## A geometrical optics polarimetric bidirectional reflectance distribution function for dielectric and metallic surfaces

Optics Express, Vol. 17, Issue 24, pp. 22138-22153 (2009)

http://dx.doi.org/10.1364/OE.17.022138

Acrobat PDF (559 KB)

### Abstract

A polarimetric bidirectional reflectance distribution function (pBRDF), based on geometrical optics, is presented. The pBRDF incorporates a visibility (shadowing/masking) function and a Lambertian (diffuse) component which distinguishes it from other geometrical optics pBRDFs in literature. It is shown that these additions keep the pBRDF bounded (and thus a more realistic physical model) as the angle of incidence or observation approaches grazing and better able to model the behavior of light scattered from rough, reflective surfaces. In this paper, the theoretical development of the pBRDF is shown and discussed. Simulation results of a rough, perfect reflecting surface obtained using an exact, electromagnetic solution and experimental Mueller matrix results of two, rough metallic samples are presented to validate the pBRDF.

© 2009 Optical Society of America

## 1. Introduction

1. D. E. Barrick, “Theory of HF and VHF propagation across the rough sea—parts I and II,” Radio Sci. **6**, 517–533 (
1971). [CrossRef]

2. C. Eckart, “The scattering of sound from the sea surface,” J. Acoust. Soc. Am. **25**, 566–570 (
1953). [CrossRef]

3. E. Y. Harper and F. M. Labianca, “Scattering of sound from a point source by a rough surface progressing over an isovelocity ocean,” J. Acoust. Soc. Am. **58(2)**, 349–364 (
1975). [CrossRef]

4. K. Krishen, “Correlation of radar backscattering cross sections with ocean wave height and wind velocity,” J. Geophys. Res. **76**, 6528–6539 (
1971). [CrossRef]

6. D. S. Kimes, “Modeling the directional reflectance from complete homogeneous vegetation canopies with various leaf-orientation distributions,” J. Opt. Soc. Am. A **1(7)**, 725–737 (
1984). [CrossRef]

7. R. Hegedüs, A. Barta, B. Bernáth, V. B. Meyer-Rochow, and G. Horváth, “Imaging polarimetry of forest canopies: how the azimuth direction of the sun, occluded by vegetation, can be assessed from the polarization pattern of the sunlit foliage,” Appl. Opt. **46(23)**, 6019–6032 (
2007). [CrossRef]

8. G. Zonios, I. Bassukas, and A. Dimou, “Comparative evaluation of two simple diffuse reflectance models for biological tissue applications,” Appl. Opt. **47(27)**, 4965–4973 (
2008). [CrossRef]

9. J. Xia and G. Yao, “Angular distribution of diffuse reflectance in biological tissue,” Appl. Opt. **46(26)**, 6552–6560 (
2007). [CrossRef]

13. M. A. Greiner, B. D. Duncan, and M. P. Dierking, “Bidirectional scattering distribution functions of maple and cottonwood leaves,” Appl. Opt. **46(25)**, 6485–6494 (
2007). [CrossRef]

14. P. Y. Ufimtsev, *Fundamentals of the Physical Theory of Diffraction* (John Wiley & Sons, Inc., Hoboken, NJ,
2007). [CrossRef]

20. K. E. Torrance and E. M. Sparrow, “Theory for off-specular reflection from roughened surfaces,” J. Opt. Soc. Am. **57(9)**, 1105–1114 (
1967). [CrossRef]

26. D. Wellems, M. Serna, S. H. Sposato, M. P. Fetrow, K. P. Bishop, S. A. Arko, and T. R. Caudill, “Spectral polarimetric BRDF model and comparison to measurements from isotropic roughened glass,” in *Workshop on Multi/Hyperspectral Sensors, Measurements, Modeling and Simulation* (U.S. Army Aviation and Missile Command, Huntsville, AL,
2000).

27. K. K. Ellis, “Polarimetric bidirectional reflectance distribution function of glossy coatings,” J. Opt. Soc. Am. A **13(8)**, 1758–1762 (
1996). [CrossRef]

24. R. G. Priest and S. R. Meier, “Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces,” Opt. Eng. **41(5)**, 988–993 (
2002). [CrossRef]

20. K. E. Torrance and E. M. Sparrow, “Theory for off-specular reflection from roughened surfaces,” J. Opt. Soc. Am. **57(9)**, 1105–1114 (
1967). [CrossRef]

28. H. G. Tompkins and E. A. Irene, *Handbook of Ellipsometry* (William Andrew, Inc., Norwich, NY,
2005). [CrossRef]

## 2. Methodology

*ω*and centered on polar angle

_{i}*θ*and azimuth angle

_{i}*ϕ*, is incident on a small area d

_{i}*A*of a larger rough surface with complex index of refraction

*η*=

*n*-j

*κ*. Light is scattered from the surface and observed within solid angle d

*ω*at polar angle

_{r}*θ*and azimuth angle

_{r}*ϕ*. The BRDF is defined as the ratio of the scattered radiance to the incident irradiance:

_{r}*L*is the scattered radiance, d

_{r}*E*is the incident irradiance, and

_{i}*ϕ*=|

*ϕ*-

_{r}*ϕ*| (i.e., surface is isotropic and homogeneous). Note that the incident irradiance is equivalent to the product of the incident radiance and the projected solid angle cos

_{i}*θ*d

_{i}*ω*[20

_{i}20. K. E. Torrance and E. M. Sparrow, “Theory for off-specular reflection from roughened surfaces,” J. Opt. Soc. Am. **57(9)**, 1105–1114 (
1967). [CrossRef]

29. R. Anderson, “Matrix description of radiometric quantities,” Appl. Opt. **30(7)**, 858–867 (
1991). [CrossRef]

35. Y. Sun, “Statistical ray method for deriving reflection models of rough surfaces,” J. Opt. Soc. Am. A **24(3)**, 724–744 (
2007). [CrossRef]

24. R. G. Priest and S. R. Meier, “Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces,” Opt. Eng. **41(5)**, 988–993 (
2002). [CrossRef]

26. D. Wellems, M. Serna, S. H. Sposato, M. P. Fetrow, K. P. Bishop, S. A. Arko, and T. R. Caudill, “Spectral polarimetric BRDF model and comparison to measurements from isotropic roughened glass,” in *Workshop on Multi/Hyperspectral Sensors, Measurements, Modeling and Simulation* (U.S. Army Aviation and Missile Command, Huntsville, AL,
2000).

