Abstract

The effects of incidence angle, geometrical shape, and optical properties of dielectric rough surfaces on reflectivity and transmissivity are discussed. Radiative properties for various surface geometries are calculated. Since the integral method is computationally expensive, a geometric optics approximation is developed. The regions of validity of the approximation compared with the integral method are quantified. Curves are presented that show these radiative properties versus the correlation length at incidence angle for a fixed rms deviation of the surface. The surface geometry, incidence angle, multiple scattering, shadowing effects, and dielectric permittivity contributions to the domains of validity of the approximation method are discussed for both TE and TM polarizations.

© 2007 Optical Society of America

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  1. K. Tang and R. O. Buckius, "Regions of validity of the geometric optics approximation for angular scattering from very rough surfaces," Int. J. Heat Mass Transfer 40, 49-59 (1997).
    [CrossRef]
  2. K. Tang and R. O. Buckius, "The geometric optics approximation for reflection from two-dimensional random rough surfaces," Int. J. Heat Mass Transfer 41, 2037-2047 (1998).
    [CrossRef]
  3. J.-J. Greffet and R. Carminati, "Radiative transfer at nanometric scale: are the usual concepts still valid?" Heat Technol. 18, 81-85 (2000).
  4. A. Khenchaf, "Bistatic scattering and depolarization by randomly rough surfaces: application to the natural rough surfaces in X-band," Waves Random Media 11, 61-89 (2001).
    [CrossRef]
  5. J. J. Greffet, R. Carminati, K. Joulain, J. Ph. Mulet, S. Mainguy, and Y. Chen, "Coherent emission of light by thermal sources," Nature (London) 416, 61 (2002).
    [CrossRef]
  6. M. Nieto-Vesperinas and J. M. Soto-Crespo, "Light-diffracted intensities from very deep gratings," Phys. Rev. B 38, 7250-7259 (1988).
    [CrossRef]
  7. R. A. Dimenna and R. O. Buckius, "Microgeometrical contour contributions to surface scattering," Therm. Sci. Eng. 2, 166-171 (1994).
  8. Y. Yang and R. O. Buckius, "Surface length scale contributions to the directional and hemispherical emissivity and reflectivity," J. Thermophys. Heat Transfer 9, 653-659 (1995).
    [CrossRef]
  9. A. J. Sant, J. C. Dainty, and M. J. Kim, "Comparison of surface scattering between identical, randomly rough metal and dielectric diffusers," Opt. Lett. 14, 1183-1185 (1989).
    [CrossRef] [PubMed]
  10. D. W. Cohn, K. Tang, and R. O. Buckius, "Comparison of theory and experiments for reflection of micro contoured Surfaces," Int. J. Heat Mass Transfer 40, 133-149 (1983).
  11. R. M. Axline and A. K. Fung, "Numerical computation of scattering from perfectly conducting random surfaces," IEEE Trans. Antennas Propag. 26, 482-488 (1978).
    [CrossRef]
  12. H. L. Chan and A. K. Fung, "A numerical study of the Kirchhoff approximation in horizontally polarized backscattering from a random surface," Radio Sci. 13, 818-881 (1978).
    [CrossRef]
  13. A. K. Fung and M. K. Chen, "Numerical simulation of scattering from simple and composite random surfaces," J. Opt. Soc. Am. A 2, 2274-2284 (1985).
    [CrossRef]
  14. E. Thorsos, "The validity of the Kirchhoff approximation for rough surface scattering using Gaussian roughness spectrum," J. Acoust. Soc. Am. 83, 78-92 (1988).
    [CrossRef]
  15. M. F. Chen and A. K. Fung, "A numerical study of the regions of validity of the Kirchhoff and small-perturbation rough surface scattering models," Radio Sci. 23, 163-170 (1988).
    [CrossRef]
  16. R. A. Dimenna and R. O. Buckius, "Quantifying specular approximations for angular scattering from perfectly conducting random rough surfaces," J. Thermophys. Heat Transfer 8, 393-399 (1994).
    [CrossRef]
  17. M. B. Matchiane, F. Ghmari, and M. S. Sifaoui, "Correction de l"approximation de Kirchhoff par la méthode intégrale reformulée: Cas des réflectivités de surfaces sinusoïdales," Can. J. Phys. 82, 303-321 (2004).
    [CrossRef]
  18. N. C. Bruce, "Scattering of light from surfaces with one-dimensional structure calculated by the ray-tracing method," J. Opt. Soc. Am. A 14, 1850-1858 (1997).
    [CrossRef]
  19. N. C. Bruce, "Calculations of the Mueller matrix for scattering of light from two-dimensional surfaces," Waves Random Media 8, 15-28 (1998).
    [CrossRef]
  20. J. A. Sanchez-Gil and M. Nieto-Vesperinas, "Light scattering from random rough dielectric surfaces," J. Opt. Soc. Am. A 8, 1270-1286 (1991).
    [CrossRef]
  21. F. Ghmari, I. Sassi, and M. S. Sifaoui, "Directional hemispherical radiative properties of random dielectric rough surfaces," Waves Random Media 15, 469-486 (2005).
    [CrossRef]
  22. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan, 1963).
  23. P. Beckmann, "Scattering by non-Gaussian surfaces," IEEE Trans. Antennas Propag. 21, 169-175 (1973).
    [CrossRef]
  24. T. Michel, A. A. Maradudin, and E. R. Méndez, "Enhanced backscattering of light from non-Gaussian random metal surface," J. Opt. Soc. Am. B 6, 2438-2446 (1989).
    [CrossRef]
  25. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), Sec. I.5.
  26. R. D. Kodis, "A note on the theory of scattering from an irregular rough surface," IEEE Trans. Antennas Propag. AP-14, 77-82 (1966).
    [CrossRef]
  27. D. E. Barrick, "Rough surface scattering based on specular point theory," IEEE Trans. Antennas Propag. AP-16, 449-454 (1968).
    [CrossRef]
  28. K. L. Williams, J. S. Stroud, and P. L. Marston, "High-frequency forward scattering Gaussian spectrum, pressure release, corrugated surfaces: catastrophe modelling," J. Acoust. Soc. Am. 96, 1687-1702 (1994).
    [CrossRef]
  29. J. A. Stratton, Electromagnetic Theory, 1st ed. (McGraw-Hill, 1941), pp. 23-131.
  30. M. Q. Brewster, Thermal Radiative Transfer and Properties (Wiley, 1992), Chaps. 2 and 4.
  31. A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, "Enhanced scattering of light from a random grating," Ann. Phys. (N.Y.) 203, 255-307 (1990).
    [CrossRef]
  32. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, 1991).
  33. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (MacMillan, 1963).
  34. J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, 1991).
  35. N. C. Bruce and J. C. Dainty, "Multiple scattering from random rough surfaces using the Kirchhoff approximation," J. Mod. Opt. 38, 597-590 (1991).
  36. N. C. Bruce and J. C. Dainty, "Multiple scattering from rough dielectric and metal surfaces using the Kirchhoff approximation," J. Mod. Opt. 38, 1471-1481 (1991).
    [CrossRef]

