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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 6 — Mar. 25, 2013
  • pp: 6807–6825
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Physical optics solution for the scattering of a partially-coherent wave from a statistically rough material surface

Milo W. Hyde, IV, Santasri Basu, Mark F. Spencer, Salvatore J. Cusumano, and Steven T. Fiorino  »View Author Affiliations


Optics Express, Vol. 21, Issue 6, pp. 6807-6825 (2013)
http://dx.doi.org/10.1364/OE.21.006807


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Abstract

The scattering of a partially-coherent wave from a statistically rough material surface is investigated via derivation of the scattered field cross-spectral density function. Two forms of the cross-spectral density are derived using the physical optics approximation. The first is applicable to smooth-to-moderately rough surfaces and is a complicated expression of source and surface parameters. Physical insight is gleaned from its analytical form and presented in this work. The second form of the cross-spectral density function is applicable to very rough surfaces and is remarkably physical. Its form is discussed at length and closed-form expressions are derived for the angular spectral degree of coherence and spectral density radii. Furthermore, it is found that, under certain circumstances, the cross-spectral density function maintains a Gaussian Schell-model form. This is consistent with published results applicable only in the paraxial regime. Lastly, the closed-form cross-spectral density functions derived here are rigorously validated with scatterometer measurements and full-wave electromagnetic and physical optics simulations. Good agreement is noted between the analytical predictions and the measured and simulated results.

© 2013 OSA

1. Introduction

Rough surface scattering has been an active area of research for a half century. The published research on the subject can generally be divided into two main groups. The first deals with rough surface scattering research performed by the RF/microwave community for synthetic aperture radar and remote sensing applications. Predominately concerned with fully-coherent, monochromatic plane-wave scattering from rough surfaces (with some exceptions), the common approaches employed by the RF/microwave community to the rough surface scattering problem are physical optics (PO) [1

1. F. T. Ulaby, R. K. Moore, and A. K. Fung, Microwave Remote Sensing: Active and Passive (Artech House, 1986, vol. 2).

7

7. M. E. Knotts, T. R. Michel, and K. A. O’Donnell, “Angular correlation functions of polarized intensities scattered from a one-dimensionally rough surface,” J. Opt. Soc. Am. A 9, 1822–1831 (1992) [CrossRef] .

], perturbation [4

4. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).

, 8

8. S. O. Rice, “Reflection of electromagnetic wave by slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951) [CrossRef] .

], and computational/full-wave [9

9. A. K. Fung and K. S. Chen, Microwave Scattering and Emission Models for Users (Artech House, 2010).

14

14. E. Bahar, “Full-wave solutions for the depolarization of the scattered radiation fields by rough surfaces of arbitrary slope,” IEEE Trans. Antennas Propag. 29, 443–454 (1981) [CrossRef] .

] methods. The interested reader is referred to Elfouhaily and Guérin [15

15. T. M. Elfouhaily and C.-A. Guérin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14, R1–R40 (2004) [CrossRef] .

], Warnick and Chew [16

16. K. F. Warnick and W. C. Chew, “Numerical simulation methods for rough surface scattering,” Waves Random Media 11, R1–R30 (2001) [CrossRef] .

], Maradudin [17

17. A. A. Maradudin, ed., Light Scattering and Nanoscale Surface Roughness (Springer, 2007) [CrossRef] .

], Ogilvy [18

18. J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (IOP Publishing, 1991).

], and Nieto-Vesperinas [19

19. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics, 2 ed. (World Scientific, 2006) [CrossRef] .

] for excellent summaries of these techniques.

The second of the two main groups deals with rough surface scattering research performed by the optics community. Initially, this work mainly dealt with how incoherent light interacted with surfaces, modeled via the bidirectional reflectance distribution function (BRDF), for applications in passive visible/near-IR remote sensing and computer graphics [20

20. K. E. Torrance and E. M. Sparrow, “Theory for off-specular reflection from roughened surfaces,” J. Opt. Soc. Am. 57, 1105–1112 (1967) [CrossRef] .

24

24. X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, “A comprehensive physical model for light reflection,” SIGGRAPH Computer Graphics 25, 175–186 (1991) [CrossRef] .

]. More recently, with the proliferation of laser-based systems (LIDAR/LADAR and directed energy), the interaction of coherent laser light with rough surfaces, in particular the statistical behaviors of the resulting speckle patterns, gained considerable interest in such fields as metrology and remote sensing. Some of the early notable research in this area was performed by Dainty [25

25. J. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 761–772 (1970) [CrossRef] .

], Goodman [26

26. J. Goodman, “Statistical properties of laser speckle patterns,” in “Laser Speckle and Related Phenomena,” vol. 9 of Topics in Applied Physics, J. C. Dainty, ed. (Springer-Verlag, 1975, pp. 9–75) [CrossRef] .

], Parry [27

27. G. Parry, “Speckle patterns in partially coherent light,” in “Laser Speckle and Related Phenomena,” vol. 9 of Topics in Applied Physics, J. C. Dainty, ed. (Springer-Verlag, 1975, pp. 77–121) [CrossRef] .

], Fujii and Asakura [28

28. H. Fujii and T. Asakura, “A contrast variation of image speckle intensity under illumination of partially coherent light,” Opt. Commun. 12, 32–38 (1974) [CrossRef] .

,29

29. H. Fujii and T. Asakura, “Statistical properties of image speckle patterns in partially coherent light,” Nouv. Rev. Opt. 6, 5–14 (1975) [CrossRef] .

], Pedersen [30

30. H. Pedersen, “The roughness dependence of partially developed, monochromatic speckle patterns,” Opt. Commun. 12, 156–159 (1974) [CrossRef] .

], and Yoshimura et al.[31

31. T. Yoshimura, K. Kato, and K. Nakagawa, “Surface-roughness dependence of the intensity correlation function under speckle-pattern illumination,” J. Opt. Soc. Am. A 7, 2254–2259 (1990) [CrossRef] .

].