29. R. Anderson, “Matrix description of radiometric quantities,” Appl. Opt. **30(7)**, 858–867 (
1991). [CrossRef]

30. D. S. Flynn and C. Alexander, “Polarized surface scattering expressed in terms of a bidirectional reflectance distribution function,” Opt. Eng. **34(6)**, 1646–1650 (
1995). [CrossRef]

33. J. R. Schott, *Fundamentals of Polarimetric Remote Sensing* (SPIE Press, Bellingham, WA,
2009). [CrossRef]

36. W. S. Bickel and W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. **53(5)**, 468–478 (
1985). [CrossRef]

37. M. G. Gartley, S. D. Brown, and J. R. Schott, “Micro-scale surface and contaminate modeling for polarimetric signature prediction,” in *Proc. SPIE*, vol. 6972 (The International Society for Optical Engineering (SPIE),
2008). [CrossRef]

*f*of Eq. (1) equals F

_{00}where the subscript 00 is the element in the first row and first column of the pBRDF Mueller matrix [33

33. J. R. Schott, *Fundamentals of Polarimetric Remote Sensing* (SPIE Press, Bellingham, WA,
2009). [CrossRef]

**57(9)**, 1105–1114 (
1967). [CrossRef]

25. D. Wellems, S. Ortega, D. Bowers, J. Boger, and M. Fetrow, “Long wave infrared polarimetric model: theory, measurements and parameters,” J. Opt. A: Pure Appl. Opt. **8(10)**, 914–925 (
2006). [CrossRef]

27. K. K. Ellis, “Polarimetric bidirectional reflectance distribution function of glossy coatings,” J. Opt. Soc. Am. A **13(8)**, 1758–1762 (
1996). [CrossRef]

33. J. R. Schott, *Fundamentals of Polarimetric Remote Sensing* (SPIE Press, Bellingham, WA,
2009). [CrossRef]

35. Y. Sun, “Statistical ray method for deriving reflection models of rough surfaces,” J. Opt. Soc. Am. A **24(3)**, 724–744 (
2007). [CrossRef]

37. M. G. Gartley, S. D. Brown, and J. R. Schott, “Micro-scale surface and contaminate modeling for polarimetric signature prediction,” in *Proc. SPIE*, vol. 6972 (The International Society for Optical Engineering (SPIE),
2008). [CrossRef]

**57(9)**, 1105–1114 (
1967). [CrossRef]

25. D. Wellems, S. Ortega, D. Bowers, J. Boger, and M. Fetrow, “Long wave infrared polarimetric model: theory, measurements and parameters,” J. Opt. A: Pure Appl. Opt. **8(10)**, 914–925 (
2006). [CrossRef]

26. D. Wellems, M. Serna, S. H. Sposato, M. P. Fetrow, K. P. Bishop, S. A. Arko, and T. R. Caudill, “Spectral polarimetric BRDF model and comparison to measurements from isotropic roughened glass,” in *Workshop on Multi/Hyperspectral Sensors, Measurements, Modeling and Simulation* (U.S. Army Aviation and Missile Command, Huntsville, AL,
2000).

*Fundamentals of Polarimetric Remote Sensing* (SPIE Press, Bellingham, WA,
2009). [CrossRef]

35. Y. Sun, “Statistical ray method for deriving reflection models of rough surfaces,” J. Opt. Soc. Am. A **24(3)**, 724–744 (
2007). [CrossRef]

37. M. G. Gartley, S. D. Brown, and J. R. Schott, “Micro-scale surface and contaminate modeling for polarimetric signature prediction,” in *Proc. SPIE*, vol. 6972 (The International Society for Optical Engineering (SPIE),
2008). [CrossRef]

**24(3)**, 724–744 (
2007). [CrossRef]

27. K. K. Ellis, “Polarimetric bidirectional reflectance distribution function of glossy coatings,” J. Opt. Soc. Am. A **13(8)**, 1758–1762 (
1996). [CrossRef]

## 2.1. Specular pBRDF component

**57(9)**, 1105–1114 (
1967). [CrossRef]

*λ*. This implies that the “roughness” of the macroscopic surface should be large compared to

*λ*. As discussed by Sun [35

**24(3)**, 724–744 (
2007). [CrossRef]

*and the surface correlation length*

_{h}*ℓ*are large compared to

*λ*, i.e., σ

*,*

_{h}*ℓ*≫

*λ*. The specular component of a microfacet model-based pBRDF takes the form

*P*is the distribution function modeling the orientation of the facets around the mean surface normal (

*z*direction in Figs. 1 and 2),

**M**is the Mueller matrix modeling the polarimetric scattering from the material surface, and

*G*is the visibility function (shadowing/masking factor) [23–26

*Workshop on Multi/Hyperspectral Sensors, Measurements, Modeling and Simulation* (U.S. Army Aviation and Missile Command, Huntsville, AL,
2000).

*Fundamentals of Polarimetric Remote Sensing* (SPIE Press, Bellingham, WA,
2009). [CrossRef]

**24(3)**, 724–744 (
2007). [CrossRef]

*α*and

*β*are derived using spherical trigonometry:

**24(3)**, 724–744 (
2007). [CrossRef]

*Fundamentals of Polarimetric Remote Sensing* (SPIE Press, Bellingham, WA,
2009). [CrossRef]

*α*in the denominator. Shell [34] notes that the impact of this factor is minimal since the BRDF magnitude rapidly decreases with increasing

*α*such that division by a decreasing cos

*α*is negligible.

