Abstract

We present an experimental investigation of backscattering enhancement arising from the excitation of polaritons on a metal surface with highly one-dimensional random roughness. The surfaces are fabricated in gold-coated photoresist with methods in which a Gaussian random process is written into the surface through its randomly phased Fourier components. Backscattering enhancement is observed in the diffusely scattered light only in p polarization, and the narrow roughness spectrum of the surface confines the effect to a well-defined range of incident and scattering angles. The dependence of the diffuse scatter on the standard deviation of surface height and on the illumination wavelength is shown to provide an additional clear indication of the involvement of the surface polaritons.

© 1995 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
    [CrossRef]
  2. V. Celli, A. A. Maradudin, A. M. Marvin, A. R. McGurn, “Some aspects of light scattering from a randomly rough metal surface,” J. Opt. Soc. Am. A 2, 2225–2239 (1985).
    [CrossRef]
  3. A. R. McGurn, A. A. Maradudin, “Localization effects in the elastic scattering of light from a randomly rough surface,” J. Opt. Soc. Am. B 4, 910–926 (1987).
    [CrossRef]
  4. A. A. Maradudin, E. R. Méndez, “Enhanced backscattering of light from weakly rough, random metal surfaces,” Appl. Opt. 32, 3335–3343 (1993).
    [CrossRef] [PubMed]
  5. P. Tran, V. Celli, “Monte Carlo calculation of backscattering enhancement for a randomly rough grating,” J. Opt. Soc. Am. A 5, 1635–1637 (1988).
    [CrossRef]
  6. T. R. Michel, “Resonant light scattering from weakly rough surfaces and imperfect gratings,” J. Opt. Soc. Am. A 11, 1874–1885 (1994).
    [CrossRef]
  7. K. Arya, Z. B. Su, J. L. Birman, “Localization of the surface plasmon polariton caused by random roughness and its role in surface-enhanced optical phenomena,” Phys. Rev. Lett. 54, 1559–1562 (1985).
    [CrossRef] [PubMed]
  8. J. A. Sanchez-Gil, M. Nieto-Vesperinas, “Resonance effects in multiple light scattering from statistically rough metallic surfaces,” Phys. Rev. B 45, 8623–8633 (1992).
    [CrossRef]
  9. A. Arsenieva, S. Feng, “Correspondence between correlation functions and enhanced backscattering peak for scattering from smooth random surfaces,” Phys. Rev. B 47, 13047–13050 (1993).
    [CrossRef]
  10. D. Maystre, M. Saillard, “Localization of light by randomly rough surfaces: concept of localiton,” J. Opt. Soc. Am. A 11, 680–690 (1994).
    [CrossRef]
  11. D. Beaglehole, O. Hunderi, “Study of the interaction of light with rough metal surfaces. I. Experiment,” Phys. Rev. B 2, 309–321 (1970).
    [CrossRef]
  12. S. O. Sari, D. K. Cohen, K. D. Scherkoske, “Study of surface plasma-wave reflectance and roughness-induced scattering in silver foils,” Phys. Rev. B 21, 2162–2174 (1980).
    [CrossRef]
  13. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, Berlin, 1988).
  14. K. A. O’Donnell, E. R. Méndez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 4, 1194–1205 (1987).
    [CrossRef]
  15. M. E. Knotts, T. R. Michel, K. A. O’Donnell, “Comparisons of theory and experiment in light scattering from a randomly rough surface,” J. Opt. Soc. Am. A 10, 928–941 (1993).
    [CrossRef]
  16. A. Steprath, D. Schumacher, A. Otto, M. Arnold, “Experimental study of enhanced backscattering from rough silver surfaces,” Phys. Lett. A 177, 433–436 (1993).
    [CrossRef]
  17. Z. H. Gu, R. S. Dummer, A. A. Maradudin, A. R. McGurn, “Experimental study of the opposition effect in the scattering of light from a randomly rough metal surface,” Appl. Opt. 28, 537–543 (1989).
    [CrossRef] [PubMed]
  18. Yu. N. Babaranenkov, Yu. A. Kravtsov, V. D. Ozrin, A. I. Saichev, “Enhanced backscattering in optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1991), Vol. 29.
    [CrossRef]
  19. M. J. Kim, J. C. Dainty, A. T. Friburg, A. J. Sant, “Experimental study of enhanced backscattering from one-and two-dimensional random rough surfaces,” J. Opt. Soc. Am. A 7, 569–577 (1990).
    [CrossRef]
  20. D. L. Jordan, F. Moreno, “Optical backscattering measurements and their possible relevance to radar scattering from the sea surface,” Waves Random Media 2, 29–38 (1992).
    [CrossRef]
  21. A. A. Maradudin, T. R. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (NY) 203, 255–307 (1990).
    [CrossRef]
  22. See S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 23, 282–332 (1944), Sect. 2.8.
  23. M. C. Hutley, Diffraction Gratings (Academic, London, 1982), Chap. 4.
  24. E. K. Popov, L. V. Tsonev, M. L. Sabeva, “Technological problems in holographic recording of plane gratings,” Opt. Eng. 31, 2168–2173 (1992).
    [CrossRef]
  25. J. W. Goodman, Introduction to Statistical Optics (Wiley, New York, 1984), Sec. 4.4.2.
  26. E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985), p. 294.
  27. Ref. 23, Chap. 5.
  28. We consider a wave with the electric-field vector in the plane of incidence to be ppolarized, with spolarization denoting the orthogonal case.
  29. K. A. O’Donnell, M. E. Knotts, “Polarization-dependence of scattering from one-dimensional rough surfaces,” J. Opt. Soc. Am. A 8, 1126–1131 (1991).
    [CrossRef]
  30. T. R. Michel, M. E. Knotts, K. A. O’Donnell, “Stokes matrix of a one-dimensional perfectly conducting rough surface,” J. Opt. Soc. Am. A 9, 585–596 (1992).
    [CrossRef]
  31. M. E. Knotts, K. A. O’Donnell, “Measurements of light scattering by a series of conducting surfaces with one-dimensional roughness,” J. Opt. Soc. Am. A 11, 697–710 (1994).
    [CrossRef]