Two forms of the scattered field CSD function are derived and discussed. The first, applicable to surfaces of smooth-to-moderate roughness, is expressed in terms of an infinite series. While its behavior depends in a complex way on source and surface parameters, its analytical form is physically intuitive. The second form of the scattered field CSD function is applicable to very rough surfaces. This form of the CSD function is incredibly physical and, under certain circumstances, maintains a GSM form, which is in agreement with the literature cited above valid in the paraxial regime. As such, closed-form expressions are derived for the angular spectral degree of coherence (SDoC) and spectral density (SD) radii. These expressions are, in general, complicated functions of both the source and surface parameters. It is demonstrated that for many scenarios of interest, the SDoC radius can be safely approximated as a function of just the source parameters and the SD radius can be simplified to a function of just the surface parameters.

To verify the theoretical analysis, experimental and Monte Carlo simulation results using a full-wave electromagnetic technique (all the physics of the wave/surface interaction are included) and one based on the PO approximation are presented and compared to the predictions of the analytical models. This paper is concluded with a summary of the work and contributions presented.

2. Methodology

The scattering geometry utilized in this analysis is shown in Fig. 1. The surface height is described by the function h(x) with mean, standard deviation, and correlation length equal to 0, σh, and h, respectively. The surface (of length 2L) is illuminated by a partially-coherent beam (parameters ws and s defined below) emanating from the source plane specified by the coordinates (u, v, z). Its origin, relative to the rough surface coordinate system, is located at the point (−xs, ys). The observation vector ρ = x +ŷy points from the rough surface origin to the observation point. The medium below the rough interface is electrically described by the permittivity ε and permeability μ; the medium above the rough interface is vacuum (ε0, μ0).

Fig. 1 Scattering geometry of a one-dimensional (the surface and source excitation are invariant in the z direction) rough surface of length 2L. The medium below the rough interface is electrically defined by the permittivity ε and permeability μ; the medium above the rough interface is vacuum (ε0, μ0). The rough surface height is described by the function h(x); the mean, standard deviation, and correlation length of the surface are 0, σh, and h, respectively. The point (−xs, ys) denotes the location of the source plane origin. The observation vector ρ = x̂x +ŷy points from the rough surface origin to the observation point. The vector ρs = xsŷys points from the source plane origin to the rough surface origin.

A GSM form for the incident field CSD is used to model the partially-coherent illumination, viz.,
Wi(u1,u2)=Ei(u1)Ei*(u2)=E02exp(u12+u224ws2)exp[(u1u2)22s2],
(1)
where ws and s are the source radius and source coherence radius, respectively and the functional dependence of Wi, E0, ws, and s on the radian frequency ω is omitted for brevity [33

33. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

, 35

35. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

]. Both s-pol and p-pol scattering solutions are derived below. Note that the rough surface and the source excitation are invariant in the z direction resulting in a two-dimensional scattering problem. As mentioned by Johnson [17

17. A. A. Maradudin, ed., Light Scattering and Nanoscale Surface Roughness (Springer, 2007) [CrossRef] .

], light scattered from one-dimensional surfaces demonstrates the same physical behaviors as light scattered from two-dimensional surfaces. The mechanisms which are not captured in two-dimensional scattering problems are cross-polarized scattering and out-of-plane scattering [17

17. A. A. Maradudin, ed., Light Scattering and Nanoscale Surface Roughness (Springer, 2007) [CrossRef] .

]. In this way, the expressions derived in this paper are equivalent to scalar-wave scattering solutions.

2.1. PO expression for the scattered field

For the sake of brevity, the analysis to follow focuses on perpendicular polarization. The solution for parallel polarization is provided at the end of the section. For perpendicular polarization, the incident field in the source plane takes the form
Ei=z^Ezi(u).
(2)
Utilizing the plane-wave spectrum representation of electromagnetic fields [54

54. T. Hansen and A. Yaghjian, Plane-Wave Theory of Time-Domain Fields: Near-Field Scanning Applications (IEEE, 1999) [CrossRef] .

], the incident electric and magnetic fields become
Ei=12πTei(kui)exp(jkiρs)exp(jki.ρ)dkuiv>0Hi=12πki×Tei(kui)ωμ0exp(jki.ρs)exp(jkiρ)dkuiv>0,
(3)
where ρs is a vector that points from the source plane origin to the rough surface origin, ki=u^kui+v^kvi, and
Tei(kui)=z^Ezi(u)exp(jkuiu)du.
(4)
Note that |ki| = k0 = 2π/λ, thus kvi=k02(kui)2.

The scattered field in the far zone can be found by utilizing the transverse components of the far-field vector potentials N and L[55

55. S. Ramo, J. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics, 3 ed. (John Wiley & Sons, 1994).

] and the equivalent electric J and magnetic M currents induced on the rough surface by the incident field, namely,
Esjk08πexp(jk0ρ)ρ[μ0ε0Nt+Lt]Nt=(ϕ^ϕ^+z^z^)CJ(ρ)exp(jk0ρ^ρ)dcLt=(ϕ^ϕ^+z^z^)CM(ρ)exp(jk0ρ^ρ)dc,
(5)
where ρ′ = x′ + ŷh(x′). Note that the integral is over the parameterized surface contour C with dc equal to the arc length, i.e., dc=dx1+[h(x)]2.

Using the PO approximation, the equivalent currents induced by the incident field on the material surface are
J(1r)n^×Hi|x=xy=h(x)M(1+r)Ei×n^|x=xy=h(x).
(6)
Here, is the unit outward normal to the surface given by
n^=y^x^h(x)1+[h(x)]2=y^x^hx1+hx2
(7)
and r is the s-pol Fresnel complex amplitude reflection coefficient. Substitution of Eqs. (6) and (7) into Eq. (5) and subsequent simplification yields the following expression for Ezs:
Ezs=exp(jk0ρ+jπ4)2π8πk0ρTezi(kui)ejkiρsLL(x,kui)ejνρdxdkui,
(8)
where ν = k0ρ̂ − ki and
(x,kui)=(1r)(kνiu^kuiv^)(x^+y^hx)+k0(1+r)ρ^(y^x^hx).
(9)
Note that r is primed because it is a function of surface coordinates (x′, h(x′)).

