et al. applied their approach to experimental
scattering data obtained when light is scattered from rough silicon surfaces
Zamani  _et al._ (2016). They compared the results for the surface-height
autocorrelation functions obtained by inversion to what was obtained directly
from the surface morphologies measured by microscopy. For the surfaces that
they considered, good agreement is reported between the two classes of
results. In their study, Zamani et al. argue that their approach is more
accurate than previous approaches, and this in particular applies to the
approach of Zhao et al. Zhao  _et al._ (1996).

All the inversion schemes reported in Refs. Chandley, 1976; Marx and
Vorburger, 1990; Zhao  _et al._ , 1996; Zamani  _et al._ , 2016 assume that
scalar wave theory is adequate to model the scattering of light from two-
dimensional randomly rough dielectric or metal surfaces. It is well known that
light is vector waves, and that the polarization states of the incident and
scattered light have to be taken into account in order to obtain a
quantitatively accurate description of the intensity of the light that is
scattered from a randomly rough surface Simonsen (2010). When the transverse
correlation length of the rough surface is significantly larger than the
wavelength of the incident light, one typically finds little difference
between the in-plane angular intensity distributions of the p-to-p and the
s-to-s scattered light [for instance, see Figs. 2(a) and 4(a)]; in such cases
a scalar wave theory may (or, may not) be sufficient. However, when the
lateral correlation length is smaller than the wavelength of the incident
light, the angular intensity distributions for in-plane p- and s-polarized
light are typically rather different [for an example, see Figs. 8(a) and
9(a)]; in these cases, scalar wave theory is inadequate and will produce
incorrect results. We also note that when the lateral correlation length of
the surface is smaller than the wavelength of the incident light, the
Kirchhoff approximation is not expected to be valid; therefore, a
reconstruction approach based on it is not expected to produce reliable
results.

Recently a vector theory was used successfully for the reconstruction of the
statistical properties of a randomly rough dielectric surface performed on the
basis of the in-plane angular dependence of s-to-s scattering data obtained
for the surface Simonsen  _et al._ (2016). The vector theory used here to
model the light that is scattered from the rough surface is second order phase
perturbation theory Shen and Maradudin (1980); Navarrete Alcalá  _et al._
(2009). The reconstructions that Simonsen et al. Simonsen  _et al._ (2016)
performed using this vector theory were otherwise preformed in a manner that
resembles how they were performed by Marx and Vorburger Marx and Vorburger
(1990). By assuming a particular form of the normalized surface-height
autocorrelation function, a model for the scattering of light derived within
phase perturbation theory, and the input scattering data, a least-square
minimization procedure was used to determine the transverse correlation length
and the rms-roughness of the surface. In this way, successful and accurate
reconstructions were demonstrated for dielectric surfaces in s polarization
for a set of different input and trial correlation functions. It ought to be
remarked that a similar approach based on phase perturbation theory and
applied to the reconstruction of surface parameters based on p-to-p scattering
data for dielectric surfaces could in principle have been performed. In
practice, however, such an approach turned out not to be very accurate due to
the Brewster angle that exists in the scattering from a planar dielectric
surface in p-polarization Kryvi, Simonsen, and Maradudin (2016). Since a
Brewster angle does not exist for the scattering of s-polarized light from a
planar dielectric surface, a similar problem does not exist for this
polarization, something that is also shown explicitly by the results reported
in Ref. Simonsen  _et al._ , 2016.

The purpose of this paper, is to use second order phase perturbation theory —
a vector theory for light — to develop the needed formalism and to use it to
perform reconstructions of the statistical properties of randomly rough
_metal_ surfaces based on the polarization dependent in-plane angular
intensity distributions. This will be possible for both p- and s-polarized
light, since for a metal surface a Brewster angle does not exist for which the
scattered intensity vanishes. First, the formalism that we propose will be
applied to the reconstruction based on computer generated scattering data for
both linear polarizations and for a set of roughness parameters. Next, the
reconstruction will be performed on the basis of experimental scattering data.
The intensity of the light scattered from weakly and moderately rough metal
surfaces, unlike light scattered from rough dielectric surfaces, can obtain,
depending on the roughness parameters of the surface, significant
contributions from the multiple scattering of surface plasmon polaritons Maier
(2007); Simonsen (2010). An important issue that we address in this work is
how multiple scattering processes, contributing to the input scattering data,
influence the quality of the height autocorrelation functions and the surface
parameters that are reconstructed by the approach that we propose. It is
demonstrated that even when multiple scattering contributions are non-
neglectable, reconstruction based on the proposed approach can produce fairly
accurate results.

The remaining part of the paper is organized in the following way: Section II
presents the scattering system that we will be concerned with, followed by
some useful elements of scattering theory on which the subsequent discussion
relies [Sec. III]. The inversion scheme that we will use for the
reconstruction of the surface roughness parameters as well as the surface-
height autocorrelation function is presented in Sec. IV. Section V presents
and discusses the results that can be obtained by applying the proposed
reconstruction procedure to a set of different scattering geometries and
computer generated and experimental scattering data. The conclusions and
outlook that can be drawn from this study is presented in Sec. VI. The paper
ends with an Appendix presenting a detailed derivation of the expressions,
central to the present work, for the first few moments of the scattering
matrix for p-to-p scattering obtained on the basis of phase perturbation
theory.

##  II The System Studied

Figure 1: Schematics of the scattering geometry considered in this work.

The scattering system that we consider in this work is depicted in Fig. 1. It
consists of vacuum in the region $x_{3}>\zeta(\mathbf{{x}}_{\parallel})$, and
a metal, characterized by a frequency dependent dielectric function
$\varepsilon(\omega)$ in the region $x_{3}<\zeta(\mathbf{{x}}_{\parallel})$.
At the angular frequency $\omega$ of the incident light, this dielectric
function has a negative real part while its imaginary part is positive or
zero. The vector $\mathbf{{x}}_{\parallel}=(x_{1},x_{2},0)$ is a position
vector in the plane $x_{3}=0$. The surface profile function
$\zeta(\mathbf{{x}}_{\parallel})$ is assumed to be a single-valued function of
$\mathbf{{x}}_{\parallel}$ that is differentiable with respect to $x_{1}$ and
$x_{2}$. It is also assumed to constitute a stationary, zero-mean, isotropic,
Gaussian random process defined by

$\displaystyle\langle\zeta({\bf x}_{\|})\rangle$ $\displaystyle=0$ (1a)
$\displaystyle\langle\zeta({\bf x}_{\|})\zeta({\bf
x}\,^{\prime}\\!\\!_{\|})\rangle$ $\displaystyle=\delta^{2}W(|{\bf
x}_{\|}-{\bf x}\,^{\prime}\\!\\!_{\|}|),$ (1b)

where the angle brackets denote an average over the ensemble of realizations
of $\zeta(\mathbf{{x}}_{\parallel})$, $\delta=\left<\zeta^{2}({\bf
x}_{\|})\right>^{1/2}$ is the root-mean-square (rms) height of the surface,
and $W(|{\bf x}_{\|}|)$ denotes the normalized surface-height autocorrelation
function, with the property that $W(0)=1$.

To aid in the later discussion, it will be convenient to introduce the
following Fourier integral representation of the surface profile function

$\displaystyle\zeta({\bf x}_{\|})$
$\displaystyle=\int\\!\frac{\mathrm{d}^{2}Q_{\parallel}}{(2\pi)^{2}}\,\hat{\zeta}({\bf
Q}_{\|})\exp(\mathrm{i}{\bf Q}_{\|}\cdot{\bf x}_{\|}),$ (2)

where ${\bf Q}_{\|}=(Q_{1},Q_{2},0)$ is a two-dimensional wave vector, so that

$\displaystyle\hat{\zeta}({\bf Q}_{\|})$
$\displaystyle=\int\\!\mathrm{d}^{2}x_{\parallel}\,\zeta({\bf
x}_{\|})\mathrm{exp}(-\mathrm{i}{\bf Q}_{\|}\cdot{\bf x}_{\|}).$ (3a) In
addition we also introduce the notation
$\displaystyle\hat{\zeta}^{(n)}(\mathbf{{Q}}_{\parallel})$
$\displaystyle=\int\\!\mathrm{d}^{2}x_{\parallel}\,\zeta^{n}({\bf
x}_{\|})\exp(-\mathrm{i}{\bf Q}_{\|}\cdot{\bf x}_{\|}),$ (3b)

where for simplicity when $n=1$ we simply will write
$\hat{\zeta}(\mathbf{{Q}}_{\parallel})\equiv\hat{\zeta}^{(1)}(\mathbf{{Q}}_{\parallel})$.

The Fourier coefficient $\hat{\zeta}({\bf Q}_{\|})$ is also a zero-mean
Gaussian random process defined by

$\displaystyle\left<\hat{\zeta}({\bf Q}_{\|})\hat{\zeta}({\bf
Q}\,^{\prime}\\!\\!_{\|})\right>$ $\displaystyle=(2\pi)^{2}\delta({\bf
Q}_{\|}+{\bf Q}\,^{\prime}\\!\\!_{\|})\,\delta^{2}g(|{\bf Q}_{\|}|),$ (4)

where $g(|{\bf Q}_{\|}|)$, the power spectrum of the surface roughness, is
defined by

$\displaystyle g(|{\bf Q}_{\|}|)$
$\displaystyle=\int\\!\mathrm{d}^{2}x_{\parallel}\,W(|{\bf
x}_{\|}|)\,\exp(-\mathrm{i}{\bf Q}_{\|}\cdot{\bf x}_{\|}).$ (5)

From Eqs. (1) and (5), it follows that $g(|{\bf Q}_{\|}|)$ is normalized to
unity,

$\displaystyle\int\frac{\\!\mathrm{d}^{2}Q_{\parallel}\,}{(2\pi)^{2}}g(|{\bf
Q}_{\|}|)$ $\displaystyle=1.$ (6)

##  III Scattering Theory

The metallic surface $x_{3}=\zeta(\mathbf{{x}}_{\parallel})$ is illuminated
from the vacuum by a plane-wave electromagnetic field of angular frequency
$\omega$. The electric field component of this (incident) field is given by

$\displaystyle\mathbf{{E}}^{(i)}(\mathbf{{x}};t)$
$\displaystyle=\Big{\\{}-\frac{c}{\omega}\left[\mathbf{\hat{k}}_{\parallel}\alpha_{0}(k_{\|})+\mathbf{\hat{x}}_{3}k_{\|}\right]B_{p}({\bf
k}_{\|})+(\mathbf{\hat{x}}_{3}\times\mathbf{\hat{k}}_{\parallel})B_{s}({\bf
k}_{\|})\Big{\\}}\,\exp\left\\{\mathrm{i}[{\bf
k}_{\|}-\mathbf{\hat{x}}_{3}\alpha_{0}(k_{\|})]\cdot\mathbf{{x}}-\mathrm{i}\omega
t\right\\},$ (7a) where $B_{p}({\bf k}_{\|})$ and $B_{s}({\bf k}_{\|})$ are
known amplitudes. The _total_ electric field in the vacuum region above the
surface consists of the sum of this incident field and a scattered field,
${\bf E}({\bf x};t)={\bf E}^{(i)}({\bf x};t)+{\bf E}^{(s)}({\bf x};t)$, where
the scattered electric field has the form
$\displaystyle\mathbf{{E}}^{(s)}(\mathbf{{x}};t)$
$\displaystyle=\int\\!\frac{\mathrm{d}^{2}q_{\|}}{(2\pi)^{2}}\,\Big{\\{}\frac{c}{\omega}\left[\mathbf{\hat{q}}_{\parallel}\alpha_{0}(q_{\|})-\mathbf{\hat{x}}_{3}q_{\|}\right]A_{p}({\bf
q}_{\|})+(\mathbf{\hat{x}}_{3}\times\mathbf{\hat{q}}_{\parallel})A_{s}({\bf
q}_{\|})\Big{\\}}\,\exp\left\\{\mathrm{i}[{\bf
q}_{\|}+\mathbf{\hat{x}}_{3}\alpha_{0}(q_{\|})]\cdot\mathbf{{x}}-\mathrm{i}\omega
t\right\\},$ (7b)

where the amplitudes $A_{p}({\bf q}_{\|})$ and $A_{s}({\bf q}_{\|})$ have to
be determined. In Eq. (7) the subscripts p and s denote the p-polarized (TM)
and s-polarized (TE) components of each of these fields, respectively, and a
caret over a vector indicates that it is a unit vector. The function
$\alpha_{0}(q_{\|})$ that appears in Eq. (7) is defined as

$\displaystyle\alpha_{0}(q_{\|})$
$\displaystyle=\left[\frac{\omega^{2}}{c^{2}}-q_{\|}^{2}\right]^{1/2},\qquad\quad\mathrm{Re}\,\alpha_{0}(q_{\|})>0,\,\,\mathrm{Im}\,\alpha_{0}(q_{\|})>0.$
(8)

Due to the linearity of the Maxwell’s equations, a linear relationship exists
between the amplitudes $A_{\alpha}({\bf q}_{\|})$ and $B_{\beta}({\bf
k}_{\|})$ of Eq. (7). We write this relation in the form
$(\alpha=p,s,\beta=p,s)$

$\displaystyle A_{\alpha}({\bf q}_{\|})$
$\displaystyle=\sum\limits_{\beta}R_{\alpha\beta}({\bf q}_{\|}|{\bf
k}_{\|})B_{\beta}({\bf k}_{\|}),$ (9)

where the quantities $\\{R_{\alpha\beta}({\bf q}_{\|}|{\bf k}_{\|})\\}$ denote
the reflection amplitudes which play a significant role in the present theory
since the mean differential reflection coefficient (DRC) is defined in terms
of them. It can be shown that the contribution to the mean DRC from the light
scattered incoherently by the rough surface is given by Simonsen  _et al._
(2016); Simonsen, Maradudin, and Leskova (2010); Simonsen (2010)

$\displaystyle\left\langle\frac{\partial R_{\alpha\beta}({\bf q}_{\|}|{\bf
k}_{\|})}{\partial\Omega_{s}}\right\rangle_{\textrm{incoh}}$
$\displaystyle=\frac{1}{{\mathcal{S}}}\left(\frac{\omega}{2\pi
c}\right)^{2}\frac{\cos^{2}\theta_{s}}{\cos\theta_{0}}\left[\left<\left|R_{\alpha\beta}({\bf
q}_{\|}|{\bf k}_{\|})\right|^{2}\right>-\left|\Big{<}R_{\alpha\beta}({\bf
q}_{\|}|{\bf k}_{\|})\Big{>}\right|^{2}\right].$ (10)

Here ${\mathcal{S}}$ denotes the area of the plane $x_{3}=0$ covered by the
rough surface, $(\theta_{0},\phi_{0})$ and $(\theta_{s},\phi_{s})$ are the
polar and azimuthal angles of incidence and scattering, respectively, as
defined in Fig. 1, and finally

$\displaystyle\mathbf{{k}}_{\parallel}$
$\displaystyle=\frac{\omega}{c}\sin\theta_{0}(\cos\phi_{0},\sin\phi_{0},0)$
(11a) and $\displaystyle\mathbf{{q}}_{\parallel}$
$\displaystyle=\frac{\omega}{c}\sin\theta_{s}(\cos\phi_{s},\sin\phi_{s},0).$
(11b)

Next we defined the scattering matrix ${\bf S}({\bf q}_{\|}|{\bf k}_{\|})$,
whose elements $\\{S_{\alpha\beta}({\bf q}_{\|}|{\bf k}_{\|})\\}$ are
expressed in terms of the elements of the matrix of reflection amplitudes
${\bf R}({\bf q}_{\|}|{\bf k}_{\|})$ by the relation Welford (1977)

$\displaystyle S_{\alpha\beta}({\bf q}_{\|}|{\bf k}_{\|})$
$\displaystyle=\frac{\alpha_{0}^{1/2}(q_{\|})}{\alpha_{0}^{1/2}(k_{\|})}R_{\alpha\beta}({\bf
q}_{\|}|{\bf k}_{\|}),$ (12)

and satisfy the reciprocity relations Saxon (1955); Jones (1964)

$\displaystyle S_{pp}({\bf q}_{\|}|{\bf k}_{\|})$ $\displaystyle=S_{pp}(-{\bf
k}_{\|}|-{\bf q}_{\|})$ (13a) $\displaystyle S_{ss}({\bf q}_{\|}|{\bf
k}_{\|})$ $\displaystyle=S_{ss}(-{\bf k}_{\|}|-{\bf q}_{\|})$ (13b)
$\displaystyle S_{ps}({\bf q}_{\|}|{\bf k}_{\|})$ $\displaystyle=-S_{sp}(-{\bf
k}_{\|}|-{\bf q}_{\|}).$ (13c)

The expression for the mean DRC in Eq. (10) can alternatively be given in
terms of the elements of the scattering matrix as

$\displaystyle\left<\frac{\partial R_{\alpha\beta}({\bf q}_{\|}|{\bf
k}_{\|})}{\partial\Omega_{s}}\right>_{\textrm{incoh}}$
$\displaystyle=\frac{1}{{\mathcal{S}}}\left(\frac{\omega}{2\pi
c}\right)^{2}\cos\theta_{s}\left[\left<\left|S_{\alpha\beta}({\bf q}_{\|}|{\bf
k}_{\|})\right|^{2}\right>-\left|\Big{<}S_{\alpha\beta}({\bf q}_{\|}|{\bf
k}_{\|})\Big{>}\right|^{2}\right].$ (14)

It is this form we will work with here for the expression for the contribution
to the mean DRC from the light scattered incoherently by the rough surface.
The advantage of using the form in Eq. (14), instead of the form in Eq. (10),
is that the the reciprocity relations (14) can be used to check the
correctness of the expressions for the mean DRC that we obtain.

