## Physical optics solution for the scattering of a partially-coherent wave from a statistically rough material surface |

Optics Express, Vol. 21, Issue 6, pp. 6807-6825 (2013)

http://dx.doi.org/10.1364/OE.21.006807

Acrobat PDF (1944 KB)

### 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

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] .

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

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] .

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] .

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

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

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

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. 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] .

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

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] .

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] .

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

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

*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] .

39. T. Leskova, A. Maradudin, and J. Munoz-Lopez, “The design of one-dimensional randomly rough surfaces that act as Collett-Wolf sources,” Opt. Commun. **242**, 123–133 (2004) [CrossRef] .

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

40. J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Condition for invariant spectral degree of coherence of an electromagnetic plane wave on scattering,” Opt. Commun. **284**, 724–728 (2011) [CrossRef] .

45. E. Wolf, J. T. Foley, and F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A **6**, 1142–1149 (1989) [CrossRef] .

46. B. G. Hoover and V. L. Gamiz, “Coherence solution for bidirectional reflectance distributions of surfaces with wavelength-scale statistics,” J. Opt. Soc. Am. A **23**, 314–328 (2006) [CrossRef] .

47. L. C. Andrews and R. L. Phillips, *Laser Beam Propagation through Random Media*, 2 ed. (SPIE, 2005) [CrossRef] .

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] .

## 2. Methodology

*h*(

*x*) with mean, standard deviation, and correlation length equal to 0,

*σ*, and

_{h}*ℓ*, respectively. The surface (of length 2

_{h}*L*) is illuminated by a partially-coherent beam (parameters

*w*and

_{s}*ℓ*defined below) emanating from the source plane specified by the coordinates (

_{s}*u*,

*v*,

*z*). Its origin, relative to the rough surface coordinate system, is located at the point (−

*x*,

_{s}*y*). The observation vector

_{s}**=**

*ρ***x̂**

*x*+

**ŷ**

*y*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 (

*ε*

_{0},

*μ*

_{0}).

*w*and

_{s}*ℓ*are the source radius and source coherence radius, respectively and the functional dependence of

_{s}*W*,

^{i}*E*

_{0},

*w*, and

_{s}*ℓ*on the radian frequency

_{s}*ω*is omitted for brevity [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 [17

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

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

### 2.1. PO expression for the scattered field

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

*ρ**is a vector that points from the source plane origin to the rough surface origin,*

_{s}**k**

*| =*

^{i}*k*

_{0}= 2

*π*/

*λ*, thus

**N**and

**L**[55] and the equivalent electric

**J**and magnetic

**M**currents induced on the rough surface by the incident field, namely,

**′ =**

*ρ***x̂**

*x*′ +

**ŷ**

*h*(

*x*′). Note that the integral is over the parameterized surface contour

*C*with d

*c*equal to the arc length, i.e.,

**n̂**is the unit outward normal to the surface given by 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

**=**

*ν**k*

_{0}

**̂ −**

*ρ***k**

*and Note that*

^{i}*r*

_{⊥}is primed because it is a function of surface coordinates (

*x*′,

*h*(

*x*′)).

*x*′ in Eq. (8) because of

*ℋ*

_{⊥}, which is a complicated function of

*h*(

*x*′) and its derivative

*h*

_{x}_{′}[1]. This integral is typically evaluated using the stationary-phase approximation in which

*h*

_{x}_{′}≈ −

*ν*/

_{x}*ν*[1,19

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

*Light Scattering and Nanoscale Surface Roughness* (Springer, 2007) [CrossRef] .

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

*Scattering and Diffraction in Physical Optics*, 2 ed. (World Scientific, 2006) [CrossRef] .

