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
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) [1–7], perturbation [4, 8], and computational/full-wave [9–14] methods. The interested reader is referred to Elfouhaily and Guérin , Warnick and Chew , Maradudin , Ogilvy , and Nieto-Vesperinas  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 [20–24]. 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 , Goodman , Parry , Fujii and Asakura [28,29], Pedersen , and Yoshimura et al..
Since the presence of speckle is typically detrimental in applications involving coherent light, techniques for suppressing speckle naturally followed. While there are many such techniques , due predominately to the work of Wolf [33–35], the use of partially-coherent light, instead of laser light, in active illumination systems is becoming tremendously popular for speckle suppression. In particular, much literature is dedicated to the properties of partially-coherent light whose cross-spectral density (CSD) function possesses a Gaussian Schell-model (GSM) form [33, 35–39]. In regards to the scattering of partially-coherent light, most of the current literature deals with scattering from low-contrast scatterers, i.e., for scatterers in which the Born approximation is valid [33, 35, 36, 40–45]. Far less work has been performed analyzing the scattering of partially-coherent light from rough metallic surfaces. Of the work that has been published, the following approaches are common: the phase-screen model [32, 46–48], ABCD matrices [47–51], and the coherent-mode representation .
The purpose of this rough surface scattering work is to extend the traditional, fully-coherent approaches to cases involving partially-coherent illumination. Previous work by the authors derived analytical forms for the scattering of a partially-coherent beam from a statistically rough perfectly-reflecting surface using the PO approximation . In this work, the previously derived scattering solutions are generalized to material surfaces and rigorously verified via experiment and simulation.
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.
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 ℓh, respectively. The surface (of length 2L) is illuminated by a partially-coherent beam (parameters ws and ℓs 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 = x̂x +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).
A GSM form for the incident field CSD is used to model the partially-coherent illumination, viz.,33, 35]. 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 , 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 . 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 form54], the incident electric and magnetic fields become
The scattered field in the far zone can be found by utilizing the transverse components of the far-field vector potentials N and L and the equivalent electric J and magnetic M currents induced on the rough surface by the incident field, namely,
Using the PO approximation, the equivalent currents induced by the incident field on the material surface areEqs. (6) and (7) into Eq. (5) and subsequent simplification yields the following expression for :
No analytical solution has yet been found for the integral over the rough surface coordinate x′ in Eq. (8) because of ℋ⊥, which is a complicated function of h(x′) and its derivative hx′. This integral is typically evaluated using the stationary-phase approximation in which hx′ ≈ −νx/νy[1,19]. This approximation physically dictates that reflection from the rough surface is locally specular, i.e., local diffraction effects are excluded [1, 17–19, 23]. Applying this approximation to Eq. (8) and rearranging the integrals results in18, 19].
2.2. Scattered field cross-spectral density function
Applying the definition of the autocorrelation of a random process to Eq. (10) yieldsEq. (1). The second autocorrelation expression in Eq. (11), after some simple manipulation, is equivalent to the joint characteristic function of the random variables h(x′1) and h(x′2) . The incident field term and the rough surface term are represented as and in the analysis to follow.
The integrals over the incident field plane-wave spectrum can be approximated using the method of stationary phase . Using the method of stationary phase to evaluate the plane-wave spectrum integrals has two implications. The first is that and is therefore approximated as for all “phase” terms (i.e., exp[j(⋯)]) and as 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 term in ℋ⊥, see Eq. (9), diminishes at the rate of 1/ρs and therefore can be neglected. Recall that this term corresponded to the v̂ 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 producesappendix.
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.,32]. Recent profilometer measurements of sandblasted metallic surfaces showed that the stretched exponential-stretched exponential (SE-SE) model  more accurately represented surfaces roughened by random industrial processes . Unfortunately, no general analytical form for the SE joint characteristic function exists. Although the results in  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: Eq. (12), performing the common variable transformation xd = x′1 − x′2 and xa = x′1 + x′2, and subsequent simplification yields
At this point it is necessary to handle the exponential term containing the surface autocorrelation function. Classically, this has been done in one of two ways. The first is to expand the exponential term in a Taylor series and proceed with the evaluation of the integrals . Mathematically, this approach is applicable to all surfaces; however, computationally (because the series is slowly convergent), this approach is limited to smooth-to-moderately rough surfaces and is investigated in the next section. The other approach involves expanding the surface autocorrelation function in a Taylor series and only retaining the first two terms, i.e., [3, 23]. This treatment is applicable to very rough surfaces and is discussed in Section 2.2.2.
