Abstract

A Lambert surface would appear equally bright from all observation directions regardless of the illumination direction. However, the reflection from a randomly scattering object generally has directional variation, which can be described in terms of the bidirectional reflectance distribution function (BRDF). We measured the BRDF of a Spectralon white reflectance standard for incoherent illumination at 405 and 680nm with unpolarized and plane-polarized light from different directions of incidence. Our measurements show deviations of the BRDF for the Spectralon white reflectance standard from that of a Lambertian reflector that depend both on the angle of incidence and the polarization states of the incident light and detected light. The non-Lambertian reflection characteristics were found to increase more toward the direction of specular reflection as the angle of incidence gets larger.

© 2011 Optical Society of America

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References

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    [CrossRef]

2011

K. Stamnes, B. Hamre, J. J. Stamnes, G. Ryzhikov, M. Biryulina, R. Mahoney, B. Haus, and A. Sei, “Modeling of radiation transport in coupled atmosphere-snow-ice-ocean systems,” J. Quant. Spectrosc. Radiat. Transfer 112, 714–726(2011).
[CrossRef]

2008

K. Fu and P. Hsu, “New regime map of the geometric optics approximation for scattering from rough surfaces,” J. Quant. Spectrosc. Radiat. Transfer 109, 180–188 (2008).
[CrossRef]

2007

B. Gordon, “Integrating sphere diffuse reflectance technology for use with UV-visible spectroscopy,” Tech. Note 51450(Thermo Fisher Scientific, 2007).

H. Parviainen and K. Muinonen, “Rough-surface shadowing of self-affine random rough surfaces,” J. Quant. Spectrosc. Radiat. Transfer 106, 398–416 (2007).
[CrossRef]

H. Noble, W. T. Lam, G. Smith, S. McClain, and R. A. Chipman, “Polarization scattering from a spectralon calibration sample,” Proc. SPIE 6682, 668219, doi:10.1117/12.747483 (2007).
[CrossRef]

G. T. Georgiev and J. J. Butler, “Long-term calibration monitoring of Spectralon diffusers BRDF in the air-ultraviolet,” Appl. Opt. 46, 7892–7898 (2007).
[CrossRef] [PubMed]

2006

K. J. Voss and H. Zhang, “Bidirectional reflectance of dry and submerged Labsphere Spectralon plaque,” Appl. Opt. 45, 7924–7927 (2006).
[CrossRef] [PubMed]

J. Jaglarz, R. Duraj, P. Szopa, J. Cisowski, and H. Czternastek, “Investigation of white standards by means of bidirectional reflection distribution function and integrated sphere methods,” Opt. Appl. XXXVI, 97–103 (2006).

L. Zhao, K. P. Nielsen, J. K. Lotsberg, E. Marken, J. J. Stamnes, and K. Stamnes, “New versatile setup for goniometric measurements of spectral radiance,” Opt. Eng. 45, 053606 (2006).
[CrossRef]

2005

G. T. Georgiev and J. J. Butler, “The effect of speckle on BRDF measurements,” Proc. SPIE 5882, 588203 (2005).
[CrossRef]

B. J. DeBoo, J. M. Sasian, and R. A. Chipman, “Depolarization of diffusely reflecting man-made objects,” Appl. Opt. 44, 5434–5444 (2005).
[CrossRef] [PubMed]

2004

G. T. Georgiev and J. J. Butler, “The effect of incident light polarization on Spectralon BRDF measurements,” Proc. SPIE 5570, 492–500 (2004).
[CrossRef]

2003

T. A. Germer, “Polarized light diffusely scattered under smooth and rough interfaces,” Proc. SPIE 5158, 193204 (2003).
[CrossRef]

2001

2000

E. A. Early, P. Y. Barnes, B. C. Johnson, J. J. Butler, C. J. Bruegge, S. F. Biggar, P. R. Spyak, and M. M. Pavlov, “Bidirectional reflectance round-robin in support of the Earth Observing System program,” Am. Met. Soc. 17, 1078–1091(2000).

1999

1998

1996

1995

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

1991

1987

K. A. O’Donell and E. R. Mendez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. 4, 1194–1204 (1987).
[CrossRef]

“Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 91–95 (1987).
[CrossRef]

1970

1966

1963

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

Alexander, C.

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

Barnes, P. Y.