2005 (1)

F. Ghmari, I. Sassi, and M. S. Sifaoui, "Directional hemispherical radiative properties of random dielectric rough surfaces," Waves Random Media 15, 469-486 (2005).
[CrossRef]

2004 (1)

M. B. Matchiane, F. Ghmari, and M. S. Sifaoui, "Correction de l"approximation de Kirchhoff par la méthode intégrale reformulée: Cas des réflectivités de surfaces sinusoïdales," Can. J. Phys. 82, 303-321 (2004).
[CrossRef]

2002 (1)

J. J. Greffet, R. Carminati, K. Joulain, J. Ph. Mulet, S. Mainguy, and Y. Chen, "Coherent emission of light by thermal sources," Nature (London) 416, 61 (2002).
[CrossRef]

2001 (1)

A. Khenchaf, "Bistatic scattering and depolarization by randomly rough surfaces: application to the natural rough surfaces in X-band," Waves Random Media 11, 61-89 (2001).
[CrossRef]

2000 (1)

J.-J. Greffet and R. Carminati, "Radiative transfer at nanometric scale: are the usual concepts still valid?" Heat Technol. 18, 81-85 (2000).

1998 (2)

K. Tang and R. O. Buckius, "The geometric optics approximation for reflection from two-dimensional random rough surfaces," Int. J. Heat Mass Transfer 41, 2037-2047 (1998).
[CrossRef]

N. C. Bruce, "Calculations of the Mueller matrix for scattering of light from two-dimensional surfaces," Waves Random Media 8, 15-28 (1998).
[CrossRef]

1997 (2)

K. Tang and R. O. Buckius, "Regions of validity of the geometric optics approximation for angular scattering from very rough surfaces," Int. J. Heat Mass Transfer 40, 49-59 (1997).
[CrossRef]

N. C. Bruce, "Scattering of light from surfaces with one-dimensional structure calculated by the ray-tracing method," J. Opt. Soc. Am. A 14, 1850-1858 (1997).
[CrossRef]

1995 (1)

Y. Yang and R. O. Buckius, "Surface length scale contributions to the directional and hemispherical emissivity and reflectivity," J. Thermophys. Heat Transfer 9, 653-659 (1995).
[CrossRef]

1994 (3)

R. A. Dimenna and R. O. Buckius, "Microgeometrical contour contributions to surface scattering," Therm. Sci. Eng. 2, 166-171 (1994).

R. A. Dimenna and R. O. Buckius, "Quantifying specular approximations for angular scattering from perfectly conducting random rough surfaces," J. Thermophys. Heat Transfer 8, 393-399 (1994).
[CrossRef]

K. L. Williams, J. S. Stroud, and P. L. Marston, "High-frequency forward scattering Gaussian spectrum, pressure release, corrugated surfaces: catastrophe modelling," J. Acoust. Soc. Am. 96, 1687-1702 (1994).
[CrossRef]

1991 (3)

N. C. Bruce and J. C. Dainty, "Multiple scattering from random rough surfaces using the Kirchhoff approximation," J. Mod. Opt. 38, 597-590 (1991).