No analytical solution has yet been found for the integral over the rough surface coordinate x′ in Eq. (8) because of , which is a complicated function of h(x′) and its derivative hx[1

1. F. T. Ulaby, R. K. Moore, and A. K. Fung, Microwave Remote Sensing: Active and Passive (Artech House, 1986, vol. 2).

]. This integral is typically evaluated using the stationary-phase approximation in which hx ≈ −νx/νy[1

1. F. T. Ulaby, R. K. Moore, and A. K. Fung, Microwave Remote Sensing: Active and Passive (Artech House, 1986, vol. 2).

,19

19. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics, 2 ed. (World Scientific, 2006) [CrossRef] .

]. This approximation physically dictates that reflection from the rough surface is locally specular, i.e., local diffraction effects are excluded [1

1. F. T. Ulaby, R. K. Moore, and A. K. Fung, Microwave Remote Sensing: Active and Passive (Artech House, 1986, vol. 2).

, 17

17. A. A. Maradudin, ed., Light Scattering and Nanoscale Surface Roughness (Springer, 2007) [CrossRef] .

19

19. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics, 2 ed. (World Scientific, 2006) [CrossRef] .

, 23

23. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, 1963).

]. Applying this approximation to Eq. (8) and rearranging the integrals results in
Ezs=exp(jk0ρ+jπ4)2π8πk0ρLL(kui)Tezi(kui)ejkiρsejνρdxdkui.
(10)
This expression represents the s-pol scattered electric field given one random incident field and one random surface realization. The only assumptions that have been made thus far are that observation is in the far field, the rough surface is such that the PO approximation for the currents holds, and shadowing/masking and multiple scattering can be safely neglected [18

18. J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (IOP Publishing, 1991).

, 19

19. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics, 2 ed. (World Scientific, 2006) [CrossRef] .

].

2.2. Scattered field cross-spectral density function

Applying the definition of the autocorrelation of a random process to Eq. (10) yields
Ws(ρ1ρ2)=Ezs(ρ1)Ezs*(ρ2)=ejk0(ρ2ρ1)32π3k0ρ1ρ2LLLL12*ej(k2ik1i)ρsTez1iTez2i*ejν1ρ1ejν2ρ2dku1idku2idx1dx2,
(11)
where the subscripts 1 and 2 imply that the associated expression is a function of x1, ρ1, ku1i or x2, ρ2, ku2i (whichever is applicable), respectively, and the expectations are computed over the ensembles of rough surface and incident field realizations. In arriving at the above expression, it has been assumed that the incident field plane-wave spectrum is statistically independent of the rough surface; this assumption is physically intuitive. Note that the autocorrelation of the incident field plane-wave spectrum is equivalent to the Fourier transform of the CSD function given in Eq. (1). The second autocorrelation expression in Eq. (11), after some simple manipulation, is equivalent to the joint characteristic function of the random variables h(x1) and h(x2) [23

23. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, 1963).

]. The incident field term and the rough surface term are represented as Φ(ku1i,ku2i) and χ(ku1i,ku2i;x1,x2) in the analysis to follow.

The integrals over the incident field plane-wave spectrum can be approximated using the method of stationary phase [33

33. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

]. Using the method of stationary phase to evaluate the plane-wave spectrum integrals has two implications. The first is that kvikui and is therefore approximated as kvik0(kui)2/(2k0) for all “phase” terms (i.e., exp[j(⋯)]) and as kvik0 for all amplitude terms. This physically implies that the incident field is highly directional being predominately directed along the v direction in Fig. 1. The second implication is that the distance from the rough surface to the source plane ρs must be much greater than L—this is generally the case. To provide some idea of how much greater ρs must be than L, ρs = 10L, ρs = 20L, and ρs = 50L are approximately 95%, 97.5%, and 99% accurate, respectively. Note that these two implications mean that the kuiv^ term in , see Eq. (9), diminishes at the rate of 1/ρs and therefore can be neglected. Recall that this term corresponded to the component of the incident magnetic field implying that the field incident on the rough surface is transverse electromagnetic (in particular, TEMv) when the method of stationary phase is utilized. Applying the method of stationary phase and substituting in Φ(ku1i,ku2i) produces
Ws(ρ1,ρ2)=E02k02ejk0(ρ2ρ1)𝒮16πρsρ1ρ2(a2b2)LLLLexp[a˜k02(x^u^)2ρs2(x12+x22)]exp[2b˜k02(x^u^)2ρs2x1x2]exp[jk0(x^u^)22ρs(x22x12)]ejk0(x^v^)(x1x2)χ(k0(x^u^)x1ρs,k0(x^u^)x2ρs;x1,x2)ejk0[(x^ρ^1)x1(x^ρ^2)x2]dx1dx2,
(12)
where the symbols used above are defined in the appendix.

To simplify Eq. (12) further, a form for the rough surface characteristic function χ must be chosen. A very common choice for the statistical distribution of the rough surface is to assume that the surface heights are Gaussian distributed and Gaussian correlated, i.e.,
pH1,H2(h1,h2)=12πσh21Γ2exp[h122Γh1h2+h222σh2(1Γ2)]Γ(x1x2)=exp[(x1x2)2h2].
(13)
Historically, Gaussian-Gaussian (G-G) models for rough surfaces were chosen for analytical convenience [32

32. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Ben Roberts & Company, 2007).

]. Recent profilometer measurements of sandblasted metallic surfaces showed that the stretched exponential-stretched exponential (SE-SE) model [56

56. C. A. Mack, “Analytic form for the power spectral density in one, two, and three dimensions,” J. Micro/Nanolithogr. MEMS MOEMS 10, 040501 (2011).

] more accurately represented surfaces roughened by random industrial processes [57

57. S. Basu, S. J. Cusumano, M. W. Hyde, M. A. Marciniak, and S. T. Fiorino, “Validity of using Gaussian Schell model for extended beacon studies,” Proc. SPIE 8380 (2012).

]. Unfortunately, no general analytical form for the SE joint characteristic function exists. Although the results in [57

57. S. Basu, S. J. Cusumano, M. W. Hyde, M. A. Marciniak, and S. T. Fiorino, “Validity of using Gaussian Schell model for extended beacon studies,” Proc. SPIE 8380 (2012).

