# Replacement of Ensemble Averaging by the Use of a Broadband Source in Scattering of Light from a One-Dimensional Randomly Rough Interface between Two Dielectric Media

Authors: Alexei A. Maradudin; Ingve Simonsen
Journal: International Journal of Antennas and Propagation
 (2018)

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## Abstract

By the use of phase perturbation theory we show that if a single realization of a one-dimensional randomly rough interface between two dielectric media is illuminated at normal incidence from either medium by a broadband Gaussian beam, it produces a scattered field whose differential reflection coefficient closely matches the result produced by averaging the differential reflection coefficient produced by a monochromatic incident beam over the ensemble of realizations of the interface profile function.


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## Body

## 1. Introduction

In theoretical calculations of some property of monochromatic light scattered from, or transmitted through, a randomly rough surface, such as the angular or spatial dependence of its intensity, what is actually calculated is the average of that property over the ensemble of realizations of the random surface profile. This procedure averages over the speckles that would be produced if monochromatic light were scattered by or transmitted through a single realization of the random surface and produces a smooth angular or spatial dependence of the property of interest.In an experiment, this property is measured for a single realization of the random surface. If the surface is illuminated by a monochromatic source, the resulting speckles have to be averaged in some way to produce the kind of smooth curve that ensemble averaging yields. This can be done by rotating or dithering the sample. However, in some cases, moving the surface is not an option. In such cases, one can exploit the fact that the speckle pattern depends on the wavelength of the incident light [1] to average over the speckles by using a broadband (polychromatic) beam to illuminate the surface instead of a monochromatic one. In an earlier paper [2], it was demonstrated that illuminating one realization of a one-dimensional randomly rough perfectly conducting surface by an s-polarized broadband Gaussian beam produced an intensity profile of the scattered field that closely matched the one produced by averaging the intensity of the scattered field produced by a monochromatic incident beam over the ensemble of realizations of the random surface profile function.In this paper, we explore the replacement of ensemble averaging by the use of an incident broadband Gaussian beam in the more realistic case where the one-dimensional rough interface between two dielectric media is illuminated at normal incidence from either medium and the differential reflection coefficient of the scattered light is sought.