E. A. Early, P. Y. Barnes, B. C. Johnson, J. J. Butler, C. J. Bruegge, S. F. Biggar, P. R. Spyak, and M. M. Pavlov, “Bidirectional reflectance round-robin in support of the Earth Observing System program,” Am. Met. Soc. 17, 1078–1091(2000).

Beckmann, P.

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

Biggar, S. F.

E. A. Early, P. Y. Barnes, B. C. Johnson, J. J. Butler, C. J. Bruegge, S. F. Biggar, P. R. Spyak, and M. M. Pavlov, “Bidirectional reflectance round-robin in support of the Earth Observing System program,” Am. Met. Soc. 17, 1078–1091(2000).

Birkebak, R. C.

Biryulina, M.

K. Stamnes, B. Hamre, J. J. Stamnes, G. Ryzhikov, M. Biryulina, R. Mahoney, B. Haus, and A. Sei, “Modeling of radiation transport in coupled atmosphere-snow-ice-ocean systems,” J. Quant. Spectrosc. Radiat. Transfer 112, 714–726(2011).
[CrossRef]

Brothers, A. M.

Bruegge, C. J.

Butler, J. J.

G. T. Georgiev and J. J. Butler, “Long-term calibration monitoring of Spectralon diffusers BRDF in the air-ultraviolet,” Appl. Opt. 46, 7892–7898 (2007).
[CrossRef] [PubMed]

G. T. Georgiev and J. J. Butler, “The effect of speckle on BRDF measurements,” Proc. SPIE 5882, 588203 (2005).
[CrossRef]

G. T. Georgiev and J. J. Butler, “The effect of incident light polarization on Spectralon BRDF measurements,” Proc. SPIE 5570, 492–500 (2004).
[CrossRef]

E. A. Early, P. Y. Barnes, B. C. Johnson, J. J. Butler, C. J. Bruegge, S. F. Biggar, P. R. Spyak, and M. M. Pavlov, “Bidirectional reflectance round-robin in support of the Earth Observing System program,” Am. Met. Soc. 17, 1078–1091(2000).

Chipman, R. A.

H. Noble, W. T. Lam, G. Smith, S. McClain, and R. A. Chipman, “Polarization scattering from a spectralon calibration sample,” Proc. SPIE 6682, 668219, doi:10.1117/12.747483 (2007).
[CrossRef]

B. J. DeBoo, J. M. Sasian, and R. A. Chipman, “Depolarization of diffusely reflecting man-made objects,” Appl. Opt. 44, 5434–5444 (2005).
[CrossRef] [PubMed]

Cisowski, J.

J. Jaglarz, R. Duraj, P. Szopa, J. Cisowski, and H. Czternastek, “Investigation of white standards by means of bidirectional reflection distribution function and integrated sphere methods,” Opt. Appl. XXXVI, 97–103 (2006).

Czternastek, H.

J. Jaglarz, R. Duraj, P. Szopa, J. Cisowski, and H. Czternastek, “Investigation of white standards by means of bidirectional reflection distribution function and integrated sphere methods,” Opt. Appl. XXXVI, 97–103 (2006).

DeBoo, B. J.

Duraj, R.

J. Jaglarz, R. Duraj, P. Szopa, J. Cisowski, and H. Czternastek, “Investigation of white standards by means of bidirectional reflection distribution function and integrated sphere methods,” Opt. Appl. XXXVI, 97–103 (2006).

Duval, V.

Early, E. A.

E. A. Early, P. Y. Barnes, B. C. Johnson, J. J. Butler, C. J. Bruegge, S. F. Biggar, P. R. Spyak, and M. M. Pavlov, “Bidirectional reflectance round-robin in support of the Earth Observing System program,” Am. Met. Soc. 17, 1078–1091(2000).

Esproles, C.

Flynn, D. S.

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

Fu, K.

K. Fu and P. Hsu, “New regime map of the geometric optics approximation for scattering from rough surfaces,” J. Quant. Spectrosc. Radiat. Transfer 109, 180–188 (2008).
[CrossRef]

Georgiev, G. T.

G. T. Georgiev and J. J. Butler, “Long-term calibration monitoring of Spectralon diffusers BRDF in the air-ultraviolet,” Appl. Opt. 46, 7892–7898 (2007).
[CrossRef] [PubMed]

G. T. Georgiev and J. J. Butler, “The effect of speckle on BRDF measurements,” Proc. SPIE 5882, 588203 (2005).
[CrossRef]

G. T. Georgiev and J. J. Butler, “The effect of incident light polarization on Spectralon BRDF measurements,” Proc. SPIE 5570, 492–500 (2004).
[CrossRef]

Germer, T. A.