] indicated that SE-SE models were superior, G-G models were fairly good approximations for the surfaces that were measured. Note that, in addition to G-G and SE-SE, numerous other surface models can be found throughout the literature. Fourier transforming the above joint probability density function yields the desired characteristic function:
χ(ku1i,ku2i;x1,x2)=exp[σh22(νy12+νy22)]exp{σh2νy1νy2exp[(x1x2)2h2]}.
(14)
Substitution of χ into Eq. (12), performing the common variable transformation xd = x1x2 and xa = x1 + x2, and subsequent simplification yields
Ws(ρ1,ρ2)=E02k02ejk0(ρ2ρ1)𝒮32πρsρ1ρ2(a2b2)exp[k02σh22(ϑy12+ϑy22)]2L2Lexp[(a˜b˜)k02(x^u^)22ρs2xa2]exp[jk02(ϑx1ϑx2)xa]|xa|2L2L|xa|exp[(a˜+b˜)k02(x^u^)22ρs2xd2]exp[k02σh2ϑy1ϑy2exp(xd2h2)]exp[jk02(ϑx1+ϑx2)xd]exp[jk0(x^u^)2xa2ρsxd]dxddxa.
(15)

2.2.1. Smooth-to-moderately rough surfaces

In order to evaluate the remaining integral and arrive at a closed-form solution, the complex error function (erf) must be properly handled. While it is possible to determine the order of the error function argument by using physical insight and examining the associated exponential functions in Eq. (17), it is much simpler to examine the integrand in Eq. (15) and determine the condition at which the xd integration limits can be extended from [|xa| − 2L, 2L − |xa|] to (−∞, ∞). A necessary condition for this approximation occurs when
exp{xd2[(a˜+b˜)k02(x^u^)22ρs2+mh2]}>δ,
(18)
where δ is a user-defined parameter and denotes the point at which the exponential function no longer has significant value. Taking note that the minimum of the exponential function’s argument occurs when m = 0 and that the maximum value xd takes on in the integration is 2L, the following condition can be derived:
L>ρsk0ws2|x^u^|lnδ.
(19)
The above condition physically means that the projected fully-coherent incident beam size must “fit” on the rough surface for the erf ≈ 1, or equivalently, the xd integration limits in Eq. (15) [|xa| − 2L, 2L − |xa|] ≈ (−∞, ∞). How well the incident beam “fits” is determined by δ—the smaller the δ, the more accurate the approximation. If the projected fully-coherent incident beam size does not “fit” on the rough surface, the erf ≠ 1 and the remaining expression must be evaluated numerically.

Assuming that the above condition holds, the remaining integral in Eq. (17) can be evaluated, namely,
Ws(ρ1,ρ2)=E02k0hρsejk0(ρ2ρ1)𝒮16|x^u^|ρ1ρ2(a2b2)exp[k02σh22(ϑy12+ϑy22)]m=0(k02σh2ϑy1ϑy2)mm!𝒜mexp[k02ρs2h24𝒟m(1ρs2h2(x^u^)2𝒜m)(ϑx12+ϑx22)]exp{ρs2[k02h2(x^u^)2b˜+mρs2]𝒜m(x^u^)2(ϑx1ϑx2)2}exp[jk0h2ρs32𝒜m(ϑx12ϑx22)]{erf[Lk0|x^u^|ρs𝒜m2𝒟m+k0|x^u^|ρs2h222𝒟m𝒜m(ϑx1+ϑx2)+jρs|x^u^|𝒟m2𝒜m(ϑx1ϑx2)]+erf[Lk0|x^u^|ρs𝒜m2𝒟mk0|x^u^|ρs2h222𝒟m𝒜m(ϑx1+ϑx2)jρs|x^u^|𝒟m2𝒜m(ϑx1ϑx2)],
(20)
where 𝒜m=4𝒟m(a˜b˜)+ρs2h2(x^u^)2. A similar analysis to that performed above can be performed for the error functions in this expression. The condition for the sum of the erf’s ≈ 2 is
L>ρs1+(2/α)2k0ws2|x^u^|lnδ,
(21)
where α = s/ws. This condition physically means that the projected partially-coherent beam size must “fit” on the rough surface in order to safely assume that the sum of the erf’s ≈ 2. This condition is more stringent than Eq. (19).

The physical interpretation of Eq. (20) is obscured somewhat by the summation; however, the exponential term on the second line of the expression generally drives the angular extent of the scattered SD. The first exponential term on the third line generally determines the angular extent over which the scattered field is correlated, i.e., the SDoC radius. The SD S and SDoC μ definitions can be found in Refs. [33

33. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

, 35

35. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

, 36

36. G. Gbur and T. Visser, “The structure of partially coherent fields,” Progress in Optics 55, 285–341 (2010) [CrossRef] .

]. When one considers a fully-coherent incident Gaussian beam (s → ∞) and quasi-monochromatic light [58

58. Under quasi-monochromatic conditions (Δωω̄), S (ρ, ω) Γ(ρ, ρ, 0)δ(ωω̄) and μ (ρ1, ρ2, ω) ≈ γ (ρ1, ρ2, τ) exp (jω̄τ). Here, Γ is the mutual coherence function and γ is the complex degree of coherence [59].

], the coherent scattered irradiance, i.e., Ics=|Ezs|2, can be determined quite easily from Eq. (20) by considering just the m = 0 term of the series and setting ρ1 = ρ2 = ρ. With some minor algebraic manipulation, one arrives at the two-dimensional form of the coherent scattered irradiance expression derived by Wang et al.[2

2. M.-J. Wang, Z.-S. Wu, and Y.-L. Li, “Investigation on the scattering characteristics of Gaussian beam from two dimensional dielectric rough surfaces based on the Kirchhoff approximation,” Prog. Electromagn. Res. B 4, 223–235 (2008) [CrossRef] .

]. One can also derive from Eq. (20) the incoherent scattered irradiance Iis for a fully-coherent incident Gaussian beam and quasi-monochromatic light. Note that Iis is equivalent to the variance of the scattered field, i.e., Iis=IsIcs, where Is=EzsEzs* is the average scattered irradiance. This expression also simplifies to the form derived by Wang et al.[2

2. M.-J. Wang, Z.-S. Wu, and Y.-L. Li, “Investigation on the scattering characteristics of Gaussian beam from two dimensional dielectric rough surfaces based on the Kirchhoff approximation,” Prog. Electromagn. Res. B 4, 223–235 (2008) [CrossRef] .