N. C. Bruce and J. C. Dainty, "Multiple scattering from rough dielectric and metal surfaces using the Kirchhoff approximation," J. Mod. Opt. 38, 1471-1481 (1991).
[CrossRef]

J. A. Sanchez-Gil and M. Nieto-Vesperinas, "Light scattering from random rough dielectric surfaces," J. Opt. Soc. Am. A 8, 1270-1286 (1991).
[CrossRef]

1990 (1)

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, "Enhanced scattering of light from a random grating," Ann. Phys. (N.Y.) 203, 255-307 (1990).
[CrossRef]

1989 (2)

1988 (3)

M. Nieto-Vesperinas and J. M. Soto-Crespo, "Light-diffracted intensities from very deep gratings," Phys. Rev. B 38, 7250-7259 (1988).
[CrossRef]

E. Thorsos, "The validity of the Kirchhoff approximation for rough surface scattering using Gaussian roughness spectrum," J. Acoust. Soc. Am. 83, 78-92 (1988).
[CrossRef]

M. F. Chen and A. K. Fung, "A numerical study of the regions of validity of the Kirchhoff and small-perturbation rough surface scattering models," Radio Sci. 23, 163-170 (1988).
[CrossRef]

1985 (1)

1983 (1)

D. W. Cohn, K. Tang, and R. O. Buckius, "Comparison of theory and experiments for reflection of micro contoured Surfaces," Int. J. Heat Mass Transfer 40, 133-149 (1983).

1978 (2)

R. M. Axline and A. K. Fung, "Numerical computation of scattering from perfectly conducting random surfaces," IEEE Trans. Antennas Propag. 26, 482-488 (1978).
[CrossRef]

H. L. Chan and A. K. Fung, "A numerical study of the Kirchhoff approximation in horizontally polarized backscattering from a random surface," Radio Sci. 13, 818-881 (1978).
[CrossRef]

1973 (1)

P. Beckmann, "Scattering by non-Gaussian surfaces," IEEE Trans. Antennas Propag. 21, 169-175 (1973).
[CrossRef]

1968 (1)

D. E. Barrick, "Rough surface scattering based on specular point theory," IEEE Trans. Antennas Propag. AP-16, 449-454 (1968).
[CrossRef]

1966 (1)

R. D. Kodis, "A note on the theory of scattering from an irregular rough surface," IEEE Trans. Antennas Propag. AP-14, 77-82 (1966).
[CrossRef]

Axline, R. M.

R. M. Axline and A. K. Fung, "Numerical computation of scattering from perfectly conducting random surfaces," IEEE Trans. Antennas Propag. 26, 482-488 (1978).
[CrossRef]

Barrick, D. E.

D. E. Barrick, "Rough surface scattering based on specular point theory," IEEE Trans. Antennas Propag. AP-16, 449-454 (1968).
[CrossRef]

Beckmann, P.

P. Beckmann, "Scattering by non-Gaussian surfaces," IEEE Trans. Antennas Propag. 21, 169-175 (1973).
[CrossRef]

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

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

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), Sec. I.5.

Brewster, M. Q.

M. Q. Brewster, Thermal Radiative Transfer and Properties (Wiley, 1992), Chaps. 2 and 4.

Bruce, N. C.

N. C. Bruce, "Calculations of the Mueller matrix for scattering of light from two-dimensional surfaces," Waves Random Media 8, 15-28 (1998).
[CrossRef]

N. C. Bruce, "Scattering of light from surfaces with one-dimensional structure calculated by the ray-tracing method," J. Opt. Soc. Am. A 14, 1850-1858 (1997).
[CrossRef]

N. C. Bruce and J. C. Dainty, "Multiple scattering from rough dielectric and metal surfaces using the Kirchhoff approximation," J. Mod. Opt. 38, 1471-1481 (1991).
[CrossRef]

N. C. Bruce and J. C. Dainty, "Multiple scattering from random rough surfaces using the Kirchhoff approximation," J. Mod. Opt. 38, 597-590 (1991).

Buckius, R. O.

K. Tang and R. O. Buckius, "The geometric optics approximation for reflection from two-dimensional random rough surfaces," Int. J. Heat Mass Transfer 41, 2037-2047 (1998).
[CrossRef]

K. Tang and R. O. Buckius, "Regions of validity of the geometric optics approximation for angular scattering from very rough surfaces," Int. J. Heat Mass Transfer 40, 49-59 (1997).
[CrossRef]

Y. Yang and R. O. Buckius, "Surface length scale contributions to the directional and hemispherical emissivity and reflectivity," J. Thermophys. Heat Transfer 9, 653-659 (1995).
[CrossRef]

R. A. Dimenna and R. O. Buckius, "Quantifying specular approximations for angular scattering from perfectly conducting random rough surfaces," J. Thermophys. Heat Transfer 8, 393-399 (1994).
[CrossRef]

R. A. Dimenna and R. O. Buckius, "Microgeometrical contour contributions to surface scattering," Therm. Sci. Eng. 2, 166-171 (1994).