Here we will be concerned with co-polarized scattering, that is, with the
$\alpha\alpha$ element of Eq. (14). To obtain them, we need
$S_{\alpha\alpha}({\bf q}_{\|}|{\bf k}_{\|})$ which we will obtain on the
basis of second-order phase perturbation theory. In the Appendix we detail
that, within second-order phase perturbation theory, the $\alpha\alpha$
element of Eq. (14) can be expressed as

$\displaystyle\left<\frac{\partial R_{\alpha\alpha}({\bf q}_{\|}|{\bf
k}_{\|})}{\partial\Omega_{s}}\right>_{\textrm{incoh}}$
$\displaystyle=\frac{\left|\varepsilon-1\right|^{2}}{2\pi}\left(\frac{\omega}{c}\right)^{2}\cos\theta_{s}\frac{|f_{\alpha}(q_{\parallel})f_{\alpha}(k_{\parallel})|}{|d_{\alpha}(q_{\parallel})d_{\alpha}(k_{\parallel})|^{2}}\exp[-2M_{\alpha}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})]$
$\displaystyle\quad\times\int\limits^{\infty}_{0}\\!\mathrm{d}u_{\parallel}\,u_{\parallel}J_{0}(|\mathbf{{q}}_{\parallel}-\mathbf{{k}}_{\parallel}|u_{\parallel})\left\\{\exp\left[4\delta^{2}\left|\frac{\alpha_{0}(q_{\parallel})\alpha_{0}(k_{\parallel})}{f_{\alpha}(q_{\parallel})f_{\alpha}(k_{\parallel})}\right|\left|H_{\alpha}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})\right|^{2}W(u_{\parallel})\right]-1\right\\},$
(15a) where $J_{0}(\cdot)$ denotes the Bessel function of the first kind and
order zero, and $\displaystyle
M_{\alpha}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})$
$\displaystyle=-2\delta^{2}\mathrm{Re}\,\left[\frac{\alpha_{0}(q_{\parallel})\alpha_{0}(k_{\parallel})}{f_{\alpha}(q_{\parallel})f_{\alpha}(k_{\parallel})}\right]^{\frac{1}{2}}\int\\!\frac{\mathrm{d}^{2}p_{\parallel}}{(2\pi)^{2}}\,F_{\alpha}(\mathbf{{q}}_{\parallel}|\mathbf{{p}}_{\parallel}|\mathbf{{k}}_{\parallel})\,g(|\mathbf{{p}}_{\parallel}-\mathbf{{k}}_{\parallel}|).$
(15b)

In writing this expression it has been assumed that the surface is isotropic
and we have introduced the functions

$\displaystyle d_{p}(q_{\parallel})$
$\displaystyle=\varepsilon\alpha_{0}(q_{\parallel})+\alpha(q_{\parallel})$
(16a) $\displaystyle d_{s}(q_{\parallel})$
$\displaystyle=\alpha_{0}(q_{\parallel})+\alpha(q_{\parallel})$ (16b)

where $\alpha_{0}(q_{\parallel})$ is defined in Eq. (8) and

$\displaystyle\alpha(q_{\|})$
$\displaystyle=\left[\varepsilon\left(\frac{\omega}{c}\right)^{2}-q_{\|}^{2}\right]^{1/2}\qquad\quad\mathrm{Re}\,\alpha(q_{\|})>0,\,\,\mathrm{Im}\,\alpha(q_{\|})>0.$
(17)

Furthermore, we have also defined the functions

$\displaystyle f_{p}(q_{\parallel})$
$\displaystyle=\varepsilon\left(\frac{\omega}{c}\right)^{2}-(\varepsilon+1)q_{\parallel}^{2}$
(18a) $\displaystyle f_{s}(q_{\parallel})$
$\displaystyle=\left(\frac{\omega}{c}\right)^{2}$ (18b)

and

$\displaystyle H_{p}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})$
$\displaystyle=\operatorname{\,sgn}(\mathbf{{\hat{q}}}_{\parallel}\cdot\mathbf{{\hat{k}}}_{\parallel})\left[\varepsilon
q_{\parallel}k_{\parallel}-\alpha(q_{\parallel})\mathbf{\hat{q}}_{\parallel}\cdot\mathbf{\hat{k}}_{\parallel}\alpha(k_{\parallel})\right]$
(19a) $\displaystyle H_{s}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})$
$\displaystyle=-\operatorname{\,sgn}(\mathbf{{\hat{q}}}_{\parallel}\cdot\mathbf{{\hat{k}}}_{\parallel})\left(\frac{\omega}{c}\right)^{2}\mathbf{\hat{q}}_{\parallel}\cdot\mathbf{\hat{k}}_{\parallel}.$
(19b)

Finally the functions
$F_{\alpha}(\mathbf{{q}}_{\parallel}|\mathbf{{p}}_{\parallel}|\mathbf{{k}}_{\parallel})$
are defined as (see the Appendix)

$\displaystyle
F_{p}(\mathbf{{q}}_{\parallel}|\mathbf{{p}}_{\parallel}|\mathbf{{k}}_{\parallel})=\operatorname{\,sgn}(\mathbf{{\hat{q}}}_{\parallel}\cdot\mathbf{{\hat{k}}}_{\parallel})$
$\displaystyle\Bigg{[}\frac{1}{2}\left[\alpha(q_{\parallel})+\alpha(k_{\parallel})\right]\left[q_{\parallel}k_{\parallel}-\alpha(q_{\parallel})\mathbf{\hat{q}}_{\parallel}\cdot\mathbf{\hat{k}}_{\parallel}\alpha(k_{\parallel})\right]$
$\displaystyle\quad+\left(\frac{\varepsilon-1}{\varepsilon}\right)\bigg{\\{}\alpha(q_{\parallel})\mathbf{\hat{q}}_{\parallel}\cdot\mathbf{\hat{p}}_{\parallel}\alpha(p_{\parallel})\mathbf{\hat{p}}_{\parallel}\cdot\mathbf{\hat{k}}_{\parallel}\alpha(k_{\parallel})$
$\displaystyle\quad\quad-\frac{\left[\varepsilon
q_{\parallel}p_{\parallel}-\alpha(q_{\parallel})\mathbf{\hat{q}}_{\parallel}\cdot\mathbf{\hat{p}}_{\parallel}\alpha(p_{\parallel})\right]\big{[}\varepsilon
p_{\parallel}k_{\parallel}-\alpha(p_{\parallel})\mathbf{\hat{p}}_{\parallel}\cdot\mathbf{\hat{k}}_{\parallel}\alpha(k_{\parallel})\big{]}}{d_{p}(p_{\parallel})}$
$\displaystyle\quad\quad-\varepsilon\left(\frac{\omega}{c}\right)^{2}\frac{\alpha(q_{\parallel})\left[\mathbf{\hat{q}}_{\parallel}\times\mathbf{\hat{p}}_{\parallel}\right]_{3}\big{[}\mathbf{\hat{p}}_{\parallel}\times\mathbf{\hat{k}}_{\parallel}\big{]}_{3}\alpha(k_{\parallel})}{d_{s}(p_{\parallel})}\bigg{\\}}\Bigg{]},$
(20a) and (see Eq. (41)) $\displaystyle
F_{s}(\mathbf{{q}}_{\parallel}|\mathbf{{p}}_{\parallel}|\mathbf{{k}}_{\parallel})=$
$\displaystyle-\operatorname{\,sgn}(\mathbf{\hat{q}}_{\parallel}\cdot\mathbf{\hat{k}}_{\parallel})\left(\frac{\omega}{c}\right)^{2}\bigg{\\{}\frac{1}{2}[\alpha(q_{\|})+\alpha(k_{\|})](\mathbf{\hat{q}}_{\parallel}\cdot\mathbf{\hat{k}}_{\parallel})+(\varepsilon-1)[\mathbf{\hat{q}}_{\parallel}\times\mathbf{\hat{p}}_{\parallel}]_{3}\frac{\alpha_{0}(p_{\|})\alpha(p_{\|})}{d_{p}(p_{\|})}[\mathbf{\hat{p}}_{\parallel}\times\mathbf{\hat{k}}_{\parallel}]_{3}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad-(\varepsilon-1)\left(\frac{\omega}{c}\right)^{2}\frac{(\mathbf{\hat{q}}_{\parallel}\cdot\mathbf{\hat{p}}_{\parallel})(\mathbf{\hat{p}}_{\parallel}\cdot\mathbf{\hat{k}}_{\parallel})}{d_{s}(p_{\|})}\bigg{\\}}.$
(20b)

Here the notation $[\cdot]_{3}$ denotes the third component of the vector
argument. It ought to be remarked that the expressions for the mean DRC in
Eqs. (15)–(20) are obtained using one, of several possible, formulations of
second-order phase perturbation theory. One alternative formulation of phase
perturbation theory is presented in Ref. Navarrete Alcalá  _et al._ , 2009
where it is used to derive the corresponding expressions for the mean DRC. It
is still not known which of the two formulations are the most accurate, or if
one formulation is more accurate for metallic scattering systems, say, and the
other for dielectric scattering systems. It might also be that the formulation
to prefer depends on the wavelength regime of interest. There are questions
that further research will have to answer.

We now turn to the calculation of the function
$M_{\alpha}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})$ that is
defined by Eqs. (15b) and (20). This calculation is facilitated by expanding
the power spectrum $g(|\mathbf{{p}}_{\parallel}-\mathbf{{k}}_{\parallel}|)$,
defined by Eq. (5), in the following way

$\displaystyle g(|\mathbf{{p}}_{\parallel}-\mathbf{{k}}_{\parallel}|)$
$\displaystyle=2\pi\sum_{n=-\infty}^{\infty}\exp\left[\mathrm{i}n(\phi_{p}-\phi_{k})\right]\int_{0}^{\infty}\\!\mathrm{d}u_{\parallel}\,u_{\parallel}W(u_{\parallel})\,J_{n}(p_{\parallel}u_{\parallel})\,J_{n}\big{(}k_{\parallel}u_{\parallel}\big{)},$
(21)

where $J_{n}(\cdot)$ denotes the Bessel function of the first kind and order
$n$. The expansion (21) is established by first introducing polar
representations of the vectors $\mathbf{{p}}_{\parallel}$,
$\mathbf{{k}}_{\parallel}$, and $\mathbf{{u}}_{\parallel}$, that is
$\mathbf{{p}}_{\parallel}=p_{\parallel}(\cos\phi_{p},\sin\phi_{p},0)$ with
$\phi_{p}$ the azimuthal angle of this vector with the positive $x_{1}$ axis
(see Fig. 1) etc., then using the Jacobi-Anger identity (Abramowitz and
Stegun, 1964, Ch. 9),
$\exp(\mathrm{i}mz\cos\phi)=\sum_{m=-\infty}^{\infty}J_{m}(z)\exp(\mathrm{i}\phi)$,
and the property of the Bessel function $J_{-m}(z)=(-1)^{m}J_{m}(z)$. When the
expansion (21) is introduced into Eq. (15b) a lengthy, but in principle
straight forward calculation, shows that the angular integration (over
$\phi_{p}$) can be performed analytically. For p-polarized light [$\alpha=p$],
the result of such a calculation is

$\displaystyle M_{p}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})$
$\displaystyle=-\delta^{2}\mathrm{sgn}\left(\mathbf{\hat{q}}_{\parallel}\cdot\mathbf{\hat{k}}_{\parallel}\right)\mathrm{Re}\,\sqrt{\frac{\alpha_{0}(q_{\|})\alpha_{0}(k_{\|})}{f_{p}(q_{\|})f_{p}(k_{\|})}}m_{p}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})$
(22a) where $\displaystyle
m_{p}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})$
$\displaystyle=q_{\parallel}[\alpha(q_{\parallel})+\alpha(k_{\parallel})]k_{\parallel}-(\mathbf{\hat{q}}_{\parallel}\cdot\mathbf{\hat{k}}_{\parallel})\alpha(q_{\parallel})\left[\alpha(q_{\parallel})+\alpha(k_{\parallel})\right]\alpha(k_{\parallel})$
$\displaystyle\quad{}+\frac{\varepsilon-1}{\varepsilon}\,\int_{0}^{\infty}\\!\mathrm{d}p_{\parallel}\,p_{\parallel}\bigg{\\{}-\frac{2\varepsilon^{2}q_{\parallel}p_{\parallel}^{2}k_{\parallel}}{d_{p}(p_{\parallel})}+(\mathbf{\hat{q}}_{\parallel}\cdot\mathbf{\hat{k}}_{\parallel})\alpha(q_{\parallel})\left[\alpha(p_{\parallel})-\frac{\alpha^{2}(p_{\parallel})}{d_{p}(p_{\parallel})}+\frac{\varepsilon(\omega/c)^{2}}{d_{s}(p_{\parallel})}\right]\alpha(k_{\parallel})\bigg{\\}}$
$\displaystyle\qquad\qquad\qquad\qquad\quad\times\int_{0}^{\infty}\\!\mathrm{d}x_{\parallel}\,x_{\parallel}W(x_{\parallel}){J_{0}(p_{\parallel}x_{\parallel})J_{0}(k_{\parallel}x_{\parallel})}$
$\displaystyle\quad{}+2(\varepsilon-1)\int_{0}^{\infty}\\!\mathrm{d}p_{\parallel}\,p_{\parallel}\frac{p_{\parallel}\alpha(p_{\parallel})}{d_{p}(p_{\parallel})}\left[q_{\parallel}\alpha(k_{\parallel})+(\mathbf{\hat{q}}_{\parallel}\cdot\mathbf{\hat{k}}_{\parallel})\alpha(q_{\parallel})k_{\parallel}\right]\int_{0}^{\infty}\\!\mathrm{d}x_{\parallel}\,x_{\parallel}W(x_{\parallel}){J_{1}(p_{\parallel}x_{\parallel})J_{1}(k_{\parallel}x_{\parallel})}$
$\displaystyle\quad{}+(\mathbf{\hat{q}}_{\parallel}\cdot\mathbf{\hat{k}}_{\parallel})\frac{\varepsilon-1}{\varepsilon}\int_{0}^{\infty}\\!\mathrm{d}p_{\parallel}\,p_{\parallel}\alpha(q_{\parallel})\left\\{\alpha(p_{\parallel})-\frac{\alpha^{2}(p_{\parallel})}{d_{p}(p_{\parallel})}-\frac{\varepsilon(\omega/c)^{2}}{d_{s}(p_{\parallel})}\right\\}\alpha(k_{\parallel})\int_{0}^{\infty}\\!\mathrm{d}x_{\parallel}\,x_{\parallel}W(x_{\parallel}){J_{2}(p_{\parallel}x_{\parallel})J_{2}(k_{\parallel}x_{\parallel})}.$
(22b)

