Abstract

A theoretical study of electromagnetic wave scattering from deep perfectly conducting one-dimensional random rough surfaces and reflection gratings is performed by means of the extinction theorem. The scattering equations are solved numerically (instead of being solved by the usual analytical procedures, which are valid only for slight corrugations). This permits us to obtain an exhaustive collection of results for the mean scattered intensity as a function of polarization and surface parameters. In particular, Lambertian scattering and enhanced backscattering are predicted for random surfaces. Also, the range of validity of the Kirchhoff approximation is established for random surfaces whose correlation length is comparable with or smaller than the wavelength. Concerning gratings, generalizations of the blaze for large angles of incidence, large periods, and arbitrary shapes are obtained. Finally, it is shown that the blaze of the antispecular order for gratings is at the root of the enhanced backscattering for random surfaces.

© 1989 Optical Society of America

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  1. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan, New York, 1963).
  2. P. Beckmann, “Scattering of light by rough surfaces,” in Progress in Optics VI, E. Wolf, ed. (North Holland, Amsterdam, 1961), pp. 55–69.
  3. F. G. Bass, I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, Oxford, 1979).
  4. J. C. Leader, “Bidirectional scattering of electromagnetic waves from rough surfaces,” J. Appl. Phys. 42, 4808–4816 (1971).
    [CrossRef]
  5. S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
    [CrossRef]
  6. J. C. Leader, “The relationship between the Kirchhoff approach and small perturbation analysis in rough surface scattering theory,”IEEE Trans. Antennas Propag. AP-19, 786–788 (1971).
    [CrossRef]
  7. G. R. Valenzuela, J. W. Wright, J. C. Leader, “Comments on the relationship between the Kirchhoff approach and small perturbation analysis in rough surface scattering theory,”IEEE Trans. Antennas Propag. AP-20, 536–539 (1972).
    [CrossRef]
  8. G. R. Valenzuela, “Depolarization of EM waves by slightly rough surfaces,”IEEE Trans. Antennas Propag. AP-15, 522–557 (1967).
  9. D. N. Pattanayak, E. Wolf, “General form and a new interpretation of the Ewald–Oseen extinction theorem,” Opt. Commun. 6, 217–220 (1972).
    [CrossRef]
  10. F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. 15, 5618–5626 (1977).
    [CrossRef]
  11. G. S. Agarwal, “Interaction of electromagnetic waves at rough dielectric surfaces,” Phys. Rev. B 15, 2371–2383 (1977).
    [CrossRef]
  12. M. Nieto-Vesperinas, N. Garcia, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
    [CrossRef]
  13. M. Nieto-Vesperinas, “Depolarization of EM waves scattered from slightly rough random surfaces: a study by means of the extinction theorem,”J. Opt. Soc. Am. 72, 539–547 (1982).
    [CrossRef]
  14. J. Shen, A. Maradudin, “Multiple scattering of waves from random rough surfaces,” Phys. Rev. B 22, 4234–4240 (1980).
    [CrossRef]
  15. D. P. Winebrenner, A. Ishimaru, “Application of the phase-perturbation technique to randomly rough surfaces,” J. Opt. Soc. Am. A 2, 2285–2294 (1985).
    [CrossRef]
  16. D. Maystre, J. P. Rossi, “Implementation of a rigorous vector theory of speckle for two-dimensional microrough surfaces,” J. Opt. Soc. Am. A 3, 1276–1282 (1986).
    [CrossRef]
  17. G. S. Brown, “A comparison of approximate theories for scattering from rough surfaces,” Wave Motion 7, 195–205 (1985).
    [CrossRef]
  18. J. A. de Santo, G. S. Brown, “Analytical techniques for multiple scattering from rough surfaces,” in Progress in Optics XXIII, E. Wolf, ed. (North-Holland, Amsterdam, 1986), pp. 3–62.
  19. E. R. Mendez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 91–95 (1987).
    [CrossRef]
  20. K. A. O’Donnell, E. R. Mendez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 4, 1194–1205 (1987).
    [CrossRef]
  21. Y. Kuga, A. Ishimaru, “Retroreflectance from a dense distribution of spherical particles,” J. Opt. Soc. Am. A 1, 831–835 (1984).
    [CrossRef]
  22. L. Tsang, A. Ishimaru, “Backscattering enhancement of random discrete scatterers,” J. Opt. Soc. Am. A 1, 836–839 (1984).
    [CrossRef]
  23. H. P. Van Albada, A. Lagendijk, “Observation of weak localization of light in a random medium,” Phys. Rev. Lett. 55, 2692–2695 (1983).
    [CrossRef]
  24. P. E. Wolf, G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
    [CrossRef] [PubMed]
  25. M. Kaveh, M. Rosenbluth, M. Edrei, I. Freund, “Weak localization and light scattering from disordered solids,” Phys. Rev. Lett. 57, 2049–2052 (1986).
    [CrossRef] [PubMed]
  26. 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); see also 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]
  27. M. Nieto-Vesperinas, J. M. Soto-Crespo, “Monte Carlo simulations for scattering of electromagnetic waves from perfectly conductive random rough surfaces,” Opt. Lett.979–981 (1987).