**24(3)**, 724–744 (
2007). [CrossRef]

24. R. G. Priest and S. R. Meier, “Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces,” Opt. Eng. **41(5)**, 988–993 (
2002). [CrossRef]

*Workshop on Multi/Hyperspectral Sensors, Measurements, Modeling and Simulation* (U.S. Army Aviation and Missile Command, Huntsville, AL,
2000).

**24(3)**, 724–744 (
2007). [CrossRef]

**57(9)**, 1105–1114 (
1967). [CrossRef]

*et al*. [25

25. D. Wellems, S. Ortega, D. Bowers, J. Boger, and M. Fetrow, “Long wave infrared polarimetric model: theory, measurements and parameters,” J. Opt. A: Pure Appl. Opt. **8(10)**, 914–925 (
2006). [CrossRef]

**M**in Eq. (6) can be found by starting with Jones vectors and matrices, i.e.,

*E*and

^{s}_{i}*E*are the s-pol (perpendicular polarization), incident and reflected, complex electric field components,

^{s}_{r}*E*and

^{p}_{i}*E*are the p-pol (parallel polarization), incident and reflected, complex electric field components, and

^{p}_{r}*r*and

_{s}*r*are the complex Fresnel field reflection coefficients for the s- and p-pol, respectively [23–26

_{p}*Workshop on Multi/Hyperspectral Sensors, Measurements, Modeling and Simulation* (U.S. Army Aviation and Missile Command, Huntsville, AL,
2000).

*Fundamentals of Polarimetric Remote Sensing* (SPIE Press, Bellingham, WA,
2009). [CrossRef]

*E*,

^{s}_{i}*E*,

^{s}_{r}*E*, and

^{p}_{i}*E*are defined with respect to the macroscopic coordinate system (see Fig. 1); however, the complex Fresnel field reflection coefficients are defined with respect to the microfacet coordinate system (see Fig. 2). Thus, it is necessary to perform coordinate system rotations to align the macroscopic planes of incidence and reflection with the microfacet planes of incidence and reflection. This fact explains the rotation matrices in Eq. (9). Relating the angles

^{p}_{r}*γ*and

_{i}*γ*to the macroscopic angles is, once again, accomplished using trigonometry [23–26

_{r}*Workshop on Multi/Hyperspectral Sensors, Measurements, Modeling and Simulation* (U.S. Army Aviation and Missile Command, Huntsville, AL,
2000).

*Fundamentals of Polarimetric Remote Sensing* (SPIE Press, Bellingham, WA,
2009). [CrossRef]

*T*,

_{ss}*T*,

_{ps}*T*, and

_{sp}*T*, can be interpreted as modeling how s- or p-pol incident light couples into s- or p-pol reflected light. For instance, the

_{pp}*T*element models how incident s-pol light couples into p-pol reflected light. The other elements can be interpreted in a similar manner. For scattering in the specular plane (i.e.,

_{ps}*ϕ*=

*π*), intuition leads one to conclude that the Jones scattering matrix in Eq. (9) becomes diagonal, i.e., incident s-pol and incident p-pol couple into reflected s-pol and reflected p-pol, respectively. This conclusion is easily confirmed by substituting

*ϕ*=

*π*into Eq. (10). Converting the Jones matrix in Eq. (9) to a Mueller matrix is performed using the analysis in Ref. [30

30. D. S. Flynn and C. Alexander, “Polarized surface scattering expressed in terms of a bidirectional reflectance distribution function,” Opt. Eng. **34(6)**, 1646–1650 (
1995). [CrossRef]

*Workshop on Multi/Hyperspectral Sensors, Measurements, Modeling and Simulation* (U.S. Army Aviation and Missile Command, Huntsville, AL,
2000).

*Fundamentals of Polarimetric Remote Sensing* (SPIE Press, Bellingham, WA,
2009). [CrossRef]

_{23}element of the above Mueller matrix models how the fourth Stokes parameter (circular polarization) couples into the third Stokes parameter (linear polarization) upon reflection.

**57(9)**, 1105–1114 (
1967). [CrossRef]

*G*determines the fraction of an illuminated microfacet which contributes to the scattered radiance. Physically, as is evident from Fig. 3,

*G*models the incident and reflected light blocked by adjacent microfacets. Most of the BRDFs/pBRDFs in literature include a shadowing/masking function of some form [11, 20

**57(9)**, 1105–1114 (
1967). [CrossRef]

**8(10)**, 914–925 (
2006). [CrossRef]

*Workshop on Multi/Hyperspectral Sensors, Measurements, Modeling and Simulation* (U.S. Army Aviation and Missile Command, Huntsville, AL,
2000).

**24(3)**, 724–744 (
2007). [CrossRef]

**41(5)**, 988–993 (
2002). [CrossRef]

*G*causes their pBRDF to asymptotically approach infinity as the angle of incidence or observation approaches grazing [25

**8(10)**, 914–925 (
2006). [CrossRef]

*Workshop on Multi/Hyperspectral Sensors, Measurements, Modeling and Simulation* (U.S. Army Aviation and Missile Command, Huntsville, AL,
2000).

*P*[Eq. (8)], the Mueller matrix

**M**[Eq. (11)], and the shadowing/masking function

*G*[Eq. (12)] into Eq. (6) [25

**8(10)**, 914–925 (
2006). [CrossRef]

*Workshop on Multi/Hyperspectral Sensors, Measurements, Modeling and Simulation* (U.S. Army Aviation and Missile Command, Huntsville, AL,
2000).