A similar calculation for s-polarized light was recently performed in Ref.
Simonsen  _et al._ , 2016 with the result that (see Eqs. (22) and (A9) in Ref.
Simonsen  _et al._ , 2016)

$\displaystyle M_{s}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})$
$\displaystyle=-\delta^{2}\left|\mathbf{\hat{q}}_{\parallel}\cdot\mathbf{\hat{k}}_{\parallel}\right|\alpha_{0}^{1/2}(q_{\|})\alpha_{0}^{1/2}(k_{\|})\mathrm{Re}\,\bigg{\\{}-\alpha(q_{\|})-\alpha(k_{\|})$
$\displaystyle\qquad\quad+(\varepsilon-1)\int_{0}^{\infty}\\!\mathrm{d}p_{\|}\,p_{\|}\left(\frac{\alpha_{0}(p_{\|})\alpha(p_{\|})}{d_{p}(p_{\|})}+\frac{(\omega/c)^{2}}{d_{s}(p_{\|})}\right)\int_{0}^{\infty}\\!\mathrm{d}x_{\|}\,x_{\|}W(x_{\|})J_{0}(p_{\|}x_{\|})J_{0}(k_{\|}x_{\|})$
$\displaystyle\qquad\quad+(\varepsilon-1)\int_{0}^{\infty}\\!\mathrm{d}p_{\|}\,p_{\|}\left(-\frac{\alpha_{0}(p_{\|})\alpha(p_{\|})}{d_{p}(p_{\|})}+\frac{(\omega/c)^{2}}{d_{s}(p_{\|})}\right)\int_{0}^{\infty}\\!\mathrm{d}x_{\|}\,x_{\|}W(x_{\|})J_{2}(p_{\|}x_{\|})J_{2}(k_{\|}x_{\|})\bigg{\\}}.$
(22c)

Note that for normal incidence, according to Eq. (11a), one has
$\mathbf{\hat{k}}_{\parallel}=(\cos\phi_{0}\sin\phi_{0},0)$ even if
$\mathbf{{k}}_{\parallel}/k_{\parallel}$ is not well defined in this case.
Below we will be concerned with in-plane scattering in which case
$|\mathbf{\hat{q}}_{\parallel}\cdot\mathbf{\hat{k}}_{\parallel}|=1$.

##  IV The Inverse Problem

The aim of this work is to determine the rms-roughness $\delta$ and the height
autocorrelation function $W(x_{\parallel})$ of the rough metal surface from
the knowledge of the in-plane angular dependence of the co-polarized light
scattered incoherently from it. This will be done by minimizing the cost
function

$\displaystyle\chi^{2}({\cal
P})=\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}}d\theta_{s}\sum_{\alpha}\bigg{[}\bigg{\langle}\frac{\partial
R_{\alpha\alpha}(\theta_{s})}{\partial\Omega_{s}}\bigg{\rangle}_{\textrm{incoh,input}}-\left\langle\frac{\partial
R_{\alpha\alpha}(\theta_{s})}{\partial\Omega_{s}}\right\rangle_{\textrm{incoh,calc}}\bigg{]}^{2},$
(23)

where we to simplify the notation have introduced

$\displaystyle\left<\frac{\partial
R_{\alpha\alpha}(\theta_{s})}{\partial\Omega_{s}}\right>_{\textrm{incoh}}\equiv\left\langle\frac{\partial
R_{\alpha\alpha}({\bf q}_{\|}|{\bf
k}_{\|})}{\partial\Omega_{s}}\right\rangle_{\textrm{incoh}}\Bigg{|}_{|\mathbf{\hat{q}}_{\parallel}\cdot\mathbf{\hat{k}}_{\parallel}|=1},$
(24)

and where ${\cal P}$ denotes the set of variational parameters used to
characterize $\langle\partial
R_{\alpha\alpha}(\theta_{s})/\partial\Omega_{s}\rangle_{\textrm{incoh,calc}}$.
The $\alpha$-summation is present in Eq. (23) to facilitate the reconstruction
of surface parameters based on the simultaneous use of scattering data in both
p and s polarization. If data for only one polarization is used in the
reconstruction, the summation is redundant. The minimization of
$\chi^{2}({\cal P})$ with respect to the elements of ${\cal P}$ was carried
out by the use of the routine “lmdif1” contained in the Fortran package
MINPACK which is part of the general purpose mathematical library SLATEC SLA
(1993). The routine lmdif1 implements a modified version of the Levenberg-
Marquardt algorithm Levenberg (1944); Marquardt (1963), and it calculates an
approximation to the Jacobian by a forward-difference approach.

The functions $\langle\partial
R_{\alpha\alpha}(\theta_{s})/\partial\Omega_{s}\rangle_{\textrm{incoh,input}}$
with $\alpha={p,s}$ were either obtained from rigorous, nonperturbative,
purely numerical solutions Leskova  _et al._ (2011); Nordam, Letnes, and
Simonsen (2013) of the reduced Rayleigh equation for the scattering of
polarized light from a two-dimensional randomly rough metal surface Brown  _et
al._ (1984) or from measurements Navarrete Alcalá  _et al._ (2009). The
calculations were carried out for an ensemble of random surfaces generated
Nordam, Letnes, and Simonsen (2013) on the basis of expressions for $W(|{\bf
x}_{\|}|)$ of the Gaussian form

$\displaystyle
W(|\mathbf{{x}}_{\parallel}|)=\exp\left\\{-\left(\frac{x_{\parallel}}{a}\right)^{2}\right\\}.$
(25)

In Eq. (25), $a$ denotes the transverse correlation length of the surface
roughness.

On the other hand, the functions $\langle\partial
R_{\alpha\alpha}(\theta_{s})/\partial\Omega_{s}\rangle_{\textrm{incoh,calc}}$
for $\alpha=p,s$ were obtained by evaluating the expression for it obtained
using phase perturbation theory [Eq. (15)] for the trial function assumed to
represent $W(|\mathbf{{x}}_{\parallel}|)$. Several forms for this trial
function were used in our calculations; either we assumed the Gaussian trial
function

$\displaystyle
W(|\mathbf{{x}}_{\parallel}|)=\exp\left\\{-\left(\frac{x_{\parallel}}{a^{\star}}\right)^{2}\right\\},$
(26)

or a stretched exponential trial function of the form

$\displaystyle W(|{\bf
x}_{\|}|)=\exp\left\\{-\left(\frac{x_{\|}}{a^{\star}}\right)^{\gamma^{\star}}\right\\}.$
(27)

When the form (26) is used for the trial function the variational parameters
of the reconstruction are $\delta^{\star}$ and $a^{\star}$. On the other hand,
when the trial function (27) is used, the variational parameters of the
reconstruction are $\delta^{\star}$, $a^{\star}$, and $\gamma^{\star}$. Note
that stretched exponential trial function reduces to the Gaussian forms (26)
when $\gamma^{\star}=2$.

##  V Results

When visible light is scattered from a weakly rough metal surface, like
silver, one of the prominent features that can be observed for some roughness
parameters in the angular distribution of the scattered light is the enhanced
backscattering phenomenon Maradudin  _et al._ (1990); Zayats, Smolyaninov, and
Maradudin (2005); Simonsen (2010) which expresses itself as a well defined
peak in the retroreflection direction. For weakly rough surfaces it is know to
result from the excitation and multiple scattering of surface waves known as
surface plasmon polaritons (SPPs) Maier (2007). Hence, when the enhanced
backscattering peaks are present in the angular distribution of the scattered
light, it is a clear signature of the contribution from multiple scattering
processes to the scattered intensity. In the case when the contribution from
single scattering processes can be neglected, the intensity at the position of
the enhanced backscattering peak is expected to be twice of the intensity of
its background Maradudin  _et al._ (1990).

###  V.1 Inversion of computer generated scattering data

The second-order phase perturbation theoretical approach that we base the
reconstruction on is primarily a single scattering approach. Even if second-
order phase perturbation theory in principle includes some (but not all)
multiple scattering processes, it is unable to predict the existence of the
enhanced backscattering phenomenon. Therefore, it is of interest to study how
well the reconstruction approach based on it is able to handle different
levels of contribution from multiple scattering to the scattered intensity. To
this end, we will illustrate the inversion (or reconstruction) procedure
developed here by applying it to the reconstruction of the rms-roughness
$\delta$ and $W(\mathbf{{x}}_{\parallel}|)$ based on computer generated
scattering data that receive various levels of contribution from multiple
scattering processes (via the excitation and multiple scattering of SPPs). In
the preceding subsection, we will apply our reconstruction approach to
experimental scattering data.

####  V.1.1 Scattering system no. 1

Figure 2: (Scattering system no. 1) Reconstruction of the rms-roughness
$\delta^{\star}$ and the transverse correlation length $a^{\star}$ from
p-polarized in-plane scattering data obtained for a Gaussian correlated rough
silver surface. The wavelength (in vacuum) of the incident light is
$\lambda=$457.90\text{\,}\mathrm{nm}$$ for which the dielectric function of
silver is $\varepsilon(\omega)=-7.50+0.24\mathrm{i}$. The surface roughness
parameters assumed in the computer simulations have the values
$\delta=\lambda/20=$22.90\text{\,}\mathrm{nm}$$ and
$a=\lambda=$457.90\text{\,}\mathrm{nm}$$. (a) The incoherent components of the
in-plane, co-polarized (p-to-p) mean differential reflection coefficient
$\left<\partial R_{pp}/\partial\Omega_{s}\right>_{\textrm{incoh}}$ as a
function of the polar angle of scattering $\theta_{s}$ obtained from computer
simulations for the polar angles of incidence $\theta_{0}=$$$ (open circles)
and $\theta_{0}=$$$ (open squares), and from second-order phase perturbation
theory for these polar angles of incidence with the use of the reconstructed
surface roughness parameters (solid lines), for a two-dimensional randomly
rough silver surface whose correlation function is defined by Eq. (25). The
mean DRC results (input data) produced by computer simulations were obtained
on the basis of $2500$ surface realizations. For normal incidence
[$\theta_{0}=$$$] the reconstructed values of the surface roughness parameters
are $\delta^{\star}=$22.72\text{\,}\mathrm{nm}$$ and
$a^{\star}=$458.61\text{\,}\mathrm{nm}$$, while for $\theta_{0}=$$$ one finds
$\delta^{\star}=$22.96\text{\,}\mathrm{nm}$$ and
$a^{\star}=$463.52\text{\,}\mathrm{nm}$$. In performing these reconstructions
the form (26) for the autocorrelation function was assumed, and the
minimization of the cost function (23) was started from the initial values
$\delta^{\star}=$8.00\text{\,}\mathrm{n}\mathrm{m}$$ and
$a^{\star}=$150.00\text{\,}\mathrm{n}\mathrm{m}$$. For both angles of
incidence, the computer simulation results for the in-plane angular dependence
of $\left<\partial R_{sp}/\partial\Omega_{s}\right>_{\textrm{incoh}}$ (p-to-s
cross-polarization) were essentially indistinguishable from zero and therefore
such results were not included. (b) For normal incidence [$\theta_{0}=$$$],
the input (open circles) and reconstructed (solid line) surface-height
autocorrelation function $W(|\mathbf{{x}}_{\parallel}|)$ for the random
surface. The shaded gray region represents the absolute difference between the
input and reconstructed surface-height autocorrelation functions. The notation
“$\times 30$” means that this difference has been multiplied by a factor $30$
for better visibility. (c) The same as Fig. 2(b) but now for the polar angle
of incidence $\theta_{0}=$$$.

The first scattering system that we consider is a silver substrate bounded by
a randomly rough surface to vacuum which is the incident medium. An
$\alpha$-polarized plane incident wave of wavelength
$\lambda=$457.90\text{\,}\mathrm{n}\mathrm{m}$$ (in vacuum) illuminates the
surface from the vacuum. At the wavelength of the incident light
[$\lambda=2\pi/\omega$] the dielectric function of silver is
$\varepsilon(\omega)=-7.50+0.24\mathrm{i}$ Palik (1997). The rms-roughness of
the surface is taken to be
$\delta=\lambda/20=$22.90\text{\,}\mathrm{n}\mathrm{m}$$ and the surface-
height autocorrelation function $W(|\mathbf{{x}}_{\parallel}|)$ is assumed to
have the Gaussian form, Eq. (25), and characterized by a transverse
correlation length $a=\lambda=$457.90\text{\,}\mathrm{n}\mathrm{m}$$. For
these roughness and geometrical parameters, the scattering problem was solved
by a rigorous, non-perturbative and direct numerical solution of the reduced
Rayleigh equation by the method of Ref. Nordam, Letnes, and Simonsen, 2013. In
this way, we calculated by computer simulations the in-plane angular
dependence of the contribution to the mean DRCs from $\alpha$-polarized light
scattered incoherently by the rough surface. When $2500$ surface realizations
were used in calculating the average, we obtained the results for
$\left<\partial R_{\alpha\alpha}/\partial\Omega_{s}\right>_{\textrm{incoh}}$
presented as open symbol in Figs. 2(a) and 4(a) for p and s polarization,
respectively; in both cases the polar angle of incidence was $\theta_{0}=$$$
or $\theta_{0}=$$$. These data constitute the input functions $\left<\partial
R_{\alpha\alpha}(\theta_{s})/\partial\Omega_{s}\right>_{\textrm{incoh,input}}$
for our first sets of reconstruction examples. It should be remarked that
computer simulation results for the in-plane, cross-polarized scattering were
essentially indistinguishable from zero on the scales of Figs. 2(a) and 4(a)
and hence such results were not presented. Since in-plane, cross-polarized
scattering is a multiple scattering effect McGurn and Maradudin (1996), we
find it reasonable to assume that the results presented in Figs. 2(a) and 4(a)
do not receive any significant contribution from multiple scattering
processes.