1994 (3)

1993 (4)

A. Arsenieva, S. Feng, “Correspondence between correlation functions and enhanced backscattering peak for scattering from smooth random surfaces,” Phys. Rev. B 47, 13047–13050 (1993).
[CrossRef]

M. E. Knotts, T. R. Michel, K. A. O’Donnell, “Comparisons of theory and experiment in light scattering from a randomly rough surface,” J. Opt. Soc. Am. A 10, 928–941 (1993).
[CrossRef]

A. Steprath, D. Schumacher, A. Otto, M. Arnold, “Experimental study of enhanced backscattering from rough silver surfaces,” Phys. Lett. A 177, 433–436 (1993).
[CrossRef]

A. A. Maradudin, E. R. Méndez, “Enhanced backscattering of light from weakly rough, random metal surfaces,” Appl. Opt. 32, 3335–3343 (1993).
[CrossRef] [PubMed]

1992 (4)

J. A. Sanchez-Gil, M. Nieto-Vesperinas, “Resonance effects in multiple light scattering from statistically rough metallic surfaces,” Phys. Rev. B 45, 8623–8633 (1992).
[CrossRef]

T. R. Michel, M. E. Knotts, K. A. O’Donnell, “Stokes matrix of a one-dimensional perfectly conducting rough surface,” J. Opt. Soc. Am. A 9, 585–596 (1992).
[CrossRef]

D. L. Jordan, F. Moreno, “Optical backscattering measurements and their possible relevance to radar scattering from the sea surface,” Waves Random Media 2, 29–38 (1992).
[CrossRef]

E. K. Popov, L. V. Tsonev, M. L. Sabeva, “Technological problems in holographic recording of plane gratings,” Opt. Eng. 31, 2168–2173 (1992).
[CrossRef]

1991 (1)

1990 (2)

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

M. J. Kim, J. C. Dainty, A. T. Friburg, A. J. Sant, “Experimental study of enhanced backscattering from one-and two-dimensional random rough surfaces,” J. Opt. Soc. Am. A 7, 569–577 (1990).
[CrossRef]

