Abstract

We present rigorous, nonperturbative, purely numerical solutions of the Rayleigh equations for the scattering of p- and s-polarized light from a two-dimensional randomly rough perfectly conducting surface. The solutions are used to calculate the reflectivity of the surface, the mean differential reflection coefficients, and the full angular distribution of the intensity of the scattered field. These results are compared with previously published rigorous numerical solutions of the Stratton–Chu equations, and very good agreement is found.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. Tsang, C. Chan, and K. Pak, “Monte Carlo simulation of a two-dimensional random rough surface using the sparse-matrix flat-surface iterative approach,” Electron. Lett. 29, 1153–1154 (1993).
    [CrossRef]
  2. L. Tsang, C. H. Chan, and K. Pak, “Backscattering enhancement of a two-dimensional random rough surface (three-dimensional scattering) based on Monte Carlo simulations,” J. Opt. Soc. Am. A 11, 711–715 (1994).
    [CrossRef]
  3. K. Pak, L. Tsang, C. H. Chan, and J. Johnson, “Backscattering enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces based on Monte Carlo simulations,” J. Opt. Soc. Am. A 12, 2491–2499 (1995).
    [CrossRef]
  4. J. T. Johnson, L. Tsang, R. T. Shin, K. Pak, C. H. Chan, A. Ishimaru, and Y. Kuga, “Backscattering enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces: a comparison of Monte Carlo simulations with experimental data,” IEEE Trans. Antennas Propag. 44, 748–756 (1996).
    [CrossRef]
  5. D. Torrungrueng, H.-T. Chou, and J. Johnson, “A novel acceleration algorithm for the computation of scattering from two-dimensional large-scale perfectly conducting random rough surfaces with the forward-backward method,” IEEE Trans. Geosci. Remote Sens. 38, 1656–1668 (2000).
    [CrossRef]
  6. G. Soriano and M. Saillard, “Scattering of electromagnetic waves from two-dimensional rough surfaces with an impedance approximation,” J. Opt. Soc. Am. A 18, 124–133 (2001).
    [CrossRef]
  7. P. Tran, V. Celli, and A. A. Maradudin, “Electromagnetic scattering from a two-dimensional, randomly rough, perfectly conducting surface: iterative methods,” J. Opt. Soc. Am. A 11, 1686–1689 (1994).
    [CrossRef]
  8. P. Tran, “Calculation of the scattering of electromagnetic waves from a two-dimensional perfectly conducting surface using the method of ordered multiple interaction,” Waves Random Media 7, 295–302 (1997).
    [CrossRef]
  9. R. Wagner, J. Song, and W. Chew, “Monte Carlo simulation of electromagnetic scattering from two-dimensional random rough surfaces,” IEEE Trans. Antennas Propag. 45, 235–245 (1997).
    [CrossRef]
  10. I. Simonsen, A. A. Maradudin, and T. A. Leskova, “Scattering of electromagnetic waves from two-dimensional randomly rough perfectly conducting surfaces: the full angular intensity distribution,” Phys. Rev. A 81, 013806 (2010).
    [CrossRef]
  11. I. Simonsen, A. A. Maradudin, and T. Leskova, “The scattering of electromagnetic waves from two-dimensional randomly rough penetrable surfaces,” Phys. Rev. Lett. 104, 223904 (2010).
    [CrossRef]
  12. A. Madrazo and A. A. Maradudin, “Numerical solutions of the reduced Rayleigh equation for the scattering of electromagnetic waves from rough dielectric films on perfectly conducting substrates,” Opt. Commun. 134, 251–263 (1997).
    [CrossRef]
  13. I. Simonsen and A. A. Maradudin, “Numerical simulation of electromagnetic wave scattering from planar dielectric films deposited on rough perfectly conducting substrates,” Opt. Commun. 162, 99–111 (1999).
    [CrossRef]
  14. K. A. O’Donnell and E. R. Mendéz, “Enhanced specular peaks in diffuse light scattering from weakly rough metal surfaces,” J. Opt. Soc. Am. A 20, 2338–2346 (2003).
    [CrossRef]
  15. I. Simonsen, “Enhanced back and forward scattering in the reflection of light from weakly rough random metal surfaces,” Phys. Status Solidi B 247, 2075–2083 (2010).
    [CrossRef]
  16. I. Simonsen, “Optics of surface disordered systems: a random walk through rough surface scattering phenomena,” Eur. J. Phys. Spec. Top. 181, 1–103 (2010).
    [CrossRef]
  17. T. M. Elfouhaily and C.-A. Guérin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14, R1–R40 (2004).
    [CrossRef]
  18. J. A. Kong, Electromagnetic Wave Theory (EMW, 2005).
  19. T. A. Leskova, P. A. Letnes, A. A. Maradudin, T. Nordam, and I. Simonsen, “The scattering of light from two-dimensional randomly rough surfaces,” Proc. SPIE 8172, 817209 (2011).
    [CrossRef]
  20. T. Nordam, P. A. Letnes, I. Simonsen, and A. A. Maradudin, “Satellite peaks in the scattering of light from the two-dimensional randomly rough surface of a dielectric film on a planar metal surface,” Opt. Express 20, 11336–11350 (2012).
    [CrossRef]
  21. P. A. Letnes, A. A. Maradudin, T. Nordam, and I. Simonsen, “Calculation of the Mueller matrix for scattering of light from two-dimensional rough surfaces,” Phys. Rev. A 86, 031803 (2012).
    [CrossRef]
  22. P. A. Letnes, T. Nordam, and I. Simonsen, “Coherent effects in the scattering of light from two-dimensional rough metal surfaces,” J. Opt. Soc. Am. A 30, 1136–1145 (2013).
    [CrossRef]
  23. A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
    [CrossRef]
  24. I. Simonsen, J. B. Kryvi, A. A. Maradudin, and T. A. Leskova, “Light scattering from anisotropic, randomly rough, perfectly conducting surfaces,” Comput. Phys. Commun. 182, 1904–1908 (2011).
    [CrossRef]
  25. T. Nordam, P. Letnes, and I. Simonsen, “Numerical simulations of scattering of light from two-dimensional surfaces using the reduced Rayleigh equation,” Front. Phys. 1, 1–15 (2013).
    [CrossRef]
  26. W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).
  27. “ScaLAPACK: Scalable linear algebra package,” www.netlib.org/scalapack/ (2012).
  28. Intel Math Kernel Library, http://software.intel.com/en-us/intel-mkl .
  29. From Fig. 1 it should be observed that the unit vector in the ϕ0 direction, ϕ^0, defines a vector normal to the plane of incidence. This way of defining the plane of incidence has the additional advantage of also being well-defined for normal incidence, and we will here assume this definition. Therefore when we for normal incidence say that the incident electromagnetic field is p polarized, it means that the electric field vector Ep(x|ω)inc lies in the x1x2 plane, but with a direction so that ϕ^0⊥Ep(x|ω)inc. Similarly, for an s-polarized field incident normally onto the surface we have ϕ^0∥Es(x|ω)inc.
  30. A. R. McGurn, A. A. Maradudin, and V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
    [CrossRef]
  31. C. S. West and K. A. O’Donnell, “Observations of backscattering enhancement from polaritons on a rough metal surface,” J. Opt. Soc. Am. A 12, 390–397 (1995).
    [CrossRef]
  32. S. Y. Kim and K. Vedam, “Analytic solution of the pseudo-Brewster angle,” J. Opt. Soc. Am. A 3, 1772–1773 (1986).
    [CrossRef]
  33. R. M. A. Azzam, “Complex reflection coefficients of p- and s-polarized light at the pseudo-Brewster angle of a dielectric–conductor interface,” J. Opt. Soc. Am. A 30, 1975–1979 (2013).
    [CrossRef]
  34. A. G. Voronovich, “Rayleigh hypothesis,” in Light Scattering and Nanoscale Surface Roughness (Springer, 2007), pp. 93–106.
  35. T. Nordam, P. A. Letnes, and I. Simonsen, “Validity of the Rayleigh hypothesis for two-dimensional randomly rough metal surfaces,” J. Phys.: Conf. Ser. 454, 012033 (2013).
    [CrossRef]
  36. A. V. Tishchenko, “Numerical demonstration of the validity of the Rayleigh hypothesis,” Opt. Express 17, 17102–17117 (2009).
    [CrossRef]