**41(5)**, 988–993 (
2002). [CrossRef]

*G*in Eq. (13). As is stated above,

*G*plays a critical role in keeping the pBRDF bounded and thus a realistic physical model. In order to demonstrate the function’s importance, consider the pBRDF predictions shown in Fig. 4. The figure shows traces comparing the F00 elements of the Priest and Germer pBRDF [23,24

**41(5)**, 988–993 (
2002). [CrossRef]

*θ*=45°, 60°, 75°, and 85° with 21/2σ

_{i}*/*

_{h}*ℓ*=0.3. The pBRDFs are evaluated in the specular plane (

*ϕ*=

*π*) and using a perfect reflecting surface, i.e., a perfect electric conductor (PEC). The figure clearly shows that the pBRDF in Eq. (13) remains bounded while the Priest and Germer pBRDF diverges as

*θ*approaches 90°. Having developed and discussed the specular pBRDF component, attention can now be turned to the diffuse component.

_{r}## 2.2. Diffuse pBRDF component

*Workshop on Multi/Hyperspectral Sensors, Measurements, Modeling and Simulation* (U.S. Army Aviation and Missile Command, Huntsville, AL,
2000).

30. D. S. Flynn and C. Alexander, “Polarized surface scattering expressed in terms of a bidirectional reflectance distribution function,” Opt. Eng. **34(6)**, 1646–1650 (
1995). [CrossRef]

*Fundamentals of Polarimetric Remote Sensing* (SPIE Press, Bellingham, WA,
2009). [CrossRef]

**24(3)**, 724–744 (
2007). [CrossRef]

*ρ*

_{DHR}≤1, otherwise the pBRDF violates the conservation of energy (assuming a passive material). The stated condition becomes an equality when the surface is a PEC. Substituting Eq. (5) into the DHR expression and applying the equality condition (a PEC surface) produces

^{d, PEC}

_{00}times M

_{00}[25

**8(10)**, 914–925 (
2006). [CrossRef]

*Workshop on Multi/Hyperspectral Sensors, Measurements, Modeling and Simulation* (U.S. Army Aviation and Missile Command, Huntsville, AL,
2000).

*→0, the facet distribution function in Eq. (13) becomes a Dirac delta function. Substituting this expression into Eq. (16) results in F*

_{h}^{d}

_{00}=0. Likewise, as surface roughness increases, one would expect light to be scattered multiple times before leaving the material surface. Mathematically this can be verified by observing that F

^{s}

_{00}→0 as σ

*→∞. Substituting F*

_{h}^{s}

_{00}=0 into Eq. (16) produces the trivial result F

^{d}

_{00}=1 (i.e., pure diffuse scattering). The second characteristic of note in favor of modeling the diffuse pBRDF component in the manner outlined above is that no fitted coefficients are required to model the strength of the diffuse pBRDF component. The use of coefficients, whose values are determined by fitting the BRDF to measured data, is a common feature in other BRDFs [11, 20

**57(9)**, 1105–1114 (
1967). [CrossRef]

*Fundamentals of Polarimetric Remote Sensing* (SPIE Press, Bellingham, WA,
2009). [CrossRef]

*Proc. SPIE*, vol. 6972 (The International Society for Optical Engineering (SPIE),
2008). [CrossRef]

## 2.3. Summary of theory

**8(10)**, 914–925 (
2006). [CrossRef]

*Workshop on Multi/Hyperspectral Sensors, Measurements, Modeling and Simulation* (U.S. Army Aviation and Missile Command, Huntsville, AL,
2000).

*θ*and

_{i}*ϕ*must be switched with

_{i}*θ*and

_{r}*ϕ*. It is easy to show that doing so produces the same expression as that in Eq. (17); thus, the pBRDF satisfies the reciprocity condition. Proving that Eq. (17) conserves energy requires one to show that

_{r}*ρ*

_{DHR}≤1. Note that the conservation of energy is enforced when finding the value of the diffuse pBRDF component (detailed above). Therefore, Eq. (17) conserves energy as well.

*G*. As discussed above,

*G*keeps Eq. (17) bounded and thus a realistic physical model. The second is the development of a diffuse pBRDF component. As previously stated, this component depends only on physical parameters and does not need to be fit to measured data. In the next section, predictions made using Eq. (17) of a rough, PEC surface are compared to Method of Moments [40] (MoM) solutions for the purpose of validating the model.

## 3. Simulation

*λ*long, random (surface height is Gaussian distributed) PEC surface. The surface is invariant in the z direction (see Fig. 5) significantly reducing the number of unknowns in the problem. Also, as is shown in Fig. 5, only s-pol is considered here. The electric field integral equation (EFIE) for the scattering problem depicted in Fig. 5 (assuming condition at the random, PEC surface, i.e.,

*E*=-

^{z}_{r}*E*:

^{z}_{i}_{0}is the intrinsic impedance of free-space (approximately 377 Ω),

**ρ**=

*x***x**+

*is the observation vector,*

**y**y*′=*

**ρ***′+*

**x**x*′ is the source vector,*

**y**y**k**

*=*

_{i}**x**sin

*θ*-

_{i}**y**cos

*θ*is the propagation vector of the incident field,

_{i}*J*is the current induced on the PEC surface by the field, and

^{z}*H*

^{(2)}

_{0}is a zeroth order Hankel function of the second kind. Note that the integral in Eq. (18) is over the parameterized surface contour denoted by

*C*′. The unknown in Eq. (18) is the surface current

*J*. Note that assuming

^{z}**J**=2

**n**×

**H**

*forms the basis of the physical optics, or Kirchoff approximation [14*

_{i}14. P. Y. Ufimtsev, *Fundamentals of the Physical Theory of Diffraction* (John Wiley & Sons, Inc., Hoboken, NJ,
2007). [CrossRef]

*J*is expanded in a set of basis functions (in this case, fixed width pulses):

^{z}*N*×

*N*matrix equation where

*N*is the number of unknowns, i.e.,

*N*element of the MoM matrix shown above models how the

*N*

^{th}source current segment contributes to the scattered field at the 2

^{nd}observation segment. The other elements of the MoM matrix can be interpreted in a similar manner. Solving Eq. (20) yields the unknown current. Once

