Abstract

We applied the bidirectional reflectance distribution function (BRDF) model consisting of diffuse, quasi-specular, and glossy components to the Monte Carlo modeling of spectral effective emissivities for nonisothermal cavities. A method for extension of a monochromatic three-component (3C) BRDF model to a continuous spectral range is proposed. The initial data for this method are the BRDFs measured in the plane of incidence at a single wavelength and several incidence angles and directional–hemispherical reflectance measured at one incidence angle within a finite spectral range. We proposed the Monte Carlo algorithm for calculation of spectral effective emissivities for nonisothermal cavities whose internal surface is described by the wavelength-dependent 3C BRDF model. The results obtained for a cylindroconical nonisothermal cavity are discussed and compared with results obtained using the conventional specular–diffuse model.

© 2012 Optical Society of America

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  1. J. Hollandt, J. Seidel, R. Klein, G. Ulm, A. Migdall, and M. Ware, “Primary sources for use in radiometry,” in Optical Radiometry, A. C. Parr, R. U. Datla, and J. L. Gardner, eds. (Academic, 2005), pp. 213–290.
  2. J. Hartmann, J. Hollandt, B. Khlevnoy, S. Morozova, S. Ogarev, and F. Sakuma, “Blackbody and other calibration sources,” in Radiometric Temperature Measurements. I. Fundamentals, Z. M. Zhang, B. K. Tsai, and G. Machin, eds. (Academic, 2010), pp. 241–295.
  3. P. J. Mohr, B. N. Taylor, and D. B. Newell, “CODATA recommended values of the fundamental physical constants: 2010,” http://physics.nist.gov/cuu/Constants/Preprints/lsa2010.pdf .
  4. S. N. Mekhontsev, A. V. Prokhorov, and L. N. Hanssen, “Experimental characterization of blackbody radiation sources,” in Radiometric Temperature Measurements. II. Applications, Z. M. Zhang, B. K. Tsai, and G. Machin, eds. (Academic, 2010), pp. 57–136.
  5. R. E. Bedford, “Calculation of effective emissivities of cavity sources of thermal radiation,” in Theory and Practice of Radiation Thermometry, D. P. DeWitt and G. D. Nutter, eds. (Wiley, 1988), pp. 653–772.
  6. A. V. Prokhorov, L. M. Hanssen, and S. N. Mekhontsev, “Calculation of the radiation characteristics of blackbody radiation sources,” in Radiometric Temperature Measurements. I. Fundamentals, Z. M. Zhang, B. K. Tsai, and G. Machin, eds. (Academic, 2010), pp. 181–240.
  7. R. P. Heinisch, E. M. Sparrow, and N. Shamsundar, “Radiant emission from baffled conical cavities,” J. Opt. Soc. Am. 63, 152–158 (1973).
    [CrossRef]
  8. A. Ono, “Calculation of the directional emissivities of cavities by the Monte Carlo method,” J. Opt. Soc. Am. 70, 547–554 (1980).
    [CrossRef]
  9. V. I. Sapritsky and A. V. Prokhorov, “Spectral effective emissivities of nonisothermal cavities calculated by the Monte Carlo method,” Appl. Opt. 34, 5645–5652 (1995).
    [CrossRef]
  10. J. Ishii, M. Kobayashi, and F. Sakuma, “Effective emissivities of black-body cavities with grooved cylinders,” Metrologia 35, 175–180 (1998).
    [CrossRef]
  11. Y. Té, P. Jeseck, C. Camy-Peyret, S. Payan, S. Briaudeau, and M. Fanjeaux, “High emissivity blackbody for radiometric calibration near ambient temperature,” Metrologia 40, 24–30 (2003).
    [CrossRef]
  12. M. J. Ballico, “Modelling of the effective emissivity of a graphite tube black body,” Metrologia 32, 259–265 (1995).
    [CrossRef]
  13. F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance,” NBS Monograph 160 (U. S. Department of Commerce, National Bureau of Standards, 1977).
  14. A. Prokhorov, “Effective emissivities of isothermal blackbody cavities calculated by the Monte Carlo method using the three-component BRDF Model,” Appl. Opt. 51, 2322–2332 (2012).
    [CrossRef]
  15. H. J. Patrick, L. M. Hanssen, J. Zeng, and T. A. Germer, “BRDF measurements of graphite used in high-temperature fixed point blackbody radiators: a multi-angle study at 405 nm and 658 nm,” Metrologia 49, S81–S92 (2012).
    [CrossRef]
  16. M. Kurt and D. Edwards, “A survey of BRDF models for computer graphics,” Comput. Graph. 43, 4 (2009).
  17. S. Liang, Quantitative Remote Sensing of Land Surfaces(Wiley, 2004).
  18. D. Geisler-Moroder and A. Dür, “A new ward BRDF model with bounded albedo,” Comput. Graph. Forum 29, 1391–1398 (2010).
    [CrossRef]
  19. C. Schlick, “An inexpensive BRDF model for physically-based rendering,” Comput. Graph. Forum 13, 233–246 (1994).
    [CrossRef]
  20. M. Clerc, Particle Swarm Optimization (ISTE, 2006).
  21. R. P. Brent, “An algorithm with guaranteed convergence for finding a zero of a function,” Comput J (Switzerland) 14, 422–425 (1971).
    [CrossRef]
  22. J. Berntsen, T. O. Espelid, and A. Genz, “An adaptive algorithm for the approximate calculation of multiple integrals,” ACM Trans. Math. Softw. 17, 437–451 (1991).
    [CrossRef]
  23. M. J. Persky, “Review of black surfaces for space-borne infrared systems,” Rev. Sci. Instrum. 70, 2193–2217 (1999).
    [CrossRef]
  24. L. M. Hanssen and A. V. Prokhorov, “Stochastic modeling of non-Lambertian surfaces for Monte Carlo computations in optical radiometry,” Proc. SPIE 7427, 742707 (2009).
    [CrossRef]
  25. Measurements were performed by Leonard M. Hanssen (NIST, Gaithersburg, Maryland).
  26. L. M. Hanssen and S. Kaplan, “Infrared diffuse reflectance instrumentation and standards at NIST,” Anal. Chim. Acta 380, 289–302 (1999).
    [CrossRef]
  27. J. Zeng and L. Hanssen, “An infrared laser-based reflectometer for low reflectance measurements of samples and cavity structures,” Proc. SPIE 7065, 70650F (2008).
    [CrossRef]