2013

P. A. Letnes, T. Nordam, and I. Simonsen, “Coherent effects in the scattering of light from two-dimensional rough metal surfaces,” J. Opt. Soc. Am. A 30, 1136–1145 (2013).
[CrossRef]

T. Nordam, P. Letnes, and I. Simonsen, “Numerical simulations of scattering of light from two-dimensional surfaces using the reduced Rayleigh equation,” Front. Phys. 1, 1–15 (2013).
[CrossRef]

R. M. A. Azzam, “Complex reflection coefficients of p- and s-polarized light at the pseudo-Brewster angle of a dielectric–conductor interface,” J. Opt. Soc. Am. A 30, 1975–1979 (2013).
[CrossRef]

T. Nordam, P. A. Letnes, and I. Simonsen, “Validity of the Rayleigh hypothesis for two-dimensional randomly rough metal surfaces,” J. Phys.: Conf. Ser. 454, 012033 (2013).
[CrossRef]

2012

T. Nordam, P. A. Letnes, I. Simonsen, and A. A. Maradudin, “Satellite peaks in the scattering of light from the two-dimensional randomly rough surface of a dielectric film on a planar metal surface,” Opt. Express 20, 11336–11350 (2012).
[CrossRef]

P. A. Letnes, A. A. Maradudin, T. Nordam, and I. Simonsen, “Calculation of the Mueller matrix for scattering of light from two-dimensional rough surfaces,” Phys. Rev. A 86, 031803 (2012).
[CrossRef]

2011

I. Simonsen, J. B. Kryvi, A. A. Maradudin, and T. A. Leskova, “Light scattering from anisotropic, randomly rough, perfectly conducting surfaces,” Comput. Phys. Commun. 182, 1904–1908 (2011).
[CrossRef]

T. A. Leskova, P. A. Letnes, A. A. Maradudin, T. Nordam, and I. Simonsen, “The scattering of light from two-dimensional randomly rough surfaces,” Proc. SPIE 8172, 817209 (2011).
[CrossRef]

2010

I. Simonsen, “Enhanced back and forward scattering in the reflection of light from weakly rough random metal surfaces,” Phys. Status Solidi B 247, 2075–2083 (2010).
[CrossRef]

I. Simonsen, “Optics of surface disordered systems: a random walk through rough surface scattering phenomena,” Eur. J. Phys. Spec. Top. 181, 1–103 (2010).
[CrossRef]

I. Simonsen, A. A. Maradudin, and T. A. Leskova, “Scattering of electromagnetic waves from two-dimensional randomly rough perfectly conducting surfaces: the full angular intensity distribution,” Phys. Rev. A 81, 013806 (2010).
[CrossRef]

I. Simonsen, A. A. Maradudin, and T. Leskova, “The scattering of electromagnetic waves from two-dimensional randomly rough penetrable surfaces,” Phys. Rev. Lett. 104, 223904 (2010).
[CrossRef]

2009

2004

T. M. Elfouhaily and C.-A. Guérin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14, R1–R40 (2004).
[CrossRef]

2003

2001

2000

D. Torrungrueng, H.-T. Chou, and J. Johnson, “A novel acceleration algorithm for the computation of scattering from two-dimensional large-scale perfectly conducting random rough surfaces with the forward-backward method,” IEEE Trans. Geosci. Remote Sens. 38, 1656–1668 (2000).
[CrossRef]