The first of these data sets that we reconstructed was for p-polarized light
and normal incidence [Fig. 2(a) open circles]. To this end, it was assumed
that the trial function $W(|\mathbf{{x}}_{\parallel}|)$ had the Gaussian form
(26), and therefore the set of variational parameters we assumed is
${\mathcal{P}}=\\{\delta^{\star},a^{\star}\\}$. The minimization procedure was
started from the initial values
$\delta^{\star}=$8.00\text{\,}\mathrm{n}\mathrm{m}$$ and
$a^{\star}=$150.00\text{\,}\mathrm{n}\mathrm{m}$$ and the minimization of the
cost function $\chi^{2}({\mathcal{P}})$ for p-polarized data, defined in Eq.
(23), resulted in the reconstructed parameters
$\delta^{\star}=$22.72\text{\,}\mathrm{n}\mathrm{m}$$ and
$a^{\star}=$458.61\text{\,}\mathrm{n}\mathrm{m}$$. When these values are
compared to the values $\delta=$22.90\text{\,}\mathrm{n}\mathrm{m}$$ and
$a=$457.90\text{\,}\mathrm{n}\mathrm{m}$$ used to generate the input data, one
finds good agreement. The corresponding reconstructed mean DRC curve is
indicated as a solid line in Fig. 2(a); the main difference between the input
and reconstructed mean DRC curves are found for the normal scattering
direction. This discrepancy we speculate is due to an inaccuracy in the input
data that is due to the use of a finite number of surface realizations for
their calculation. The input and reconstructed surface-height autocorrelation
function $W(|\mathbf{{x}}_{\parallel}|)$ are shown in Fig. 2(b) as open
symbols and a solid line, respectively. The absolute difference between these
two functions are presented as the gray shaded regions in Fig. 2(b) where
$\times 5$ indicates that this difference is multiplied by a factor $5$ for
reasons of clarity.

Next we performed reconstruction of p-polarized scattering data corresponding
to the polar angle of incidence $\theta_{0}=$$$; this input data set is
presented as open squares in Fig. 2(a). When the reconstruction was performed
in a completely equivalent way to how it was done for the scattering data
corresponding to normal incidence, for instance by assuming the same initial
values for the variational parameters, we obtained the following values for
the reconstructed parameters
$\delta^{\star}=$22.96\text{\,}\mathrm{n}\mathrm{m}$$ and
$a^{\star}=$463.52\text{\,}\mathrm{n}\mathrm{m}$$. These values for the
reconstructed parameters are also in excellent agreements with the values
assumed for input roughness parameters. It is found that the reconstructed
transverse correlation length $a^{\star}$ deviates slightly more from the
corresponding input value $a$ when $\theta_{0}=$$$ than what is found when
data for normal incidence is used in the reconstruction; this is reflected in
a larger absolute difference between the input and reconstructed
$W(|\mathbf{{x}}_{\parallel}|)$ as presented in Fig. 2(c).

For the same input data set, that is, for p polarization and $\theta_{0}=$$$
[Figs. 2(a) and 3(a)], we also performed reconstruction based on the
assumption that the surface-height autocorrelation function
$W(|\mathbf{{x}}_{\parallel}|)$ had the stretched exponential form (27). The
set of variational parameters is therefore
${\mathcal{P}}=\\{\delta^{\star},a^{\star},\gamma^{\star}\\}$. When the
minimization was started from the values
$\delta^{\star}=$8.00\text{\,}\mathrm{n}\mathrm{m}$$,
$a^{\star}=$150.00\text{\,}\mathrm{n}\mathrm{m}$$ and $\gamma^{\star}=1.00$,
where the initial values of the two former parameters are identical to the
initial values assumed in producing the results in Fig. 2, the reconstruction
procedure produced the values
$\delta^{\star}=$23.11\text{\,}\mathrm{n}\mathrm{m}$$,
$a^{\star}=$459.74\text{\,}\mathrm{n}\mathrm{m}$$ and $\gamma^{\star}=1.95$
[Fig. 3]. These values are rather close to the input values assumed when
generating the scattering data; recall that the value of $\gamma$ in Eq. (27)
that corresponds to the Gaussian correlation function (25) is $\gamma=2.00$.
Moreover, when basing the reconstruction on the stretched exponential (27), it
is interesting to note that the value obtained of $\delta^{\star}$ only is
marginally larger than what is obtained when assuming the Gaussian form for
$W(|\mathbf{{x}}_{\parallel}|)$, while the value obtained for $a^{\star}$ is
better by several nanometers. A summary of the reconstructed roughness
parameters for p-polarized light can be found in the upper half of Table 1.

Figure 3: (Scattering system no. 1) Reconstruction of the rms-roughness $\delta^{\star}$, the transverse correlation length $a^{\star}$, and the exponent $\gamma^{\star}$ from in-plane p-to-p scattering data obtained for a Gaussian correlated silver surface. The parameters defining the scattering system, including the surface roughness, are identical to those given in the caption of Fig. 2. (a) The incoherent components of the in-plane, co-polarized (p-to-p) mean differential reflection coefficient $\left<\partial R_{pp}/\partial\Omega_{s}\right>_{\textrm{incoh}}$ as a function of the polar angle of scattering $\theta_{s}$ obtained from computer simulations for the polar angles of incidence $\theta_{0}=$$$ (open circles) [same data as in Fig. 2(a)], and from second-order phase perturbation theory with the use of the reconstructed surface roughness parameters (solid lines) for a two-dimensional randomly rough silver surface whose correlation function is assumed to have the form (27). The values of the reconstructed surface roughness parameters are $\delta^{\star}=$23.11\text{\,}\mathrm{nm}$$, $a^{\star}=$459.74\text{\,}\mathrm{nm}$$, and $\gamma^{\star}=1.95$. In obtaining these results, the minimization was started from the values $\delta^{\star}=$8.00\text{\,}\mathrm{n}\mathrm{m}$$, $a^{\star}=$150.00\text{\,}\mathrm{n}\mathrm{m}$$ and $\gamma^{\star}=1.00$, (b) The input (open circles) and reconstructed (solid line) surface-height autocorrelation function $W(|\mathbf{{x}}_{\parallel}|)$ for the random surface. The shaded gray region represents the absolute difference between the input and reconstructed surface-height autocorrelation functions. The notation “$\times 5$” means that this difference has been multiplied by a factor $5$ for better visibility.  Table 1:  (Scattering system no. 1) Summary of the scattering system parameters obtained during the different reconstruction scenarios based on in-plane and co-polarized scattering data corresponding to an Gaussianly correlated silver surface; $\delta^{\star}$, $a^{\star}$, and $\gamma^{\star}$. The incident light, of wavelength $\lambda=$457.90\text{\,}\mathrm{n}\mathrm{m}$$, was $\alpha$-polarized and the polar angle of incidence was $\theta_{0}=$$$ or $\theta_{0}=$$$. The dielectric function of silver was $\varepsilon(\omega)=-7.50+0.24\mathrm{i}$ Palik (1997). The surface roughness parameters assumed in generating the input data were: $\delta=\lambda/20=$22.90\text{\,}\mathrm{n}\mathrm{m}$$ and $a=\lambda=$457.90\text{\,}\mathrm{n}\mathrm{m}$$. The last column indicates the relevant figure where the results of the reconstruction in question is presented. The symbol “—” indicates that the corresponding variable was not reconstructed and instead had the value assumed in the input data (numerical simulations). When $\gamma^{\star}$ was among the reconstructed parameters, the trial correlation function of the form (27) was assumed; in all other cases the Gaussian form (26) was assumed for this function. Note that a Gaussian correlation function (25) corresponds to the exponent $\gamma^{\star}=$2$$ for the stretched exponential in Eq. (27). The initial values for $\\{\delta^{\star},a^{\star},\gamma^{\star}\\}$ used in performing the reconstructions were $\\{$8\text{\,}\mathrm{nm}$,$150\text{\,}\mathrm{nm}$,1\\}$.  $\alpha$ |  $\theta_{0}$[deg] | $\delta^{\star}\;$\mathrm{[}\mathrm{n}\mathrm{m}\mathrm{]}$$ | $a^{\star}\;$\mathrm{[}\mathrm{n}\mathrm{m}\mathrm{]}$$ | $\gamma^{\star}$ | Comments
---|---|---|---|---|---
p | 0 | $22.72$ | $458.61$ | — | Fig. 2
p | 40 | $22.96$ | $463.52$ | — | Fig. 2
p | 40 | $23.11$ | $459.74$ | $1.95$ | Fig. 3
s | 0 | $22.99$ | $454.34$ | — | Fig. 4
s | 40 | $22.81$ | $457.10$ | — | Fig. 4
s | 40 | $23.53$ | $439.63$ | $1.84$ | Fig. 5

We now turn to the reconstruction based on scattering data obtained when
s-polarized light, instead of p-polarized light, is scattered from this rough
silver surface. The input data sets for s polarization corresponding to
$\theta_{0}=$$$ and $\theta_{0}=$$$ are presented in Fig. 4(a) as open circles
and squares, respectively. One observes that these scattering data are not
very different from the corresponding scattering data produced when the
incident light is p-polarized [see Fig. 2(a)]; this is due to the relatively
long correlation length that was assumed in generating them. Performing
reconstruction on the basis of these s-polarized scattering data, done in a
completely equivalent manner to how it was performed for the corresponding
p-polarized data sets, produced the results presented in Figs. 4 and 5. In
this way we obtained the values for the reconstructed roughness parameters
listed for s-polarization [$\alpha=s$] in the lower half of Table 1. By
comparing the values presented in this table, or by comparing the results
presented in Figs. 2–5, uncovers that the results obtained by basing the
reconstruction of the surface roughness parameters on p- or s-polarized
scattering data produce rather similar results; at least this is the case for
the angles of incidence and roughness parameters that were assumed here. The
most noticeable difference in the values of the reconstructed parameters
obtained when basing the inversion on p- and s-polarized scattering data is
found when the stretched exponential form for $W(|\mathbf{{x}}_{\parallel}|)$
is used; in this case, the reconstruction based on p-polarized scattering data
produce the most accurate results.

It is remarked that we have also performed reconstruction based on unpolarized
incident light, joint inversion using input data for several polar angles of
incidence, and joint inversion based on simultaneously use of input data
corresponding to p- and s-polarized incident light (see Ref. Simonsen  _et
al._ (2016)). The values obtained in this way for the reconstructed parameters
did only improve marginally relative to the results reported in Table 1;
hence, we do not give additional details here.

Figure 4: (Scattering system no. 1) Reconstruction of the rms-roughness
$\delta^{\star}$ and transverse correlation length $a^{\star}$ from in-plane
s-to-s scattering data obtained for a Gaussian correlated silver surface. The
same as Fig. 2 but for s-polarized light. The values of the reconstructed
surface roughness parameters are $\delta^{\star}=$22.99\text{\,}\mathrm{nm}$$
and $a^{\star}=$454.34\text{\,}\mathrm{nm}$$ for $\theta_{0}=$$$; and
$\delta^{\star}=$22.81\text{\,}\mathrm{nm}$$ and
$a^{\star}=$457.10\text{\,}\mathrm{nm}$$ for $\theta_{0}=$$$. [Input values:
$\delta=\lambda/20=$22.90\text{\,}\mathrm{nm}$$ and
$a=\lambda=$457.90\text{\,}\mathrm{nm}$$].

Figure 5: (Scattering system no. 1) Reconstruction of the rms-roughness
$\delta^{\star}$, the transverse correlation length $a^{\star}$, and the
exponent $\gamma^{\star}$ from in-plane s-to-s scattering data. The same as
Fig. 3 but for s-polarized light. The values of the reconstructed surface
roughness parameters are $\delta^{\star}=$23.53\text{\,}\mathrm{nm}$$,
$a^{\star}=$439.63\text{\,}\mathrm{nm}$$, and $\gamma^{\star}=$1.84$$. [Input
values: $\delta=$22.90\text{\,}\mathrm{nm}$$, $a=$457.90\text{\,}\mathrm{nm}$$
and $\gamma=2$].

####  V.1.2 Scattering system no. 2

The second scattering system that we consider is identical to the first except
that the correlation length of the rough surface now is half of what it was
for the first scattering system; hence, the rms-roughness and correlation
length for this scattering system are assumed to be
$\delta=\lambda/20=$22.90\text{\,}\mathrm{n}\mathrm{m}$$ and
$a=\lambda/2=$228.95\text{\,}\mathrm{n}\mathrm{m}$$, respectively. For these
roughness parameters, a computer simulation approach Nordam, Letnes, and
Simonsen (2013) was again used to generate scattering data. By averaging the
results obtained on the basis of $2500$ surface realizations, we calculated
the contributions to the mean DRC from the light scattered incoherently by the
rough surface. When the polar angle of incidence is $\theta_{0}=$$$, such
results for co-polarized scattering are presented for p- and s-polarized light
in Figs. 6(a) and 7(a), respectively. These scattering data constitute the
input functions $\left<\partial
R_{\alpha\alpha}(\theta_{s})/\partial\Omega_{s}\right>_{\textrm{incoh,input}}$
for the reconstruction we will perform based on these data. In Figs. 6(a) and
7(a) the vertical dashed lines indicate the backscattering direction and the
simulation results show a weak enhanced backscattering peak. To confirm for
this scattering system that multiple scattering processes contribute to the
in-plane scattering, we have also in Figs. 6(a) and 7(a) included the cross-
polarized components of the mean DRCs (blue dashed lines and shaded regions);
for both p- and s-polarized incident light we observe that the in-plane,
cross-polarized components of the mean DRCs do not vanish. This, in addition
to the presence of a weak enhanced backscattering peak, is a clear indication
that multiple scattering contributes to the in-plane scattered intensity. It
is stressed that in the plane of incidence, but not outside of it, the cross-
polarized component of the mean DRC is zero within single scattering Navarrete
Alcalá  _et al._ (2009); finding such a component to be non-zero signals
multiple scattering processes.

As our first example of reconstruction based on scattering data for this
scattering geometry, we assume p-polarized incident light and a trial function
$W(|\mathbf{{x}}_{\parallel}|)$ of a Gaussian form given by Eq. (26). The set
of variational parameters is therefore
${\mathcal{P}}=\\{\delta^{\star},a^{\star}\\}$. The minimization procedure of
the cost function $\chi^{2}({\mathcal{P}})$, Eq. (23), was started from the
values $\delta^{\star}=$8.00\text{\,}\mathrm{n}\mathrm{m}$$ and
$a^{\star}=$75.00\text{\,}\mathrm{n}\mathrm{m}$$. In this way, it was found
that the minimum of $\chi^{2}({\mathcal{P}})$ occurred for the values
$\delta^{\star}=$25.69\text{\,}\mathrm{nm}$$ and
$a^{\star}=$223.42\text{\,}\mathrm{nm}$$, that should be compared with the
values $\delta=$22.90\text{\,}\mathrm{n}\mathrm{m}$$ and
$a=$228.95\text{\,}\mathrm{n}\mathrm{m}$$ used to generate the input data. The
reconstruction of the surface roughness parameters is fair but less accurate
than what we found when basing the reconstruction on scattering data for the
first scattering system where multiple scattering processes contributed less.
The absolute relative errors in the reconstruction of the surface roughness
parameters $\delta$ and $a$ are $12.2\text{\,}\mathrm{\char 37\relax}$ and
$2.4\text{\,}\mathrm{\char 37\relax}$, respectively. The corresponding
reconstructed correlation function is compared to the input correlation
function in Fig. 6(b).

By performing a similar reconstruction based on the corresponding input data
for s-polarized light [Fig. 7(a)] we obtained the following values for the
reconstructed parameters $\delta^{\star}=$29.15\text{\,}\mathrm{nm}$$ and
$a^{\star}=$211.29\text{\,}\mathrm{nm}$$. These values are less accurate than
what was found when basing the inversion on p-polarized scattering data; this
is the same conclusion that we drew when reconstructing scattering data from
our first scattering geometry. The corresponding reconstructed correlation
function is compared to the input correlation function are compared in Fig.
7(b). The absolute relative error obtained in the reconstruction of $\delta$
and $a$ when basing it on s-polarized scattering data is
$27.3\text{\,}\mathrm{\char 37\relax}$ and $7.7\text{\,}\mathrm{\char
37\relax}$, respectively. Also when the incident light is s-polarized the rms-
roughness is found to be reconstructed less accurately than the correlation
length. Moreover, for the roughness and geometrical parameters we assume, we
find that the reconstruction based on p-polarized scattering data overall
produces a more accurate reconstruction than when basing it on s-polarized
scattering data.