2012 (2)

H. J. Patrick, L. M. Hanssen, J. Zeng, and T. A. Germer, “BRDF measurements of graphite used in high-temperature fixed point blackbody radiators: a multi-angle study at 405 nm and 658 nm,” Metrologia 49, S81–S92 (2012).
[CrossRef]

A. Prokhorov, “Effective emissivities of isothermal blackbody cavities calculated by the Monte Carlo method using the three-component BRDF Model,” Appl. Opt. 51, 2322–2332 (2012).
[CrossRef]

2010 (1)

D. Geisler-Moroder and A. Dür, “A new ward BRDF model with bounded albedo,” Comput. Graph. Forum 29, 1391–1398 (2010).
[CrossRef]

2009 (2)

M. Kurt and D. Edwards, “A survey of BRDF models for computer graphics,” Comput. Graph. 43, 4 (2009).

L. M. Hanssen and A. V. Prokhorov, “Stochastic modeling of non-Lambertian surfaces for Monte Carlo computations in optical radiometry,” Proc. SPIE 7427, 742707 (2009).
[CrossRef]

2008 (1)

J. Zeng and L. Hanssen, “An infrared laser-based reflectometer for low reflectance measurements of samples and cavity structures,” Proc. SPIE 7065, 70650F (2008).
[CrossRef]

2003 (1)

Y. Té, P. Jeseck, C. Camy-Peyret, S. Payan, S. Briaudeau, and M. Fanjeaux, “High emissivity blackbody for radiometric calibration near ambient temperature,” Metrologia 40, 24–30 (2003).
[CrossRef]

1999 (2)

L. M. Hanssen and S. Kaplan, “Infrared diffuse reflectance instrumentation and standards at NIST,” Anal. Chim. Acta 380, 289–302 (1999).
[CrossRef]

M. J. Persky, “Review of black surfaces for space-borne infrared systems,” Rev. Sci. Instrum. 70, 2193–2217 (1999).
[CrossRef]

1998 (1)

J. Ishii, M. Kobayashi, and F. Sakuma, “Effective emissivities of black-body cavities with grooved cylinders,” Metrologia 35, 175–180 (1998).
[CrossRef]