T. A. Germer, “Polarized light diffusely scattered under smooth and rough interfaces,” Proc. SPIE 5158, 193204 (2003).
[CrossRef]

Gordon, B.

B. Gordon, “Integrating sphere diffuse reflectance technology for use with UV-visible spectroscopy,” Tech. Note 51450(Thermo Fisher Scientific, 2007).

Hamre, B.

K. Stamnes, B. Hamre, J. J. Stamnes, G. Ryzhikov, M. Biryulina, R. Mahoney, B. Haus, and A. Sei, “Modeling of radiation transport in coupled atmosphere-snow-ice-ocean systems,” J. Quant. Spectrosc. Radiat. Transfer 112, 714–726(2011).
[CrossRef]

Haner, D. A.

Haus, B.

K. Stamnes, B. Hamre, J. J. Stamnes, G. Ryzhikov, M. Biryulina, R. Mahoney, B. Haus, and A. Sei, “Modeling of radiation transport in coupled atmosphere-snow-ice-ocean systems,” J. Quant. Spectrosc. Radiat. Transfer 112, 714–726(2011).
[CrossRef]

Hsu, P.

K. Fu and P. Hsu, “New regime map of the geometric optics approximation for scattering from rough surfaces,” J. Quant. Spectrosc. Radiat. Transfer 109, 180–188 (2008).
[CrossRef]

Jaglarz, J.

J. Jaglarz, R. Duraj, P. Szopa, J. Cisowski, and H. Czternastek, “Investigation of white standards by means of bidirectional reflection distribution function and integrated sphere methods,” Opt. Appl. XXXVI, 97–103 (2006).

Johnson, B. C.

E. A. Early, P. Y. Barnes, B. C. Johnson, J. J. Butler, C. J. Bruegge, S. F. Biggar, P. R. Spyak, and M. M. Pavlov, “Bidirectional reflectance round-robin in support of the Earth Observing System program,” Am. Met. Soc. 17, 1078–1091(2000).

Johnson, J. T.

Lam, W. T.

H. Noble, W. T. Lam, G. Smith, S. McClain, and R. A. Chipman, “Polarization scattering from a spectralon calibration sample,” Proc. SPIE 6682, 668219, doi:10.1117/12.747483 (2007).
[CrossRef]

Leskova, T. A.

I. Simonsen, A. A. Maradudin, and T. A. Leskova, “The scattering of electromagnetic waves from two dimensional random rough penetrable surfaces,” Phys. Rev. Lett. (to be published).

Lotsberg, J. K.

L. Zhao, K. P. Nielsen, J. K. Lotsberg, E. Marken, J. J. Stamnes, and K. Stamnes, “New versatile setup for goniometric measurements of spectral radiance,” Opt. Eng. 45, 053606 (2006).
[CrossRef]

Macaskill, C.

Mahoney, R.

K. Stamnes, B. Hamre, J. J. Stamnes, G. Ryzhikov, M. Biryulina, R. Mahoney, B. Haus, and A. Sei, “Modeling of radiation transport in coupled atmosphere-snow-ice-ocean systems,” J. Quant. Spectrosc. Radiat. Transfer 112, 714–726(2011).
[CrossRef]

Maradudin, A. A.

I. Simonsen, A. A. Maradudin, and T. A. Leskova, “The scattering of electromagnetic waves from two dimensional random rough penetrable surfaces,” Phys. Rev. Lett. (to be published).

Marken, E.

L. Zhao, K. P. Nielsen, J. K. Lotsberg, E. Marken, J. J. Stamnes, and K. Stamnes, “New versatile setup for goniometric measurements of spectral radiance,” Opt. Eng. 45, 053606 (2006).
[CrossRef]

McClain, S.

H. Noble, W. T. Lam, G. Smith, S. McClain, and R. A. Chipman, “Polarization scattering from a spectralon calibration sample,” Proc. SPIE 6682, 668219, doi:10.1117/12.747483 (2007).
[CrossRef]

McGuckin, B. T.

Mendez, E. R.

K. A. O’Donell and E. R. Mendez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. 4, 1194–1204 (1987).
[CrossRef]

Menzies, R. T.

Muinonen, K.