1989 (1)

1988 (1)

1987 (2)

1985 (3)

K. Arya, Z. B. Su, J. L. Birman, “Localization of the surface plasmon polariton caused by random roughness and its role in surface-enhanced optical phenomena,” Phys. Rev. Lett. 54, 1559–1562 (1985).
[CrossRef] [PubMed]

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

V. Celli, A. A. Maradudin, A. M. Marvin, A. R. McGurn, “Some aspects of light scattering from a randomly rough metal surface,” J. Opt. Soc. Am. A 2, 2225–2239 (1985).
[CrossRef]

1980 (1)

S. O. Sari, D. K. Cohen, K. D. Scherkoske, “Study of surface plasma-wave reflectance and roughness-induced scattering in silver foils,” Phys. Rev. B 21, 2162–2174 (1980).
[CrossRef]

1970 (1)

D. Beaglehole, O. Hunderi, “Study of the interaction of light with rough metal surfaces. I. Experiment,” Phys. Rev. B 2, 309–321 (1970).
[CrossRef]

1944 (1)

See S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 23, 282–332 (1944), Sect. 2.8.

Arnold, M.

A. Steprath, D. Schumacher, A. Otto, M. Arnold, “Experimental study of enhanced backscattering from rough silver surfaces,” Phys. Lett. A 177, 433–436 (1993).
[CrossRef]

Arsenieva, A.

A. Arsenieva, S. Feng, “Correspondence between correlation functions and enhanced backscattering peak for scattering from smooth random surfaces,” Phys. Rev. B 47, 13047–13050 (1993).
[CrossRef]

Arya, K.

K. Arya, Z. B. Su, J. L. Birman, “Localization of the surface plasmon polariton caused by random roughness and its role in surface-enhanced optical phenomena,” Phys. Rev. Lett. 54, 1559–1562 (1985).
[CrossRef] [PubMed]

Babaranenkov, Yu. N.

Yu. N. Babaranenkov, Yu. A. Kravtsov, V. D. Ozrin, A. I. Saichev, “Enhanced backscattering in optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1991), Vol. 29.
[CrossRef]

Beaglehole, D.

D. Beaglehole, O. Hunderi, “Study of the interaction of light with rough metal surfaces. I. Experiment,” Phys. Rev. B 2, 309–321 (1970).
[CrossRef]

Birman, J. L.

K. Arya, Z. B. Su, J. L. Birman, “Localization of the surface plasmon polariton caused by random roughness and its role in surface-enhanced optical phenomena,” Phys. Rev. Lett. 54, 1559–1562 (1985).
[CrossRef] [PubMed]

Celli, V.

Cohen, D. K.

S. O. Sari, D. K. Cohen, K. D. Scherkoske, “Study of surface plasma-wave reflectance and roughness-induced scattering in silver foils,” Phys. Rev. B 21, 2162–2174 (1980).
[CrossRef]

Dainty, J. C.

Dummer, R. S.

Feng, S.

A. Arsenieva, S. Feng, “Correspondence between correlation functions and enhanced backscattering peak for scattering from smooth random surfaces,” Phys. Rev. B 47, 13047–13050 (1993).
[CrossRef]

Friburg, A. T.

Goodman, J. W.

J. W. Goodman, Introduction to Statistical Optics (Wiley, New York, 1984), Sec. 4.4.2.

Gu, Z. H.

Hunderi, O.

D. Beaglehole, O. Hunderi, “Study of the interaction of light with rough metal surfaces. I. Experiment,” Phys. Rev. B 2, 309–321 (1970).
[CrossRef]

Hutley, M. C.

M. C. Hutley, Diffraction Gratings (Academic, London, 1982), Chap. 4.

Jordan, D. L.

D. L. Jordan, F. Moreno, “Optical backscattering measurements and their possible relevance to radar scattering from the sea surface,” Waves Random Media 2, 29–38 (1992).
[CrossRef]

Kim, M. J.

Knotts, M. E.

Kravtsov, Yu. A.