2.2.1. Smooth-to-moderately rough surfaces
As briefly discussed in the previous section, further progress can be made on Eq. (15) by expanding the exponential term containing the surface autocorrelation function in a Taylor series, specifically,Eq. (15) and subsequent evaluation of the integral over xd results in
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 whenEq. (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,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, 35, 36]. When one considers a fully-coherent incident Gaussian beam (ℓs → ∞) and quasi-monochromatic light , the coherent scattered irradiance, i.e., , 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.. One can also derive from Eq. (20) the incoherent scattered irradiance for a fully-coherent incident Gaussian beam and quasi-monochromatic light. Note that is equivalent to the variance of the scattered field, i.e., , where is the average scattered irradiance. This expression also simplifies to the form derived by Wang et al..
2.2.2. Very rough surfaces
Consider the form of the joint characteristic function:Eq. (15) and carrying out the integrations, one arrives at the desired result for the scattered field CSD function: 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., . 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.
Because of the magnitude of the argument of the “correlation” exponential (specifically, the term), , implying that , for the exponential to have a significant value. This implies that the correlation exponential is approximately a function of . Since this term is predominately responsible for the behavior of the SDoC, the SDoC is also a function (approximately) of . This is in agreement with the findings of previous studies which were restricted to the paraxial regime.
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.22, 23]. 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
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 . Note that when θr ≈ θi, 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.
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)  at the Air Force Institute of Technology was used. Measurements of the scattered SD Ss versus θr from a LabSphere Infragold  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 . 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 ℓh were 11.09 μm, 0.2441 rad, and 116.9 μm, respectively.
For the full-wave electromagnetic simulations, the Method of Moments (MoM)  was used to solve the following impedance-boundary-condition-based s-pol electric field integral equation for the unknown electric current J:64]. 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 . 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 . The 500 rough surface realizations were generated using the method outlined by Yura and Hanson .
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 . For the random surface, 500 rough surface instances were generated using the technique detailed in Ref. . 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  with varying degrees of surface roughness. In addition, the properties of the incident field (size and coherence) were also varied to examine their effects.
Figure 2 shows the normalized Ss CASI measurement, MoM simulation, PO analytical solution, and PO simulation results versus θr. The simulations were setup to best match the CASI experimental scenario, i.e., ρs = 185 cm and ws ≈ 0.5 mm. For MoM computational reasons, L = 2.5 mm, which resulted in an incident beam which over-illuminated the rough surface by less than 2%. While all the traces possess a Gaussian-like shape, there is a significant difference between the measured and MoM trace widths versus the PO results. As briefly discussed in Section 2.2, previous work had found that the SE-SE model more accurately represented surfaces roughened by random industrial processes . If the Infragold sample measured and analyzed here adhered to the G-G model of rough surfaces, rad. The measured σh′ was roughly twice that value. Thus, in this case, the G-G model is not an ideal choice. Note that in these MoM simulation results, SE-SE surfaces were used, hence the good agreement with the CASI results. SE-SE surfaces could have been used in the PO simulations; however, as discussed previously, it would not have been possible to present analytical results. These results are presented for two main reasons. The first is to verify the analytical solutions which, based on the excellent agreement between the PO simulation and PO analytical results, is clearly accomplished. The other is to demonstrate the validity of the ubiquitous G-G model of rough surfaces. As can be seen, the model tends to underpredict the width of the scattered SD. Nevertheless, considering that analytical scattering solutions are possible using the G-G model, this is likely an acceptable drawback.
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 ℓh 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 ℓh 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.
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 ℓh = 20λ, , 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 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.
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 ℓh, 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.
The scattering of a partially-coherent field from a statistically rough surface was investigated. This work significantly extended previous efforts by incorporating the effects of material parameters (ε and μ) in the analysis and by rigorously validating the derived solutions via experiment and extensive simulation. Two forms of the scattered-field CSD function were derived in this work using the PO approximation. The first, applicable to smooth-to-moderately rough surfaces, was represented as an infinite series. While being a rather complicated expression dependent on both source and surface parameters, physical insight was gleaned from its analytical form. The second form of the scattered field CSD function was applicable to very rough surfaces. This expression was examined at length to include derivations of the angular SDoC and SD radii. It was noted that under certain circumstances, this form of the CSD function maintained a GSM form in agreement with published results valid only in the paraxial regime. Lastly, the closed-form expressions for the scattered field CSD functions were rigorously validated with scatterometer measurements, as well as, MoM and PO Monte Carlo simulations. The analytical predictions were found to be in good agreement with the measurement and simulation results.
The symbols first used in Eq. (12) are
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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