1995 (2)

1994 (1)

C. Schlick, “An inexpensive BRDF model for physically-based rendering,” Comput. Graph. Forum 13, 233–246 (1994).
[CrossRef]

1991 (1)

J. Berntsen, T. O. Espelid, and A. Genz, “An adaptive algorithm for the approximate calculation of multiple integrals,” ACM Trans. Math. Softw. 17, 437–451 (1991).
[CrossRef]

1980 (1)

1973 (1)

1971 (1)

R. P. Brent, “An algorithm with guaranteed convergence for finding a zero of a function,” Comput J (Switzerland) 14, 422–425 (1971).
[CrossRef]

Ballico, M. J.

M. J. Ballico, “Modelling of the effective emissivity of a graphite tube black body,” Metrologia 32, 259–265 (1995).
[CrossRef]

Bedford, R. E.

R. E. Bedford, “Calculation of effective emissivities of cavity sources of thermal radiation,” in Theory and Practice of Radiation Thermometry, D. P. DeWitt and G. D. Nutter, eds. (Wiley, 1988), pp. 653–772.

Berntsen, J.

J. Berntsen, T. O. Espelid, and A. Genz, “An adaptive algorithm for the approximate calculation of multiple integrals,” ACM Trans. Math. Softw. 17, 437–451 (1991).
[CrossRef]

Brent, R. P.

R. P. Brent, “An algorithm with guaranteed convergence for finding a zero of a function,” Comput J (Switzerland) 14, 422–425 (1971).
[CrossRef]

Briaudeau, S.

Y. Té, P. Jeseck, C. Camy-Peyret, S. Payan, S. Briaudeau, and M. Fanjeaux, “High emissivity blackbody for radiometric calibration near ambient temperature,” Metrologia 40, 24–30 (2003).
[CrossRef]

Camy-Peyret, C.

Y. Té, P. Jeseck, C. Camy-Peyret, S. Payan, S. Briaudeau, and M. Fanjeaux, “High emissivity blackbody for radiometric calibration near ambient temperature,” Metrologia 40, 24–30 (2003).
[CrossRef]

Clerc, M.

M. Clerc, Particle Swarm Optimization (ISTE, 2006).

Dür, A.

D. Geisler-Moroder and A. Dür, “A new ward BRDF model with bounded albedo,” Comput. Graph. Forum 29, 1391–1398 (2010).
[CrossRef]

Edwards, D.

M. Kurt and D. Edwards, “A survey of BRDF models for computer graphics,” Comput. Graph. 43, 4 (2009).

Espelid, T. O.

J. Berntsen, T. O. Espelid, and A. Genz, “An adaptive algorithm for the approximate calculation of multiple integrals,” ACM Trans. Math. Softw. 17, 437–451 (1991).
[CrossRef]

Fanjeaux, M.

Y. Té, P. Jeseck, C. Camy-Peyret, S. Payan, S. Briaudeau, and M. Fanjeaux, “High emissivity blackbody for radiometric calibration near ambient temperature,” Metrologia 40, 24–30 (2003).
[CrossRef]

Geisler-Moroder, D.

D. Geisler-Moroder and A. Dür, “A new ward BRDF model with bounded albedo,” Comput. Graph. Forum 29, 1391–1398 (2010).
[CrossRef]

Genz, A.

J. Berntsen, T. O. Espelid, and A. Genz, “An adaptive algorithm for the approximate calculation of multiple integrals,” ACM Trans. Math. Softw. 17, 437–451 (1991).
[CrossRef]

Germer, T. A.

H. J. Patrick, L. M. Hanssen, J. Zeng, and T. A. Germer, “BRDF measurements of graphite used in high-temperature fixed point blackbody radiators: a multi-angle study at 405 nm and 658 nm,” Metrologia 49, S81–S92 (2012).
[CrossRef]

Ginsberg, I. W.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance,” NBS Monograph 160 (U. S. Department of Commerce, National Bureau of Standards, 1977).

Hanssen, L.

J. Zeng and L. Hanssen, “An infrared laser-based reflectometer for low reflectance measurements of samples and cavity structures,” Proc. SPIE 7065, 70650F (2008).
[CrossRef]

Hanssen, L. M.