].

2.2.2. Very rough surfaces

Consider the form of the joint characteristic function:
χ=exp[k02σh22(ϑy12+ϑy22)]exp[k02σh2ϑy1ϑy2exp(xd2h2)]=exp{k02σh22ϑy1ϑy2[ϑy1ϑy2+ϑy2ϑy12exp(xd2h2)]}.
(22)
In the case of very rough surfaces, i.e., k02σh2ϑy1ϑy2/21, χ only has significant value when ϑy1/ϑy2+ϑy2/ϑy12exp(xd2/h2)0. Considering that ϑy1/ϑy2ϑy2/ϑy1, ϑy1+ϑy2+ϑy2/ϑy12exp(xd2/h2)0 is only possible for small xd. Thus, it makes sense to expand exp(xd2/h2) and retain the first two terms, i.e., exp(xd2/h2)1xd2/h2. Substituting this into Eq. (15) and carrying out the integrations, one arrives at the desired result for the scattered field CSD function:
Ws(ρ1,ρ2)=E02k0hρsejk0(ρ2ρ1)𝒮16|x^u^|ρ1ρ2(a2b2)exp[k02σh22(ϑy1ϑy2)2]1𝒜˜exp[k02ρs2h24𝒟˜(1ρs2h2(x^u^)2𝒜˜)(ϑx12+ϑx22)]exp[jk0h2ρs32𝒜˜(ϑx12ϑx22)]exp{k02ρs2[h2(x^u^)2b˜+σh2ρs2ϑy1ϑy2]𝒜˜(x^u^)2(ϑx1ϑx2)2}{erf[Lk0|x^u^|ρs𝒜˜2𝒟˜+k0|x^u^|ρs2h222𝒟˜𝒜˜(ϑx1+ϑx2)+jρs|x^u^|𝒟˜2𝒜˜(ϑx1ϑx2)]+erf[Lk0|x^u^|ρs𝒜˜2𝒟˜k0|x^u^|ρs2h222𝒟˜𝒜˜(ϑx1+ϑx2)jρs|x^u^|𝒟˜2𝒜˜(ϑx1ϑx2)]},
(23)
where 𝒟˜=k02(x^u^)2h2(a˜+b˜)+2k02σh2ϑy1ϑy2ρs2 and 𝒜˜=4𝒟˜(a˜b˜)+ρs2h2(x^u^)2. If Eq. (21) is satisfied, then the sum of the erf’s ≈ 2.

The above expression for the scattered field CSD function is remarkably physical. The first exponential term on the second line of Eq. (23) is predominately responsible for the angular extent of the scattered SD. This exponential term is a function of the sum of the squares of the projected observation angles ϑx1 and ϑx2. The exponential term on the third line of Eq. (23) determines the SDoC radius. Note that this term is a function of the difference of the projected observation angles, i.e., |ϑx1ϑx2|=|sinθ1rsinθ2r|. Thus, it could be stated that the scattered field CSD function maintains its GSM form with respect to ϑx1 and ϑx2 if Eq. (21) is satisfied.

2.3. Angular spectral degree of coherence radius

In this section, a theoretical form for the angular SDoC radius is derived from Eq. (23). Note that because of the summation in Eq. (20), it is not possible to derive an expression for the angular SDoC radius using that form of the scattered field CSD function.

Assuming Eq. (21) is satisfied and setting the exponential term on line three of Eq. (23) equal to 1/e results in the following expression:
|ϑx1ϑx2|1/e=|sinθ1rsinθ2r|1/e|x^u^|ρs8ws2s2s2+4ws2+2ρs2h2(x^u^)2k021h2ws2(x^u^)2+2ρs2σh2ϑy1ϑy2.
(24)
As discussed above, the magnitude of the argument of the correlation exponential term is very large; thus, θ1rθ2r for the correlation term to have a significant value. Note that this physically implies that the scattered field is correlated for observation points separated by very small angles. The fact that θ1rθ2r motivates setting θ1r=θ2r+Δθr and expanding sinθ1r in a Taylor series about θ2r=θr. After some simple algebra, the expression for the angular SDoC radius becomes
|Δθr|1/ecosθiρscosθr8ws2s2s2+4ws2+2k021σh2(1+cosθr/cosθi)211+[Ωσh(1+cosθr/cosθi)]2,
(25)
where α = s/ws (previously defined), Ω = ws/ρs is the half-angle subtended by the source viewed from the rough surface, and σh=2σh/h is the surface slope standard deviation [22

22. Y. Sun, “Statistical ray method for deriving reflection models of rough surfaces,” J. Opt. Soc. Am. A 24, 724–744 (2007) [CrossRef] .

, 23

23. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, 1963).

]. Note that the same procedure applied to ϑx1 and ϑx2 (discussed above) has been applied to ϑy1 and ϑy2 in deriving the above expression.

The term involving Ω and σh can be neglected when one considers that typical values for these parameters range from approximately 10−4 rad and 0.05–0.5 rad, respectively. Note that the first term under the radical contains only source parameters, while the second term contains mostly rough surface parameters. In many scenarios of interest the source term is much greater than the rough surface term; thus, factoring out the source term and expanding the resulting radical in a Taylor series yields
|Δθr|1/e2Ω21+(2/α)2cosθicosθr+Ω2k02ws21+(2/α)24σh2(1+cosθr/cosθi)2cosθicosθr.
(26)

The first term is entirely composed of source parameters and therefore represents the source contribution to the angular SDoC radius. Because of the presence of σh, the second term provides a small correction to the angular SDoC radius due to the roughness of the surface. In most cases, the angular SDoC radius can be safely approximated by utilizing only the first term making the expression dependent only on the properties of the source illumination. This is consistent with the classic narrow-band, fully-coherent result derived by Goodman [26

26. J. Goodman, “Statistical properties of laser speckle patterns,” in “Laser Speckle and Related Phenomena,” vol. 9 of Topics in Applied Physics, J. C. Dainty, ed. (Springer-Verlag, 1975, pp. 9–75) [CrossRef] .