1999

I. Simonsen and A. A. Maradudin, “Numerical simulation of electromagnetic wave scattering from planar dielectric films deposited on rough perfectly conducting substrates,” Opt. Commun. 162, 99–111 (1999).
[CrossRef]

1997

A. Madrazo and A. A. Maradudin, “Numerical solutions of the reduced Rayleigh equation for the scattering of electromagnetic waves from rough dielectric films on perfectly conducting substrates,” Opt. Commun. 134, 251–263 (1997).
[CrossRef]

P. Tran, “Calculation of the scattering of electromagnetic waves from a two-dimensional perfectly conducting surface using the method of ordered multiple interaction,” Waves Random Media 7, 295–302 (1997).
[CrossRef]

R. Wagner, J. Song, and W. Chew, “Monte Carlo simulation of electromagnetic scattering from two-dimensional random rough surfaces,” IEEE Trans. Antennas Propag. 45, 235–245 (1997).
[CrossRef]

1996

J. T. Johnson, L. Tsang, R. T. Shin, K. Pak, C. H. Chan, A. Ishimaru, and Y. Kuga, “Backscattering enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces: a comparison of Monte Carlo simulations with experimental data,” IEEE Trans. Antennas Propag. 44, 748–756 (1996).
[CrossRef]

1995

1994

1993

L. Tsang, C. Chan, and K. Pak, “Monte Carlo simulation of a two-dimensional random rough surface using the sparse-matrix flat-surface iterative approach,” Electron. Lett. 29, 1153–1154 (1993).
[CrossRef]

1990

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

1986

1985

A. R. McGurn, A. A. Maradudin, and V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

Azzam, R. M. A.

Celli, V.

P. Tran, V. Celli, and A. A. Maradudin, “Electromagnetic scattering from a two-dimensional, randomly rough, perfectly conducting surface: iterative methods,” J. Opt. Soc. Am. A 11, 1686–1689 (1994).
[CrossRef]

A. R. McGurn, A. A. Maradudin, and V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

Chan, C.

L. Tsang, C. Chan, and K. Pak, “Monte Carlo simulation of a two-dimensional random rough surface using the sparse-matrix flat-surface iterative approach,” Electron. Lett. 29, 1153–1154 (1993).
[CrossRef]

Chan, C. H.

Chew, W.

R. Wagner, J. Song, and W. Chew, “Monte Carlo simulation of electromagnetic scattering from two-dimensional random rough surfaces,” IEEE Trans. Antennas Propag. 45, 235–245 (1997).
[CrossRef]

Chou, H.-T.

D. Torrungrueng, H.-T. Chou, and J. Johnson, “A novel acceleration algorithm for the computation of scattering from two-dimensional large-scale perfectly conducting random rough surfaces with the forward-backward method,” IEEE Trans. Geosci. Remote Sens. 38, 1656–1668 (2000).
[CrossRef]

Elfouhaily, T. M.

T. M. Elfouhaily and C.-A. Guérin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14, R1–R40 (2004).
[CrossRef]

Flannery, B.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).

Guérin, C.-A.

T. M. Elfouhaily and C.-A. Guérin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14, R1–R40 (2004).
[CrossRef]

Ishimaru, A.

J. T. Johnson, L. Tsang, R. T. Shin, K. Pak, C. H. Chan, A. Ishimaru, and Y. Kuga, “Backscattering enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces: a comparison of Monte Carlo simulations with experimental data,” IEEE Trans. Antennas Propag. 44, 748–756 (1996).
[CrossRef]

Johnson, J.

D. Torrungrueng, H.-T. Chou, and J. Johnson, “A novel acceleration algorithm for the computation of scattering from two-dimensional large-scale perfectly conducting random rough surfaces with the forward-backward method,” IEEE Trans. Geosci. Remote Sens. 38, 1656–1668 (2000).
[CrossRef]

K. Pak, L. Tsang, C. H. Chan, and J. Johnson, “Backscattering enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces based on Monte Carlo simulations,” J. Opt. Soc. Am. A 12, 2491–2499 (1995).
[CrossRef]

Johnson, J. T.

J. T. Johnson, L. Tsang, R. T. Shin, K. Pak, C. H. Chan, A. Ishimaru, and Y. Kuga, “Backscattering enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces: a comparison of Monte Carlo simulations with experimental data,” IEEE Trans. Antennas Propag. 44, 748–756 (1996).
[CrossRef]

Kim, S. Y.

Kong, J. A.

J. A. Kong, Electromagnetic Wave Theory (EMW, 2005).

Kryvi, J. B.

I. Simonsen, J. B. Kryvi, A. A. Maradudin, and T. A. Leskova, “Light scattering from anisotropic, randomly rough, perfectly conducting surfaces,” Comput. Phys. Commun. 182, 1904–1908 (2011).
[CrossRef]

Kuga, Y.

J. T. Johnson, L. Tsang, R. T. Shin, K. Pak, C. H. Chan, A. Ishimaru, and Y. Kuga, “Backscattering enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces: a comparison of Monte Carlo simulations with experimental data,” IEEE Trans. Antennas Propag. 44, 748–756 (1996).
[CrossRef]

Leskova, T.

I. Simonsen, A. A. Maradudin, and T. Leskova, “The scattering of electromagnetic waves from two-dimensional randomly rough penetrable surfaces,” Phys. Rev. Lett. 104, 223904 (2010).
[CrossRef]

Leskova, T. A.

T. A. Leskova, P. A. Letnes, A. A. Maradudin, T. Nordam, and I. Simonsen, “The scattering of light from two-dimensional randomly rough surfaces,” Proc. SPIE 8172, 817209 (2011).
[CrossRef]

I. Simonsen, J. B. Kryvi, A. A. Maradudin, and T. A. Leskova, “Light scattering from anisotropic, randomly rough, perfectly conducting surfaces,” Comput. Phys. Commun. 182, 1904–1908 (2011).
[CrossRef]

I. Simonsen, A. A. Maradudin, and T. A. Leskova, “Scattering of electromagnetic waves from two-dimensional randomly rough perfectly conducting surfaces: the full angular intensity distribution,” Phys. Rev. A 81, 013806 (2010).
[CrossRef]

Letnes, P.