H. J. Patrick, L. M. Hanssen, J. Zeng, and T. A. Germer, “BRDF measurements of graphite used in high-temperature fixed point blackbody radiators: a multi-angle study at 405 nm and 658 nm,” Metrologia 49, S81–S92 (2012).
[CrossRef]

L. M. Hanssen and A. V. Prokhorov, “Stochastic modeling of non-Lambertian surfaces for Monte Carlo computations in optical radiometry,” Proc. SPIE 7427, 742707 (2009).
[CrossRef]

L. M. Hanssen and S. Kaplan, “Infrared diffuse reflectance instrumentation and standards at NIST,” Anal. Chim. Acta 380, 289–302 (1999).
[CrossRef]

A. V. Prokhorov, L. M. Hanssen, and S. N. Mekhontsev, “Calculation of the radiation characteristics of blackbody radiation sources,” in Radiometric Temperature Measurements. I. Fundamentals, Z. M. Zhang, B. K. Tsai, and G. Machin, eds. (Academic, 2010), pp. 181–240.

Hanssen, L. N.

S. N. Mekhontsev, A. V. Prokhorov, and L. N. Hanssen, “Experimental characterization of blackbody radiation sources,” in Radiometric Temperature Measurements. II. Applications, Z. M. Zhang, B. K. Tsai, and G. Machin, eds. (Academic, 2010), pp. 57–136.

Hanssen, Leonard M.

Measurements were performed by Leonard M. Hanssen (NIST, Gaithersburg, Maryland).

Hartmann, J.

J. Hartmann, J. Hollandt, B. Khlevnoy, S. Morozova, S. Ogarev, and F. Sakuma, “Blackbody and other calibration sources,” in Radiometric Temperature Measurements. I. Fundamentals, Z. M. Zhang, B. K. Tsai, and G. Machin, eds. (Academic, 2010), pp. 241–295.

Heinisch, R. P.

Hollandt, J.

J. Hollandt, J. Seidel, R. Klein, G. Ulm, A. Migdall, and M. Ware, “Primary sources for use in radiometry,” in Optical Radiometry, A. C. Parr, R. U. Datla, and J. L. Gardner, eds. (Academic, 2005), pp. 213–290.

J. Hartmann, J. Hollandt, B. Khlevnoy, S. Morozova, S. Ogarev, and F. Sakuma, “Blackbody and other calibration sources,” in Radiometric Temperature Measurements. I. Fundamentals, Z. M. Zhang, B. K. Tsai, and G. Machin, eds. (Academic, 2010), pp. 241–295.

Hsia, J. J.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance,” NBS Monograph 160 (U. S. Department of Commerce, National Bureau of Standards, 1977).

Ishii, J.

J. Ishii, M. Kobayashi, and F. Sakuma, “Effective emissivities of black-body cavities with grooved cylinders,” Metrologia 35, 175–180 (1998).
[CrossRef]

Jeseck, P.

Y. Té, P. Jeseck, C. Camy-Peyret, S. Payan, S. Briaudeau, and M. Fanjeaux, “High emissivity blackbody for radiometric calibration near ambient temperature,” Metrologia 40, 24–30 (2003).
[CrossRef]

Kaplan, S.

L. M. Hanssen and S. Kaplan, “Infrared diffuse reflectance instrumentation and standards at NIST,” Anal. Chim. Acta 380, 289–302 (1999).
[CrossRef]

Khlevnoy, B.

J. Hartmann, J. Hollandt, B. Khlevnoy, S. Morozova, S. Ogarev, and F. Sakuma, “Blackbody and other calibration sources,” in Radiometric Temperature Measurements. I. Fundamentals, Z. M. Zhang, B. K. Tsai, and G. Machin, eds. (Academic, 2010), pp. 241–295.

Klein, R.

J. Hollandt, J. Seidel, R. Klein, G. Ulm, A. Migdall, and M. Ware, “Primary sources for use in radiometry,” in Optical Radiometry, A. C. Parr, R. U. Datla, and J. L. Gardner, eds. (Academic, 2005), pp. 213–290.

Kobayashi, M.

J. Ishii, M. Kobayashi, and F. Sakuma, “Effective emissivities of black-body cavities with grooved cylinders,” Metrologia 35, 175–180 (1998).
[CrossRef]

Kurt, M.

M. Kurt and D. Edwards, “A survey of BRDF models for computer graphics,” Comput. Graph. 43, 4 (2009).

Liang, S.