Figure 6: (Scattering system no. 2) Reconstruction of the rms-roughness
$\delta^{\star}$ and the transverse correlation length $a^{\star}$ from in-
plane p-to-p scattering data obtained for a Gaussian correlated silver
surface. The wavelength (in vacuum) of the incident light is
$\lambda=$457.90\text{\,}\mathrm{nm}$$ for which the dielectric function of
silver is $\varepsilon(\omega)=-7.50+0.24\mathrm{i}$. The surface roughness
parameters assumed in the computer simulations have the values
$\delta=\lambda/20=$22.90\text{\,}\mathrm{nm}$$ and
$a=\lambda/2=$228.95\text{\,}\mathrm{nm}$$. (a) The incoherent components of
the in-plane, co-polarized (p-to-p) mean differential reflection coefficient
$\left<\partial R_{pp}/\partial\Omega_{s}\right>_{\textrm{incoh}}$ as a
function of the polar angle of scattering $\theta_{s}$ obtained from computer
simulations for the polar angles of incidence $\theta_{0}=$$$ (open circles),
and from second-order phase perturbation theory with the use of the
reconstructed surface roughness parameters (solid line), for a two-dimensional
randomly rough silver surface whose correlation function is defined by Eq.
(26). The values of the reconstructed surface roughness parameters are
$\delta^{\star}=$25.69\text{\,}\mathrm{nm}$$ and
$a^{\star}=$223.42\text{\,}\mathrm{nm}$$. Computer simulation results for the
in-plane angular dependence of $\left<\partial
R_{sp}/\partial\Omega_{s}\right>_{\textrm{incoh}}$ (p-to-s cross-polarization)
is shown as a dashed line; this contribution is a result of multiple
scattering. The vertical dashed line represents the backscattering direction
$\theta_{s}=-\theta_{0}$. The mean DRC results produced by computer
simulations were obtained on the basis of $2500$ surface realizations. (b) The
input (open circles) and reconstructed (solid line) surface-height
autocorrelation function $W(|\mathbf{{x}}_{\parallel}|)$ for the random
surface. The shaded gray region represents the absolute difference between the
input and reconstructed surface-height autocorrelation functions. The notation
“$\times 2$” means that this difference has been multiplied by a factor $2$
for better visibility.

Figure 7: (Scattering system no. 2) The same as Fig. 6 but for s-polarized
incident light. The values of the reconstructed surface roughness parameters
are $\delta^{\star}=$29.15\text{\,}\mathrm{nm}$$ and
$a^{\star}=$211.29\text{\,}\mathrm{nm}$$. [Input values:
$\delta=\lambda/20=$22.90\text{\,}\mathrm{n}\mathrm{m}$$ and
$a=\lambda/2=$228.95\text{\,}\mathrm{n}\mathrm{m}$$].

####  V.1.3 Scattering system no. 3

The final scattering system that we will use for the purpose of generating
scattering data by computer simulation is again a rough silver surface.
However, in this case the wavelength of the incident light is
$\lambda=$632.80\text{\,}\mathrm{n}\mathrm{m}$$ (in vacuum) and the dielectric
function of silver at this wavelength we take to be
$\varepsilon(\omega)=-18.28+0.48\mathrm{i}$ Johnson and Christy (1972). The
surface roughness parameters that we assume are
$\delta=\lambda/40=$15.82\text{\,}\mathrm{n}\mathrm{m}$$ and
$a=\lambda/4=$158.20\text{\,}\mathrm{n}\mathrm{m}$$; they are identical to
those assumed in a recent study of the reconstruction based on scattering data
from rough dielectric surfaces Simonsen  _et al._ (2016). For these parameters
computer simulation, averaged over $2500$ surface realizations, were performed
Nordam, Letnes, and Simonsen (2013) to generate scattering data for p- and
s-polarized incident light for a polar angle of incidence $\theta_{0}=$$$.
Figures 8(a) and 9(a) present the results of such simulations for the in-plane
angular distributions of co-polarized (blue open symbols) and cross-polarized
(gray filled symbols) scattering; moreover, for better clarity thin sold lines
connects the data points of each of the data sets. For both polarizations and
for both co- and cross-polarized scattering, $\left<\partial
R_{\alpha\beta}/\partial\Omega_{s}\right>_{\textrm{incoh}}$ show enhanced
backscattering peaks at the polar angle of scattering $\theta_{s}=$$$ (dashed
vertical lines in the figures). These enhanced backscattering peaks, which are
the most pronounced for in-plane, co-polarized scattering, appear due to the
multiple scattering of SPPs propagating along the weakly rough silver surface.
From the results presented in Figs. 8(a) and 9(a) it is readily observed that
cross-polarized scattering is significant for the whole angular interval
$-$$<\theta_{s}<$$$. Furthermore, in this case, there is a readily detectable
difference between the in-plane angular dependence of the co-polarized
scattering data corresponding to p- and s-polarized incident light; this
difference is mainly due to the smaller value of the lateral correlation
length that characterize the surface roughness [$a=\lambda/4$]. One also
observes that the amplitude of the enhanced backscattering peak over its
background is higher in p-to-p scattering than what it is in s-to-s
scattering.

In order to reconstruct the parameters of the surface roughness for this
scattering system, we started by considering the data corresponding to
p-polarized incident light. Since the reconstruction is based on co-polarized
scattering, the input data for this reconstruction is the data set marked by
blue open symbols in Fig. 8(a) and, thus, it constitutes the input function
$\left<\partial
R_{pp}(\theta_{s})/\partial\Omega_{s}\right>_{\textrm{incoh,input}}$ that we
will use. The trial function was again assumed to have the Gaussian form (26)
and hence the variational parameter set is
${\mathcal{P}}=\\{\delta^{\star},a^{\star}\\}$. With initial values
$\delta^{\star}=$2.00\text{\,}\mathrm{nm}$$ and
$a^{\star}=$75.00\text{\,}\mathrm{nm}$$, the minimization procedure of the
cost function $\chi^{2}({\mathcal{P}})$ produced the surface roughness
parameter values $\delta^{\star}=$16.54\text{\,}\mathrm{nm}$$ and
$a^{\star}=$168.50\text{\,}\mathrm{nm}$$. These values compare favorably to
the surface roughness parameters used to generate the input scattering data.
For the relative difference between the reconstructed and input parameters we
find $4.5\text{\,}\mathrm{\char 37\relax}$ for the rms-roughness and
$6.5\text{\,}\mathrm{\char 37\relax}$ for the lateral correlation length.
Given the amount of contribution from multiple scattering that is present in
the input scattering data, we find this result quite encouraging. The
resulting co-polarized mean DRC and $W(|\mathbf{{x}}_{\parallel}|)$ obtained
on the basis of the reconstructed surface roughness parameters are presented
as solid lines in the two panels of Fig. 8.

Figure 8: (Scattering system no. 3) Reconstruction of the rms-roughness
$\delta^{\star}$ and the transverse correlation length $a^{\star}$ from in-
plane p-to-p scattering data obtained for a Gaussian correlated silver
surface. The wavelength (in vacuum) of the incident light is
$\lambda=$632.80\text{\,}\mathrm{nm}$$ for which the dielectric function of
silver is $\varepsilon(\omega)=-18.28+0.48\mathrm{i}$. The surface roughness
parameters assumed in the computer simulations have the values
$\delta=\lambda/40=$15.82\text{\,}\mathrm{nm}$$ and
$a=\lambda/4=$158.20\text{\,}\mathrm{nm}$$. (a) The incoherent components of
the in-plane, co-polarized (p-to-p) mean differential reflection coefficient
$\left<\partial R_{pp}/\partial\Omega_{s}\right>_{\textrm{incoh}}$ as a
function of the polar angle of scattering $\theta_{s}$ obtained from computer
simulations for the polar angles of incidence $\theta_{0}=$$$ (open circles),
and from second-order phase perturbation theory with the use of the
reconstructed surface roughness parameters (solid line), for a two-dimensional
randomly rough silver surface whose correlation function is defined by Eq.
(26). The values of the reconstructed surface roughness parameters are
$\delta^{\star}=$16.54\text{\,}\mathrm{nm}$$ and
$a^{\star}=$168.50\text{\,}\mathrm{nm}$$. The blue dashed line (gray shaded
region) represents the in-plane and cross-polarized (p-to-s) scattering data
which non-zero value is due to multiple scattering processes. The vertical
dashed line represents the backscattering direction $\theta_{s}=-\theta_{0}$.
The mean DRC results produced by computer simulations were obtained on the
basis of $2500$ surface realizations. (b) The input (open circles) and
reconstructed (solid line) surface-height autocorrelation function
$W(|\mathbf{{x}}_{\parallel}|)$ for the random surface. The shaded gray region
represents the absolute difference between the input and reconstructed
surface-height autocorrelation functions.

When a completely equivalent reconstruction to what was just done for p
polarization is performed based on the corresponding s-to-s scattering data
shown in Fig. 9(a), one obtains the reconstructed values
$\delta^{\star}=$16.78\text{\,}\mathrm{nm}$$ and
$a^{\star}=$168.84\text{\,}\mathrm{nm}$$ for the surface roughness parameters.
These values are essentially indistinguishable from the values obtained when
reconstructing p-to-p scattering data; the difference between them being a
fraction of a nanometer which is below the expected uncertainty one has on the
reconstructed values. The solid lines in the two panels of Fig. 9 represent
the ss component of the mean DRC, calculated on the basis of Eq. (15), and the
correlation function $W(|\mathbf{{x}}_{\parallel}|)$, calculated on the basis
of Eq. (26), when the values of the surface roughness parameters reconstructed
on the basis of s-to-s scattering data are used. A fairly good agreement are
found between these functions and those corresponding to the input functions.

It should be apparent from the results in Figs. 8(a) and 9(a) that the second-
order phase perturbation theory that we base the inversion on is not capable
to predict the enhanced backscattering peak that input data in these figures
show. Except for this multiple scattering feature of the input data, the phase
perturbation theoretical results seems to agree well with the input data over
the whole range of scattering angles $\theta_{s}$; in particular, the
difference in width of angular dependencies of the co-polarized scattered
intensities for p and s polarized light is well captured by it.

Figure 9: (Scattering system no. 3) The same as Fig. 8 but for s-polarized
incident light. The values of the reconstructed surface roughness parameters
are $\delta^{\star}=$16.78\text{\,}\mathrm{nm}$$ and
$a^{\star}=$168.84\text{\,}\mathrm{nm}$$. [Input values:
$\delta=\lambda/40=$15.82\text{\,}\mathrm{n}\mathrm{m}$$ and
$a=\lambda/4=$158.20\text{\,}\mathrm{n}\mathrm{m}$$].

We end this subsection with the remark that there is one important message we
hope the reader will take away from the different reconstruction examples
presented here, and in particular, in Figs. 8 and 9. Even if the input
scattering data receive significant contributions from multiple scattering,
the proposed reconstruction procedure for the surface roughness parameters
based on second-order phase perturbation theory is in principle capable of
obtaining useful, and often rather accurate, results for the value of these
parameters; at least, this was the case for the scattering systems and the
roughness and geometrical parameters that we have assumed.

###  V.2 Inversion of experimental scattering data

So far in this and our recent related work on the reconstruction of surface
roughness parameters based on scattering data for dielectric surfaces Simonsen
_et al._ (2016), we have exclusively based the inversion on scattering data
obtained by rigorous computer simulations. In generating such data, the
statistical properties of the used randomly rough surfaces were well
controlled both when it comes to the form of the correlation function and the
height distribution and the parameters that define these functions. Now we
address the more practically relevant case where the inversion is performed on
the basis of _experimental_ scattering data obtained when polarized light is
scattered from a rough metal surface.

In a series of experiments, Navarrete Alcalá et al. Navarrete Alcalá  _et al._
(2009) performed measurements of the in-plane and co-polarized angular
dependence of the mean DRC for incident light of wavelength
$\lambda=$10.6\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ (in vacuum) that was
scattered from two-dimensional randomly rough gold surfaces (also see Ref.
Navarrete  _et al._ , 2002). At this wavelength, the value of the dielectric
function of gold is $\varepsilon(\omega)=$-2489.77$+\mathrm{i}$2817.36$$ Palik
(1997). The rough surfaces used in these experiments were fabricated by the
method described in Refs. Gray, 1978; O’Donnell and Méndez, 1987 and
characterized by a Gaussian distribution of heights and an isotropic surface-
height autocorrelation function of an approximately Gaussian form. The
difference between the samples produced were therefore, in principle, found in
the parameters defining these functions, that is, the values of the roughness
parameters $\delta$ and $a$ of each sample. After production, each of the
rough surface were given a morphological characterization and the experimental
values for the roughness parameters $\delta$ and $a$ were obtained (see Refs.
Navarrete Alcalá  _et al._ , 2009; Navarrete  _et al._ , 2002).

Figure 10: (Sample 7047) Reconstruction of the rms-roughness $\delta^{\star}$
and transverse correlation length $a^{\star}$ from measured scattering data in
s-polarization obtained for a Gaussian correlated gold surface. The wavelength
of the incident light (in vacuum) is
$\lambda=$10.6\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ for which the
dielectric function of gold is assumed to be
$\varepsilon(\omega)=$-2489.77$+\mathrm{i}$2817.36$$ Palik (1997). The values
of the surface roughness parameters of the sample used in the experiment were
determined to be $\delta=$1.60\pm 0.05\text{\,}\mathrm{\SIUnitSymbolMicro m}$$
and $a=$9.50\pm 1.30\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ (or
$\delta\approx\lambda/7$ and $a\approx\lambda$) Navarrete Alcalá  _et al._
(2009). (a) The mean differential reflection coefficient $\left<\partial
R_{ss}/\partial\Omega_{s}\right>_{\textrm{incoh}}$ as a function of the polar
angle of scattering $\theta_{s}$ obtained experimentally for the polar angles
of incidence $\theta_{0}=$$$ (symbols), and from second-order phase
perturbation theory for $\theta_{0}=$$$ with the use of the reconstructed
surface roughness parameters (solid lines), for a two-dimensional randomly
rough gold surface whose correlation function is defined by Eq. (26). The
experimental data points (symbols) are taken from Ref. Navarrete Alcalá  _et
al._ , 2009 for sample 7047. When performing the reconstruction, however, only
the experimental data points marked by blue open symbols were included. The
values of the reconstructed surface roughness parameters are
$\delta^{\star}=$1.62\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and
$a^{\star}=$9.46\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ when the minimization
procedure was started from the values
$\delta^{\star}=$0.50\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and
$a^{\star}=$1.00\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. The dash-dotted and
dashed vertical lines indicate the specular and anti-specular (backscattering)
directions respectively. (b) The input (open circles) and reconstructed (solid
line) surface-height autocorrelation function $W(|\mathbf{{x}}_{\parallel}|)$
for the random surface; the former was obtained from Eq. (25) using the
experimentally determined value for $a$. The shaded gray region represents the
absolute difference between the input and reconstructed surface-height
autocorrelation functions. The notation “$\times 5$” means that this
difference has been multiplied by a factor $5$ for better visibility.