H. Parviainen and K. Muinonen, “Rough-surface shadowing of self-affine random rough surfaces,” J. Quant. Spectrosc. Radiat. Transfer 106, 398–416 (2007).
[CrossRef]

Nicodemus, F. E.

Nielsen, K. P.

L. Zhao, K. P. Nielsen, J. K. Lotsberg, E. Marken, J. J. Stamnes, and K. Stamnes, “New versatile setup for goniometric measurements of spectral radiance,” Opt. Eng. 45, 053606 (2006).
[CrossRef]

Noble, H.

H. Noble, W. T. Lam, G. Smith, S. McClain, and R. A. Chipman, “Polarization scattering from a spectralon calibration sample,” Proc. SPIE 6682, 668219, doi:10.1117/12.747483 (2007).
[CrossRef]

O’Donell, K. A.

K. A. O’Donell and E. R. Mendez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. 4, 1194–1204 (1987).
[CrossRef]

Parviainen, H.

H. Parviainen and K. Muinonen, “Rough-surface shadowing of self-affine random rough surfaces,” J. Quant. Spectrosc. Radiat. Transfer 106, 398–416 (2007).
[CrossRef]

Pavlov, M. M.

E. A. Early, P. Y. Barnes, B. C. Johnson, J. J. Butler, C. J. Bruegge, S. F. Biggar, P. R. Spyak, and M. M. Pavlov, “Bidirectional reflectance round-robin in support of the Earth Observing System program,” Am. Met. Soc. 17, 1078–1091(2000).

Ryzhikov, G.

K. Stamnes, B. Hamre, J. J. Stamnes, G. Ryzhikov, M. Biryulina, R. Mahoney, B. Haus, and A. Sei, “Modeling of radiation transport in coupled atmosphere-snow-ice-ocean systems,” J. Quant. Spectrosc. Radiat. Transfer 112, 714–726(2011).
[CrossRef]

Sasian, J. M.

Sei, A.

K. Stamnes, B. Hamre, J. J. Stamnes, G. Ryzhikov, M. Biryulina, R. Mahoney, B. Haus, and A. Sei, “Modeling of radiation transport in coupled atmosphere-snow-ice-ocean systems,” J. Quant. Spectrosc. Radiat. Transfer 112, 714–726(2011).
[CrossRef]

Simonsen, I.

I. Simonsen, A. A. Maradudin, and T. A. Leskova, “The scattering of electromagnetic waves from two dimensional random rough penetrable surfaces,” Phys. Rev. Lett. (to be published).

Smith, G.

H. Noble, W. T. Lam, G. Smith, S. McClain, and R. A. Chipman, “Polarization scattering from a spectralon calibration sample,” Proc. SPIE 6682, 668219, doi:10.1117/12.747483 (2007).
[CrossRef]

Sparrow, E. M.

Spizzichino, A.

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

Spyak, P. R.

E. A. Early, P. Y. Barnes, B. C. Johnson, J. J. Butler, C. J. Bruegge, S. F. Biggar, P. R. Spyak, and M. M. Pavlov, “Bidirectional reflectance round-robin in support of the Earth Observing System program,” Am. Met. Soc. 17, 1078–1091(2000).

Stamnes, J. J.

K. Stamnes, B. Hamre, J. J. Stamnes, G. Ryzhikov, M. Biryulina, R. Mahoney, B. Haus, and A. Sei, “Modeling of radiation transport in coupled atmosphere-snow-ice-ocean systems,” J. Quant. Spectrosc. Radiat. Transfer 112, 714–726(2011).
[CrossRef]

L. Zhao, K. P. Nielsen, J. K. Lotsberg, E. Marken, J. J. Stamnes, and K. Stamnes, “New versatile setup for goniometric measurements of spectral radiance,” Opt. Eng. 45, 053606 (2006).
[CrossRef]

Stamnes, K.

K. Stamnes, B. Hamre, J. J. Stamnes, G. Ryzhikov, M. Biryulina, R. Mahoney, B. Haus, and A. Sei, “Modeling of radiation transport in coupled atmosphere-snow-ice-ocean systems,” J. Quant. Spectrosc. Radiat. Transfer 112, 714–726(2011).
[CrossRef]

L. Zhao, K. P. Nielsen, J. K. Lotsberg, E. Marken, J. J. Stamnes, and K. Stamnes, “New versatile setup for goniometric measurements of spectral radiance,” Opt. Eng. 45, 053606 (2006).
[CrossRef]

G. Thomas and K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge University, 1999).
[CrossRef]

Szopa, P.