S. Liang, Quantitative Remote Sensing of Land Surfaces(Wiley, 2004).

Limperis, T.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance,” NBS Monograph 160 (U. S. Department of Commerce, National Bureau of Standards, 1977).

Mekhontsev, S. N.

S. N. Mekhontsev, A. V. Prokhorov, and L. N. Hanssen, “Experimental characterization of blackbody radiation sources,” in Radiometric Temperature Measurements. II. Applications, Z. M. Zhang, B. K. Tsai, and G. Machin, eds. (Academic, 2010), pp. 57–136.

A. V. Prokhorov, L. M. Hanssen, and S. N. Mekhontsev, “Calculation of the radiation characteristics of blackbody radiation sources,” in Radiometric Temperature Measurements. I. Fundamentals, Z. M. Zhang, B. K. Tsai, and G. Machin, eds. (Academic, 2010), pp. 181–240.

Migdall, A.

J. Hollandt, J. Seidel, R. Klein, G. Ulm, A. Migdall, and M. Ware, “Primary sources for use in radiometry,” in Optical Radiometry, A. C. Parr, R. U. Datla, and J. L. Gardner, eds. (Academic, 2005), pp. 213–290.

Morozova, S.

J. Hartmann, J. Hollandt, B. Khlevnoy, S. Morozova, S. Ogarev, and F. Sakuma, “Blackbody and other calibration sources,” in Radiometric Temperature Measurements. I. Fundamentals, Z. M. Zhang, B. K. Tsai, and G. Machin, eds. (Academic, 2010), pp. 241–295.

Nicodemus, F. E.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance,” NBS Monograph 160 (U. S. Department of Commerce, National Bureau of Standards, 1977).

Ogarev, S.

J. Hartmann, J. Hollandt, B. Khlevnoy, S. Morozova, S. Ogarev, and F. Sakuma, “Blackbody and other calibration sources,” in Radiometric Temperature Measurements. I. Fundamentals, Z. M. Zhang, B. K. Tsai, and G. Machin, eds. (Academic, 2010), pp. 241–295.

Ono, A.

Patrick, H. J.

H. J. Patrick, L. M. Hanssen, J. Zeng, and T. A. Germer, “BRDF measurements of graphite used in high-temperature fixed point blackbody radiators: a multi-angle study at 405 nm and 658 nm,” Metrologia 49, S81–S92 (2012).
[CrossRef]

Payan, S.

Y. Té, P. Jeseck, C. Camy-Peyret, S. Payan, S. Briaudeau, and M. Fanjeaux, “High emissivity blackbody for radiometric calibration near ambient temperature,” Metrologia 40, 24–30 (2003).
[CrossRef]

Persky, M. J.

M. J. Persky, “Review of black surfaces for space-borne infrared systems,” Rev. Sci. Instrum. 70, 2193–2217 (1999).
[CrossRef]

Prokhorov, A.

Prokhorov, A. V.

L. M. Hanssen and A. V. Prokhorov, “Stochastic modeling of non-Lambertian surfaces for Monte Carlo computations in optical radiometry,” Proc. SPIE 7427, 742707 (2009).
[CrossRef]

V. I. Sapritsky and A. V. Prokhorov, “Spectral effective emissivities of nonisothermal cavities calculated by the Monte Carlo method,” Appl. Opt. 34, 5645–5652 (1995).
[CrossRef]

S. N. Mekhontsev, A. V. Prokhorov, and L. N. Hanssen, “Experimental characterization of blackbody radiation sources,” in Radiometric Temperature Measurements. II. Applications, Z. M. Zhang, B. K. Tsai, and G. Machin, eds. (Academic, 2010), pp. 57–136.

A. V. Prokhorov, L. M. Hanssen, and S. N. Mekhontsev, “Calculation of the radiation characteristics of blackbody radiation sources,” in Radiometric Temperature Measurements. I. Fundamentals, Z. M. Zhang, B. K. Tsai, and G. Machin, eds. (Academic, 2010), pp. 181–240.

Richmond, J. C.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance,” NBS Monograph 160 (U. S. Department of Commerce, National Bureau of Standards, 1977).

Sakuma, F.