D. W. Cohn, K. Tang, and R. O. Buckius, "Comparison of theory and experiments for reflection of micro contoured Surfaces," Int. J. Heat Mass Transfer 40, 133-149 (1983).

Carminati, R.

J. J. Greffet, R. Carminati, K. Joulain, J. Ph. Mulet, S. Mainguy, and Y. Chen, "Coherent emission of light by thermal sources," Nature (London) 416, 61 (2002).
[CrossRef]

J.-J. Greffet and R. Carminati, "Radiative transfer at nanometric scale: are the usual concepts still valid?" Heat Technol. 18, 81-85 (2000).

Chan, H. L.

H. L. Chan and A. K. Fung, "A numerical study of the Kirchhoff approximation in horizontally polarized backscattering from a random surface," Radio Sci. 13, 818-881 (1978).
[CrossRef]

Chen, M. F.

M. F. Chen and A. K. Fung, "A numerical study of the regions of validity of the Kirchhoff and small-perturbation rough surface scattering models," Radio Sci. 23, 163-170 (1988).
[CrossRef]

Chen, M. K.

Chen, Y.

J. J. Greffet, R. Carminati, K. Joulain, J. Ph. Mulet, S. Mainguy, and Y. Chen, "Coherent emission of light by thermal sources," Nature (London) 416, 61 (2002).
[CrossRef]

Cohn, D. W.

D. W. Cohn, K. Tang, and R. O. Buckius, "Comparison of theory and experiments for reflection of micro contoured Surfaces," Int. J. Heat Mass Transfer 40, 133-149 (1983).

Dainty, J. C.

N. C. Bruce and J. C. Dainty, "Multiple scattering from rough dielectric and metal surfaces using the Kirchhoff approximation," J. Mod. Opt. 38, 1471-1481 (1991).
[CrossRef]

N. C. Bruce and J. C. Dainty, "Multiple scattering from random rough surfaces using the Kirchhoff approximation," J. Mod. Opt. 38, 597-590 (1991).

A. J. Sant, J. C. Dainty, and M. J. Kim, "Comparison of surface scattering between identical, randomly rough metal and dielectric diffusers," Opt. Lett. 14, 1183-1185 (1989).
[CrossRef] [PubMed]

Dimenna, R. A.

R. A. Dimenna and R. O. Buckius, "Microgeometrical contour contributions to surface scattering," Therm. Sci. Eng. 2, 166-171 (1994).

R. A. Dimenna and R. O. Buckius, "Quantifying specular approximations for angular scattering from perfectly conducting random rough surfaces," J. Thermophys. Heat Transfer 8, 393-399 (1994).
[CrossRef]

Fung, A. K.

M. F. Chen and A. K. Fung, "A numerical study of the regions of validity of the Kirchhoff and small-perturbation rough surface scattering models," Radio Sci. 23, 163-170 (1988).
[CrossRef]

A. K. Fung and M. K. Chen, "Numerical simulation of scattering from simple and composite random surfaces," J. Opt. Soc. Am. A 2, 2274-2284 (1985).
[CrossRef]

R. M. Axline and A. K. Fung, "Numerical computation of scattering from perfectly conducting random surfaces," IEEE Trans. Antennas Propag. 26, 482-488 (1978).
[CrossRef]

H. L. Chan and A. K. Fung, "A numerical study of the Kirchhoff approximation in horizontally polarized backscattering from a random surface," Radio Sci. 13, 818-881 (1978).
[CrossRef]

Ghmari, F.

F. Ghmari, I. Sassi, and M. S. Sifaoui, "Directional hemispherical radiative properties of random dielectric rough surfaces," Waves Random Media 15, 469-486 (2005).
[CrossRef]

M. B. Matchiane, F. Ghmari, and M. S. Sifaoui, "Correction de l"approximation de Kirchhoff par la méthode intégrale reformulée: Cas des réflectivités de surfaces sinusoïdales," Can. J. Phys. 82, 303-321 (2004).
[CrossRef]

Greffet, J. J.

J. J. Greffet, R. Carminati, K. Joulain, J. Ph. Mulet, S. Mainguy, and Y. Chen, "Coherent emission of light by thermal sources," Nature (London) 416, 61 (2002).
[CrossRef]

Greffet, J.-J.

J.-J. Greffet and R. Carminati, "Radiative transfer at nanometric scale: are the usual concepts still valid?" Heat Technol. 18, 81-85 (2000).

Joulain, K.

J. J. Greffet, R. Carminati, K. Joulain, J. Ph. Mulet, S. Mainguy, and Y. Chen, "Coherent emission of light by thermal sources," Nature (London) 416, 61 (2002).
[CrossRef]

Khenchaf, A.