The first sample we consider was named sample 7047 by the authors of Ref.
Navarrete Alcalá  _et al._ , 2009. Their morphological characterization
confirmed that surface roughness of the sample was approximately characterized
by a Gaussian height distribution and a Gaussian correlation function. The
experimental values for the rms-roughness and the lateral correlation length
were determined to be $\delta=$1.60\pm 0.05\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$ and $a=$9.50\pm 1.30\text{\,}\mathrm{\SIUnitSymbolMicro m}$$,
respectively. In terms of the wavelength of the incident light, this means
that $\delta\approx\lambda/7$ and $a\approx\lambda$. The results of the
measurements for the in-plane dependence of the s-to-s component of the mean
DRC for sample 7047 when the light is incident at a polar angle
$\theta_{0}=$$$ are presented as symbols in Fig. 10(a); the dash-dotted and
dashed lines represent the specular and backscattering (anti-specular)
directions, respectively. For the purpose of the inversion, we are primarily
interested in the contribution to the mean DRC from the light that has been
scattered incoherently by the surface. Hence, only the experimental data
points presented as blue open symbols in Fig. 10(a) have been used for the
inversion; the data points shown as gray filled symbols around the specular
and anti-specular directions in this figure have been excluded. The reason
that we do not include the data points around the backscattering direction is
in the inversion is that here reliable measurements cannot be performed
Navarrete Alcalá  _et al._ (2009).

The set of data points shown as blue open symbols in Fig. 10(a) constitute the
input function $\left<\partial
R_{ss}(\theta_{s})/\partial\Omega_{s}\right>_{\textrm{incoh,input}}$ for our
first reconstruction example based on experimental data. To perform the
reconstruction of this data set, we assumed the trial function
$W(|\mathbf{{x}}_{\parallel}|)$ of the Gaussian form (26), and the set of
variational parameters is therefore
${\mathcal{P}}=\\{\delta^{\star},a^{\star}\\}$. The reconstruction procedure
was started with the values
$\delta^{\star}=$0.50\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and
$a^{\star}=$1.00\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and the values of
these parameters that minimize the cost function $\chi^{2}({\mathcal{P}})$,
Eq. (23), were found to be
$\delta^{\star}=$1.62\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and
$a^{\star}=$9.46\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. These values are in
excellent agreement with the corresponding values found from the analysis of
the surface topography map. This agreement is reflected in the good
correspondence between the functions $\left<\partial
R_{ss}(\theta_{s})/\partial\Omega_{s}\right>_{\textrm{incoh,input}}$ [open
symbols in Fig. 10(a)] and the function $\left<\partial
R_{ss}(\theta_{s})/\partial\Omega_{s}\right>_{\textrm{incoh,calc}}$ calculated
on the basis of Eq. (15) (for $\alpha=s$) using the reconstructed values for
the surface roughness parameters [solid line in Fig. 10(a)]. Figure 10(b)
compares the input and reconstructed surface-height autocorrelation function
$W(|\mathbf{{x}}_{\parallel}|)$. It should be mentioned that to arrive at the
results presented in Fig. 10 it was assumed that the polar angle of incidence
was $\theta_{0}=$$$, not $\theta_{0}=$$$ for which the experiments were
claimed to have been performed. We speculate that this behavior could be due
to an alignment issue of the sample; for instance, it is observed from the
experimental data in Fig. 10(a) that the specular peak is not at at $None$.

Figure 11: (Sample 8053) The same as Fig. 10 but for sample 8053 from Ref.
Navarrete Alcalá  _et al._ , 2009 for the polar angle of incidence
$\theta_{0}=$$$. In this publication, this sample is reported to have
approximately Gaussian statistics and characterized by the rms-roughness
$\delta=$0.75\pm 0.04\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and the lateral
correlation length $a=$3.00\pm 0.20\text{\,}\mathrm{\SIUnitSymbolMicro m}$$.
The reconstruction based on the experimental data points (open symbols in Fig.
11(a)) and the form (26) for $W(|\mathbf{{x}}_{\parallel}|)$, resulted in the
values $\delta^{\star}=$0.92\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and
$a^{\star}=$2.58\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ when the minimization
procedure was started from the values
$\delta^{\star}=$0.10\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and
$a^{\star}=$0.50\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. The region bounded
by the two dashed gray lines represents the variation in
$W(|\mathbf{{x}}_{\parallel}|)$, when assuming the form (26), that is due to
the experimental uncertainty on the value of the lateral correlation length
$a$.

The second experimental sample that we consider is sample 8053 from Ref.
Navarrete Alcalá  _et al._ , 2009. This is also a gold sample and the
morphological characterization of it revealed the rms-roughness
$\delta=$0.75\pm 0.04\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and the lateral
correlation length $a=$3.00\pm 0.20\text{\,}\mathrm{\SIUnitSymbolMicro m}$$
Navarrete Alcalá  _et al._ (2009); in terms of the wavelength of the incident
light [$\lambda=$10.6\text{\,}\mathrm{\SIUnitSymbolMicro m}$$] one has
$\delta\approx\lambda/14$ and $a\approx 2\lambda/7$. Also for this sample both
the height distribution and the correlation function were found to be well
described by Gaussian forms.

The symbols in Fig. 11(a) are the measured data points for the in-plane
dependence of the co-polarized component of the mean DRC when s-polarized
light is incident (from vacuum) on the surface of sample 8053 at the polar
angle of incidence $\theta=$$$. As was done when basing the reconstruction on
data for sample 7047, we neglected some of the measured data points around the
specular and anti-specular directions (filled symbols) so that only the
experimental data points shown as blue open symbols in Fig. 11(a) are
considered part of the function $\left<\partial
R_{ss}(\theta_{s})/\partial\Omega_{s}\right>_{\textrm{incoh,input}}$. Using
these input data, the minimization of the cost function (23) with respect to
the set of variational parameters
${\mathcal{P}}=\\{\delta^{\star},a^{\star}\\}$, resulted in the reconstructed
surface roughness parameter values
$\delta^{\star}=$0.92\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and
$a^{\star}=$2.58\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. These results were
obtained by assuming a Gaussian form (26) for $W(\mathbf{{x}}_{\parallel}|)$
and the initial values
$\delta^{\star}=$0.10\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and
$a^{\star}=$0.50\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. Moreover, for this
data set, the polar angle of incidence assumed in the reconstruction of it is
the same as the one reported for the experimental input data
[$\theta_{0}=$$$].

For sample 8053 the agreement between the experimental values for
$\delta^{\star}$ and $a^{\star}$ and the values obtained by reconstruction of
these parameters is less good than what was found previously for sample 7047.
However, the agreement is still fair; the relative errors in the surface
roughness parameters $\delta$ and $a$ are found to be
$16.5\text{\,}\mathrm{\char 37\relax}$ and $7.9\text{\,}\mathrm{\char
37\relax}$, respectively. The results of Fig. 11(a) show a good agreement
between the input and reconstructed in-plane angular dependence of the mean
DRC. Figure 11(b) shows that the lateral correlation length is underestimated
as compared to what is found by a direct examination of the surface
topography. In this figure the region between the two dashed gray lines
represents the variation in $W(|\mathbf{{x}}_{\parallel}|)$, when assuming the
form (26), that is due to the experimental uncertainty on the value of the
lateral correlation length $a$.

##  VI Conclusions

A detailed derivation of the expressions for the in-plane, p-to-p incoherent
light scattering from a two-dimensional randomly rough metal surface is
presented within second-order phase perturbation theory. Similar expressions
was recently obtained in Ref. Simonsen  _et al._ , 2016 for the scattering of
s-polarized light. On the basis of second-order phase perturbation theory, it
is shown that the mean differential reflection coefficient can be calculated
accurately for a wide range of two-dimensional randomly rough metal surfaces.
In particular, this is also the case for p-to-p scattering close to the
“pseudo-Brewster” scattering angle for which the reflectivity of the
corresponding planar metal surface has a minimum. In contrast, in a recent
study focusing on p-to-p incoherent light scattering from dielectric surfaces,
it was shown that the same theory produces unreliable results for polar
scattering angles around the Brewster angle Kryvi, Simonsen, and Maradudin
(2016).

An inversion approach based on in-plane, co-polarized scattering data is
introduced for the normalized surface-height autocorrelation function
$W(|\mathbf{{x}}_{\parallel}|)$ and the rms-roughness of a two-dimensional
randomly rough metal surface. To this end, a cost function is defined as the
in-plane angular integral of the square difference between the input mean DRC
and similar quantity calculated within second-order phase perturbation theory
for the same scattering angles. By minimizing this cost-function, the surface
rms-roughness and the correlation function $W(|\mathbf{{x}}_{\parallel}|)$
could be obtain for an assumed parametrized form of the latter function. The
forms for $W(|\mathbf{{x}}_{\parallel}|)$ that we considered were the Gaussian
and stretched exponential forms.

In this way, several sets of inversions were performed on two classes of
scattering data. The first class of scattering data were produced by rigorous
computer simulations based on the non-perturbative numerical solution of the
reduced Rayleigh equation Nordam, Letnes, and Simonsen (2013). For several
sets of roughness parameters, normal and non-normal incidence, and p and s
polarization of the incident light, inversion based on such scattering data
resulted in quite accurate reconstruction of both the correlation function and
the rms-roughness. This was also the case when we deliberately chose the
roughness parameters such that the scattering data displayed well-pronounced
enhanced scattering peaks which are signatures of multiple scattering. Even if
second-order phase perturbation theory is not able to produce such peaks, its
use in the inversion of the in-plane scattering data still produced rather
useful, and often accurate, results. At least, this was the case for the
roughness parameters we assumed in producing the scattering data on which the
inversion was performed.

The second class of scattering data were obtained by experimentally measuring
the intensity of the light that is diffusely scattered in-plane from two-
dimensional randomly rough gold surfaces that were specially prepared to be
characterized by Gaussian correlation functions. The reconstruction of the
first scattering data set we considered produced statistical properties of the
randomly rough surface that agreed quite well with those assumed in
fabricating the rough surface of the sample. The second sample we considered
assumed roughness parameters for which second-order phase perturbation theory
is expected to be less reliable due to the higher local slopes that the
surface has. Hence, it is not surprising that the results that we obtain by
the inversion are less accurate than for the first sample. Even for this
sample, the obtained results are useful. The obtained relative error in the
parameters characterizing the surface roughness of the sample, relative to the
input values, are found to be just below and above $10\text{\,}\mathrm{\char
37\relax}$ for the rms-roughness and the correlation length, respectively.

The inversion approach that we have presented is practical, rather accurate
(in many cases), computational efficient and therefore fast to perform.
However, the drawback of the present formulation of the approach is that it
relies on a parametrization for the autocorrelation function
$W(|\mathbf{{x}}_{\parallel}|)$. Further research should, therefore, focus on
the non-parametric reconstruction of the autocorrelation of the surface
profile function. Moreover, one also needs to determining the wavelength,
ranges of roughness and correlation lengths for which the proposed approach
can produce reliable results. These issues will be addressed in future work.

###### Acknowledgements.

The authors are indebted to Dr. A.G. Navarrete Alcalá and Dr. E.I. Chaikina
for providing the experimental scattering data from Ref. Navarrete Alcalá  _et
al._ , 2009 and to allow us to use them in this study. This research was
supported in part by NTNU and the Norwegian metacenter for High Performance
Computing (NOTUR) by the allocation of computer time. The research of I.S. was
supported in part by The Research Council of Norway (Contract No. 216699) and
the French National Research Agency (ANR-15-CHIN-0003).

*

##  Appendix A Derivation of Eq. (15)

The expressions derived in this Appendix are valid for a scattering geometry
where the substrate is either metallic or dielectric. To simplify the
notation, $\varepsilon$ will here denote the dielectric function of the
substrate suppressing any explicit reference to the frequency dependence of
this quantity.

The starting point for our derivation of the pp element of Eq. (15) is the pp
elements of Eqs. (12), (15), and (16)–(19) from Ref. McGurn and Maradudin,
1996, and the definition (12) of the scattering matrix
$S(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})$ in terms of the matrix
of reflection amplitudes ${\bf
R}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})$. From these equations
we obtain for the pp element of the scattering matrix the expansion

$\displaystyle S_{pp}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})$
$\displaystyle=S^{(0)}_{pp}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})-\mathrm{i}S^{(1)}_{pp}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})-\frac{1}{2}S^{(2)}_{pp}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})+\cdots$
(28)

where the superscript denotes the order of the corresponding term in the
surface profile function $\zeta(\mathbf{{x}}_{\parallel})$. The coefficient
$S^{(0)}_{pp}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})$ is

$\displaystyle
S^{(0)}_{pp}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})$
$\displaystyle=(2\pi)^{2}\delta(\mathbf{{q}}_{\parallel}-\mathbf{{k}}_{\parallel})\frac{\varepsilon\alpha_{0}(k_{\parallel})-\alpha(k_{\parallel})}{\varepsilon\alpha_{0}(k_{\parallel})+\alpha(k_{\parallel})}$
$\displaystyle=(2\pi)^{2}\delta(\mathbf{{q}}_{\parallel}-\mathbf{{k}}_{\parallel})\frac{\left[\varepsilon\alpha_{0}(q_{\parallel})-\alpha(q_{\parallel})\right]^{\frac{1}{2}}}{\left[\varepsilon\alpha_{0}(q_{\parallel})+\alpha(q_{\parallel})\right]^{\frac{1}{2}}}\frac{\left[\varepsilon\alpha_{0}(k_{\parallel})-\alpha(k_{\parallel})\right]^{\frac{1}{2}}}{\left[\varepsilon\alpha_{0}(k_{\parallel})+\alpha(k_{\parallel})\right]^{\frac{1}{2}}}$
$\displaystyle=(2\pi)^{2}\delta(\mathbf{{q}}_{\parallel}-\mathbf{{k}}_{\parallel})(\varepsilon-1)\frac{f_{p}^{\frac{1}{2}}(q_{\parallel})}{d_{p}(q_{\parallel})}\frac{f_{p}^{\frac{1}{2}}(k_{\parallel})}{d_{p}(k_{\parallel})},$
(29)

where the $f_{p}(q_{\parallel})$, $\alpha_{0}(q_{\parallel})$ and
$\alpha(q_{\parallel})$ are defined by Eqs. (18a), (8) and (17), respectively.
The coefficient
$S^{(1)}_{pp}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})$ is obtained
as

$\displaystyle
S^{(1)}_{pp}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})=$
$\displaystyle-2(\varepsilon-1)\alpha^{\frac{1}{2}}_{0}(q_{\parallel})\alpha^{\frac{1}{2}}_{0}(k_{\parallel})\frac{\varepsilon
q_{\parallel}k_{\parallel}-\alpha(q_{\parallel})\mathbf{\hat{q}}_{\parallel}\cdot\
\mathbf{\hat{k}}_{\parallel}\alpha(k_{\parallel})}{d_{p}(q_{\parallel})d_{p}(k_{\parallel})}\hat{\zeta}(\mathbf{{q}}_{\parallel}-\mathbf{{k}}_{\parallel}),$
(30)

while the coefficient
$S^{(2)}_{pp}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})$ is given by