J. Ishii, M. Kobayashi, and F. Sakuma, “Effective emissivities of black-body cavities with grooved cylinders,” Metrologia 35, 175–180 (1998).
[CrossRef]

J. Hartmann, J. Hollandt, B. Khlevnoy, S. Morozova, S. Ogarev, and F. Sakuma, “Blackbody and other calibration sources,” in Radiometric Temperature Measurements. I. Fundamentals, Z. M. Zhang, B. K. Tsai, and G. Machin, eds. (Academic, 2010), pp. 241–295.

Sapritsky, V. I.

Schlick, C.

C. Schlick, “An inexpensive BRDF model for physically-based rendering,” Comput. Graph. Forum 13, 233–246 (1994).
[CrossRef]

Seidel, J.

J. Hollandt, J. Seidel, R. Klein, G. Ulm, A. Migdall, and M. Ware, “Primary sources for use in radiometry,” in Optical Radiometry, A. C. Parr, R. U. Datla, and J. L. Gardner, eds. (Academic, 2005), pp. 213–290.

Shamsundar, N.

Sparrow, E. M.

Té, Y.

Y. Té, P. Jeseck, C. Camy-Peyret, S. Payan, S. Briaudeau, and M. Fanjeaux, “High emissivity blackbody for radiometric calibration near ambient temperature,” Metrologia 40, 24–30 (2003).
[CrossRef]

Ulm, G.

J. Hollandt, J. Seidel, R. Klein, G. Ulm, A. Migdall, and M. Ware, “Primary sources for use in radiometry,” in Optical Radiometry, A. C. Parr, R. U. Datla, and J. L. Gardner, eds. (Academic, 2005), pp. 213–290.

Ware, M.

J. Hollandt, J. Seidel, R. Klein, G. Ulm, A. Migdall, and M. Ware, “Primary sources for use in radiometry,” in Optical Radiometry, A. C. Parr, R. U. Datla, and J. L. Gardner, eds. (Academic, 2005), pp. 213–290.

Zeng, J.

H. J. Patrick, L. M. Hanssen, J. Zeng, and T. A. Germer, “BRDF measurements of graphite used in high-temperature fixed point blackbody radiators: a multi-angle study at 405 nm and 658 nm,” Metrologia 49, S81–S92 (2012).
[CrossRef]

J. Zeng and L. Hanssen, “An infrared laser-based reflectometer for low reflectance measurements of samples and cavity structures,” Proc. SPIE 7065, 70650F (2008).
[CrossRef]

ACM Trans. Math. Softw. (1)

J. Berntsen, T. O. Espelid, and A. Genz, “An adaptive algorithm for the approximate calculation of multiple integrals,” ACM Trans. Math. Softw. 17, 437–451 (1991).
[CrossRef]

Anal. Chim. Acta (1)

L. M. Hanssen and S. Kaplan, “Infrared diffuse reflectance instrumentation and standards at NIST,” Anal. Chim. Acta 380, 289–302 (1999).
[CrossRef]

Appl. Opt. (2)

Comput J (Switzerland) (1)

R. P. Brent, “An algorithm with guaranteed convergence for finding a zero of a function,” Comput J (Switzerland) 14, 422–425 (1971).
[CrossRef]

Comput. Graph. (1)

M. Kurt and D. Edwards, “A survey of BRDF models for computer graphics,” Comput. Graph. 43, 4 (2009).

Comput. Graph. Forum (2)

D. Geisler-Moroder and A. Dür, “A new ward BRDF model with bounded albedo,” Comput. Graph. Forum 29, 1391–1398 (2010).
[CrossRef]

C. Schlick, “An inexpensive BRDF model for physically-based rendering,” Comput. Graph. Forum 13, 233–246 (1994).
[CrossRef]

J. Opt. Soc. Am. (2)

Metrologia (4)

J. Ishii, M. Kobayashi, and F. Sakuma, “Effective emissivities of black-body cavities with grooved cylinders,” Metrologia 35, 175–180 (1998).
[CrossRef]

Y. Té, P. Jeseck, C. Camy-Peyret, S. Payan, S. Briaudeau, and M. Fanjeaux, “High emissivity blackbody for radiometric calibration near ambient temperature,” Metrologia 40, 24–30 (2003).
[CrossRef]

M. J. Ballico, “Modelling of the effective emissivity of a graphite tube black body,” Metrologia 32, 259–265 (1995).
[CrossRef]