A. Khenchaf, "Bistatic scattering and depolarization by randomly rough surfaces: application to the natural rough surfaces in X-band," Waves Random Media 11, 61-89 (2001).
[CrossRef]

Kim, M. J.

Kodis, R. D.

R. D. Kodis, "A note on the theory of scattering from an irregular rough surface," IEEE Trans. Antennas Propag. AP-14, 77-82 (1966).
[CrossRef]

Mainguy, S.

J. J. Greffet, R. Carminati, K. Joulain, J. Ph. Mulet, S. Mainguy, and Y. Chen, "Coherent emission of light by thermal sources," Nature (London) 416, 61 (2002).
[CrossRef]

Maradudin, A. A.

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, "Enhanced scattering of light from a random grating," Ann. Phys. (N.Y.) 203, 255-307 (1990).
[CrossRef]

T. Michel, A. A. Maradudin, and E. R. Méndez, "Enhanced backscattering of light from non-Gaussian random metal surface," J. Opt. Soc. Am. B 6, 2438-2446 (1989).
[CrossRef]

Marston, P. L.

K. L. Williams, J. S. Stroud, and P. L. Marston, "High-frequency forward scattering Gaussian spectrum, pressure release, corrugated surfaces: catastrophe modelling," J. Acoust. Soc. Am. 96, 1687-1702 (1994).
[CrossRef]

Matchiane, M. B.

M. B. Matchiane, F. Ghmari, and M. S. Sifaoui, "Correction de l"approximation de Kirchhoff par la méthode intégrale reformulée: Cas des réflectivités de surfaces sinusoïdales," Can. J. Phys. 82, 303-321 (2004).
[CrossRef]

McGurn, A. R.

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, "Enhanced scattering of light from a random grating," Ann. Phys. (N.Y.) 203, 255-307 (1990).
[CrossRef]

Méndez, E. R.

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, "Enhanced scattering of light from a random grating," Ann. Phys. (N.Y.) 203, 255-307 (1990).
[CrossRef]

T. Michel, A. A. Maradudin, and E. R. Méndez, "Enhanced backscattering of light from non-Gaussian random metal surface," J. Opt. Soc. Am. B 6, 2438-2446 (1989).
[CrossRef]

Michel, T.

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, "Enhanced scattering of light from a random grating," Ann. Phys. (N.Y.) 203, 255-307 (1990).
[CrossRef]

T. Michel, A. A. Maradudin, and E. R. Méndez, "Enhanced backscattering of light from non-Gaussian random metal surface," J. Opt. Soc. Am. B 6, 2438-2446 (1989).
[CrossRef]

Mulet, J. Ph.

J. J. Greffet, R. Carminati, K. Joulain, J. Ph. Mulet, S. Mainguy, and Y. Chen, "Coherent emission of light by thermal sources," Nature (London) 416, 61 (2002).
[CrossRef]

Nieto-Vesperinas, M.

J. A. Sanchez-Gil and M. Nieto-Vesperinas, "Light scattering from random rough dielectric surfaces," J. Opt. Soc. Am. A 8, 1270-1286 (1991).
[CrossRef]

M. Nieto-Vesperinas and J. M. Soto-Crespo, "Light-diffracted intensities from very deep gratings," Phys. Rev. B 38, 7250-7259 (1988).
[CrossRef]

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, 1991).

Ogilvy, J. A.

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, 1991).

Sanchez-Gil, J. A.

Sant, A. J.

Sassi, I.

F. Ghmari, I. Sassi, and M. S. Sifaoui, "Directional hemispherical radiative properties of random dielectric rough surfaces," Waves Random Media 15, 469-486 (2005).
[CrossRef]

Sifaoui, M. S.

F. Ghmari, I. Sassi, and M. S. Sifaoui, "Directional hemispherical radiative properties of random dielectric rough surfaces," Waves Random Media 15, 469-486 (2005).
[CrossRef]

M. B. Matchiane, F. Ghmari, and M. S. Sifaoui, "Correction de l"approximation de Kirchhoff par la méthode intégrale reformulée: Cas des réflectivités de surfaces sinusoïdales," Can. J. Phys. 82, 303-321 (2004).
[CrossRef]

Soto-Crespo, J. M.

M. Nieto-Vesperinas and J. M. Soto-Crespo, "Light-diffracted intensities from very deep gratings," Phys. Rev. B 38, 7250-7259 (1988).
[CrossRef]

Spizzichino, A.

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

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

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory, 1st ed. (McGraw-Hill, 1941), pp. 23-131.

Stroud, J. S.

K. L. Williams, J. S. Stroud, and P. L. Marston, "High-frequency forward scattering Gaussian spectrum, pressure release, corrugated surfaces: catastrophe modelling," J. Acoust. Soc. Am. 96, 1687-1702 (1994).
[CrossRef]

Tang, K.