]. Note that when θrθi, i.e., when observation is in the specular direction, the dependence of the expression on observation and incident angles disappears.

2.4. Angular spectral density radius

As is the case with the angular SDoC radius, an expression for the angular SD radius can only be derived using the very rough surface form of the scattered field CSD function in Eq. (23). Furthermore, the expression can only be derived for near-normal incidence.

Assuming Eq. (21) is satisfied, letting θi = 0 and θ1r=θ2r=θ1/er, and setting the first exponential term on line two of Eq. (23) equal to 1/e yields the following expression after some simplification:
sin2θ1/er=2Ω2+2σh2(1+cosθ1/er)2+1+(2/α)22k02ws2,
(27)
where all symbols have been previously defined. With liberal use of trigonometric identities and some algebra, one arrives at
cos4(θ1/er/2)11+2σh2cos2(θ1/er/2)+2Ω2+1+(2/α)22k02ws24+8σh2=0.
(28)
One quickly recognizes that the above expression is a quadratic equation in terms of cos2(θ1/er/2). Letting ψ=cos2(θ1/er/2), implying that θ1/er=2cos1ψ (only the positive ψ root makes physical sense), and utilizing the quadratic formula, yields the following roots for ψ:
ψ=12+4σh2[1±1(1+2σh2)(2Ω2+(1+(2/α)2)2k02ws2)].
(29)
The second term in the radicand is much less than one; therefore, only the “plus” root has significant value. Choosing the “plus” root, expanding the radical in a Taylor series, and applying θ1/er=2cos1ψ produces
θ1/er2cos1[11+2σh2114(1+2σh2)(2Ω2+1+(2/α)22k02ws2)]2cos1[11+2σh2181+2σh2(2Ω2+1+(2/α)22k02ws2)].
(30)

In Eq. (30), the first term in the argument of the inverse cosine depends solely on the characteristics of the rough surface and it physically represents the rough surface contribution to the angular extent of the SD. The second argument contains terms dealing with the source parameters and provides a small correction to the angular SD radius due to the size and coherence of the source. In most cases, the angular SD radius can be approximated using only the first term making the expression solely dependent on the roughness of the surface.

2.5. Parallel-polarization cross-spectral density function

Considering that the rough surface’s features are large compared to wavelength, intuition dictates that the p-pol scattered field should have the same general behavior as the s-pol scattered field. This is the case. The only difference between the s-pol and p-pol CSD functions is the parameter 𝒮. Setting 𝒮 = 𝒮 in Eqs. (20) and (23) yields the p-pol scattered field CSD functions. Expressions for 𝒮 and 𝒮 can be found in the appendix.

3. Validation of analytical solutions

In this section, the analytical solutions for the scattered field CSD functions derived above are validated via experiments, full-wave electromagnetic simulations, and PO simulations. For the experimental validation, the Complete Angle Scatter Instrument (CASI) [60

60. B. Balling, “A comparative study of the bidirectional reflectance distribution function of several surfaces as a mid-wave infrared diffuse reflectance standard,” Master’s thesis, Graduate School of Engineering and Management, Air Force Institute of Technology (AETC), Wright-Patterson AFB OH (2009). http://oai.dtic.mil/oai/oai?verb=getRecord&metadataPrefix=html&identifier=ADA495933.

] at the Air Force Institute of Technology was used. Measurements of the scattered SD Ss versus θr from a LabSphere Infragold [61

61. LabSphere Inc., “A guide to reflectance coatings and materials” (LabSphere, Inc., 2012). http://www.labsphere.com/uploads/technical-guides/a-guide-to-reflectance-materials-and-coatings.pdf.

] 5.08 cm × 5.08 cm coupon were made at λ = 3.39 μm (HeNe MWIR laser) and θi = 20°, 40°, and 60°. Before the scattering measurements were made, the surface statistics of the Infragold coupon were determined using a KLA Tencor Alpha-Step IQ surface profiler [62

62. KLA-Tencor Corporation, “Alpha-Step IQ Surface Profiler” (KLA-Tencor Corporation, 2012). http://www.kla-tencor.com/surface-profiling/alpha-step-iq.html.

]. Four 1 cm scans (step size 0.2 μm), performed along different directions, were taken and analyzed. The measured surface height standard deviation σh, surface slope standard deviation σh, and correlation length h were 11.09 μm, 0.2441 rad, and 116.9 μm, respectively.

For the full-wave electromagnetic simulations, the Method of Moments (MoM) [63

63. R. Harrington, Field Computation by Moment Methods (IEEE, 1993) [CrossRef] .

] was used to solve the following impedance-boundary-condition-based s-pol electric field integral equation for the unknown electric current J:
Ezi(ρ)=μεJz(ρ)+jk0μ0ε0Az(ρ)+(Fy(ρ)xFx(ρ)y)ρC,
(31)
where A and F are the magnetic and electric vector potentials, respectively [64

64. A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics (IEEE, 1998).

]. Once J had been found, the scattered field was determined via convolution with the far-field form of the free-space Green’s function. A narrow-band, fully-coherent Gaussian beam was used for the incident field Ezi. Ss versus θr and the modulus of the SDoC |μs| versus Δθr were calculated using the scattered field predicted from 500 rough surface realizations at λ = 3.39 μm, the Infragold surface statistics stated above, and a complex index of refraction n − jκ for gold of 1.995 − j20.95 [65

65. RefractiveIndex.INFO, “RefractiveIndex.INFO Refractive index database” (RefractiveIndex.INFO, 2012). http://refractiveindex.info.

]. The 500 rough surface realizations were generated using the method outlined by Yura and Hanson [66

66. H. T. Yura and S. G. Hanson, “Digital simulation of an arbitrary stationary stochastic process by spectral representation,” J. Opt. Soc. Am. A 28, 675–685 (2011) [CrossRef] .

].