T. Nordam, P. Letnes, and I. Simonsen, “Numerical simulations of scattering of light from two-dimensional surfaces using the reduced Rayleigh equation,” Front. Phys. 1, 1–15 (2013).
[CrossRef]

Letnes, P. A.

T. Nordam, P. A. Letnes, and I. Simonsen, “Validity of the Rayleigh hypothesis for two-dimensional randomly rough metal surfaces,” J. Phys.: Conf. Ser. 454, 012033 (2013).
[CrossRef]

P. A. Letnes, T. Nordam, and I. Simonsen, “Coherent effects in the scattering of light from two-dimensional rough metal surfaces,” J. Opt. Soc. Am. A 30, 1136–1145 (2013).
[CrossRef]

T. Nordam, P. A. Letnes, I. Simonsen, and A. A. Maradudin, “Satellite peaks in the scattering of light from the two-dimensional randomly rough surface of a dielectric film on a planar metal surface,” Opt. Express 20, 11336–11350 (2012).
[CrossRef]

P. A. Letnes, A. A. Maradudin, T. Nordam, and I. Simonsen, “Calculation of the Mueller matrix for scattering of light from two-dimensional rough surfaces,” Phys. Rev. A 86, 031803 (2012).
[CrossRef]

T. A. Leskova, P. A. Letnes, A. A. Maradudin, T. Nordam, and I. Simonsen, “The scattering of light from two-dimensional randomly rough surfaces,” Proc. SPIE 8172, 817209 (2011).
[CrossRef]

Madrazo, A.

A. Madrazo and A. A. Maradudin, “Numerical solutions of the reduced Rayleigh equation for the scattering of electromagnetic waves from rough dielectric films on perfectly conducting substrates,” Opt. Commun. 134, 251–263 (1997).
[CrossRef]

Maradudin, A. A.

P. A. Letnes, A. A. Maradudin, T. Nordam, and I. Simonsen, “Calculation of the Mueller matrix for scattering of light from two-dimensional rough surfaces,” Phys. Rev. A 86, 031803 (2012).
[CrossRef]

T. Nordam, P. A. Letnes, I. Simonsen, and A. A. Maradudin, “Satellite peaks in the scattering of light from the two-dimensional randomly rough surface of a dielectric film on a planar metal surface,” Opt. Express 20, 11336–11350 (2012).
[CrossRef]

I. Simonsen, J. B. Kryvi, A. A. Maradudin, and T. A. Leskova, “Light scattering from anisotropic, randomly rough, perfectly conducting surfaces,” Comput. Phys. Commun. 182, 1904–1908 (2011).
[CrossRef]

T. A. Leskova, P. A. Letnes, A. A. Maradudin, T. Nordam, and I. Simonsen, “The scattering of light from two-dimensional randomly rough surfaces,” Proc. SPIE 8172, 817209 (2011).
[CrossRef]

I. Simonsen, A. A. Maradudin, and T. A. Leskova, “Scattering of electromagnetic waves from two-dimensional randomly rough perfectly conducting surfaces: the full angular intensity distribution,” Phys. Rev. A 81, 013806 (2010).
[CrossRef]

I. Simonsen, A. A. Maradudin, and T. Leskova, “The scattering of electromagnetic waves from two-dimensional randomly rough penetrable surfaces,” Phys. Rev. Lett. 104, 223904 (2010).
[CrossRef]

I. Simonsen and A. A. Maradudin, “Numerical simulation of electromagnetic wave scattering from planar dielectric films deposited on rough perfectly conducting substrates,” Opt. Commun. 162, 99–111 (1999).
[CrossRef]

A. Madrazo and A. A. Maradudin, “Numerical solutions of the reduced Rayleigh equation for the scattering of electromagnetic waves from rough dielectric films on perfectly conducting substrates,” Opt. Commun. 134, 251–263 (1997).
[CrossRef]

P. Tran, V. Celli, and A. A. Maradudin, “Electromagnetic scattering from a two-dimensional, randomly rough, perfectly conducting surface: iterative methods,” J. Opt. Soc. Am. A 11, 1686–1689 (1994).
[CrossRef]

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

A. R. McGurn, A. A. Maradudin, and V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

McGurn, A. R.

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

A. R. McGurn, A. A. Maradudin, and V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

Mendéz, E. R.

Méndez, E. R.

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

Michel, T.

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

Nordam, T.

P. A. Letnes, T. Nordam, and I. Simonsen, “Coherent effects in the scattering of light from two-dimensional rough metal surfaces,” J. Opt. Soc. Am. A 30, 1136–1145 (2013).
[CrossRef]

T. Nordam, P. Letnes, and I. Simonsen, “Numerical simulations of scattering of light from two-dimensional surfaces using the reduced Rayleigh equation,” Front. Phys. 1, 1–15 (2013).
[CrossRef]

T. Nordam, P. A. Letnes, and I. Simonsen, “Validity of the Rayleigh hypothesis for two-dimensional randomly rough metal surfaces,” J. Phys.: Conf. Ser. 454, 012033 (2013).
[CrossRef]

T. Nordam, P. A. Letnes, I. Simonsen, and A. A. Maradudin, “Satellite peaks in the scattering of light from the two-dimensional randomly rough surface of a dielectric film on a planar metal surface,” Opt. Express 20, 11336–11350 (2012).
[CrossRef]

P. A. Letnes, A. A. Maradudin, T. Nordam, and I. Simonsen, “Calculation of the Mueller matrix for scattering of light from two-dimensional rough surfaces,” Phys. Rev. A 86, 031803 (2012).
[CrossRef]

T. A. Leskova, P. A. Letnes, A. A. Maradudin, T. Nordam, and I. Simonsen, “The scattering of light from two-dimensional randomly rough surfaces,” Proc. SPIE 8172, 817209 (2011).
[CrossRef]

O’Donnell, K. A.