K. Tang and R. O. Buckius, "The geometric optics approximation for reflection from two-dimensional random rough surfaces," Int. J. Heat Mass Transfer 41, 2037-2047 (1998).
[CrossRef]

K. Tang and R. O. Buckius, "Regions of validity of the geometric optics approximation for angular scattering from very rough surfaces," Int. J. Heat Mass Transfer 40, 49-59 (1997).
[CrossRef]

D. W. Cohn, K. Tang, and R. O. Buckius, "Comparison of theory and experiments for reflection of micro contoured Surfaces," Int. J. Heat Mass Transfer 40, 133-149 (1983).

Thorsos, E.

E. Thorsos, "The validity of the Kirchhoff approximation for rough surface scattering using Gaussian roughness spectrum," J. Acoust. Soc. Am. 83, 78-92 (1988).
[CrossRef]

Williams, K. L.

K. L. Williams, J. S. Stroud, and P. L. Marston, "High-frequency forward scattering Gaussian spectrum, pressure release, corrugated surfaces: catastrophe modelling," J. Acoust. Soc. Am. 96, 1687-1702 (1994).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), Sec. I.5.

Yang, Y.

Y. Yang and R. O. Buckius, "Surface length scale contributions to the directional and hemispherical emissivity and reflectivity," J. Thermophys. Heat Transfer 9, 653-659 (1995).
[CrossRef]

Ann. Phys. (N.Y.) (1)

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, "Enhanced scattering of light from a random grating," Ann. Phys. (N.Y.) 203, 255-307 (1990).
[CrossRef]

Can. J. Phys. (1)

M. B. Matchiane, F. Ghmari, and M. S. Sifaoui, "Correction de l"approximation de Kirchhoff par la méthode intégrale reformulée: Cas des réflectivités de surfaces sinusoïdales," Can. J. Phys. 82, 303-321 (2004).
[CrossRef]

Heat Technol. (1)

J.-J. Greffet and R. Carminati, "Radiative transfer at nanometric scale: are the usual concepts still valid?" Heat Technol. 18, 81-85 (2000).

IEEE Trans. Antennas Propag. (4)

R. M. Axline and A. K. Fung, "Numerical computation of scattering from perfectly conducting random surfaces," IEEE Trans. Antennas Propag. 26, 482-488 (1978).
[CrossRef]

R. D. Kodis, "A note on the theory of scattering from an irregular rough surface," IEEE Trans. Antennas Propag. AP-14, 77-82 (1966).
[CrossRef]

D. E. Barrick, "Rough surface scattering based on specular point theory," IEEE Trans. Antennas Propag. AP-16, 449-454 (1968).
[CrossRef]

P. Beckmann, "Scattering by non-Gaussian surfaces," IEEE Trans. Antennas Propag. 21, 169-175 (1973).
[CrossRef]

Int. J. Heat Mass Transfer (3)

K. Tang and R. O. Buckius, "Regions of validity of the geometric optics approximation for angular scattering from very rough surfaces," Int. J. Heat Mass Transfer 40, 49-59 (1997).
[CrossRef]

K. Tang and R. O. Buckius, "The geometric optics approximation for reflection from two-dimensional random rough surfaces," Int. J. Heat Mass Transfer 41, 2037-2047 (1998).
[CrossRef]

D. W. Cohn, K. Tang, and R. O. Buckius, "Comparison of theory and experiments for reflection of micro contoured Surfaces," Int. J. Heat Mass Transfer 40, 133-149 (1983).

J. Acoust. Soc. Am. (2)

K. L. Williams, J. S. Stroud, and P. L. Marston, "High-frequency forward scattering Gaussian spectrum, pressure release, corrugated surfaces: catastrophe modelling," J. Acoust. Soc. Am. 96, 1687-1702 (1994).
[CrossRef]

E. Thorsos, "The validity of the Kirchhoff approximation for rough surface scattering using Gaussian roughness spectrum," J. Acoust. Soc. Am. 83, 78-92 (1988).
[CrossRef]

J. Mod. Opt. (2)

N. C. Bruce and J. C. Dainty, "Multiple scattering from random rough surfaces using the Kirchhoff approximation," J. Mod. Opt. 38, 597-590 (1991).

N. C. Bruce and J. C. Dainty, "Multiple scattering from rough dielectric and metal surfaces using the Kirchhoff approximation," J. Mod. Opt. 38, 1471-1481 (1991).
[CrossRef]

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (1)

J. Thermophys. Heat Transfer (2)

Y. Yang and R. O. Buckius, "Surface length scale contributions to the directional and hemispherical emissivity and reflectivity," J. Thermophys. Heat Transfer 9, 653-659 (1995).
[CrossRef]

R. A. Dimenna and R. O. Buckius, "Quantifying specular approximations for angular scattering from perfectly conducting random rough surfaces," J. Thermophys. Heat Transfer 8, 393-399 (1994).
[CrossRef]

Nature (London) (1)