*J*is computed, the scattered field can be found at any observation point by

^{z}*C*′

*is the segment of the parameterized contour represented by the*

_{n}*n*

^{th}pulse and

*ρ*=(

*x*

^{2}+

*y*

^{2})

^{1/2}is the Euclidean distance from the origin. The second line of the above expression assumes that the observation point

*ρ*is in the far-field as defined by Fraunhofer [15]. Note that the MoM solution shown above is a coherent field solution. Since the incoherent solution is the one desired, the 15,000

*λ*surface is divided up into

*M*=100 partitions and the scattered field from each partition is summed incoherently [41

41. R. M. Axline and A. K. Fung, “Numerical computation of scattering from a perfectly conducting random surface,” IEEE Trans. Antennas Propag. **AP-26(3)**, 482–488 (
1978). [CrossRef]

44. M. F. Chen and S. Y. Bai, “Computer simulation of wave scattering from a dielectric random surface in two dimensions—cylindrical case,” J. Electromagn. Waves Appl. **4(10)**, 963–982 (
1990). [CrossRef]

*J*is windowed using a Gaussian taper:

^{z}*m*represents the

*m*

^{th}surface partition,

*x*is the center of the

_{m}*m*

^{th}partition, and

*w*is the taper width [41

41. R. M. Axline and A. K. Fung, “Numerical computation of scattering from a perfectly conducting random surface,” IEEE Trans. Antennas Propag. **AP-26(3)**, 482–488 (
1978). [CrossRef]

44. M. F. Chen and S. Y. Bai, “Computer simulation of wave scattering from a dielectric random surface in two dimensions—cylindrical case,” J. Electromagn. Waves Appl. **4(10)**, 963–982 (
1990). [CrossRef]

*E*

^{z}_{r,m}is the scattered field from the

*m*

^{th}partition [41

41. R. M. Axline and A. K. Fung, “Numerical computation of scattering from a perfectly conducting random surface,” IEEE Trans. Antennas Propag. **AP-26(3)**, 482–488 (
1978). [CrossRef]

44. M. F. Chen and S. Y. Bai, “Computer simulation of wave scattering from a dielectric random surface in two dimensions—cylindrical case,” J. Electromagn. Waves Appl. **4(10)**, 963–982 (
1990). [CrossRef]

**AP-26(3)**, 482–488 (
1978). [CrossRef]

**4(10)**, 963–982 (
1990). [CrossRef]

*λ*long, random (surface height is Gaussian distributed) PEC surface. The Gaussian surface is generated as shown in Ref. [43

43. A. K. Fung and M. F. Chen, “Numerical simulation of scattering from simple and composite random surfaces,” J. Opt. Soc. Am. A **2(12)**, 2274–2284 (
1985). [CrossRef]

^{1/2}σ

*/*

_{h}*ℓ*=0.3. The traces on the figure are far-field reflectance distributions for

*θ*=10°, 30°, 45°, 60°, and 75°. Note that the reflectance distributions in the figure are normalized with respect to their values at the specular angles (

_{i}*θ*=

_{i}*θ*), and observation for both the MoM and pBRDF predictions is in the specular plane (

_{r}*ϕ*=

*π*). Overall, the pBRDF predictions match very well with the exact, MoM solutions. At some incident angles, the pBRDF predictions deviate from the MoM solutions; however, there is almost unanimous agreement between the pBRDF and MoM solutions on the locations and magnitudes of reflectance maxima. Note that ripples are visible in the MoM solution traces for

*θ*=45°, 60°, and 75°. These ripples are caused by constructive and destructive interference in the scattered field. Incoherent scatter should not interfere; however, as discussed above, the MoM solution is coherent. The incoherent scatter is being approximated by summing the scattered field incoherently over 100 partitions of the entire 15,000

_{i}*λ*surface. Although this should be sufficient [41

**AP-26(3)**, 482–488 (
1978). [CrossRef]

## 4. Mueller matrix measurement results

28. H. G. Tompkins and E. A. Irene, *Handbook of Ellipsometry* (William Andrew, Inc., Norwich, NY,
2005). [CrossRef]

*θ*,

_{i}*θ*) to be measured. The polarization optics used in the PSA are identical to and are contained within the same type of mechanical housing as that of the PSG. This allows the PSA to analyze four independent polarization states. Overall, the instrument is able to make 16 independent, polarimetric measurements of a MUT, thereby providing all the necessary information to deduce the MUT’s Mueller matrix.

_{r}45. E. Compain, S. Poirier, and B. Drevillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and Mueller-matrix ellipsometers,” Appl. Opt. **38(16)**, 3490–3502 (
1999). [CrossRef]

*et al*. [45

45. E. Compain, S. Poirier, and B. Drevillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and Mueller-matrix ellipsometers,” Appl. Opt. **38(16)**, 3490–3502 (
1999). [CrossRef]

45. E. Compain, S. Poirier, and B. Drevillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and Mueller-matrix ellipsometers,” Appl. Opt. **38(16)**, 3490–3502 (
1999). [CrossRef]

**S**is a 4×4 matrix of measured (ellipsometric) irradiances,

**W**is the experimental Mueller matrix of the PSG, and

**A**is the experimental Mueller matrix of the PSA. In the measurement results presented here, the standards used to calibrate the ellipsometer are a no sample measurement, a linear polarizer, and a quarter-wave plate. The linear polarizer is measured at 0°, 60°, and 120°; the quarter-wave plate is measured at 45°.