Pak, K.

J. T. Johnson, L. Tsang, R. T. Shin, K. Pak, C. H. Chan, A. Ishimaru, and Y. Kuga, “Backscattering enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces: a comparison of Monte Carlo simulations with experimental data,” IEEE Trans. Antennas Propag. 44, 748–756 (1996).
[CrossRef]

K. Pak, L. Tsang, C. H. Chan, and J. Johnson, “Backscattering enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces based on Monte Carlo simulations,” J. Opt. Soc. Am. A 12, 2491–2499 (1995).
[CrossRef]

L. Tsang, C. H. Chan, and K. Pak, “Backscattering enhancement of a two-dimensional random rough surface (three-dimensional scattering) based on Monte Carlo simulations,” J. Opt. Soc. Am. A 11, 711–715 (1994).
[CrossRef]

L. Tsang, C. Chan, and K. Pak, “Monte Carlo simulation of a two-dimensional random rough surface using the sparse-matrix flat-surface iterative approach,” Electron. Lett. 29, 1153–1154 (1993).
[CrossRef]

Press, W.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).

Saillard, M.

Shin, R. T.

J. T. Johnson, L. Tsang, R. T. Shin, K. Pak, C. H. Chan, A. Ishimaru, and Y. Kuga, “Backscattering enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces: a comparison of Monte Carlo simulations with experimental data,” IEEE Trans. Antennas Propag. 44, 748–756 (1996).
[CrossRef]

Simonsen, I.

P. A. Letnes, T. Nordam, and I. Simonsen, “Coherent effects in the scattering of light from two-dimensional rough metal surfaces,” J. Opt. Soc. Am. A 30, 1136–1145 (2013).
[CrossRef]

T. Nordam, P. Letnes, and I. Simonsen, “Numerical simulations of scattering of light from two-dimensional surfaces using the reduced Rayleigh equation,” Front. Phys. 1, 1–15 (2013).
[CrossRef]

T. Nordam, P. A. Letnes, and I. Simonsen, “Validity of the Rayleigh hypothesis for two-dimensional randomly rough metal surfaces,” J. Phys.: Conf. Ser. 454, 012033 (2013).
[CrossRef]

P. A. Letnes, A. A. Maradudin, T. Nordam, and I. Simonsen, “Calculation of the Mueller matrix for scattering of light from two-dimensional rough surfaces,” Phys. Rev. A 86, 031803 (2012).
[CrossRef]

T. Nordam, P. A. Letnes, I. Simonsen, and A. A. Maradudin, “Satellite peaks in the scattering of light from the two-dimensional randomly rough surface of a dielectric film on a planar metal surface,” Opt. Express 20, 11336–11350 (2012).
[CrossRef]

I. Simonsen, J. B. Kryvi, A. A. Maradudin, and T. A. Leskova, “Light scattering from anisotropic, randomly rough, perfectly conducting surfaces,” Comput. Phys. Commun. 182, 1904–1908 (2011).
[CrossRef]

T. A. Leskova, P. A. Letnes, A. A. Maradudin, T. Nordam, and I. Simonsen, “The scattering of light from two-dimensional randomly rough surfaces,” Proc. SPIE 8172, 817209 (2011).
[CrossRef]

I. Simonsen, “Enhanced back and forward scattering in the reflection of light from weakly rough random metal surfaces,” Phys. Status Solidi B 247, 2075–2083 (2010).
[CrossRef]

I. Simonsen, “Optics of surface disordered systems: a random walk through rough surface scattering phenomena,” Eur. J. Phys. Spec. Top. 181, 1–103 (2010).
[CrossRef]

I. Simonsen, A. A. Maradudin, and T. A. Leskova, “Scattering of electromagnetic waves from two-dimensional randomly rough perfectly conducting surfaces: the full angular intensity distribution,” Phys. Rev. A 81, 013806 (2010).
[CrossRef]

I. Simonsen, A. A. Maradudin, and T. Leskova, “The scattering of electromagnetic waves from two-dimensional randomly rough penetrable surfaces,” Phys. Rev. Lett. 104, 223904 (2010).
[CrossRef]

I. Simonsen and A. A. Maradudin, “Numerical simulation of electromagnetic wave scattering from planar dielectric films deposited on rough perfectly conducting substrates,” Opt. Commun. 162, 99–111 (1999).
[CrossRef]

Song, J.

R. Wagner, J. Song, and W. Chew, “Monte Carlo simulation of electromagnetic scattering from two-dimensional random rough surfaces,” IEEE Trans. Antennas Propag. 45, 235–245 (1997).
[CrossRef]

Soriano, G.

Teukolsky, S.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).

Tishchenko, A. V.

Torrungrueng, D.

D. Torrungrueng, H.-T. Chou, and J. Johnson, “A novel acceleration algorithm for the computation of scattering from two-dimensional large-scale perfectly conducting random rough surfaces with the forward-backward method,” IEEE Trans. Geosci. Remote Sens. 38, 1656–1668 (2000).
[CrossRef]

Tran, P.

P. Tran, “Calculation of the scattering of electromagnetic waves from a two-dimensional perfectly conducting surface using the method of ordered multiple interaction,” Waves Random Media 7, 295–302 (1997).
[CrossRef]

P. Tran, V. Celli, and A. A. Maradudin, “Electromagnetic scattering from a two-dimensional, randomly rough, perfectly conducting surface: iterative methods,” J. Opt. Soc. Am. A 11, 1686–1689 (1994).
[CrossRef]

Tsang, L.