Yu. N. Babaranenkov, Yu. A. Kravtsov, V. D. Ozrin, A. I. Saichev, “Enhanced backscattering in optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1991), Vol. 29.
[CrossRef]

Maradudin, A. A.

Marvin, A. M.

Maystre, D.

McGurn, A. R.

Méndez, E. R.

Michel, T. R.

Moreno, F.

D. L. Jordan, F. Moreno, “Optical backscattering measurements and their possible relevance to radar scattering from the sea surface,” Waves Random Media 2, 29–38 (1992).
[CrossRef]

Nieto-Vesperinas, M.

J. A. Sanchez-Gil, M. Nieto-Vesperinas, “Resonance effects in multiple light scattering from statistically rough metallic surfaces,” Phys. Rev. B 45, 8623–8633 (1992).
[CrossRef]

O’Donnell, K. A.

Otto, A.

A. Steprath, D. Schumacher, A. Otto, M. Arnold, “Experimental study of enhanced backscattering from rough silver surfaces,” Phys. Lett. A 177, 433–436 (1993).
[CrossRef]

Ozrin, V. D.

Yu. N. Babaranenkov, Yu. A. Kravtsov, V. D. Ozrin, A. I. Saichev, “Enhanced backscattering in optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1991), Vol. 29.
[CrossRef]

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985), p. 294.

Popov, E. K.

E. K. Popov, L. V. Tsonev, M. L. Sabeva, “Technological problems in holographic recording of plane gratings,” Opt. Eng. 31, 2168–2173 (1992).
[CrossRef]

Raether, H.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, Berlin, 1988).

Rice, S. O.

See S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 23, 282–332 (1944), Sect. 2.8.

Sabeva, M. L.

E. K. Popov, L. V. Tsonev, M. L. Sabeva, “Technological problems in holographic recording of plane gratings,” Opt. Eng. 31, 2168–2173 (1992).
[CrossRef]

Saichev, A. I.

Yu. N. Babaranenkov, Yu. A. Kravtsov, V. D. Ozrin, A. I. Saichev, “Enhanced backscattering in optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1991), Vol. 29.
[CrossRef]

Saillard, M.

Sanchez-Gil, J. A.

J. A. Sanchez-Gil, M. Nieto-Vesperinas, “Resonance effects in multiple light scattering from statistically rough metallic surfaces,” Phys. Rev. B 45, 8623–8633 (1992).
[CrossRef]

Sant, A. J.

Sari, S. O.

S. O. Sari, D. K. Cohen, K. D. Scherkoske, “Study of surface plasma-wave reflectance and roughness-induced scattering in silver foils,” Phys. Rev. B 21, 2162–2174 (1980).
[CrossRef]

Scherkoske, K. D.

S. O. Sari, D. K. Cohen, K. D. Scherkoske, “Study of surface plasma-wave reflectance and roughness-induced scattering in silver foils,” Phys. Rev. B 21, 2162–2174 (1980).
[CrossRef]

Schumacher, D.

A. Steprath, D. Schumacher, A. Otto, M. Arnold, “Experimental study of enhanced backscattering from rough silver surfaces,” Phys. Lett. A 177, 433–436 (1993).
[CrossRef]

Steprath, A.

A. Steprath, D. Schumacher, A. Otto, M. Arnold, “Experimental study of enhanced backscattering from rough silver surfaces,” Phys. Lett. A 177, 433–436 (1993).
[CrossRef]

Su, Z. B.

K. Arya, Z. B. Su, J. L. Birman, “Localization of the surface plasmon polariton caused by random roughness and its role in surface-enhanced optical phenomena,” Phys. Rev. Lett. 54, 1559–1562 (1985).
[CrossRef] [PubMed]

Tran, P.

Tsonev, L. V.

E. K. Popov, L. V. Tsonev, M. L. Sabeva, “Technological problems in holographic recording of plane gratings,” Opt. Eng. 31, 2168–2173 (1992).
[CrossRef]

Ann. Phys. (NY) (1)

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

Appl. Opt. (2)

Bell Syst. Tech. J. (1)

See S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 23, 282–332 (1944), Sect. 2.8.