Both the experimental and full-wave simulations used narrow-band, fully-coherent incident fields. To verify the predictions of the analytical solutions for partially-coherent incident fields, PO Monte Carlo simulations were also performed. To model the random nature of the incident field, 500 source plane field instances were generated using the phase screen method described by Xiao and Voelz [67

67. X. Xiao and D. Voelz, “Wave optics simulation approach for partial spatially coherent beams,” Opt. Express 14, 6986–6992 (2006) [CrossRef] [PubMed] .

]. For the random surface, 500 rough surface instances were generated using the technique detailed in Ref. [66

66. H. T. Yura and S. G. Hanson, “Digital simulation of an arbitrary stationary stochastic process by spectral representation,” J. Opt. Soc. Am. A 28, 675–685 (2011) [CrossRef] .

]. The 500 source plane field instances were propagated to and evaluated at each rough surface instance. The scattered fields were then computed numerically using the relations in Eq. (5). The end result of this process was a set of 5002 scattered fields, from which |μs| and Ss were computed. These simulations were performed at λ = 3.39 μm using hypothetical rough aluminum surfaces n − jκ = 5.366 − j33.33 [65

65. RefractiveIndex.INFO, “RefractiveIndex.INFO Refractive index database” (RefractiveIndex.INFO, 2012). http://refractiveindex.info.

] with varying degrees of surface roughness. In addition, the properties of the incident field (size and coherence) were also varied to examine their effects.

3.1. Results

Fig. 2 CASI measurement, MoM simulation, analytical PO solution, and PO simulation results of normalized Ss versus θr for Infragold—(a) θi = 20°, (a) θi = 40°, and (c) θi = 60°.

To truly validate the PO analytical solutions, MoM simulations were also performed using hypothetical Infragold G-G surfaces. These results, along with PO simulation and PO analytical solution results, are shown in Fig. 3. The source distance ρs, source radius ws, and surface 1/2 length L were the same as above. The figure reports the normalized Ss versus θr in Fig. 3(a) and |μs| versus Δθr in Figs. 3(b) and 3(c). In Figs. 3(a) and 3(b), the simulated Infragold surface possessed the same σh and h values reported above. Since this hypothetical surface qualifies as a very rough surface, the theoretical angular SD and SDoC radii, given in Eqs. (30) and (26), respectively, are also plotted as vertical dashed line in Figs. 3(a) and 3(b). Note the excellent agreement between the MoM simulation, PO simulation, and analytical PO solution results. In Fig. 3(c), |μs| is shown for hypothetical Infragold G-G surfaces which possessed σh = 0.05λ, 0.1λ, and 0.25λ with h held constant at 8λ. Surfaces with these parameters do not qualify as very rough surfaces and therefore, the angular SDoC radii are not shown. Again, note the very good agreement among the results.

Fig. 3 MoM simulation, analytical PO solution, and PO simulation results of normalized Ss versus θr [(a)] and |μs| versus Δθr [(b) and (c)] for hypothetical Infragold G-G surfaces. In (a) and (b), the hypothetical Infragold surface possessed the measured σh and h values of 11.09 μm and 116.9 μm, respectively. Since this hypothetical surface qualifies as a very rough surface, θ1/er and |Δθr|1/e, given in Eqs. (30) and (26), are also plotted as vertical dashed line in (a) and (b), respectively. In (c), σh is varied while h is held constant at 8λ.

Having validated the PO analytical solutions (as well as the PO simulation procedure) via experiment and full-wave electromagnetic simulations which utilized narrow-band, fully-coherent incident fields, attention can now be turned to validating the PO solutions via PO simulations using partially-coherent incident fields. These results are reported in Figs. 4 and 5. Figure 4 shows the theoretical (solid curves) and simulated (circles) Ss for σh = 0.01 rad, 0.05 rad, 0.1 rad, and 0.25 rad (with h = 20λ, σh=hσh/2, and L = 0.35 m) versus θr at normal incidence. Figures 4(a)–4(d) depict these curves for α = 2, 1, 0.5, and 0.25 (with ws = 5 mm), respectively, i.e., a coherent source to a relatively incoherent source. The vertical dashed lines in the figures mark the locations of θ1/er given by Eq. (30). Note the excellent agreement between the simulated and theoretical predictions. Also note that although the coherence properties of the source are very different for the curves plotted in the figures, the Ss for the very rough surface curves σh = 0.05 rad, 0.1 rad, and 0.25 rad are nearly identical. This physically implies that Ss (or the angular spread of the spectral density) for very rough surfaces is driven by surface properties not source parameters. Recall that this was theoretically predicted.

Fig. 4 Ss versus θr at normal incidence, i.e., θi = 0°—(a) α = 2, (b) α = 1, (c) α = 0.5, and (d) α = 0.25. The solid traces are the PO analytical predictions, i.e., Eq. (20) or Eq. (23) (whichever is applicable); the circles are the PO simulation results. The vertical dashed lines in the figures mark the locations of θ1/er, namely, Eq. (30).
Fig. 5 |μs| versus Δθr—(a) σh = 0.01 rad, (b) σh = 0.05 rad, (c) σh = 0.1 rad, and (d) σh = 0.25 rad. The solid traces are the PO analytical predictions, i.e., Eq. (20) or Eq. (23) (whichever is applicable); the circles are the PO simulation results. The vertical dashed lines in the figures mark the locations of |Δθr|1/e, namely, Eq. (26).

Figure 5 shows the theoretical (solid traces) and simulated (circles) |μs| for α = 2, 1, 0.5, and 0.25 (with the same ws as above) versus Δθr. Figures 5(a)–5(d) show these curves for σh = 0.01 rad, 0.05 rad, 0.1 rad, and 0.25 rad (with the same h, σh, and L as above), respectively, i.e., a smooth surface to a very rough surface. The vertical dashed lines in the figures mark the locations of |Δθr|1/e given by Eq. (26). Recall that Eq. (26) is applicable to very rough surfaces, a condition not met by the surfaces whose results are shown in Fig. 5(a). Note the excellent agreement between the Monte Carlo simulation results and the theoretical predictions. Also note that |Δθr|1/e depend almost exclusively on the coherence properties of the source α, as predicted.