H. J. Patrick, L. M. Hanssen, J. Zeng, and T. A. Germer, “BRDF measurements of graphite used in high-temperature fixed point blackbody radiators: a multi-angle study at 405 nm and 658 nm,” Metrologia 49, S81–S92 (2012).
[CrossRef]

Proc. SPIE (2)

J. Zeng and L. Hanssen, “An infrared laser-based reflectometer for low reflectance measurements of samples and cavity structures,” Proc. SPIE 7065, 70650F (2008).
[CrossRef]

L. M. Hanssen and A. V. Prokhorov, “Stochastic modeling of non-Lambertian surfaces for Monte Carlo computations in optical radiometry,” Proc. SPIE 7427, 742707 (2009).
[CrossRef]

Rev. Sci. Instrum. (1)

M. J. Persky, “Review of black surfaces for space-borne infrared systems,” Rev. Sci. Instrum. 70, 2193–2217 (1999).
[CrossRef]

Other (10)

Measurements were performed by Leonard M. Hanssen (NIST, Gaithersburg, Maryland).

M. Clerc, Particle Swarm Optimization (ISTE, 2006).

S. Liang, Quantitative Remote Sensing of Land Surfaces(Wiley, 2004).

J. Hollandt, J. Seidel, R. Klein, G. Ulm, A. Migdall, and M. Ware, “Primary sources for use in radiometry,” in Optical Radiometry, A. C. Parr, R. U. Datla, and J. L. Gardner, eds. (Academic, 2005), pp. 213–290.

J. Hartmann, J. Hollandt, B. Khlevnoy, S. Morozova, S. Ogarev, and F. Sakuma, “Blackbody and other calibration sources,” in Radiometric Temperature Measurements. I. Fundamentals, Z. M. Zhang, B. K. Tsai, and G. Machin, eds. (Academic, 2010), pp. 241–295.

P. J. Mohr, B. N. Taylor, and D. B. Newell, “CODATA recommended values of the fundamental physical constants: 2010,” http://physics.nist.gov/cuu/Constants/Preprints/lsa2010.pdf .

S. N. Mekhontsev, A. V. Prokhorov, and L. N. Hanssen, “Experimental characterization of blackbody radiation sources,” in Radiometric Temperature Measurements. II. Applications, Z. M. Zhang, B. K. Tsai, and G. Machin, eds. (Academic, 2010), pp. 57–136.

R. E. Bedford, “Calculation of effective emissivities of cavity sources of thermal radiation,” in Theory and Practice of Radiation Thermometry, D. P. DeWitt and G. D. Nutter, eds. (Wiley, 1988), pp. 653–772.

A. V. Prokhorov, L. M. Hanssen, and S. N. Mekhontsev, “Calculation of the radiation characteristics of blackbody radiation sources,” in Radiometric Temperature Measurements. I. Fundamentals, Z. M. Zhang, B. K. Tsai, and G. Machin, eds. (Academic, 2010), pp. 181–240.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance,” NBS Monograph 160 (U. S. Department of Commerce, National Bureau of Standards, 1977).

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Figures (11)

Fig. 1.
Fig. 1.

Plots of the 3C BRDF in spherical coordinates for three incidence angles θi; kd=0.7, kqs=0, kg=0.3, σg=0.1, Rd=Rg(0°)=0.5. All BRDFs are normalized by dividing by their maxima. The incident and specularly reflected rays are shown.

Fig. 2.
Fig. 2.

Schematics of cavity section, viewing conditions, and axial temperature distributions used for numerical experiments.

Fig. 3.
Fig. 3.

3C BRDF model (lines) fitted to Z302 black paint BRDFs measured at 10.6 μm (symbols).

Fig. 4.
Fig. 4.

Spectral DHR (measured, smoothed, and sampled from the smoothed curve) and partial spectral DHRs for diffuse, quasi-specular, and glossy components. Sampled values were used for numerical experiments.

Fig. 5.
Fig. 5.

Spectral effective emissivities at normal viewing conditions for rb=0.25 (upper graph) and rb=0.75 (lower graph) computed for isothermal cavity and for three temperature distributions.

Fig. 6.
Fig. 6.

Dependences of the radiance temperatures at 10.6 μm on the viewing beam radius rb of normal viewing conditions computed for three temperature distributions.

Fig. 7.
Fig. 7.

Spectral effective emissivities at conical viewing conditions for zf=2.5 (upper graph) and zf=2.5 (lower graph) computed for isothermal cavity and for three temperature distributions.