J. Opt. Soc. Am. A (10)

M. J. Kim, J. C. Dainty, A. T. Friburg, A. J. Sant, “Experimental study of enhanced backscattering from one-and two-dimensional random rough surfaces,” J. Opt. Soc. Am. A 7, 569–577 (1990).
[CrossRef]

K. A. O’Donnell, M. E. Knotts, “Polarization-dependence of scattering from one-dimensional rough surfaces,” J. Opt. Soc. Am. A 8, 1126–1131 (1991).
[CrossRef]

T. R. Michel, M. E. Knotts, K. A. O’Donnell, “Stokes matrix of a one-dimensional perfectly conducting rough surface,” J. Opt. Soc. Am. A 9, 585–596 (1992).
[CrossRef]

M. E. Knotts, K. A. O’Donnell, “Measurements of light scattering by a series of conducting surfaces with one-dimensional roughness,” J. Opt. Soc. Am. A 11, 697–710 (1994).
[CrossRef]

K. A. O’Donnell, E. R. Méndez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 4, 1194–1205 (1987).
[CrossRef]

M. E. Knotts, T. R. Michel, K. A. O’Donnell, “Comparisons of theory and experiment in light scattering from a randomly rough surface,” J. Opt. Soc. Am. A 10, 928–941 (1993).
[CrossRef]

D. Maystre, M. Saillard, “Localization of light by randomly rough surfaces: concept of localiton,” J. Opt. Soc. Am. A 11, 680–690 (1994).
[CrossRef]

P. Tran, V. Celli, “Monte Carlo calculation of backscattering enhancement for a randomly rough grating,” J. Opt. Soc. Am. A 5, 1635–1637 (1988).
[CrossRef]

T. R. Michel, “Resonant light scattering from weakly rough surfaces and imperfect gratings,” J. Opt. Soc. Am. A 11, 1874–1885 (1994).
[CrossRef]

V. Celli, A. A. Maradudin, A. M. Marvin, A. R. McGurn, “Some aspects of light scattering from a randomly rough metal surface,” J. Opt. Soc. Am. A 2, 2225–2239 (1985).
[CrossRef]

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

Opt. Eng. (1)

E. K. Popov, L. V. Tsonev, M. L. Sabeva, “Technological problems in holographic recording of plane gratings,” Opt. Eng. 31, 2168–2173 (1992).
[CrossRef]

Phys. Lett. A (1)

A. Steprath, D. Schumacher, A. Otto, M. Arnold, “Experimental study of enhanced backscattering from rough silver surfaces,” Phys. Lett. A 177, 433–436 (1993).
[CrossRef]

Phys. Rev. B (5)

D. Beaglehole, O. Hunderi, “Study of the interaction of light with rough metal surfaces. I. Experiment,” Phys. Rev. B 2, 309–321 (1970).
[CrossRef]

S. O. Sari, D. K. Cohen, K. D. Scherkoske, “Study of surface plasma-wave reflectance and roughness-induced scattering in silver foils,” Phys. Rev. B 21, 2162–2174 (1980).
[CrossRef]

J. A. Sanchez-Gil, M. Nieto-Vesperinas, “Resonance effects in multiple light scattering from statistically rough metallic surfaces,” Phys. Rev. B 45, 8623–8633 (1992).
[CrossRef]

A. Arsenieva, S. Feng, “Correspondence between correlation functions and enhanced backscattering peak for scattering from smooth random surfaces,” Phys. Rev. B 47, 13047–13050 (1993).
[CrossRef]

A. R. McGurn, A. A. Maradudin, 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. (1)

K. Arya, Z. B. Su, J. L. Birman, “Localization of the surface plasmon polariton caused by random roughness and its role in surface-enhanced optical phenomena,” Phys. Rev. Lett. 54, 1559–1562 (1985).
[CrossRef] [PubMed]

Waves Random Media (1)

D. L. Jordan, F. Moreno, “Optical backscattering measurements and their possible relevance to radar scattering from the sea surface,” Waves Random Media 2, 29–38 (1992).
[CrossRef]

Other (7)

M. C. Hutley, Diffraction Gratings (Academic, London, 1982), Chap. 4.

J. W. Goodman, Introduction to Statistical Optics (Wiley, New York, 1984), Sec. 4.4.2.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985), p. 294.