## 2. Scattering Theory

The system we study consists of a dielectric medium whose dielectric constant isε1 in the region x3 > ζ(x1) and a dielectric medium whose dielectric constant is ε2 in the region x3 < ζ(x1) (Figure 1). Both ε1 and ε2 are assumed to be real, positive, and frequency independent. The interface profile function ζ(x1) is assumed to be a single-valued function of x1 that is differentiable and constitutes a random process. This interface is illuminated at normal incidence from the region x3 > ζ(x1) by a p- or s-polarized broadband Gaussian beam of light of angular frequency ω, whose plane of incidence is the x1x3 plane. The single nonzero component of its electromagnetic field is a weighted superposition of incoming plane waves,
(1)Fvx1,x3;tinc=∫−∞∞dω2π∫−ε1ωcε1ωcdk2πWk,ωexpikx1−iα1k,ωx3−iωt,where Fν(x1,x3;t)inc is H2(x1,x3;t)inc when ν = p and is E2(x1,x3;t) when ν = s. The function α1(k,ω) is defined by α1k,ω=ε1ω/c2−k21/2 with Reα1(k,ω) > 0 and Imα1(k,ω) > 0, where c is the speed of light in vacuum. The weight function W(k,ω) has the factored form
(2)Wk,ω=GkFω,where
(3)Gk=2πα1k,ωε1wω2cexp−ε1wω2c2arcsin2kcε1ω,while F(ω) is a random function that possesses the properties
(4)FωF∗ω′F=2πδω−ω′S0ω,FωFω′F=0.The angle brackets ⋯F here denote an average over the ensemble of realizations of the field [3]. An incident field of this nature is produced by a superluminescent diode [4], for example. We assume that the spectral density of the incident field S0(ω) has a Gaussian form with a central frequency ω0 and a 1/e halfwidth ∆ω,
(5)S0ω=1πΔωexp−ω−ω0Δω2.In the following, it will be assumed that the halfwidth is small enough that the spectral density of the incident light can be regarded as zero when ω < 0. Moreover, for convenience, the function F(ω) will be regarded as zero when ω < 0.Figure 1
Schematics of the scattering geometry.In the limit thatε1wω/2c≫1, that will be assumed here, (1) represents a Gaussian beam of 1/e halfwidth w that is incident normally on the rough interface. To see this, one will make the change of variable k=ε1ω/csinθ in (1) which results in a Gaussian integral on the right-hand side of this equation that is evaluated analytically to produce
(6)Fvx1,x3;tinc=∫−∞∞dω2πFωexp−x1w2−iε1ωcx3−iωt.Due to the linearity of the scattering problem, the scattered field can be written as(7)Fvx1,x3;tsc=∫−∞∞dω2πFω∫−ε1ωcε1ωcdk2πGk∫−∞∞dq2πRvq∣kexpiqx1+iα1q,ωx3−iωt,where Rν(q|k) is the scattering amplitude that is obtained when the incident field is given by expikx1−iα1k,ωx3−iωt or
(8)Fvr,θs;tsc=∫−∞∞dω2πFω∫−ε1ωcε1ωcdk2πGk∫−∞∞dq2πRvq∣kexpiqrsinθs+ia1q,ωrcosθs−iωt,where r=x12+x32 and x1=rsinθs, x3=rcosθs. It can be calculated by any of the several approaches such as small amplitude perturbation theory [5], the Kirchhoff approximation [6], phase-perturbation theory [7], and by rigorous numerical solutions of the equations of scattering theory [8]. We will use here the first-order phase perturbation theory expression for Rvq∣k, due to its simplicity and because it interpolates between small-amplitude perturbation theory and the Kirchhoff approximation.In small-amplitude perturbation theory, the expression obtained forRvq∣k to the lowest nonzero order in the interface profile function ζ(x1) is
(9)Rvq∣k=Rvk2πδq−k+iΦvq∣kζ^q−k+⋯,where
(10)ζ^Q=∫−∞∞dx1ζx1exp−iQx1,while
(11)Rpk=ε2α1k,ω−ε1α2k,ωε2α1k,ω+ε1α2k,ω,Φpq∣k=ε2−ε1ε2α1q,ω+ε1α2q,ωε2qk−ε1α2q,ωα2k,ω2α1k,ωε2a1k,ω−ε1α2k,ω,and
(12)Rsk=α1k,ω−α2k,ωα1k,ω+α2k,ω,Φsq∣k=α2q,ω−α1q,ω2α1k,ωα1k,ω−α2k,ω,with α2k,ω=ε2ω/c2−k21/2, Reα2k,ω>0, and Imα2k,ω>0. We can rewrite the right-hand side of (9) as a Fourier integral,
(13)Rνq∣k=Rνk∫−∞∞dx1exp−iq−kx11+iΦνq∣kζx1+⋯.On exponentiating the expression in brackets in the integrand in this expression, we obtain the first-order phase perturbation theory expression for Rvq∣k,