J. J. Greffet, R. Carminati, K. Joulain, J. Ph. Mulet, S. Mainguy, and Y. Chen, "Coherent emission of light by thermal sources," Nature (London) 416, 61 (2002).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. B (1)

M. Nieto-Vesperinas and J. M. Soto-Crespo, "Light-diffracted intensities from very deep gratings," Phys. Rev. B 38, 7250-7259 (1988).
[CrossRef]

Radio Sci. (2)

M. F. Chen and A. K. Fung, "A numerical study of the regions of validity of the Kirchhoff and small-perturbation rough surface scattering models," Radio Sci. 23, 163-170 (1988).
[CrossRef]

H. L. Chan and A. K. Fung, "A numerical study of the Kirchhoff approximation in horizontally polarized backscattering from a random surface," Radio Sci. 13, 818-881 (1978).
[CrossRef]

Therm. Sci. Eng. (1)

R. A. Dimenna and R. O. Buckius, "Microgeometrical contour contributions to surface scattering," Therm. Sci. Eng. 2, 166-171 (1994).

Waves Random Media (3)

A. Khenchaf, "Bistatic scattering and depolarization by randomly rough surfaces: application to the natural rough surfaces in X-band," Waves Random Media 11, 61-89 (2001).
[CrossRef]

N. C. Bruce, "Calculations of the Mueller matrix for scattering of light from two-dimensional surfaces," Waves Random Media 8, 15-28 (1998).
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F. Ghmari, I. Sassi, and M. S. Sifaoui, "Directional hemispherical radiative properties of random dielectric rough surfaces," Waves Random Media 15, 469-486 (2005).
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P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan, 1963).

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, 1991).

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

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, 1991).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), Sec. I.5.

J. A. Stratton, Electromagnetic Theory, 1st ed. (McGraw-Hill, 1941), pp. 23-131.

M. Q. Brewster, Thermal Radiative Transfer and Properties (Wiley, 1992), Chaps. 2 and 4.

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Figures (11)

Fig. 1
Fig. 1

Realization of surfaces for the same surface parameters τ = 1 λ , σ = 1 λ : (a) z = h 1 ( x ) , (b) z = h 2 ( x ) .

Fig. 2
Fig. 2

Scattered electromagnetic waves.

Fig. 3
Fig. 3

Comparison of exact method (IM) with the GOA for the directional hemispherical reflectivity at θ 0 = 1 ° for periodic rough surfaces h 1 ( x ) with ϵ = 2.04 .

Fig. 4
Fig. 4

Comparison of exact method (IM) with the GOA for the directional hemispherical reflectivity at θ 0 = 1 ° for periodic rough surfaces h 2 ( x ) with ϵ = 2.04 .

Fig. 5
Fig. 5

Regions of validity of the GOA in terms of the reflectivity of two periodic dielectric surfaces with ϵ = 2.04 at θ 0 = 1 ° . (a) h 1 ( x ) , (b) h 2 ( x ) .

Fig. 6
Fig. 6

Regions of validity of the GOA in terms of the reflectivity of two periodic dielectric surfaces with ϵ = 7.5 at incidence angle θ 0 = 1 ° . (a) TM polarization, (b) TE polarization.

Fig. 7
Fig. 7

Regions of validity of the GOA in terms of the directional hemispherical reflection function of periodic dielectric surface z = h 1 ( x ) at incidence angles θ 0 = ( a ) 1°, (b) 60° for two values of permittivity ϵ = 2.04 and ϵ = 7.5 and TE polarization.

Fig. 8
Fig. 8

Regions of validity of the GOA in terms of the reflectivity of periodic and random dielectric surfaces ( ϵ = 2.04 ) at incidence angle θ 0 = 1 ° , Cr lim = 6 % . (a) TM polarization, (b) TE polarization.

Fig. 9
Fig. 9

Regions of validity of the GOA in terms of the reflectivity of periodic and random dielectric surfaces ( ϵ = 7.5 ) at incidence angle θ 0 = 1 ° . (a), (b) TM polarization; (c), (d) TE polarization.

Fig. 10
Fig. 10

Comparison of the regions of validity of the GOA in terms of the reflectivity of periodic and random dielectric surfaces ( ϵ = 2.04 ) at incidence angles θ 0 = 60 ° and θ 0 = 1 ° , Cr lim = 7 % , for TE polarization (a) h 1 ( x ) , (b) R. R. surface.

Fig. 11
Fig. 11

Range of validity of GOA for transmission for TM waves on rough dielectric surfaces at incidence angle θ 0 = 60 ° and Cr lim = 1 % . (a) h 1 ( x ) , (b) R. R. surface.