J. Jaglarz, R. Duraj, P. Szopa, J. Cisowski, and H. Czternastek, “Investigation of white standards by means of bidirectional reflection distribution function and integrated sphere methods,” Opt. Appl. XXXVI, 97–103 (2006).

Thomas, G.

G. Thomas and K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge University, 1999).
[CrossRef]

Torrance, K. E.

Torrungrueng, D.

Voss, K. J.

Zhang, H.

Zhao, L.

L. Zhao, K. P. Nielsen, J. K. Lotsberg, E. Marken, J. J. Stamnes, and K. Stamnes, “New versatile setup for goniometric measurements of spectral radiance,” Opt. Eng. 45, 053606 (2006).
[CrossRef]

Am. Met. Soc.

E. A. Early, P. Y. Barnes, B. C. Johnson, J. J. Butler, C. J. Bruegge, S. F. Biggar, P. R. Spyak, and M. M. Pavlov, “Bidirectional reflectance round-robin in support of the Earth Observing System program,” Am. Met. Soc. 17, 1078–1091(2000).

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Quant. Spectrosc. Radiat. Transfer

K. Stamnes, B. Hamre, J. J. Stamnes, G. Ryzhikov, M. Biryulina, R. Mahoney, B. Haus, and A. Sei, “Modeling of radiation transport in coupled atmosphere-snow-ice-ocean systems,” J. Quant. Spectrosc. Radiat. Transfer 112, 714–726(2011).
[CrossRef]

H. Parviainen and K. Muinonen, “Rough-surface shadowing of self-affine random rough surfaces,” J. Quant. Spectrosc. Radiat. Transfer 106, 398–416 (2007).
[CrossRef]

K. Fu and P. Hsu, “New regime map of the geometric optics approximation for scattering from rough surfaces,” J. Quant. Spectrosc. Radiat. Transfer 109, 180–188 (2008).
[CrossRef]

Opt. Appl.

J. Jaglarz, R. Duraj, P. Szopa, J. Cisowski, and H. Czternastek, “Investigation of white standards by means of bidirectional reflection distribution function and integrated sphere methods,” Opt. Appl. XXXVI, 97–103 (2006).

Opt. Commun.

“Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 91–95 (1987).
[CrossRef]

Opt. Eng.

L. Zhao, K. P. Nielsen, J. K. Lotsberg, E. Marken, J. J. Stamnes, and K. Stamnes, “New versatile setup for goniometric measurements of spectral radiance,” Opt. Eng. 45, 053606 (2006).
[CrossRef]

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

Proc. SPIE

G. T. Georgiev and J. J. Butler, “The effect of incident light polarization on Spectralon BRDF measurements,” Proc. SPIE 5570, 492–500 (2004).
[CrossRef]

T. A. Germer, “Polarized light diffusely scattered under smooth and rough interfaces,” Proc. SPIE 5158, 193204 (2003).
[CrossRef]

G. T. Georgiev and J. J. Butler, “The effect of speckle on BRDF measurements,” Proc. SPIE 5882, 588203 (2005).
[CrossRef]

H. Noble, W. T. Lam, G. Smith, S. McClain, and R. A. Chipman, “Polarization scattering from a spectralon calibration sample,” Proc. SPIE 6682, 668219, doi:10.1117/12.747483 (2007).
[CrossRef]

Other

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

I. Simonsen, A. A. Maradudin, and T. A. Leskova, “The scattering of electromagnetic waves from two dimensional random rough penetrable surfaces,” Phys. Rev. Lett. (to be published).

“A guide to reflectance coatings & materials,” http://www.pro-lite.co.uk/File/Tech_guide_coatings_&_materials.pdf.

B. Gordon, “Integrating sphere diffuse reflectance technology for use with UV-visible spectroscopy,” Tech. Note 51450(Thermo Fisher Scientific, 2007).

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Figures (10)

Fig. 1
Fig. 1

Geometry and symbols for the definition of the BRDF. The normal to the surface area d A of a diffusely scattering object is taken to be along the z axis, and the incident beam along the unit vector Ω ^ ( θ , ϕ ) is taken to lie in the x z plane with azimuth angle ϕ = 0 ° . The observation direction is along the unit vector Ω ^ ( θ , ϕ ) , and α is the backscattering angle, which is given by α = π Θ , where cos Θ = Ω ^ · Ω ^ , Θ being the scattering angle. When the azimuth angle ϕ = 0 ° , the scattering plane coincides with the plane of incidence (the x z plane).