(14)Rvq∣k=Rvk∫−∞∞dx1exp−iq−kx1expiΦvq∣kζx1.We will use this expression here due to its simplicity. After interchanging the order of the k and q integrations in (7), it follows that the scattering amplitude for a Gaussian monochromatic beam of frequency ω can be expressed as
(15)Rvq,ω=∫−ε1ωcε1ωcdk2πRvq∣kGk,so that the scattered field becomes
(16)Fvx1,x3;tsc=∫−∞∞dω2πFω∫−∞∞dq2πRvq,ωexpiqx1+iα1q,ωx3−iωt.The scattering amplitude Rvq,ω enters the definition of the differential reflection coefficient which is defined as the fraction of the power flux incident onto the rough interface that is scattered into an angular interval of width dθs about the scattering angle θs [9]. For an illumination of the random interface by a normally incident Gaussian beam of frequency ω, the expression for the differential reflection coefficient in the wide beam limit ε1wω/2c≫1 reads [8, 10]
(17)∂Rv∂θsθs=ε12π3/2ωcwcos2θsRq,ω2,where q=ε1ω/csinθs. When the differential reflection coefficient from (17), for monochromatic illumination at frequency ω0, is averaged over an ensemble of realizations of the random interface, the mean differential reflection coefficient is obtained and we denote it ∂Rv/∂θs in the following. On the other hand, if the illumination of the random interface is done by a broadband source and characterized by the center frequency ω0 and the halfwidth is ∆ω, the “broadband” differential reflection coefficient ∂Rv/∂θsF is obtained.We now turn to the calculation of a simple expression for the scattering amplitudeRvq,ω. On substituting into (15), the results from (14) and (3) and making the change of variable k=ε1ω/csinθ in the resulting expression, one gets
(18)Rvq,ω=ε1πwω2c∫−π/2π/2dθexp−ε1wω2c2θ2Rvε1ωcsinθ×∫−∞∞dx1exp−iqx1+iε1ωcsinθx1expiΦvq∣ε1ωcsinθζx1.On passing to the limit ε1wω/2c≫1, corresponding to a wide Gaussian beam, one may take advantage of the approximations sinθ≈0 and cosθ≈1, with the consequence that Φν becomes a linear function of the variable θ,
(19)Φvq∣ε1ωcsinθ≈ϕv0q,ω+ϕv1q,ωθ,with
(20)ϕv0q,ω=Φvq∣0,and
(21)ϕp1q,ω=2ωcε1ε2ε1+ε2qε2α1q,ω+ε1α2q,ωϕs1q,ω=0.Moreover, in the limit ε1wω/2c≫1, the θ integral in (18) takes the Gaussian form, due to the results in (19)–(21) and can hence be evaluated analytically with the result that
(22)Rvq,ω=Rv0∫−∞∞dx1exp−x1w+ϕv1q,ωζx1ε1ω/cw2−iqx1+iϕv0q,ωζx1.This is the simplified expression for the scattering amplitude derived from phase perturbation theory that we will use to produce the results to be presented later in this paper. It represents a significant simplification relative to, for instance, obtaining the scattering amplitude by rigorous means [8], which requires solving a linear system of equations that becomes time-consuming when the interface becomes long.The interface profile functionζ(x1) is assumed to be a single-valued function of x1 that is differentiable and constitutes a zero-mean, stationary Gaussian random process defined by
(23)ζx1ζx1′=δ2W∣x1−x1′∣.The angle brackets here denote an average over the ensemble of realizations of ζx1,δ=ζ2x11/2 is the root mean-square height of the interface, and W(|x1|) is the normalized interface height auto-correlation function.The power spectrum of the interface roughness,g(|k|), is the Fourier transform of W(|x1|),
(24)gk=∫−∞∞dx1Wx1exp−ikx1.In the calculations carried out in this work, W(|x1|), will be assumed to have the Gaussian form
(25)Wx1=exp−x12a2,where the characteristic length a is the transverse correlation length if the interface roughness. The power spectrum g(|k|) in this case also has the Gaussian form
(26)gk=πaexp−k2a24.A single realization of the interface profile function is given by [11]
(27)ζx1=δ2L∑m=1∞g2πmL1/2ξ2m−1sin2πmx1L+ξ2mcos2πmx1L.In this expression, the {ξm} are independent Gaussian random deviates with zero mean and unit variance:
(28)ξm=0,ξm2=1.The function defined by (27) is a periodic function of x1 with a period L. To avoid edge effects, only the portion of this function in the interval −L/2 < x1 < L/2 where L=ℒ/2 is used in calculations.