J. T. Johnson, L. Tsang, R. T. Shin, K. Pak, C. H. Chan, A. Ishimaru, and Y. Kuga, “Backscattering enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces: a comparison of Monte Carlo simulations with experimental data,” IEEE Trans. Antennas Propag. 44, 748–756 (1996).
[CrossRef]

K. Pak, L. Tsang, C. H. Chan, and J. Johnson, “Backscattering enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces based on Monte Carlo simulations,” J. Opt. Soc. Am. A 12, 2491–2499 (1995).
[CrossRef]

L. Tsang, C. H. Chan, and K. Pak, “Backscattering enhancement of a two-dimensional random rough surface (three-dimensional scattering) based on Monte Carlo simulations,” J. Opt. Soc. Am. A 11, 711–715 (1994).
[CrossRef]

L. Tsang, C. Chan, and K. Pak, “Monte Carlo simulation of a two-dimensional random rough surface using the sparse-matrix flat-surface iterative approach,” Electron. Lett. 29, 1153–1154 (1993).
[CrossRef]

Vedam, K.

Vetterling, W.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).

Voronovich, A. G.

A. G. Voronovich, “Rayleigh hypothesis,” in Light Scattering and Nanoscale Surface Roughness (Springer, 2007), pp. 93–106.

Wagner, R.

R. Wagner, J. Song, and W. Chew, “Monte Carlo simulation of electromagnetic scattering from two-dimensional random rough surfaces,” IEEE Trans. Antennas Propag. 45, 235–245 (1997).
[CrossRef]

West, C. S.

Ann. Phys.

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

Comput. Phys. Commun.

I. Simonsen, J. B. Kryvi, A. A. Maradudin, and T. A. Leskova, “Light scattering from anisotropic, randomly rough, perfectly conducting surfaces,” Comput. Phys. Commun. 182, 1904–1908 (2011).
[CrossRef]

Electron. Lett.

L. Tsang, C. Chan, and K. Pak, “Monte Carlo simulation of a two-dimensional random rough surface using the sparse-matrix flat-surface iterative approach,” Electron. Lett. 29, 1153–1154 (1993).
[CrossRef]

Eur. J. Phys. Spec. Top.

I. Simonsen, “Optics of surface disordered systems: a random walk through rough surface scattering phenomena,” Eur. J. Phys. Spec. Top. 181, 1–103 (2010).
[CrossRef]

Front. Phys.

T. Nordam, P. Letnes, and I. Simonsen, “Numerical simulations of scattering of light from two-dimensional surfaces using the reduced Rayleigh equation,” Front. Phys. 1, 1–15 (2013).
[CrossRef]

IEEE Trans. Antennas Propag.

R. Wagner, J. Song, and W. Chew, “Monte Carlo simulation of electromagnetic scattering from two-dimensional random rough surfaces,” IEEE Trans. Antennas Propag. 45, 235–245 (1997).
[CrossRef]

J. T. Johnson, L. Tsang, R. T. Shin, K. Pak, C. H. Chan, A. Ishimaru, and Y. Kuga, “Backscattering enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces: a comparison of Monte Carlo simulations with experimental data,” IEEE Trans. Antennas Propag. 44, 748–756 (1996).
[CrossRef]

IEEE Trans. Geosci. Remote Sens.

D. Torrungrueng, H.-T. Chou, and J. Johnson, “A novel acceleration algorithm for the computation of scattering from two-dimensional large-scale perfectly conducting random rough surfaces with the forward-backward method,” IEEE Trans. Geosci. Remote Sens. 38, 1656–1668 (2000).
[CrossRef]

J. Opt. Soc. Am. A

G. Soriano and M. Saillard, “Scattering of electromagnetic waves from two-dimensional rough surfaces with an impedance approximation,” J. Opt. Soc. Am. A 18, 124–133 (2001).
[CrossRef]

P. Tran, V. Celli, and A. A. Maradudin, “Electromagnetic scattering from a two-dimensional, randomly rough, perfectly conducting surface: iterative methods,” J. Opt. Soc. Am. A 11, 1686–1689 (1994).
[CrossRef]

L. Tsang, C. H. Chan, and K. Pak, “Backscattering enhancement of a two-dimensional random rough surface (three-dimensional scattering) based on Monte Carlo simulations,” J. Opt. Soc. Am. A 11, 711–715 (1994).
[CrossRef]

K. Pak, L. Tsang, C. H. Chan, and J. Johnson, “Backscattering enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces based on Monte Carlo simulations,” J. Opt. Soc. Am. A 12, 2491–2499 (1995).
[CrossRef]

K. A. O’Donnell and E. R. Mendéz, “Enhanced specular peaks in diffuse light scattering from weakly rough metal surfaces,” J. Opt. Soc. Am. A 20, 2338–2346 (2003).
[CrossRef]

C. S. West and K. A. O’Donnell, “Observations of backscattering enhancement from polaritons on a rough metal surface,” J. Opt. Soc. Am. A 12, 390–397 (1995).
[CrossRef]

S. Y. Kim and K. Vedam, “Analytic solution of the pseudo-Brewster angle,” J. Opt. Soc. Am. A 3, 1772–1773 (1986).
[CrossRef]

R. M. A. Azzam, “Complex reflection coefficients of p- and s-polarized light at the pseudo-Brewster angle of a dielectric–conductor interface,” J. Opt. Soc. Am. A 30, 1975–1979 (2013).
[CrossRef]

P. A. Letnes, T. Nordam, and I. Simonsen, “Coherent effects in the scattering of light from two-dimensional rough metal surfaces,” J. Opt. Soc. Am. A 30, 1136–1145 (2013).
[CrossRef]

J. Phys.: Conf. Ser.

T. Nordam, P. A. Letnes, and I. Simonsen, “Validity of the Rayleigh hypothesis for two-dimensional randomly rough metal surfaces,” J. Phys.: Conf. Ser. 454, 012033 (2013).
[CrossRef]

Opt. Commun.