Fig. 2
Fig. 2

Schematic of collimated beam scattering geometry. The surface of a diffusely reflecting object lies in the vertical x y plane, the incident beam is along the unit vector Ω ^ 0 ( θ 0 , ϕ 0 = 0 ) , and the angle of incidence is θ 0 [ 0 θ 0 90 ° ] . The observation direction is along the unit vector Ω ^ ( θ , ϕ ) , and the polar angle of observation is θ [ 90 ° θ 90 ° ] . When the azimuth angle of observation ϕ = 0 , the observation direction lies in the plane of incidence (the x z plane). Note that θ = α θ 0 = π Θ θ 0 , where Θ is the scattering angle and α is the backscattering angle. The unit vectors e ^ and the e ^ are parallel and perpendicular, respectively, to the plane of incidence.

Fig. 3
Fig. 3

Sketch of optical setup in the experiment: IF, interference filter; L, lens; P, polarizer; CA, circular aperture; M, plane mirror; A , temporary analyzer; A, analyzer; D, detector. Note that the Spectralon scattering surface is aligned parallel to the vertical plane.

Fig. 4
Fig. 4

Spectralon BRDF curves ( ρ u u ) measured at 670 nm for unpolarized illumination from a xenon lamp ( ) and for illumination from a 670 nm diode laser ( * ) at θ 0 = 0 ° [see Figure 2]. The curves are normalized to 1 at θ = 0 ° .

Fig. 5
Fig. 5

Raw data averages and rms values obtained from test measurements at normal incidence ( θ 0 = 0 ° in Fig. 2) to examine the repeatability of measured results. Bullets (•) and crosses (×) indicate measured values between which there is linear interpolation, except between the values for θ = 20 ° and θ = + 20 ° , in which case spline interpolation was used. The units on the left vertical axis represent the percentage rms values and the detected average output power in nanowatts.

Fig. 6
Fig. 6

Raw data averages and rms values for measurements at normal incidence ( θ 0 = 0 ° ) on the Spectralon test material at five different areas. Bullets (•) and crosses (×) indicate measured values between which there is linear interpolation, except between the values for θ = 20 ° and θ = + 20 ° , in which case spline interpolation was used. The units on the left vertical axis represent the percentage rms values and the detected output power in nanowatts.

Fig. 7
Fig. 7

Spectralon BRDF curves measured in the plane of incidence at a wavelength of 680 nm for angles of incidence θ 0 as indicated in each of parts (a)–(d). Curves with blue or red color represent s or p polarization, respectively, of the incident light. For each polarization state of the incident light, both the copolarized component (indicated by −) and the cross-polarized component (indicated by ) are presented. The green curves represent unpolarized detected light, i.e., measurements in which there was no analyzer in front of the detector, and in this case the symbols − and are used to indicate s and p polarization, respectively, of the incident light. The black symbols × along the horizontal axis show the Lambertian BRDF, and the black − curves show the BRDF for the case u u of unpolarized incident and detected light. Thus, [ u u / 2 (black)], [ s s (blue)], [ s p (blue)], [ p p (red)], [ p s (red)], [ s u / 2 (green)], and [ p u / 2 (green)]. For comparison, BRDF curves at 632.8 nm , measured by Haner et al. [9] are represented by [ s s (blue)], [ s p (blue)], [ p p (red)], [ p s (red)]. Note that the test sample of Haner et al. [9] was different from ours.

Fig. 8
Fig. 8

Spectralon BRDF curves measured in an observation plane that was 6 ° above the plane of incidence at a wavelength of 680 nm for angles of incidence θ 0 as indicated in each of parts (a)–(f). Curves with blue or red color represent s or p polarization, respectively, of the incident light. For each of these polarization states of the incident light, both the copolarized component (indicated by −) and the cross-polarized component (indicated by ) are presented. The green curves represent unpolarized detected light, i.e., measurements in which there was no polarizer in front of the detector, and in this case the symbols − and are used to indicate s and p polarization, respectively, of the incident light. The black symbols × along the horizontal axis show the Lambertian BRDF, and the black − curves show the BRDF for the u u case of unpolarized incident and detected light. Thus, [ u u / 2 (black)], [ s s (blue)], [ s p (blue)], [ p p (red)], [ p s (red)], [ s u / 2 (green)], and [ p u / 2 (green)].