Equations (35)

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Pr 1 , h 1 ( x ) = σ 4 cos ( 4 π τ ) x σ cos ( 2 π τ ) x ,
Pr 2 , h 2 ( x ) = σ cos ( 4 π τ ) x σ cos ( 2 π τ ) x .
U inc ( x , z ) = exp [ i ( α 0 x β 0 z ) ] ,
2 U 1 ( x , z ) + k 0 2 U 1 ( x , z ) = 0 , z > h ( x ) ,
2 U 2 ( x , z ) + k 0 2 ϵ ( ω ) U 2 ( x , z ) = 0 , z < h ( x ) ,
U 1 ( x , z ) z = h + ( x ) = U 2 ( x , z ) z = h ( x ) ,
U 1 ( x , z ) n z = h + ( x ) = 1 C U 2 ( x , z ) n z = h ( x ) ,
H ( x ) = H 1 ( x , z ) z h + ( x ) ,
L ( x ) = [ h ( x ) x + z ] H 1 ( x , z ) z = h + ( x ) ;
E ( x ) = E 1 ( x , z ) z = h + ( x ) ,
F ( x ) = [ h ( x ) x + z ] E ( 1 ) ( x , z ) z = h + ( x ) .
H ( x ) = H inc [ x , h ( x ) ] + 1 4 π + { H ( x ) [ h ( x ) x + z ] × G 0 ( x , z ; x , z ) z = h ( x ) G 0 ( x , z ; x , z ) z = h ( x ) L ( x ) } d x ,
0 = 1 4 π + { H ( x ) [ h ( x ) x + z ] G ϵ ( x , z ; x , z ) z = h ( x ) ϵ G e ( x , z ; x , z ) z = h ( x ) L ( x ) } d x ,
E ( x ) = E inc [ x , h ( x ) ] + 1 4 π + { E ( x ) [ h ( x ) x + z ] × G 0 ( x , z ; x , z ) z = h ( x ) G 0 ( x , z ; x , z ) z = h ( x ) F ( x ) } d x ,
0 = 1 4 π + { E ( x ) [ h ( x ) x + z ] G ϵ ( x , z ; x , z ) z = h ( x ) G ϵ ( x , z ; x , z ) z = h ( x ) F ( x ) } d x .
ρ λ ( θ 0 , θ r ) = ( π cos θ ) r ( d Φ r d Ω r ) d Φ 0 d Ω 0 ,
τ λ ( θ 0 , θ t ) = ( π cos θ ) ( d Φ t d Ω t ) d Φ 0 d Ω 0 ,
ρ λ , s ( θ 0 , θ r ) = 1 8 k 0 L x cos θ 0 cos θ r r ( s ) ( θ r ) 2 ,
r s ( θ r ) = Δ x n = 1 n = N { j k 0 [ h ( x n ) sin θ r cos θ r ] E ( x n ) F ( x n ) } exp { j k 0 [ x n sin θ r + h ( x n ) cos θ r ] } ;
ρ λ , p ( θ 0 , θ r ) = 1 8 k 0 L x cos θ 0 cos θ r r p ( θ r ) 2 ,
r p ( θ r ) = Δ x n = 1 n = N { j k 0 [ h ( x n ) sin θ r cos θ r ] H ( x n ) L ( x n ) } × exp { j k 0 [ x n sin θ r + h ( x n ) cos θ r ] } .
τ λ , s ( θ 0 , θ t ) = 1 8 k 0 L x cos θ 0 cos θ r t ( s ) ( θ t ) 2 ,
t s ( θ t ) = Δ x n = 1 n = N ( { j k 0 ϵ [ h ( x n ) sin θ t + cos θ t ] E ( x n ) + F ( x n ) } × exp { j ϵ k 0 [ x n sin θ t + h ( x n ) cos θ t ] } ) ;
τ λ , p ( θ 0 , θ t ) = 1 8 k 0 1 ϵ L x cos θ 0 cos θ t t ( p ) ( θ t ) 2 ,
t p ( θ t ) = Δ x n = 1 n = N ( { j k 0 ϵ [ h ( x n ) sin θ t + cos θ t ] H ( x n ) + ϵ L ( x n ) } × exp { j ϵ k 0 [ x n sin θ t + h ( x n ) cos θ t ] } ) .
ρ λ ( θ 0 ) = π 2 π 2 ρ λ ( θ 0 , θ r ) cos θ r d θ r ,
τ λ ( θ 0 ) = π 2 π 2 τ λ ( θ 0 , θ t ) cos θ t d θ t .
ρ λ + τ λ = 1 .
ρ TE , λ ( ξ ) = ( p cos ξ ) 2 ( p + cos ξ ) 2 ,
ρ TM , λ ( ξ ) = [ ( p 2 + sin 2 ξ ) cos ξ p ] 2 [ ( p 2 + sin 2 ξ ) cos ξ + p ] 2 ,
p = ( ϵ sin 2 ξ ) 1 2 .
ρ λ ( θ 0 ) = 1 N j = 1 N [ i = 1 i = N j r λ ( ξ i ) ] .
{ a n = ( 1 ) n cos Φ n b n = ( 1 ) n sin Φ n } ,
z diff , n ( x k ) = z n ( x k ) h ( x k ) ,
φ n = arctan ( a n + b n p f n b n a n p f n ) ,

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