Fig. 8.
Fig. 8.

Spectral effective emissivities at conical viewing conditions for zf=6 (upper graph) and zf=14.5 (lower graph) computed for isothermal cavity and for three temperature distributions.

Fig. 9.
Fig. 9.

Dependences of radiance temperatures on the focal point position zf of conical viewing conditions computed for three temperature distributions.

Fig. 10.
Fig. 10.

Deviation of radiance temperatures at 10.6 μm computed using the USD reflection model from those computed using 3C BRDF model. Difference in radiance temperatures is plotted as a function of the viewing beam radius rb of normal viewing conditions.

Fig. 11.
Fig. 11.

Deviation of radiance temperatures at 10.6 μm computed using the USD reflection model from those computed using 3C BRDF model. Difference in radiance temperatures is plotted as a function of the focal point position zf of conical viewing conditions.

Equations (23)

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εe(λ,ξ,ω,Tref)=Lλ(λ,ξ,ω)Lλ,bb(λ,Tref),
Lλ,bb(λ,Tref)=c1{π·λ5[exp(c2λ·Tref)1]}1,
εe(λ,ξ,ω,Tref,Tbg)=εe(λ,ξ,ω,Tref)+[1εe(λ,ξ,ω)]exp(C2λTref)1exp(C2λTbg)1.
TS(λ,ξ,ω,Tref)=c2{λln[1+exp(c2λTref)1εe(λ,ξ,ω,Tref)]}1.
fr(λ,θi,ϕi,θv,ϕv)=dLλ,v(λ,θi,ϕi,θv,ϕv)dEλ,i(λ,θi,ϕi)=dLλ,v(λ,θi,ϕi,θv,ϕv)dLλ,i(λ,θi,ϕi)dΩi[sr1],
ρ(θi)=02πdϕ0π2fr(θi,ϕi,θv,ϕv)sinθvcosθvdθv.
fr3C,=kdfr,d+kqsfr,qs+kgfr,g,
kd+kqs+kg=1.
fr,d=Rd/π.
fr,qs,g(θi,θv,ϕ)=Rqs,g(θh)πσqs,gexp[(tanθhσqs,g)2]2[1+cosθicosθvsinθisinθvcosϕ](cosθicosθv)4,
Rqs,g(θi)={Rqs,g,0+(1Rqs,g,0)(1cosθh)5ifRqs,g,0>00ifRqs,g,0=0,
ρ3C(θi)=ρd+ρqs(θi)+ρg(θi),
ρd=kdRd,
ρqs,g(θi)=kqs,gϕ=02πθv=0π/2fr,qs,g(Rqs,g,0,θi,,θv,ϕ)sinθvcosθvdθvdϕ.
{γd=ρd(λ0)/ρ*(θi,0,λ0)γqs(θi,0)=ρqs(θi,0,λ0)/ρ*(θi,0,λ0)γg(θi,0)=ρg(θi,0,λ0)/ρ*(θi,0,λ0),
γdρ*(θi,0,λ)=kdRd(λ),
γqsρ*(θi,0,λ)=kqsϕ=02πθv=0π/2fr,qs(Rqs,0(λ0),θi,0,θv,ϕ)sinθvcosθvdθvdϕ,
γgρ*(θi,0,λ)=kgϕ=02πθv=0π/2fr,g(Rg,0(λ0),θi,0,θv,ϕ)sinθvcosθvdθvdϕ.
Rd(λ)=γdρ*(θi,0,λ)/kd.
wj,k=wj1,k×{Rd(λ)for diffuse reflection2Rqs(λ,θi,j1,k)(1+cosθi,j1,k/cosθv,j1,k)for quasi-specular reflection2Rg(λ,θi,j1,k)(1+cosθi,j1,k/cosθv,j1,k)for glossy reflection,
Lλ,j,k(λ)=(1wj+1,k)Lλ,bb(λ,Tj+1,k)+wj+1,kLλ,j+1,k,
Lλ,k(λ)=[1w1,k(λ)]Lλ,bb(λ,T1,k)+j=2mk[1wj,k(λ)]Lλ,bb(λ,Tj,k)l=1j1wl,k(λ),
εe(λ,Tref)=1nLλ,bb(λ,Tref)k=1nLλ,k(λ).

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