46. LabSphere, Inc., “A guide to reflectance coatings and materials,” http://www.labsphere.com/tecdocs.aspx.

*η*=0.285-j7.3523) is obtained from Ref. [47

47. Luxpop, Inc. http://www.luxpop.com/.

_{01}, M

_{11}, M

_{22}, and M

_{23}elements compared to predictions made using the pBRDF in Eq. (17). Note that all measurements are made in the specular plane (

*ϕ*=

*π*), and the Mueller matrix elements are normalized with respect to theM00 element as annotated on the figure. The plotted values for the measured Mueller matrix elements of LabSphere Infragold are the means of 256 irradiance measurements. The bars on the figure represent ±1σ, i.e., one standard deviation of those 256 measurements. Note that the LabSphere Infragold results are consistent with those published by Priest and Meier [24

**41(5)**, 988–993 (
2002). [CrossRef]

*η*=1.226-j10.413 [47

47. Luxpop, Inc. http://www.luxpop.com/.

*ϕ*=

*π*), and the Mueller matrix elements are normalized with respect to the M

_{00}element. Once again, the plotted values for the measured Mueller matrix elements of FSA are the means of 256 irradiance measurements. The bars on the figure represent ±1σ of those 256 measurements. As is the case in Fig. 8, discrepancies do exist between the measured data and the pBRDF predictions; however, the pBRDF, once again, captures the trend of the measured data.

*η*for gold and aluminum are accurate for the LabSphere Infragold and FSA samples measured in this experiment. As noted in Ref. [47

47. Luxpop, Inc. http://www.luxpop.com/.

*η*can vary greatly depending on sample quality, sample preparation, or measurement technique. If, instead of using the published values for

*η*, the best values for

*η*are found via nonlinear least squares, one obtains for LabSphere Infragold

*η*=0.4364-j5.2526 and for FSA

*η*=0.8886-j3.4602. While it is possible that the best-fit index for LabSphere Infragold could be more representative of the true

*η*value for the specimen (considering how LabSphere Infragold is manufactured [46

46. LabSphere, Inc., “A guide to reflectance coatings and materials,” http://www.labsphere.com/tecdocs.aspx.

_{2}O

_{3}). This hypothesis would explain the sharper than predicted rise in the measured M

_{22}/M

_{00}element in Fig. 9.

## 5. Conclusion

**57(9)**, 1105–1114 (
1967). [CrossRef]

## Acknowledgments

## References and links

1. | D. E. Barrick, “Theory of HF and VHF propagation across the rough sea—parts I and II,” Radio Sci. |

2. | C. Eckart, “The scattering of sound from the sea surface,” J. Acoust. Soc. Am. |

3. | E. Y. Harper and F. M. Labianca, “Scattering of sound from a point source by a rough surface progressing over an isovelocity ocean,” J. Acoust. Soc. Am. |

4. | K. Krishen, “Correlation of radar backscattering cross sections with ocean wave height and wind velocity,” J. Geophys. Res. |

5. | B. W. Hapke, “A theoretical photometric function for the lunar surface,” J. Geophys. Res. |

6. | D. S. Kimes, “Modeling the directional reflectance from complete homogeneous vegetation canopies with various leaf-orientation distributions,” J. Opt. Soc. Am. A |

7. | R. Hegedüs, A. Barta, B. Bernáth, V. B. Meyer-Rochow, and G. Horváth, “Imaging polarimetry of forest canopies: how the azimuth direction of the sun, occluded by vegetation, can be assessed from the polarization pattern of the sunlit foliage,” Appl. Opt. |

8. | G. Zonios, I. Bassukas, and A. Dimou, “Comparative evaluation of two simple diffuse reflectance models for biological tissue applications,” Appl. Opt. |

9. | J. Xia and G. Yao, “Angular distribution of diffuse reflectance in biological tissue,” Appl. Opt. |

10. | J. F. Blinn, “Models of light reflection for computer synthesized pictures,” in |

11. | R. L. Cook and K. E. Torrance, “A reflectance model for computer graphics,” in |

12. | X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, “A comprehensive physical model for light reflection,” in |

13. | M. A. Greiner, B. D. Duncan, and M. P. Dierking, “Bidirectional scattering distribution functions of maple and cottonwood leaves,” Appl. Opt. |

14. | P. Y. Ufimtsev, |

15. | M. Born and E. Wolf, |

16. | P. Beckmann and A. Spizzichino, |

17. | A. Ishimaru, |

18. | C.-H. An and K. J. Zeringue, “Polarization scattering from rough surfaces based on the vector Kirchoff diffraction model,” in |

19. | D. A. McNamara, C. W. I. Pistorius, and J. A. G. Malherbe, |

20. | K. E. Torrance and E. M. Sparrow, “Theory for off-specular reflection from roughened surfaces,” J. Opt. Soc. Am. |

21. | B. P. Sandford and D. C. Robertson, “Infrared reflectance properties of aircraft paints,” in Proceedings of IRIS Targets, Backgrounds and Discrimination ( 1985). |

22. | M. P. Fetrow, D. Wellems, S. H. Sposato, K. P. Bishop, T. R. Caudill, M. L. Davis, and E. R. Simrell, “Results of a new polarization simulation,” in |

23. | R. G. Priest and T. A. Germer, “Polarimetric BRDF in the microfacet model: theory and measurements,” in |

24. | R. G. Priest and S. R. Meier, “Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces,” Opt. Eng. |

25. | D. Wellems, S. Ortega, D. Bowers, J. Boger, and M. Fetrow, “Long wave infrared polarimetric model: theory, measurements and parameters,” J. Opt. A: Pure Appl. Opt. |

26. | D. Wellems, M. Serna, S. H. Sposato, M. P. Fetrow, K. P. Bishop, S. A. Arko, and T. R. Caudill, “Spectral polarimetric BRDF model and comparison to measurements from isotropic roughened glass,” in |

27. | K. K. Ellis, “Polarimetric bidirectional reflectance distribution function of glossy coatings,” J. Opt. Soc. Am. A |

28. | H. G. Tompkins and E. A. Irene, |

29. | R. Anderson, “Matrix description of radiometric quantities,” Appl. Opt. |

30. | D. S. Flynn and C. Alexander, “Polarized surface scattering expressed in terms of a bidirectional reflectance distribution function,” Opt. Eng. |