### 2.2. Scattered field cross-spectral density function

*x*′

_{1},

*ρ*_{1},

*x*′

_{2},

*ρ*_{2},

*h*(

*x*′

_{1}) and

*h*(

*x*′

_{2}) [23]. The incident field term and the rough surface term are represented as

*v*direction in Fig. 1. The second implication is that the distance from the rough surface to the source plane

*ρ*must be much greater than

_{s}*L*—this is generally the case. To provide some idea of how much greater

*ρ*must be than

_{s}*L*,

*ρ*= 10

_{s}*L*,

*ρ*= 20

_{s}*L*, and

*ρ*= 50

_{s}*L*are approximately 95%, 97.5%, and 99% accurate, respectively. Note that these two implications mean that the

*ℋ*

_{⊥}, see Eq. (9), diminishes at the rate of 1/

*ρ*and therefore can be neglected. Recall that this term corresponded to the

_{s}**v̂**component of the incident magnetic field implying that the field incident on the rough surface is transverse electromagnetic (in particular, TEM

*) when the method of stationary phase is utilized. Applying the method of stationary phase and substituting in*

^{v}#### 2.2.1. Smooth-to-moderately rough surfaces

*x*results in

_{d}*x*integration limits can be extended from [|

_{d}*x*| − 2

_{a}*L*, 2

*L*− |

*x*|] to (−∞, ∞). A necessary condition for this approximation occurs when where

_{a}*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

*x*takes on in the integration is 2

_{d}*L*, the following condition can be derived: 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

*x*integration limits in Eq. (15) [|

_{d}*x*| − 2

_{a}*L*, 2

*L*− |

*x*|] ≈ (−∞, ∞). How well the incident beam “fits” is determined by

_{a}*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.

*α*=

*ℓ*/

_{s}*w*. 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).

_{s}#### 2.2.2. Very rough surfaces

*χ*only has significant value when

*ϑ*

_{y}_{1}/

*ϑ*

_{y}_{2}≈

*ϑ*

_{y}_{2}/

*ϑ*

_{y}_{1},

*x*. Thus, it makes sense to expand

_{d}*ϑ*

_{x1}and

*ϑ*

_{x2}. 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.,

*ϑ*

_{x1}and

*ϑ*

_{x2}if Eq. (21) is satisfied.

### 2.3. Angular spectral degree of coherence radius

*σ*

_{h}_{′}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

*σ*

_{h}_{′}, 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] .

*θ*≈

^{r}*θ*, i.e., when observation is in the specular direction, the dependence of the expression on observation and incident angles disappears.

^{i}### 2.4. Angular spectral density radius

*θ*= 0 and

^{i}*e*yields the following expression after some simplification: where all symbols have been previously defined. With liberal use of trigonometric identities and some algebra, one arrives at One quickly recognizes that the above expression is a quadratic equation in terms of

*ψ*root makes physical sense), and utilizing the quadratic formula, yields the following roots for

*ψ*: 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

### 2.5. Parallel-polarization cross-spectral density function

*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

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.

*S*versus

^{s}*θ*from a LabSphere Infragold [61

^{r}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.

*λ*= 3.39

*μ*m (HeNe MWIR laser) and

*θ*= 20

^{i}*°*, 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.

*μ*m), performed along different directions, were taken and analyzed. The measured surface height standard deviation

*σ*, surface slope standard deviation

_{h}*σ*

_{h}_{′}, and correlation length

*ℓ*were 11.09

_{h}*μ*m, 0.2441 rad, and 116.9

*μ*m, respectively.

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

**J**: where

**A**and

**F**are the magnetic and electric vector potentials, respectively [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

*S*versus

^{s}*θ*and the modulus of the SDoC |

^{r}*μ*| versus

^{s}*Δθ*were calculated using the scattered field predicted from 500 rough surface realizations at

^{r}*λ*= 3.39

*μ*m, the Infragold surface statistics stated above, and a complex index of refraction

*n*− j

*κ*for gold of 1.995 − j20.95 [65

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] .

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] .

^{2}scattered fields, from which |

*μ*| and

^{s}*S*were computed. These simulations were performed at

^{s}*λ*= 3.39

*μ*m using hypothetical rough aluminum surfaces

*n*− j

*κ*= 5.366 − j33.33 [65

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

### 3.1. Results

*S*CASI measurement, MoM simulation, PO analytical solution, and PO simulation results versus

^{s}*θ*. The simulations were setup to best match the CASI experimental scenario, i.e.,

^{r}*ρ*= 185 cm and

_{s}*w*≈ 0.5 mm. For MoM computational reasons,

_{s}*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 [57]. If the Infragold sample measured and analyzed here adhered to the G-G model of rough surfaces,

*σ*

_{h}_{′}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.