A. Madrazo and A. A. Maradudin, “Numerical solutions of the reduced Rayleigh equation for the scattering of electromagnetic waves from rough dielectric films on perfectly conducting substrates,” Opt. Commun. 134, 251–263 (1997).
[CrossRef]

I. Simonsen and A. A. Maradudin, “Numerical simulation of electromagnetic wave scattering from planar dielectric films deposited on rough perfectly conducting substrates,” Opt. Commun. 162, 99–111 (1999).
[CrossRef]

Opt. Express

Phys. Rev. A

P. A. Letnes, A. A. Maradudin, T. Nordam, and I. Simonsen, “Calculation of the Mueller matrix for scattering of light from two-dimensional rough surfaces,” Phys. Rev. A 86, 031803 (2012).
[CrossRef]

I. Simonsen, A. A. Maradudin, and T. A. Leskova, “Scattering of electromagnetic waves from two-dimensional randomly rough perfectly conducting surfaces: the full angular intensity distribution,” Phys. Rev. A 81, 013806 (2010).
[CrossRef]

Phys. Rev. B

A. R. McGurn, A. A. Maradudin, and V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

Phys. Rev. Lett.

I. Simonsen, A. A. Maradudin, and T. Leskova, “The scattering of electromagnetic waves from two-dimensional randomly rough penetrable surfaces,” Phys. Rev. Lett. 104, 223904 (2010).
[CrossRef]

Phys. Status Solidi B

I. Simonsen, “Enhanced back and forward scattering in the reflection of light from weakly rough random metal surfaces,” Phys. Status Solidi B 247, 2075–2083 (2010).
[CrossRef]

Proc. SPIE

T. A. Leskova, P. A. Letnes, A. A. Maradudin, T. Nordam, and I. Simonsen, “The scattering of light from two-dimensional randomly rough surfaces,” Proc. SPIE 8172, 817209 (2011).
[CrossRef]

Waves Random Media

T. M. Elfouhaily and C.-A. Guérin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14, R1–R40 (2004).
[CrossRef]

P. Tran, “Calculation of the scattering of electromagnetic waves from a two-dimensional perfectly conducting surface using the method of ordered multiple interaction,” Waves Random Media 7, 295–302 (1997).
[CrossRef]

Other

J. A. Kong, Electromagnetic Wave Theory (EMW, 2005).

A. G. Voronovich, “Rayleigh hypothesis,” in Light Scattering and Nanoscale Surface Roughness (Springer, 2007), pp. 93–106.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).

“ScaLAPACK: Scalable linear algebra package,” www.netlib.org/scalapack/ (2012).

Intel Math Kernel Library, http://software.intel.com/en-us/intel-mkl .

From Fig. 1 it should be observed that the unit vector in the ϕ0 direction, ϕ^0, defines a vector normal to the plane of incidence. This way of defining the plane of incidence has the additional advantage of also being well-defined for normal incidence, and we will here assume this definition. Therefore when we for normal incidence say that the incident electromagnetic field is p polarized, it means that the electric field vector Ep(x|ω)inc lies in the x1x2 plane, but with a direction so that ϕ^0⊥Ep(x|ω)inc. Similarly, for an s-polarized field incident normally onto the surface we have ϕ^0∥Es(x|ω)inc.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1.
Fig. 1.

Scattering geometry used in this paper.

Fig. 2.
Fig. 2.

Angular dependence of the incoherent component of the mean differential reflection coefficients for angles of incidence (θ0,ϕ0)=(0°,45°) and various combinations of the polarizations of the incident and scattered light. The results were obtained by numerically solving the Rayleigh equations [Eq. (17)]. The surface roughness had both a Gaussian height distribution and height–height correlation function, as defined by Eqs. (1) and (2), with δ=λ/10 and a=3λ/5, where λ is the wavelength of the incident radiation (in vacuum). An ensemble of 5000 surface realizations was used to produce these results. Both edges of the surfaces were L=15λ, and the surface was discretized at 321×321 points.

Fig. 3.
Fig. 3.

Same as Fig. 2, but now for the angles of incidence (θ0,ϕ0)=(22°,45°).

Fig. 4.
Fig. 4.

Reflectivity, Rα(θ0), as defined in Eq. (30), as a function of polar angle of incidence θ0 and for given polarization α of the incident light. The roughness parameters assumed in obtaining these results are identical to those assumed in producing the results of Fig. 2, but the Rayleigh equation has here been solved for a set of 13 slightly different surface lengths, in order to achieve the desired resolution in the angles close to 90°.

Fig. 5.
Fig. 5.

Unitarity, U(θ0,ϕ0)=[Up(θ0,ϕ0)+Us(θ0,ϕ0)]/2, as a function of the correlation length a, or, equivalently, the rms slope s of the surface, for the angles of incidence (θ0,ϕ0)=(0°,45°). Since the surfaces used to produce these results had both Gaussian height distributions and Gaussian correlation functions, the rms slope of the surfaces is given by s=2δ/a [16,23]. The inset depicts a semi-logarithmic scale |U1| as a function of correlation length in order to better identify the deviation of U from unity. In the simulations performed to produce the data presented in this figure, we kept the rms height of the surface fixed at δ=λ/10 while varying its transverse correlation length a. For each set of roughness parameters the results were averaged over an ensemble consisting of 20 surface realizations, which was sufficient to produce converged results. The remaining parameters used in obtaining these results are given in the caption of Fig. 2.

Fig. 6.
Fig. 6.

Comparison of the results for the incoherent component of the mean differential reflection coefficient as obtained by numerically solving the Rayleigh equations (lines) and those resulting from applying the (rigorous) Green’s function surface integral method [10] (open symbols) to the same scattering system. Both the co- and cross-polarized components of the mean differential reflection coefficients coming from the light that has been scattered incoherently by the surface are presented, but only their in-plane (left panel) and out-of-plane (right panel) dependencies are shown. The incident light was p polarized, and the remaining roughness parameters were identical to those assumed in producing the results of Fig. 2. The curves corresponding to the Rayleigh equation approach were actually obtained by making appropriate cuts through the corresponding angular intensity distributions of Fig. 2. When applying the Green’s function surface integral approach, the simulation parameters were identical to those used in the Rayleigh approach (see caption of Fig. 2) with the following exceptions: the spatial discretization interval was Δx=2λ/17, the incident beam had a finite size of full width w=4λ, and the number of surface realizations used was Np=10000.