## 3. Results and Discussion

In Figure2(a), we present a plot of the speckle pattern of the scattered field as a function of the scattering angle θs produced by a monochromatic p-polarized Gaussian beam of frequency ω0, given by (6) with F(ω) = 2πδ(ω − ω0), incident on a single realization of a one-dimensional randomly rough interface generated with the use of (27). In Figure 2(b), we present the differential reflection coefficient of the scattered field given by (16) when the realization of the interface profile function used in obtaining Figure 2(a) is illuminated by a broadband Gaussian beam whose center frequency is ω0 with a halfwidth ∆ω=0.2ω0. To obtain this result, calculations of the differential reflection coefficient were carried out for several values of ∆ω, ranging from 0.2ω0 to 0.4ω0. The results did not differ in any significant way so we chose to use the smallest of these ∆ω values. It should be remarked that when a different realization of the surface was illuminated by the same broadband beam used to produce the result in Figure 2(b), the result was essentially ∂Rp/∂θsF from the same figure except for some small-amplitude fine details. Finally, in Figure 2(c), we plot the mean differential reflection coefficient ∂Rp/∂θs obtained by averaging the results from Np = 10,000 realizations of the interface profile function generated by (27) when the interface is illuminated by the same monochromatic Gaussian beam of frequency ω0 used in obtaining Figure 2(a). The values of the theoretical and experimental parameters assumed in obtaining these results were ε1=1, ε2=2.25, λ0=2πc/ω0=632.8nm, and w=L/4 where L is the length of the surface. The parameters defining the interface roughness were δ=0.15λ0, a=1.50λ0, and L=104λ0. The sampling interval used was ∆x1 = λ0/10 so that the interface was discretized onto N = 105 points. Figure 3 presents corresponding results for s-polarized incident beams.Figure 2
The differential reflection coefficient∂Rp/∂θs defined by (17) for a vacuum-glass system [ε1=1 and ε2=2.25] illuminated from the vacuum side by different p-polarized incident beams: (a) a Gaussian monochromatic beam of halfwidth w and frequency ω0; (b) a broadband Gaussian beam of center frequency ω0 and frequency bandwidth ∆ω=0.2ω0; and (c) a monochromatic beam as in Figure 2(a) but with an average performed over an ensemble of Np = 10,000 realizations of the interface profile function. The angle of incidence was θ0=0° and the wavelength was λ0 = 2πc/ω0 = 632.8 nm. The randomly rough interface was characterized by the parameters δ=0.15λ0, a=1.50λ0, and L=104λ0, and the sampling interval used to discretize the surface was ∆x1=λ0/10. For the width of the Gaussian beam, the value w=L/4 was used. The same random interface was used in producing the results of the first two panels of this figure.Figure 3
Same as Figure2 but for s-polarized incident beams, the single realization of the rough interface used to obtain the results of the first two panels of this figure is the same one used in Figure 2.The results presented in Figures2 and 3 show that the use of a broadband beam in illuminating a single realization of a one-dimensional randomly rough interface averages over the speckles produced by a monochromatic beam. It therefore produces a differential reflection coefficient that closely matches the one produced by a monochromatic beam when the resulting differential reflection coefficient is averaged over the ensemble of realizations of the interface profile function.To facilitate the comparison of∂Rv/∂θsF and ∂Rv/∂θs, in Figures 4(a) and 4(c), we plot simultaneously these quantities on a semilogarithmic scale. It is observed that the agreement between them is rather good even in the tails of the scattered intensity distributions.Figure 4
The quantities∂Rv/∂θsF and ∂Rv/∂θs obtained for a vacuum-glass system [ε1=1 and ε2=2.25] plotted on semilogarithmic scales for ν = p [Figure 4(a)] and ν = s [Figure 4(c)]. Comparison of the mean differential reflection coefficients ∂Rv/∂θs obtained by either a rigorous simulation approach or on the basis of phase perturbation theory (PPT) for p-polarized (Figure 4(b)) or s-polarized (Figure 4(d)) illumination. All ensemble averaged quantities were obtained on the basis of Np = 10,000 realizations of the interface profile function. The remaining parameters are identical to those of Figure 2.The results presented in Figures2 and 3 were, for convenience, all obtained under the assumption of phase perturbation theory. In Figure 4, we compare the mean differential reflection coefficients ∂Rv/∂θs from Figures 2(c) and 3(c) to rigorous computer simulation results obtained by solving the equations of scattering theory [8]. The agreement between the two sets of results is rather convincing. In passing, it should be noted that in performing the rigorous simulations the length used for the rough interface was L′=102λ0 and the width of the Gaussian beam was L′/4. This length of the rough interface is two orders of magnitude shorter than the length used in obtaining the results based on phase perturbation theory.We now turn to a scattering geometry where the medium of incidence is the optically denser medium. Here, we assume a glass-vacuum system characterized byε1=2.25 and ε2=1. Physically, this corresponds to the light being incident from the opposite side of the rough interface relative to the system we previously studied. The results for the differential reflection coefficients for the glass-vacuum system are presented in Figures 5, 6, and 7. In these figures, the angles for which θs≥θs∗ have been indicated as shaded regions, where θs∗=arcsinε2/ε1=41.81° denotes the critical angle for total internal reflection. Due to the assumptions underlying phase perturbation theory, it is expected not to work well when the angles of incidence and/or scattering in absolute value are larger than this critical angle. Hence, in Figures 5, 6, and 7, we have plotted the results obtained on the basis of phase perturbation theory only for θs<θs∗.Figure 5
Same as Figure2, that is, p-polarized illumination, but for a system where the medium of incidence is the optically denser medium (ε1=2.25 and ε2=1), the shaded regions correspond to θs≥θs∗ where θs∗=arcsinε2/ε1=41.81∘ denotes the critical angle for total internal reflection. The results are presented only for θs<θs∗.Figure 6
Same as Figure5 (ε1=2.25 and ε2=1) but assuming s-polarized illumination.Figure 7
Same as Figure4 but for a glass-vacuum system (ε1=2.25 and ε2=1).On the basis of the results presented in Figures5, 6, and 7, it is concluded that also for systems where the medium of incidence is the optically denser medium, one finds that the differential reflection coefficients ∂Rv/∂θsF and ∂Rv/∂θs match each other rather well in the angular interval θs<θs∗.


## 4. Conclusions

The scattering of normally incident p- or s-polarized light from a one-dimensional randomly rough interface between two dielectric media is studied. Based on phase perturbation theory, it is demonstrated that using a broadband Gaussian beam to illuminate the surface produces a differential reflection coefficient that closely matches the one produced by a monochromatic Gaussian beam when the resulting differential reflection coefficient is averaged over the ensemble of realizations of the interface profile function. This result is obtained since the broadband beam averages over the speckles produced by a monochromatic beam.The confirmation of the conjecture prompting the present work by these proof-of-concept calculations encourages additional calculations using rigorous computer simulation methods instead of the phase perturbation theory approach, to explore the efficacy of a broadband source in calculations of rough surface scattering phenomena and in experimental studies of them.



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