4. Conclusion

Appendix

The symbols first used in Eq. (12) are
𝒮=12*1,2=(x^u^ϑx1,2ϑy1,2y^u^+y^ρ^1,2+ϑx1,2ϑy1,2x^ρ^1,2)r1,2(x^u^ϑx1,2ϑy1,2y^u^y^ρ^1,2ϑx1,2ϑy1,2x^ρ^1,2),
(32)
where
ϑx1,2=x^ρ^1,2x^v^ϑy1,2=y^ρ^1,2y^v^a=14ws2+12s2b=12s2a˜=a4(a2b2)b˜=b4(a2b2).
(33)
For the p-pol case,
𝒮||=||1||2*||1,2=(y^ρ^1,2+ϑx1,2ϑy1,2x^ρ^1,2x^u^+ϑx1,2ϑy1,2y^u^)r||1,2(y^ρ^1,2+ϑx1,2ϑy1,2x^ρ^1,2+x^u^ϑx1,2ϑy1,2y^u^),
(34)
where r is the p-pol Fresnel complex amplitude reflection coefficient. Note that r⊥,‖ = −1 and 1 for a perfect electric conductor (i.e., a perfect reflector) and a perfect magnetic conductor, respectively.

Acknowledgments

This research was supported in part by an appointment to the Postgraduate Research Participation Program at the Air Force Institute of Technology administered by the Oak Ridge Institute for Science and Education through an interagency agreement between the U.S. Department of Energy and AFIT.

The views expressed in this paper are those of the authors and do not reflect the official policy or position of the U.S. Air Force, the Department of Defense, or the U.S. Government.

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51.

S. Sahin, Z. Tong, and O. Korotkova, “Sensing of semi-rough targets embedded in atmospheric turbulence by means of stochastic electromagnetic beams,” Opt. Commun. 283, 4512–4518 (2010) [CrossRef] .

52.

J. Huttunen, A. T. Friberg, and J. Turunen, “Scattering of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E 52, 3081–3092 (1995) [CrossRef] .

53.

M. W. Hyde, S. Basu, S. J. Cusumano, and M. F. Spencer, “Scalar wave solution for the scattering of a partially coherent beam from a statistically rough metallic surface,” Proc. SPIE 8550 (2012) [CrossRef] .

54.

T. Hansen and A. Yaghjian, Plane-Wave Theory of Time-Domain Fields: Near-Field Scanning Applications (IEEE, 1999) [CrossRef] .

55.

S. Ramo, J. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics, 3 ed. (John Wiley & Sons, 1994).

56.

C. A. Mack, “Analytic form for the power spectral density in one, two, and three dimensions,” J. Micro/Nanolithogr. MEMS MOEMS 10, 040501 (2011).

57.

S. Basu, S. J. Cusumano, M. W. Hyde, M. A. Marciniak, and S. T. Fiorino, “Validity of using Gaussian Schell model for extended beacon studies,” Proc. SPIE 8380 (2012).

58.

Under quasi-monochromatic conditions (Δωω̄), S (ρ, ω) Γ(ρ, ρ, 0)δ(ωω̄) and μ (ρ1, ρ2, ω) ≈ γ (ρ1, ρ2, τ) exp (jω̄τ). Here, Γ is the mutual coherence function and γ is the complex degree of coherence [59].

59.

A. T. Friberg and E. Wolf, “Relationships between the complex degrees of coherence in the space-time and in the space-frequency domains,” Opt. Lett. 20, 623–625 (1995) [CrossRef] [PubMed] .

60.

B. Balling, “A comparative study of the bidirectional reflectance distribution function of several surfaces as a mid-wave infrared diffuse reflectance standard,” Master’s thesis, Graduate School of Engineering and Management, Air Force Institute of Technology (AETC), Wright-Patterson AFB OH (2009). http://oai.dtic.mil/oai/oai?verb=getRecord&metadataPrefix=html&identifier=ADA495933.

61.

LabSphere Inc., “A guide to reflectance coatings and materials” (LabSphere, Inc., 2012). http://www.labsphere.com/uploads/technical-guides/a-guide-to-reflectance-materials-and-coatings.pdf.

62.

KLA-Tencor Corporation, “Alpha-Step IQ Surface Profiler” (KLA-Tencor Corporation, 2012). http://www.kla-tencor.com/surface-profiling/alpha-step-iq.html.

63.

R. Harrington, Field Computation by Moment Methods (IEEE, 1993) [CrossRef] .

64.

A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics (IEEE, 1998).

65.

RefractiveIndex.INFO, “RefractiveIndex.INFO Refractive index database” (RefractiveIndex.INFO, 2012). http://refractiveindex.info.

66.

H. T. Yura and S. G. Hanson, “Digital simulation of an arbitrary stationary stochastic process by spectral representation,” J. Opt. Soc. Am. A 28, 675–685 (2011) [CrossRef] .

67.

X. Xiao and D. Voelz, “Wave optics simulation approach for partial spatially coherent beams,” Opt. Express 14, 6986–6992 (2006) [CrossRef] [PubMed] .

OCIS Codes
(030.0030) Coherence and statistical optics : Coherence and statistical optics
(240.5770) Optics at surfaces : Roughness
(260.2110) Physical optics : Electromagnetic optics
(290.5880) Scattering : Scattering, rough surfaces

ToC Category:
Coherence and Statistical Optics

History
Original Manuscript: January 28, 2013
Revised Manuscript: February 13, 2013
Manuscript Accepted: February 16, 2013
Published: March 12, 2013

Citation
Milo W. Hyde, Santasri Basu, Mark F. Spencer, Salvatore J. Cusumano, and Steven T. Fiorino, "Physical optics solution for the scattering of a partially-coherent wave from a statistically rough material surface," Opt. Express 21, 6807-6825 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-6-6807


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  58. Under quasi-monochromatic conditions (Δω ≪ ω̄), S (ρ, ω) ≈ Γ(ρ, ρ, 0)δ(ω − ω̄) and μ (ρ1, ρ2, ω) ≈ γ (ρ1, ρ2, τ) exp (jω̄τ). Here, Γ is the mutual coherence function and γ is the complex degree of coherence [59].
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  67. X. Xiao and D. Voelz, “Wave optics simulation approach for partial spatially coherent beams,” Opt. Express14, 6986–6992 (2006). [CrossRef] [PubMed]

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