31. | F. E. Nicodemus, “Radiance,” Am. J. Phys. |

32. | F. E. Nicodemus, “Directional reflectance and emissivity of an opaque surface,” Appl. Opt. |

33. | J. R. Schott, |

34. | J. R. Shell, “Polarimetric Remote Sensing in the Visible to Near Infrared,” Ph.D. dissertation, Chester F. Carslon Center for Imaging Science, Rochester Institute of Technology, Rochester, NY ( 2005). |

35. | Y. Sun, “Statistical ray method for deriving reflection models of rough surfaces,” J. Opt. Soc. Am. A |

36. | W. S. Bickel and W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. |

37. | M. G. Gartley, S. D. Brown, and J. R. Schott, “Micro-scale surface and contaminate modeling for polarimetric signature prediction,” in |

38. | J. R. Maxwell, J. Beard, S. Weiner, D. Ladd, and S. Ladd, “Bidirectional Reflectance Model Validation and Utilization,” Tech. Rep. AFAL-TR-73-303, Air Force Avionics Laboratory, Wright-Patterson Air Force Base, OH ( 1973). |

39. | M. G. Gartley, “Polarimetric Modeling of Remotely Sensed Scenes in the Thermal Infrared,” Ph.D. dissertation, Chester F. Carslon Center for Imaging Science, Rochester Institute of Technology, Rochester, NY ( 2007). |

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

41. | R. M. Axline and A. K. Fung, “Numerical computation of scattering from a perfectly conducting random surface,” IEEE Trans. Antennas Propag. |

42. | E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. |

43. | A. K. Fung and M. F. Chen, “Numerical simulation of scattering from simple and composite random surfaces,” J. Opt. Soc. Am. A |

44. | M. F. Chen and S. Y. Bai, “Computer simulation of wave scattering from a dielectric random surface in two dimensions—cylindrical case,” J. Electromagn. Waves Appl. |

45. | E. Compain, S. Poirier, and B. Drevillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and Mueller-matrix ellipsometers,” Appl. Opt. |

46. | LabSphere, Inc., “A guide to reflectance coatings and materials,” http://www.labsphere.com/tecdocs.aspx. |

47. | Luxpop, Inc. http://www.luxpop.com/. |

**OCIS Codes**

(160.3900) Materials : Metals

(290.5880) Scattering : Scattering, rough surfaces

(290.1483) Scattering : BSDF, BRDF, and BTDF

(290.5855) Scattering : Scattering, polarization

**ToC Category:**

Scattering

**History**

Original Manuscript: October 6, 2009

Manuscript Accepted: November 5, 2009

Published: November 18, 2009

**Citation**

M. W. Hyde, J. D. Schmidt, and M. J. Havrilla, "A geometrical optics polarimetric bidirectional reflectance distribution function for dielectric and metallic surfaces," Opt. Express **17**, 22138-22153 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-24-22138