Ref. 23, Chap. 5.

We consider a wave with the electric-field vector in the plane of incidence to be ppolarized, with spolarization denoting the orthogonal case.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, Berlin, 1988).

Yu. N. Babaranenkov, Yu. A. Kravtsov, V. D. Ozrin, A. I. Saichev, “Enhanced backscattering in optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1991), Vol. 29.
[CrossRef]

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

Excitation of polaritons on a rough metal surface. The incident and the scattering angles θi and θs are positive, as shown.

Fig. 2
Fig. 2

Results from surface profilometry. Top plots: segments of typical unprocessed profilometer data for surfaces A (σ = 10.9 nm) and B (σ = 8.3 nm). The vertical scale of each segment is ±50 nm, and the ticks on the horizontal axis are spaced by 5000 nm; these scales exaggerate surface slopes by a factor of 15. Bottom plot: roughness spectrum S(k) for surfaces A (solid curve) and B (dashed curve); normalization is such that σ2 is 0 S ( k ) d k. The two arrows denote the desired spectral limits kmin and kmax; noise levels are consistent with averaging over the limited number of scans.

Fig. 3
Fig. 3

Diffuse intensities Ip and Is for surface A plotted as a function of the scattering angle θs. The wavelength is 612 nm, the backscattering direction (θs = −θi) is denoted by the arrow, and the angles of incidence θi are as noted.

Fig. 4
Fig. 4

Detail of the diffuse intensities Ip for surface A for the small scattering angles θs; the angles of incidence θi are as noted. The ticks on the y axis are spaced by 0.05.

Fig. 5
Fig. 5

Top: diffuse intensity Ip for surface A plotted as a function of the incident and the scattering angles θi and θs. Backscattering enhancement persists at θs = −θi; the break in the data at θs = θi surrounds the coherent reflection that rises above the plot’s scale. Bottom: powers of the coherent reflection Pp and Ps in p and s polarization as a function of the angle of incidence θi. The wavelength is 612 nm.

Fig. 6
Fig. 6

Diffuse intensity Ip for surface A, with θi = 4°, with wavelengths λ as shown. The dashed vertical lines denote the outward coupling limits of +ksp (dense lines) and −ksp (sparse lines); these limits overlap at λ = 612 nm. The coupling limits assume that ksp = 1.077, 1.061, 1.059, 1.054 ω/c for λ = 543, 594, 612, 633 nm, respectively, as determined from Eq. (2) and Ref. 26. The ticks on the y axis are spaced by 0.05.

Fig. 7
Fig. 7

Intensities Ip and Is for surface B plotted as a function of the scattering angle θs; the backscattering direction (θs = −θi) is denoted by the arrow. The angle of incidence θi is 10°, and the wavelength is 612 nm.

Equations (13)

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

+ k sp = ω c sin θ i + k 1 , - k sp = ω c sin θ i - k 2 ,
± k sp = ± ω c Re ( + 1 ) ,
ω c sin θ s = + k sp - k 3 , ω c sin θ s = - k sp + k 4 ,
k 1 = k min = k sp - ω c sin θ max , k 2 = k max = k sp + ω c sin θ max .
h ( x ) = α n = 1 N I n [ 1 + M n cos ( k n x - ϕ n ) ] Δ t n ,
h ( x ) = α n = 1 N I n Δ t n ,
Δ h ( x ) = n = 1 N c n cos ( k n x - ϕ n ) ,
c n = [ 2 S ( k n ) Δ k ] 1 / 2 ,
k n = k min + ( n - 1 ) Δ k ,
Δ k = ( k max - k min ) ( N - 1 ) ,
σ = 1 2 N h ,
ω c sin θ s = ω c sin θ i + n k r ,
ω c sin θ max = + k sp - k min , ω c sin θ min = + k sp - k max .

Metrics