$\displaystyle
S^{(2)}_{pp}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})$
$\displaystyle=-4(\varepsilon-1)\frac{\alpha^{\frac{1}{2}}_{0}(q_{\parallel})\alpha^{\frac{1}{2}}_{0}(k_{\parallel})}{d_{p}(q_{\parallel})d_{p}(k_{\parallel})}\int\\!\frac{\mathrm{d}^{2}p_{\parallel}}{(2\pi)^{2}}\,\hat{\zeta}(\mathbf{{q}}_{\parallel}-\mathbf{{p}}_{\parallel})\hat{\zeta}(\mathbf{{p}}_{\parallel}-\mathbf{{k}}_{\parallel})$
$\displaystyle\quad\times\bigg{\\{}\frac{1}{2}\left[\alpha(q_{\parallel})+\alpha(k_{\parallel})\right]\left[q_{\parallel}k_{\parallel}-\alpha(q_{\parallel})\mathbf{\hat{q}}_{\parallel}\cdot\mathbf{\hat{k}}_{\parallel}\alpha(k_{\parallel})\right]$
$\displaystyle\quad\qquad+\left(\frac{\varepsilon-1}{\varepsilon}\right)\bigg{[}\alpha(q_{\parallel})\mathbf{\hat{q}}_{\parallel}\cdot\mathbf{\hat{p}}_{\parallel}\alpha(p_{\parallel})\mathbf{\hat{p}}_{\parallel}\cdot\mathbf{\hat{k}}_{\parallel}\alpha(k_{\parallel})$
$\displaystyle\quad\qquad\qquad\qquad\qquad-\frac{\left[\varepsilon
q_{\parallel}p_{\parallel}-\alpha(q_{\parallel})\mathbf{\hat{q}}_{\parallel}\cdot\mathbf{\hat{p}}_{\parallel}\alpha(p_{\parallel})\right]\left[\varepsilon
p_{\parallel}k_{\parallel}-\alpha(p_{\parallel})\mathbf{\hat{p}}_{\parallel}\cdot\mathbf{\hat{k}}_{\parallel}\alpha(k_{\parallel})\right]}{d_{p}(p_{\parallel})}$
$\displaystyle\quad\qquad\qquad\qquad\qquad-\varepsilon\left(\frac{\omega}{c}\right)^{2}\frac{\alpha(q_{\parallel})\big{[}\mathbf{\hat{q}}_{\parallel}\times\mathbf{\hat{p}}_{\parallel}\big{]}_{3}\big{[}\mathbf{\hat{p}}_{\parallel}\times\mathbf{\hat{k}}_{\parallel}\big{]}_{3}\alpha(k_{\parallel})}{d_{s}(p_{\parallel})}\bigg{]}\bigg{\\}}.$
(31)

On substituting Eqs. (29), (30), and (31) into Eq. (28) we find that through
terms of second order in the surface profile function
$S_{pp}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})$ is given by

$\displaystyle S_{pp}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})=$
$\displaystyle\operatorname{\,sgn}(\mathbf{{\hat{q}}}_{\parallel}\cdot\mathbf{{\hat{k}}}_{\parallel})(\varepsilon-1)\frac{f_{p}^{\frac{1}{2}}(q_{\parallel})f_{p}^{\frac{1}{2}}(k_{\parallel})}{d_{p}(q_{\parallel})d_{p}(k_{\parallel})}\bigg{\\{}(2\pi)^{2}\delta(\mathbf{{q}}_{\parallel}-\mathbf{{k}}_{\parallel})\operatorname{\,sgn}(\mathbf{{\hat{q}}}_{\parallel}\cdot\mathbf{{\hat{k}}}_{\parallel})$
$\displaystyle\quad+2\mathrm{i}\frac{\alpha^{\frac{1}{2}}_{0}(q_{\parallel})\alpha^{\frac{1}{2}}_{0}(k_{\parallel})}{f_{p}^{\frac{1}{2}}(q_{\parallel})f_{p}^{\frac{1}{2}}(k_{\parallel})}\operatorname{\,sgn}(\mathbf{{\hat{q}}}_{\parallel}\cdot\mathbf{{\hat{k}}}_{\parallel})\left[\varepsilon
q_{\parallel}k_{\parallel}-\alpha(q_{\parallel})\mathbf{\hat{q}}_{\parallel}\cdot\mathbf{\hat{k}}_{\parallel}\alpha(k_{\parallel})\right]\hat{\zeta}(\mathbf{{q}}_{\parallel}-\mathbf{{k}}_{\parallel})$
$\displaystyle\quad+2\frac{\alpha^{\frac{1}{2}}_{0}(q_{\parallel})\alpha^{\frac{1}{2}}_{0}(k_{\parallel})}{f_{p}^{\frac{1}{2}}(q_{\parallel})f_{p}^{\frac{1}{2}}(k_{\parallel})}\operatorname{\,sgn}(\mathbf{{\hat{q}}}_{\parallel}\cdot\mathbf{{\hat{k}}}_{\parallel})\int\\!\frac{\mathrm{d}^{2}p_{\parallel}}{(2\pi)^{2}}\,\hat{\zeta}(\mathbf{{q}}_{\parallel}-\mathbf{{p}}_{\parallel})\hat{\zeta}(\mathbf{{p}}_{\parallel}-\mathbf{{k}}_{\parallel})$
$\displaystyle\quad\quad\times\bigg{[}\frac{1}{2}\left[\alpha(q_{\parallel})+\alpha(k_{\parallel})\right]\left(q_{\parallel}k_{\parallel}-\alpha(q_{\parallel})\mathbf{\hat{q}}_{\parallel}\cdot\mathbf{\hat{k}}_{\parallel}\alpha(k_{\parallel})\right)$
$\displaystyle\quad\qquad+\left(\frac{\varepsilon-1}{\varepsilon}\right)\bigg{(}\alpha(q_{\parallel})\mathbf{\hat{q}}_{\parallel}\cdot\mathbf{\hat{p}}_{\parallel}\alpha(p_{\parallel})\mathbf{\hat{p}}_{\parallel}\cdot\mathbf{\hat{k}}_{\parallel}\alpha(k_{\parallel})$
$\displaystyle\quad\qquad\quad-\frac{\left[\varepsilon
q_{\parallel}p_{\parallel}-\alpha(q_{\parallel})\mathbf{\hat{q}}_{\parallel}\cdot\mathbf{\hat{p}}_{\parallel}\alpha(p_{\parallel})\right]\big{[}\varepsilon
p_{\parallel}k_{\parallel}-\alpha(p_{\parallel})\mathbf{\hat{p}}_{\parallel}\cdot\mathbf{\hat{k}}_{\parallel}\alpha(k_{\parallel})\big{]}}{d_{p}(p_{\parallel})}$
$\displaystyle\quad\qquad\quad-\varepsilon\left(\frac{\omega}{c}\right)^{2}\frac{\alpha(q_{\parallel})\big{[}\mathbf{\hat{q}}_{\parallel}\times\mathbf{\hat{p}}_{\parallel}\big{]}_{3}\big{[}\mathbf{\hat{p}}_{\parallel}\times\mathbf{\hat{k}}_{\parallel}\big{]}_{3}\alpha(k_{\parallel})}{d_{s}(p_{\parallel})}\bigg{)}\bigg{]}\bigg{\\}}.$
(32)

This expression for
$S_{pp}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})$ is manifestly
reciprocal, i.e. it satisfies Eq. (13). In writing the expression (32), we
have for reasons of later convenience factored out a phase
$\\!\operatorname{\,sgn}(\mathbf{{\hat{q}}}_{\parallel}\cdot\mathbf{{\hat{k}}}_{\parallel})$,
where $\operatorname{\,sgn}(\cdot)$ denotes the signum (or sign) function
defined by $x=\operatorname{\,sgn}(x)|x|$.

We next express Eq. (32) in the from of a Fourier integral,

$\displaystyle S_{pp}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})$
$\displaystyle=\operatorname{\,sgn}(\mathbf{{\hat{q}}}_{\parallel}\cdot\mathbf{{\hat{k}}}_{\parallel})(\varepsilon-1)\frac{\left[f_{p}(q_{\parallel})f_{p}(k_{\parallel})\right]^{\frac{1}{2}}}{d_{p}(q_{\parallel})d_{p}(k_{\parallel})}\int\\!\mathrm{d}^{2}x_{\parallel}\,\,\exp\left[-\mathrm{i}(\mathbf{{q}}_{\parallel}-\mathbf{{k}}_{\parallel})\cdot\mathbf{{x}}_{\parallel}\right]$
$\displaystyle\quad\times\bigg{\\{}1+2\mathrm{i}\bigg{[}\frac{\alpha_{0}(q_{\parallel})\alpha_{0}(k_{\parallel})}{f_{p}(q_{\parallel})f_{p}(k_{\parallel})}\bigg{]}^{\frac{1}{2}}H_{p}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})\zeta(\mathbf{{x}}_{\parallel})+2\bigg{[}\frac{\alpha_{0}(q_{\parallel})\alpha_{0}(k_{\parallel})}{f_{p}(q_{\parallel})f_{p}(k_{\parallel})}\bigg{]}^{\frac{1}{2}}$
$\displaystyle\quad\qquad\times\int\\!\frac{\mathrm{d}^{2}p_{\parallel}}{(2\pi)^{2}}\,F_{p}(\mathbf{{q}}_{\parallel}|\mathbf{{p}}_{\parallel}|\mathbf{{k}}_{\parallel})\int\\!\mathrm{d}^{2}u_{\parallel}\,\exp\left[-\mathrm{i}(\mathbf{{p}}_{\parallel}-\mathbf{{k}}_{\parallel})\cdot\mathbf{{u}}_{\parallel}\right]\zeta(\mathbf{{x}}_{\parallel})\zeta(\mathbf{{x}}_{\parallel}+\mathbf{{u}}_{\parallel})\bigg{\\}},$
(33)

where the functions $H_{p}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})$
and
$F_{p}(\mathbf{{q}}_{\parallel}|\mathbf{{p}}_{\parallel}|\mathbf{{k}}_{\parallel})$
are defined by the expressions in Eqs. (19a) and (20), respectively. For
polarization $\alpha=p,s$, the functions
$F_{\alpha}(\mathbf{{q}}_{\parallel}|\mathbf{{p}}_{\parallel}|\mathbf{{k}}_{\parallel})$,
defined by Eq. (20), satisfy the relation
$F_{\alpha}(\mathbf{{q}}_{\parallel}|\mathbf{{p}}_{\parallel}|\mathbf{{k}}_{\parallel})=F_{\alpha}(-\mathbf{{k}}_{\parallel}|-\mathbf{{p}}_{\parallel}|-\mathbf{{q}}_{\parallel})$.
Moreover, the quantities
$F_{\alpha}(\mathbf{{q}}_{\parallel}|\mathbf{{p}}_{\parallel}|\mathbf{{k}}_{\parallel})$
are continuous functions of their first argument $\mathbf{{q}}_{\parallel}$
when the wave vector of the scattered light
$\mathbf{{q}}=\mathbf{{q}}_{\parallel}+\alpha_{0}(q_{\parallel})\mathbf{\hat{x}}_{3}$
varies in the plane of incidence.

From Eq. (33) and with the use of Eq. (1), we find

$\displaystyle\left<S_{pp}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})\right>$
$\displaystyle=\operatorname{\,sgn}(\mathbf{{\hat{q}}}_{\parallel}\cdot\mathbf{{\hat{k}}}_{\parallel})(\varepsilon-1)\frac{[f_{p}(q_{\parallel})f_{p}(k_{\parallel})]^{\frac{1}{2}}}{d_{p}(q_{\parallel})d_{p}(k_{\parallel})}\int\\!\mathrm{d}^{2}x_{\parallel}\,\exp\left[-\mathrm{i}(\mathbf{{q}}_{\parallel}-\mathbf{{k}}_{\parallel})\cdot\mathbf{{x}}_{\parallel}\right]$
$\displaystyle\qquad\times\bigg{\\{}1+2\delta^{2}\bigg{[}\frac{\alpha_{0}(q_{\parallel})\alpha_{0}(k_{\parallel})}{f_{p}(q_{\parallel})f_{p}(k_{\parallel})}\bigg{]}^{\frac{1}{2}}\int\\!\frac{\mathrm{d}^{2}p_{\parallel}}{(2\pi)^{2}}\,F_{p}(\mathbf{{q}}_{\parallel}|\mathbf{{p}}_{\parallel}|\mathbf{{k}}_{\parallel})\,g(|\mathbf{{p}}_{\parallel}-\mathbf{{k}}_{\parallel}|)\bigg{\\}}$
$\displaystyle\cong\operatorname{\,sgn}(\mathbf{{\hat{q}}}_{\parallel}\cdot\mathbf{{\hat{k}}}_{\parallel})(\varepsilon-1)\frac{[f_{p}(q_{\parallel})f_{p}(k_{\parallel})]^{\frac{1}{2}}}{d_{p}(q_{\parallel})d_{p}(k_{\parallel})}\int\\!\mathrm{d}^{2}x_{\parallel}\,\exp\left[-\mathrm{i}(\mathbf{{q}}_{\parallel}-\mathbf{{k}}_{\parallel})\cdot\mathbf{{x}}_{\parallel}\right]$
$\displaystyle\qquad\times\exp\bigg{\\{}2\delta^{2}\bigg{[}\frac{\alpha_{0}(q_{\parallel})\alpha_{0}(k_{\parallel})}{f_{p}(q_{\parallel})f_{p}(k_{\parallel})}\bigg{]}^{\frac{1}{2}}\int\\!\frac{\mathrm{d}^{2}p_{\parallel}}{(2\pi)^{2}}\,F_{p}(\mathbf{{q}}_{\parallel}|\mathbf{{p}}_{\parallel}|\mathbf{{k}}_{\parallel})\,g(|\mathbf{{p}}_{\parallel}-\mathbf{{k}}_{\parallel}|)\bigg{\\}}.$
(34)

It follows that

$\displaystyle\left|\left<S_{pp}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})\right>\right|^{2}$
$\displaystyle=\left|\varepsilon-1\right|^{2}\frac{|f_{p}(q_{\parallel})f_{p}(k_{\parallel})|}{|d_{p}(q_{\parallel})d_{p}(k_{\parallel})|^{2}}\exp\left[-2M_{p}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})\right]\int\\!\mathrm{d}^{2}x_{\parallel}\,\\!\\!\int\\!\mathrm{d}^{2}x_{\parallel}^{\prime}\,\exp\left[-\mathrm{i}(\mathbf{{q}}_{\parallel}-\mathbf{{k}}_{\parallel})\cdot(\mathbf{{x}}_{\parallel}-\mathbf{{x}}_{\parallel}^{\prime})\right],$
(35)

where we have introduced the function
$M_{p}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})$ defined by Eq.
(15b) with $\alpha=p$.