Sort: Year | Journal | Reset

### References

- D. E. Barrick, "Theory of HF and VHF propagation across the rough sea—parts I and II," Radio Sci. 6, 517-533 (1971). [CrossRef]
- C. Eckart, "The scattering of sound from the sea surface," J. Acoust. Soc. Am. 25, 566-570 (1953). [CrossRef]
- E. Y. Harper and F. M. Labianca, "Scattering of sound from a point source by a rough surface progressing over an isovelocity ocean," J. Acoust. Soc. Am. 58(2), 349-364 (1975). [CrossRef]
- K. Krishen, "Correlation of radar backscattering cross sections with ocean wave height and wind velocity," J. Geophys. Res. 76, 6528-6539 (1971). [CrossRef]
- B. W. Hapke, "A theoretical photometric function for the lunar surface," J. Geophys. Res. 68(15), 4571-4586 (1963).
- D. S. Kimes, "Modeling the directional reflectance from complete homogeneous vegetation canopies with various leaf-orientation distributions," J. Opt. Soc. Am. A 1(7), 725-737 (1984). [CrossRef]
- R. Hegedüs, A. Barta, B. Bernáth, V. B. Meyer-Rochow, and G. Horváth, "Imaging polarimetry of forest canopies: how the azimuth direction of the sun, occluded by vegetation, can be assessed from the polarization pattern of the sunlit foliage," Appl. Opt. 46(23), 6019-6032 (2007). [CrossRef]
- G. Zonios, I. Bassukas, and A. Dimou, "Comparative evaluation of two simple diffuse reflectance models for biological tissue applications," Appl. Opt. 47(27), 4965-4973 (2008). [CrossRef]
- J. Xia and G. Yao, "Angular distribution of diffuse reflectance in biological tissue," Appl. Opt. 46(26), 6552-6560 (2007). [CrossRef]
- J. F. Blinn, "Models of light reflection for computer synthesized pictures," in SIGGRAPH 1977 Proceedings, vol. 11, pp. 192-198, Special Interest Group on Graphics and Interactive Techniques (Computer Graphics, 1977).
- R. L. Cook and K. E. Torrance, "A reflectance model for computer graphics," in SIGGRAPH 1981 Proceedings, vol. 15, pp. 307-316, Special Interest Group on Graphics and Interactive Techniques (Computer Graphics, 1981).
- X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, "A comprehensive physical model for light reflection," in SIGGRAPH 1991 Proceedings, vol. 25, pp. 175-186, Special Interest Group on Graphics and Interactive Techniques (Computer Graphics, 1991).
- M. A. Greiner, B. D. Duncan, and M. P. Dierking, "Bidirectional scattering distribution functions of maple and cottonwood leaves," Appl. Opt. 46(25), 6485-6494 (2007). [CrossRef]
- P. Y. Ufimtsev, Fundamentals of the Physical Theory of Diffraction (John Wiley & Sons, Inc., Hoboken, NJ, 2007). [CrossRef]
- M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, New York, NY, 1999).
- P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, Inc., Norwood, MA, 1963).
- A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, New York, NY, 1997).
- C.-H. An and K. J. Zeringue, "Polarization scattering from rough surfaces based on the vector Kirchoff diffraction model," in Proc. SPIE, vol. 5158, pp. 205-216 (The International Society for Optical Engineering (SPIE), 2003).
- D. A. McNamara, C. W. I. Pistorius, and J. A. G. Malherbe, Introduction to the Uniform Geometrical Theory of Diffraction (Artech House, Inc., Norwood, MA, 1990).
- K. E. Torrance and E. M. Sparrow, "Theory for off-specular reflection from roughened surfaces," J. Opt. Soc. Am. 57(9), 1105-1114 (1967). [CrossRef]
- B. P. Sandford and D. C. Robertson, "Infrared reflectance properties of aircraft paints," in Proceedings of IRIS Targets, Backgrounds and Discrimination (1985).
- M. P. Fetrow, D. Wellems, S. H. Sposato, K. P. Bishop, T. R. Caudill, M. L. Davis, and E. R. Simrell, "Results of a new polarization simulation," in Proc. SPIE, vol. 4481, pp. 149-162 (The International Society for Optical Engineering (SPIE), 2002).
- R. G. Priest and T. A. Germer, "Polarimetric BRDF in the microfacet model: theory and measurements," in Proceedings of the 2000 Meeting of the Military Sensing Symposia Specialty Group on Passive Sensors, pp. 169-181 (Infrared Information Analysis Center, 2000).
- R. G. Priest and S. R. Meier, "Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces," Opt. Eng. 41(5), 988-993 (2002). [CrossRef]
- D. Wellems, S. Ortega, D. Bowers, J. Boger, and M. Fetrow, "Long wave infrared polarimetric model: theory, measurements and parameters," J. Opt. A: Pure Appl. Opt. 8(10), 914-925 (2006). [CrossRef]
- D. Wellems, M. Serna, S. H. Sposato, M. P. Fetrow, K. P. Bishop, S. A. Arko, and T. R. Caudill, "Spectral polarimetric BRDF model and comparison to measurements from isotropic roughened glass," in Workshop on Multi/Hyperspectral Sensors, Measurements, Modeling and Simulation (U.S. Army Aviation and Missile Command, Huntsville, AL, 2000).
- K. K. Ellis, "Polarimetric bidirectional reflectance distribution function of glossy coatings," J. Opt. Soc. Am. A 13(8), 1758-1762 (1996). [CrossRef]
- H. G. Tompkins and E. A. Irene, Handbook of Ellipsometry (William Andrew, Inc., Norwich, NY, 2005). [CrossRef]
- R. Anderson, "Matrix description of radiometric quantities," Appl. Opt. 30(7), 858-867 (1991). [CrossRef]
- D. S. Flynn and C. Alexander, "Polarized surface scattering expressed in terms of a bidirectional reflectance distribution function," Opt. Eng. 34(6), 1646-1650 (1995). [CrossRef]
- F. E. Nicodemus, "Radiance," Am. J. Phys. 31, 368-377 (1963). [CrossRef]
- F. E. Nicodemus, "Directional reflectance and emissivity of an opaque surface," Appl. Opt. 4(7), 368-377 (1965).
- J. R. Schott, Fundamentals of Polarimetric Remote Sensing (SPIE Press, Bellingham, WA, 2009). [CrossRef]
- J. R. Shell, "Polarimetric Remote Sensing in the Visible to Near Infrared," Ph.D. dissertation, Chester F. Carslon Center for Imaging Science, Rochester Institute of Technology, Rochester, NY (2005).
- Y. Sun, "Statistical ray method for deriving reflection models of rough surfaces," J. Opt. Soc. Am. A 24(3), 724-744 (2007). [CrossRef]
- W. S. Bickel and W. M. Bailey, "Stokes vectors, Mueller matrices, and polarized scattered light," Am. J. Phys. 53(5), 468-478 (1985). [CrossRef]
- M. G. Gartley, S. D. Brown, and J. R. Schott, "Micro-scale surface and contaminate modeling for polarimetric signature prediction," in Proc. SPIE, vol. 6972 (The International Society for Optical Engineering (SPIE), 2008). [CrossRef]
- J. R. Maxwell, J. Beard, S. Weiner, D. Ladd, and S. Ladd, "Bidirectional Reflectance Model Validation and Utilization," Tech. Rep. AFAL-TR-73-303, Air Force Avionics Laboratory, Wright-Patterson Air Force Base, OH (1973).
- M. G. Gartley, "Polarimetric Modeling of Remotely Sensed Scenes in the Thermal Infrared," Ph.D. dissertation, Chester F. Carslon Center for Imaging Science, Rochester Institute of Technology, Rochester, NY (2007).
- A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics (IEEE Press, New York, NY, 1998).
- R. M. Axline and A. K. Fung, "Numerical computation of scattering from a perfectly conducting random surface," IEEE Trans. Antennas Propag. AP-26(3), 482-488 (1978). [CrossRef]
- E. I. Thorsos, "The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum," J. Acoust. Soc. Am. 83(1), 78-92 (1988). [CrossRef]
- A. K. Fung and M. F. Chen, "Numerical simulation of scattering from simple and composite random surfaces," J. Opt. Soc. Am. A 2(12), 2274-2284 (1985). [CrossRef]
- M. F. Chen and S. Y. Bai, "Computer simulation of wave scattering from a dielectric random surface in two dimensions—cylindrical case," J. Electromagn. Waves Appl. 4(10), 963-982 (1990). [CrossRef]
- E. Compain, S. Poirier, and B. Drevillon, "General and self-consistent method for the calibration of polarization modulators, polarimeters, and Mueller-matrix ellipsometers," Appl. Opt. 38(16), 3490-3502 (1999). [CrossRef]
- LabSphere, Inc., "A guide to reflectance coatings and materials," http://www.labsphere.com/tecdocs.aspx.
- Luxpop, Inc.http://www.luxpop.com/.

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