Fig. 9
Fig. 9

Symbols and legends are the same as in Fig. 8 except that the wavelength of the incident light was 405 nm for the results shown in this figure.

Fig. 10
Fig. 10

Degree of reduced linear polarization P (in percent) versus observation angle θ for different angles of incidence θ 0 as indicated in panels (a)–(f). The blue and red curves represent incident light at 405 and 680 nm , respectively, and the degree of reduced linear polarization P is represented by the symbol − for s-polarized incident light and by for p-polarized incident light. Thus, s -polarized incident light at 405 nm , p -polarized incident light at 405 nm , s -polarized incident light of 680 nm , and p -polarized incident light at 680 nm .

Equations (18)

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ρ ( ν , Ω ^ , Ω ^ ) = L ( ν , Ω ^ , Ω ^ ) E ( ν , Ω ^ ) = L ( ν , Ω ^ , Ω ^ ) L ( ν , Ω ^ ) cos θ d ω [ sr 1 ] .
L ( ν , Ω ^ ) = 2 π d ω cos θ ρ ( ν , Ω ^ , Ω ^ ) L ( ν , Ω ^ ) ,
L ( ν , Ω ^ ) = ρ L ( ν ) 2 π d ω cos θ L ( ν , Ω ^ ) = ρ L ( ν ) E ( ν ) .
L s ( ν , Ω ^ ) = E s ( ν ) δ ( Ω ^ Ω ^ 0 ) = E s ( ν ) δ ( cos θ cos θ 0 ) δ ( ϕ ϕ 0 ) ,
E ( ν , Ω 0 ^ ) = 2 π d ω cos θ L s ( ν , Ω ^ ) = E s ( ν ) cos θ 0 .
E ( ν , Ω ^ 0 ) = 2 π d ω cos θ L ( ν , Ω ^ ) = π ρ L ( ν ) E s ( ν ) cos θ 0 .
ρ ( ν , Ω ^ 0 , 2 π ) = E ( ν , Ω ^ 0 ) E s ( ν ) cos θ 0 = π ρ L ( ν ) ,
E ( ν , z ^ ) = 2 π 0 π / 2 L meas ( ν , z ^ , θ ) cos θ sin θ d θ ,
ρ ( ν , z ^ , 2 π ) = E ( ν , z ^ ) E s ( ν ) = 0.98 ,
ρ ( ν , z ^ , Ω ^ ) = L ( ν , z ^ , θ , ϕ ) E s ( ν ) = 0.98 L meas ( ν , z ^ , θ ) E ( ν , z ^ ) .
ρ ( ν , Ω ^ 0 , Ω ^ ) = ρ ( ν , Ω ^ , Ω ^ 0 ) .
ρ ( ν , Ω ^ 0 , + z ^ ) = ρ ( ν , z ^ , Ω ^ 0 ) = ρ ( ν , z ^ , θ 0 ) = 0.98 L meas ( ν , z ^ , θ 0 ) E ( ν , z ^ ) ,
ρ ( ν , Ω ^ 0 , Ω ^ ) = ρ ( ν , Ω ^ 0 , + z ^ ) L meas ( ν , Ω ^ 0 , z ^ ) + L meas ( ν , Ω ^ 0 , Ω ^ ) ,
ρ ( ν , Ω ^ 0 , Ω ^ ) = 0.98 L meas ( ν , z ^ , θ 0 ) E ( ν , z ^ ) L meas ( ν , Ω ^ 0 , Ω ^ ) L meas ( ν , Ω ^ 0 , z ^ ) ,
ρ j k ( Ω ^ 0 , Ω ^ , ν ) = L k ( Ω ^ 0 , Ω ^ ) E j s cos θ 0 [ sr 1 ] ,
P = | Q sc | L sc = | L sc L sc | L sc + L sc ,
θ = { arccos ( cos θ g cos ϕ g ) for     θ g 0 arccos ( cos θ g cos ϕ g ) for     θ g < 0 ,
ϕ = { arcsin [ sin ϕ g [ 1 ( cos θ g cos ϕ g ) 2 ] 1 / 2 ] for     θ g 0 π + arcsin [ sin ϕ g [ 1 ( cos θ g cos ϕ g ) 2 ] 1 / 2 ] for     θ g < 0 .

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