We next find that

$\displaystyle\left<\left|S_{pp}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})\right|^{2}\right>$
$\displaystyle=\left|\varepsilon-1\right|^{2}\frac{|f_{p}(q_{\parallel})f_{p}(k_{\parallel})|}{|d_{p}(q_{\parallel})d_{p}(k_{\parallel})|^{2}}\int\\!\mathrm{d}^{2}x_{\parallel}\,\\!\\!\int\\!\mathrm{d}^{2}x_{\parallel}^{\prime}\,\exp\left[-\mathrm{i}(\mathbf{{q}}_{\parallel}-\mathbf{{k}}_{\parallel})\cdot(\mathbf{{x}}_{\parallel}-\mathbf{{x}}_{\parallel}^{\prime})\right]$
$\displaystyle\quad\times\bigg{\\{}1+4\left|\frac{\alpha_{0}(q_{\parallel})\alpha_{0}(k_{\parallel})}{f_{p}(q_{\parallel})f_{p}(k_{\parallel})}\right|\left|H_{p}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})\right|^{2}\delta^{2}W(|\mathbf{{x}}_{\parallel}-\mathbf{{x}}_{\parallel}^{\prime}|)$
$\displaystyle\quad\qquad+4\delta^{2}\mathrm{Re}\,\left[\frac{\alpha_{0}(q_{\parallel})\alpha_{0}(k_{\parallel})}{f_{p}(q_{\parallel})f_{p}(k_{\parallel})}\right]^{\frac{1}{2}}\int\\!\frac{\mathrm{d}^{2}p_{\parallel}}{(2\pi)^{2}}\,F_{p}(\mathbf{{q}}_{\parallel}|\mathbf{{p}}_{\parallel}|\mathbf{{k}}_{\parallel})\,g(|\mathbf{{p}}_{\parallel}-\mathbf{{k}}_{\parallel}|)\bigg{\\}}$
$\displaystyle\cong\left|\varepsilon-1\right|^{2}\frac{|f_{p}(q_{\parallel})f_{p}(k_{\parallel})|}{|d_{p}(q_{\parallel})d_{p}(k_{\parallel})|^{2}}\int\\!\mathrm{d}^{2}x_{\parallel}\,\\!\\!\int\\!\mathrm{d}^{2}x_{\parallel}^{\prime}\,\exp\left[-\mathrm{i}(\mathbf{{q}}_{\parallel}-\mathbf{{k}}_{\parallel})\cdot(\mathbf{{x}}_{\parallel}-\mathbf{{x}}_{\parallel}^{\prime})\right]$
$\displaystyle\quad\times\exp\bigg{\\{}4\delta^{2}\mathrm{Re}\,\left[\frac{\alpha_{0}(q_{\parallel})\alpha_{0}(k_{\parallel})}{f_{p}(q_{\parallel})f_{p}(k_{\parallel})}\right]^{\frac{1}{2}}\int\\!\frac{\mathrm{d}^{2}p_{\parallel}}{(2\pi)^{2}}\,F_{p}(\mathbf{{q}}_{\parallel}|\mathbf{{p}}_{\parallel}|\mathbf{{k}}_{\parallel})\,g(|\mathbf{{p}}_{\parallel}-\mathbf{{k}}_{\parallel}|)$
$\displaystyle\quad\qquad\quad+4\delta^{2}\left|\frac{\alpha_{0}(q_{\parallel})\alpha_{0}(k_{\parallel})}{f_{p}(q_{\parallel})f_{p}(k_{\parallel})}\right|\left|H_{p}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})\right|^{2}W(|\mathbf{{x}}_{\parallel}-\mathbf{{x}}_{\parallel}^{\prime}|)\bigg{\\}}$
$\displaystyle=\left|\varepsilon-1\right|^{2}\frac{|f_{p}(q_{\parallel})f_{p}(k_{\parallel})|}{|d_{p}(q_{\parallel})d_{p}(k_{\parallel})|^{2}}\exp[-2M_{p}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})]\int\\!\mathrm{d}^{2}x_{\parallel}\,\\!\\!\int\\!\mathrm{d}^{2}x_{\parallel}^{\prime}\,\exp\left[-\mathrm{i}(\mathbf{{q}}_{\parallel}-\mathbf{{k}}_{\parallel})\cdot(\mathbf{{x}}_{\parallel}-\mathbf{{x}}_{\parallel}^{\prime})\right]$
$\displaystyle\quad\times\exp\bigg{\\{}4\delta^{2}\left|\frac{\alpha_{0}(q_{\parallel})\alpha_{0}(k_{\parallel})}{f_{p}(q_{\parallel})f_{p}(k_{\parallel})}\right|\left|H_{p}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})\right|^{2}W(|\mathbf{{x}}_{\parallel}-\mathbf{{x}}_{\parallel}^{\prime}|)\bigg{\\}}.$
(36)

On subtracting the expression in Eq. (35) from the expression in Eq. (36) we
obtain

$\displaystyle\left<\left|S_{pp}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})\right|^{2}\right>$
$\displaystyle-\left|\left<S_{pp}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})\right>\right|^{2}$
$\displaystyle=\left|\varepsilon-1\right|^{2}\frac{|f_{p}(q_{\parallel})f_{p}(k_{\parallel})|}{|d_{p}(q_{\parallel})d_{p}(k_{\parallel})|^{2}}\exp[-2M_{p}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})]\int\\!\mathrm{d}^{2}x_{\parallel}\,\\!\\!\int\\!\mathrm{d}^{2}x_{\parallel}^{\prime}\,\exp\left[-\mathrm{i}(\mathbf{{q}}_{\parallel}-\mathbf{{k}}_{\parallel})\cdot(\mathbf{{x}}_{\parallel}-\mathbf{{x}}_{\parallel}^{\prime})\right]$
$\displaystyle\quad\times\left\\{\exp\left[4\delta^{2}\left|\frac{\alpha_{0}(q_{\parallel})\alpha_{0}(k_{\parallel})}{f_{p}(q_{\parallel})f_{p}(k_{\parallel})}\right|\left|H_{p}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})\right|^{2}W(|\mathbf{{x}}_{\parallel}-\mathbf{{x}}_{\parallel}^{\prime}|)\right]-1\right\\}.$
(37)

Since the integrand on the right-hand-side of Eq. (37) depends on
$\mathbf{{x}}_{\parallel}$ and $\mathbf{{x}}_{\parallel}^{\prime}$ only
through the difference
$\mathbf{{x}}_{\parallel}-\mathbf{{x}}_{\parallel}^{\prime}$, it is convenient
to make the change of variable
$\mathbf{{x}}_{\parallel}^{\prime}=\mathbf{{x}}_{\parallel}-\mathbf{{u}}_{\parallel}$
in this equation. The integral over $\mathbf{{x}}_{\parallel}$ that the
resulting equation contains gives the area ${\mathcal{S}}$ of the plane
$x_{3}=0$ covered by the randomly rough surface. In this way we obtain

$\displaystyle\left<\left|S_{pp}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})\right|^{2}\right>$
$\displaystyle-\left|\left<S_{pp}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})\right>\right|^{2}$
$\displaystyle={\mathcal{S}}\left|\varepsilon-1\right|^{2}\frac{|f_{p}(q_{\parallel})f_{p}(k_{\parallel})|}{|d_{p}(q_{\parallel})d_{p}(k_{\parallel})|^{2}}\exp[-2M_{p}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})]$
$\displaystyle\quad\times\int\\!\mathrm{d}^{2}u_{\parallel}\,\exp\left[-\mathrm{i}(\mathbf{{q}}_{\parallel}-\mathbf{{k}}_{\parallel})\cdot\mathbf{{u}}_{\parallel}\right]\left\\{\exp\left[4\delta^{2}\left|\frac{\alpha_{0}(q_{\parallel})\alpha_{0}(k_{\parallel})}{f_{p}(q_{\parallel})f_{p}(k_{\parallel})}\right|\left|H_{p}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})\right|^{2}W(|\mathbf{{u}}_{\parallel}|)\right]-1\right\\}.$
(38)

The double integral over $\mathbf{{u}}_{\parallel}$ that this equation
contains can be transformed to a single integral over the length of this
vector by noting that the part of the integrand in the curly brackets in Eq.
(38) is a function of $u_{\parallel}=|\mathbf{{u}}_{\parallel}|$ only. This
enables us to perform the directional integration analytically to produce a
result depending on the Bessel function of the first kind and order zero
[denoted $J_{0}(\cdot)$ below]. In this way we find that Eq. (38) can
alternatively be written in the equivalent form

$\displaystyle\left<\left|S_{pp}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})\right|^{2}\right>$
$\displaystyle-\left|\left<S_{pp}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})\right>\right|^{2}$
$\displaystyle=2\pi{\mathcal{S}}\left|\varepsilon-1\right|^{2}\frac{|f_{p}(q_{\parallel})f_{p}(k_{\parallel})|}{|d_{p}(q_{\parallel})d_{p}(k_{\parallel})|^{2}}\exp[-2M_{p}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})]$
$\displaystyle\quad\times\int\limits^{\infty}_{0}\\!\mathrm{d}u_{\parallel}\,u_{\parallel}J_{0}(|\mathbf{{q}}_{\parallel}-\mathbf{{k}}_{\parallel}|u_{\parallel})\left\\{\exp\left[4\delta^{2}\left|\frac{\alpha_{0}(q_{\parallel})\alpha_{0}(k_{\parallel})}{f_{p}(q_{\parallel})f_{p}(k_{\parallel})}\right|\left|H_{p}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})\right|^{2}W(u_{\parallel})\right]-1\right\\}.$
(39)

The substitution of the result from Eq. (39), or equivalent [Eq. (38)], into
Eq. (14) for $\alpha=\beta=p$ yields Eq. (15) for $\alpha=p$.

We now turn to the case of s polarization. In the Appendix of Ref. Simonsen
_et al._ , 2016, Eq. (A5), an expression for
$S_{ss}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})$ was recently
derived within second-order phase perturbation theory. In order to harmonize
the notation for the expressions for
$S_{\alpha\alpha}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})$ for p-
and s-polarized light, we here recast Eq. (A5) of Ref. Simonsen  _et al._ ,
2016 in the alternative form

$\displaystyle S_{ss}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})$
$\displaystyle=-\operatorname{\,sgn}(\mathbf{{\hat{q}}}_{\parallel}\cdot\mathbf{{\hat{k}}}_{\parallel})(\varepsilon-1)\frac{\left[f_{s}(q_{\parallel})f_{s}(k_{\parallel})\right]^{\frac{1}{2}}}{d_{s}(q_{\parallel})d_{s}(k_{\parallel})}\int\\!\mathrm{d}^{2}x_{\parallel}\,\,\exp\left[-\mathrm{i}(\mathbf{{q}}_{\parallel}-\mathbf{{k}}_{\parallel})\cdot\mathbf{{x}}_{\parallel}\right]$
$\displaystyle\quad\times\bigg{\\{}1+2\mathrm{i}\bigg{[}\frac{\alpha_{0}(q_{\parallel})\alpha_{0}(k_{\parallel})}{f_{s}(q_{\parallel})f_{s}(k_{\parallel})}\bigg{]}^{\frac{1}{2}}H_{s}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})\zeta(\mathbf{{x}}_{\parallel})+2\bigg{[}\frac{\alpha_{0}(q_{\parallel})\alpha_{0}(k_{\parallel})}{f_{s}(q_{\parallel})f_{s}(k_{\parallel})}\bigg{]}^{\frac{1}{2}}$
$\displaystyle\quad\qquad\times\int\\!\frac{\mathrm{d}^{2}p_{\parallel}}{(2\pi)^{2}}\,F_{s}(\mathbf{{q}}_{\parallel}|\mathbf{{p}}_{\parallel}|\mathbf{{k}}_{\parallel})\int\\!\mathrm{d}^{2}u_{\parallel}\,\exp\left[-\mathrm{i}(\mathbf{{p}}_{\parallel}-\mathbf{{k}}_{\parallel})\cdot\mathbf{{u}}_{\parallel}\right]\zeta(\mathbf{{x}}_{\parallel})\zeta(\mathbf{{x}}_{\parallel}+\mathbf{{u}}_{\parallel})\bigg{\\}},$
(40)

with $f_{s}(q_{\parallel})$ and
$H_{s}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})$ defined by Eqs.
(18b) and (19b), respectively, and

$\displaystyle
F_{s}(\mathbf{{q}}_{\parallel}|\mathbf{{p}}_{\parallel}|\mathbf{{k}}_{\parallel})=$
$\displaystyle-\left(\frac{\omega}{c}\right)^{2}F(\mathbf{{q}}_{\parallel}|\mathbf{{p}}_{\parallel}|\mathbf{{k}}_{\parallel}).$
(41)

Here the function
$F(\mathbf{{q}}_{\parallel}|\mathbf{{p}}_{\parallel}|\mathbf{{k}}_{\parallel})$
is defined by Eq. (A6) of Ref. Simonsen  _et al._ , 2016 so that the
expression for
$F_{s}(\mathbf{{q}}_{\parallel}|\mathbf{{p}}_{\parallel}|\mathbf{{k}}_{\parallel})$
is given by Eq. (20). The expression for
$S_{ss}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})$ in Eq. (40) is, up
to a phase factor, form equivalent to the expression for
$S_{pp}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})$ given in Eq. (33);
we only have to make sure that we use the functions
$d_{\alpha}(q_{\parallel})$, $f_{\alpha}(q_{\parallel})$,
$H_{\alpha}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})$, and
$F_{\alpha}(\mathbf{{q}}_{\parallel}|\mathbf{{p}}_{\parallel}|\mathbf{{k}}_{\parallel})$
when writing the expression for
$S_{\alpha\alpha}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})$. The
consequence of this is that the expressions in Eqs. (34)–(42) derived for
p-to-p scattering also should hold for s-to-s scattering given that the
polarization indices p are replaced by the indices s, and hence the expression
within phase perturbation theory for the $\left<\partial
R_{ss}/\partial\Omega_{s}\right>_{\textrm{incoh}}$ is given by Eq. (15) with
$\alpha=s$.

Before closing this Appendix, we would like to make a few final remarks. For
some forms of the correlation function $W(|\mathbf{{u}}_{\parallel}|)$, like
the Gaussian form, it can be of advantage to expand in powers of $\delta^{2}$
the exponential function present in the integrands of Eq. (38) or (39) and
integrate the resulting series term-by-term. Applying this procedure to the
integrand in Eq. (39) (also valid for $\alpha=s$) will, for instance, produce

$\displaystyle\left<\left|S_{\alpha\alpha}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})\right|^{2}\right>$
$\displaystyle-\left|\left<S_{\alpha\alpha}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})\right>\right|^{2}$
$\displaystyle=2\pi{\mathcal{S}}\left|\varepsilon-1\right|^{2}\frac{\left|f_{\alpha}(q_{\parallel})f_{\alpha}(k_{\parallel})\right|}{\left|d_{\alpha}(q_{\parallel})d_{\alpha}(k_{\parallel})\right|^{2}}\exp\left[-2M_{\alpha}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})\right]$
$\displaystyle\quad\times\sum\limits^{\infty}_{n=1}\frac{1}{n!}\left[4\delta^{2}\left|\frac{\alpha_{0}(q_{\parallel})\alpha_{0}(k_{\parallel})}{f_{\alpha}(q_{\parallel})f_{\alpha}(k_{\parallel})}\right|\left|H_{\alpha}(\mathbf{{q}}_{\parallel}|\mathbf{{k}}_{\parallel})\right|^{2}\right]^{n}\int\limits^{\infty}_{0}\\!\mathrm{d}u_{\parallel}\,u_{\parallel}J_{0}(|\mathbf{{q}}_{\parallel}-\mathbf{{k}}_{\parallel}|u_{\parallel})W^{n}(u_{\|}).$
(42)

The expression in Eq. (42) may represent an advantage over the expression in
Eq. (39) if the integrals contained in the former equation can be calculated
analytically while the integral in the latter has to be calculated
numerically. For instance, if the correlation function
$W(|\mathbf{{u}}_{\parallel}|)$ has the Gaussian form (25), it is
straightforward to show that for $n$ a positive integer

$\displaystyle\int\\!\mathrm{d}^{2}u_{\parallel}\,\exp\left[-i(\mathbf{{q}}_{\parallel}-\mathbf{{k}}_{\parallel})\cdot\mathbf{{u}}_{\parallel}\right]W^{n}(|\mathbf{{u}}_{\parallel}|)$
$\displaystyle=\frac{\pi
a^{2}}{n}\exp\left[-\frac{a^{2}}{4n}\left(\mathbf{{q}}_{\parallel}-\mathbf{{k}}_{\parallel}\right)^{2}\right],$
(43)

which is the integral that will result after expanding the exponential
function of the integrand in Eq. (38).

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