Fig. 7.
Fig. 7.

Same as Fig. 6, but now the incident light is s polarized and the polar angle of incidence is θ0=22°.

Tables (1)

Tables Icon

Table 1. Values for Uβ(θ0,ϕ0) and Uβincoh(θ0,ϕ0) as Defined by Eqs. (32) and (33) for the Simulation Results Presented in Figs. 2 and 3

Equations (40)

Equations on this page are rendered with MathJax. Learn more.

ζ(x)ζ(x)=δ2W(|xx|).
W(|x|)=exp(x2/a2),
E(x;t)=[E(x|ω)inc+E(x|ω)sc]exp(iωt),
E(x|ω)inc=E0(k)exp[ik·xiα0(k)x3],
E(x|ω)sc=d2q(2π)2A(q)×exp[iq·x+iα0(q)x3].
E0(k)=cω[k^α0(k)+x^3k]E0p(k)+(x^3×k^)E0s(k),
A(q)=cω[q^α0(q)x^3q]Ap(q)+(x^3×q^)As(q).
n×[E(x|ω)inc+E(x|ω)sc]|x3=ζ(x)=0,
n=[ζ1(x),ζ2(x),1],
{n×[E(x|ω)inc+E(x|ω)sc]|x3=ζ(x)}1=0,
{n×[E(x|ω)inc+E(x|ω)sc]|x3=ζ(x)}2=0,
{n×[E(x|ω)inc+E(x|ω)sc]|x3=ζ(x)}3=0.
{[cωα0(k)k^2+cωkζ2(x)]E0p(k)k^1E0s(k)}exp[ik·xiα0(k)ζ(x)]+d2q(2π)2{[cωα0(q)q^2+cωqζ2(x)]Ap(q)q^1As(q)}exp[iq·x+iα0(q)ζ(x)]=0
{[cωα0(k)k^1cωkζ1(x)]E0p(k)k^1E0s(k)}exp[ik·xiα0(k)ζ(x)]+d2q(2π)2{[cωα0(q)q^1cωqζ1(x)]Ap(q)q^2As(q)}exp[iq·x+iα0(q)ζ(x)]=0.
exp[iγζ(x)]=d2Q(2π)2I(γ|Q)exp(iQ·x),
ζμ(x)exp[iγζ(x)]=d2Q(2π)2QμγI(γ|Q)exp(iQ·x).
I(γ|Q)=d2xexp(iQ·x)exp[iγζ(x)].
I(α0(k)|pk){cω(ω/c)2k^2pkp^2α0(k)E0p(k)k^1E0s(k)}+d2q(2π)2I(α0(q)|pq){cω(ω/c)2q^2pqp^2α0(q)Ap(q)q^1As(q)}=0,
I(α0(k)|pk){cω(ω/c)2k^1+pkp^1α0(k)E0p(k)k^2E0s(k)}+d2q(2π)2I(α0(q)|pq){cω(ω/c)2q^1pqp^1α0(q)Ap(q)q^2As(q)}=0,
I(α0(k)|pk)[cω(ω/c)2p^·k^pkα0(k)E0p(k)+[p^×k^]3E0s(k)]+d2q(2π)2I(α0(q)|pq)[cω(ω/c)2(p^·q^)+pqα0(q)Ap(q)+[p^×q^]3As(q)]=0.
I(α0(k)|pk)[ωc[p×k]3α0(k)E0p(k)+(p^·k^)E0s(k)]+d2q(2π)2I(α0(q)|pq)[ωc[p×q]3α0(q)Ap(q)+(p^·q^)As(q)]=0.
Aα(q)=βRαβ(q|k)E0β(k).
d2q(2π)2M(p|q)R(q|k)=N(p|k),
R(q|k)=(Rpp(q|k)Rps(q|k)Rsp(q|k)Rss(q|k)),
M(p|q)=I(α0(q|pq)(cωpq(ω/c)2(p^·q^)α0(q)[p^×q^]3ωc[p^×q^]3α0(q)p^·q^),
N(p|k)=I(α0(k)|pk)(cωpk(ω/c)2(p^·k^)α0(k)[p^×k^]3ωc[p^×k^]3α0(k)p^·k^).
RαβΩs=1S(ω2πc)2cos2θscosθ0|Rαβ(q|k)|2.
Rαβ(q|k)=Rαβ(q|k)+[Rαβ(q|k)Rαβ(q|k)],
RαβΩs=RαβΩscoh+RαβΩsincoh,
RαβΩscoh=1S(ω2πc)2cos2θscosθ0|Rαβ(q|k)|2,
RαβΩsincoh=1S(ω2πc)2cos2θscosθ0[|Rαβ(q|k)|2|Rαβ(q|k)|2].
Rαβ(q|k)=(2π)2δ(qk)δαβRα(k).
RαβΩscoh=δ(θsθ0)sinθ0δ(ϕsϕ0)δαβ|Rα(k)|2.
(2π)2δ(0)=S,
δ(qk)=(cω)2δ(θsθ0)δ(ϕsϕ0)sinθ0cosθ0,
k=(ω/c)sinθ0,q=(ω/c)sinθs.
Rα(θ0)=|Rα((ω/c)sinθ0)|2.
Rα(k)=1SRαα(k|k).
Uβ(θ0,ϕ0)=α=p,sdΩsRαβΩs=1,
Uβincoh(θ0,ϕ0)=α=p,sdΩsRαβΩsincoh,

Metrics