*ρ*, source radius

_{s}*w*, and surface 1/2 length

_{s}*L*were the same as above. The figure reports the normalized

*S*versus

^{s}*θ*in Fig. 3(a) and

^{r}**|**

*μ*| versus Δ

^{s}*θ*in Figs. 3(b) and 3(c). In Figs. 3(a) and 3(b), the simulated Infragold surface possessed the same

^{r}*σ*and

_{h}*ℓ*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), |

_{h}*μ*| is shown for hypothetical Infragold G-G surfaces which possessed

^{s}*σ*= 0.05

_{h}*λ*, 0.1

*λ*, and 0.25

*λ*with

*ℓ*held constant at 8

_{h}*λ*. 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.

*S*for

^{s}*σ*

_{h}_{′}= 0.01 rad, 0.05 rad, 0.1 rad, and 0.25 rad (with

*ℓ*= 20

_{h}*λ*,

*L*= 0.35 m) versus

*θ*at normal incidence. Figures 4(a)–4(d) depict these curves for

^{r}*α*= 2, 1, 0.5, and 0.25 (with

*w*= 5 mm), respectively, i.e., a coherent source to a relatively incoherent source. The vertical dashed lines in the figures mark the locations of

_{s}*S*for the very rough surface curves

^{s}*σ*

_{h}_{′}= 0.05 rad, 0.1 rad, and 0.25 rad are nearly identical. This physically implies that

*S*(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.

^{s}*μ*| for

^{s}*α*= 2, 1, 0.5, and 0.25 (with the same

*w*as above) versus

_{s}*Δθ*. Figures 5(a)–5(d) show these curves for

^{r}*σ*

_{h}_{′}= 0.01 rad, 0.05 rad, 0.1 rad, and 0.25 rad (with the same

*ℓ*,

_{h}*σ*, and

_{h}*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 |

*Δθ*|

^{r}_{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 |

*Δθ*|

^{r}_{1/e}depend almost exclusively on the coherence properties of the source

*α*, as predicted.

## 4. Conclusion

*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.

## Appendix

*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

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42. | T. Shirai and T. Asakura, “Multiple light scattering from spatially random media under the second-order Born approximation,” Opt. Commun. |

43. | G. Gbur and E. Wolf, “Determination of density correlation functions from scattering of polychromatic light,” Opt. Commun. |

44. | F. Gori, C. Palma, and M. Santarsiero, “A scattering experiment with partially coherent light,” Opt. Commun. |

45. | E. Wolf, J. T. Foley, and F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A |

46. | B. G. Hoover and V. L. Gamiz, “Coherence solution for bidirectional reflectance distributions of surfaces with wavelength-scale statistics,” J. Opt. Soc. Am. A |

47. | L. C. Andrews and R. L. Phillips, |

48. | O. Korotkova, |

49. | R. S. Hansen, H. T. Yura, and S. G. Hanson, “First-order speckle statistics: an analytic analysis using ABCD matrices,” J. Opt. Soc. Am. A |

50. | H. Yura and S. Hanson, “Variance of intensity for Gaussian statistics and partially developed speckle in complex ABCD optical systems,” Opt. Commun. |

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. |

52. | J. Huttunen, A. T. Friberg, and J. Turunen, “Scattering of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E |

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 |

54. | T. Hansen and A. Yaghjian, |

55. | S. Ramo, J. Whinnery, and T. Van Duzer, |

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

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 |

58. | Under quasi-monochromatic conditions ( ω) ≈ Γ(, ρ, 0)ρδ(ω − ω̄) and μ (ρ_{1}, ρ_{2}, ω) ≈ γ (ρ_{1}, ρ_{2}, 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. |

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, |

64. | A. F. Peterson, S. L. Ray, and R. Mittra, |

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 |

67. | X. Xiao and D. Voelz, “Wave optics simulation approach for partial spatially coherent beams,” Opt. Express |

**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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