    [CrossRef] [PubMed]
  28. E. Bahar, M. A. Fitzwater, “Full wave theory and controlled optical experiments for enhanced scatter and depolarization by random rough surfaces,” Opt. Commun. 63, 355–360 (1987).
    [CrossRef]
  29. R. Petit, ed., “Electromagnetic theory of gratings,” Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  30. N. Garcia, N. Cabrera, “New method for solving the scattering of waves from a periodic hard surface,” Phys. Rev. B 18, 576–589 (1978).
    [CrossRef]
  31. See Ref. 29, pp. 159 – 225 .
  32. D. Y. Tseng, A. Hessel, A. A. Oliner, “Scattering by a multimode corrugated structure with application to p type Wood anomalies,” Alta Freq. 38 (special issue: URSI Symposium on EM Waves), 82–88 (1969).
  33. A. Hessel, A. A. Oliner, “A new theory of Wood’s anomalies on optical gratings,” Appl. Opt. 4, 1275–1279 (1965).
    [CrossRef]
  34. A. Hessel, J. Schmoys, D. Y. Tseng, “Bragg-angle blazing of diffraction gratings,”J. Opt. Soc. Am. 65, 380–384 (1975).
    [CrossRef]
  35. N. Garcia, A. A. Maradudin, “Exact calculations of the diffraction of s-polarized electromagnetic radiation from large-amplitude dielectric gratings,” Opt. Commun. 45, 301–306 (1983).
    [CrossRef]
  36. A. Hessel, A. A. Oliner, “Wood’s anomaly effects on gratings of large amplitude,” Opt. Commun. 59, 327–330 (1986).
    [CrossRef]
  37. M. Nieto-Vesperinas, J. M. Soto-Crespo, “Light diffracted intensities from deep gratings,” Phys. Rev. B 38, 7250–7259 (1988).
    [CrossRef]
  38. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 364.
  39. R. M. Axline, A. K. Fung, “Numerical computation of scattering from a perfectly conducting random surface,”IEEE Trans. Antennas Propag. AP-16, 482–488 (1978); H. L. Chan, A. K. Fung, “A numerical study of the Kirchhoff approximation in horizontally polarized backscattering from a random surface,” Radio Sci. 13, 811–818 (1978); A. K. Fung, M. F. Chen, “Numerical simulation of scattering from simple and composite random surfaces,” J. Opt. Soc. Am. A 2, 2274–2284 (1985).
    [CrossRef]
  40. N. Garcia, “Exact calculations of p-polarized electromagnetic fields incident on grating surfaces: Surface polaritons resonances,” Opt. Commun. 45, 307–310 (1983), and references therein.
    [CrossRef]
  41. J. C. Dainty, M. J. Kim, Imperial College, London, UK (personal communication, 1988).
  42. J. Renau, J. A. Collinson, “Measurements of electromagnetic backscattering from known, rough surfaces,” Bell Syst. Tech. J. 44, 2203–2226 (1965).
  43. M. Nieto-Vesperinas, “Radiometry of rough surfaces,” Opt. Acta 29, 961–971 (1982).
    [CrossRef]
  44. J. M. Soto-Crespo, M. Nieto-Vesperinas, “Enhancement of all antispecular orders from deep gratings,” Opt. Commun. (to be published).
  45. D. Maystre, “Electromagnetic scattering from perfectly conducting rough surfaces in the resonance region,”IEEE Trans. Antennas. Propag. AP-31, 885–895 (1983).
    [CrossRef]
  46. V. Garibaldi, A. C. Levi, R. Spadacini, G. E. Tommei, “Quantum theory of atom-surface scattering: diffraction and rainbow,” Surf. Sci. 48, 649–675 (1975).
    [CrossRef]
  47. E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,”J. Acoust. Soc. Am. 83, 78–92 (1988).
    [CrossRef]
  48. M. G. Andreasen, “Scattering from parallel metallic cylinders with arbitrary cross sections,”IEEE Trans. Antennas Propag. AP-12, 746–754 (1964).
    [CrossRef]
  49. D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics XXI, E. Wolf, ed. (North-Holland, Amsterdam, 1984).
    [CrossRef]
  50. See Ref. 38, p. 823 .
  51. N. R. Hill, V. Celli, “Multiple hits in atom-surface diffraction,” Surf. Sci. 75, 577–591 (1978).
    [CrossRef]
  52. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978), p. 214.

1988 (2)

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

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

1987 (4)

M. Nieto-Vesperinas, J. M. Soto-Crespo, “Monte Carlo simulations for scattering of electromagnetic waves from perfectly conductive random rough surfaces,” Opt. Lett.979–981 (1987).
[CrossRef] [PubMed]

E. Bahar, M. A. Fitzwater, “Full wave theory and controlled optical experiments for enhanced scatter and depolarization by random rough surfaces,” Opt. Commun. 63, 355–360 (1987).
[CrossRef]

E. R. Mendez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 91–95 (1987).
[CrossRef]

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

1986 (3)

D. Maystre, J. P. Rossi, “Implementation of a rigorous vector theory of speckle for two-dimensional microrough surfaces,” J. Opt. Soc. Am. A 3, 1276–1282 (1986).
[CrossRef]

M. Kaveh, M. Rosenbluth, M. Edrei, I. Freund, “Weak localization and light scattering from disordered solids,” Phys. Rev. Lett. 57, 2049–2052 (1986).
[CrossRef] [PubMed]

A. Hessel, A. A. Oliner, “Wood’s anomaly effects on gratings of large amplitude,” Opt. Commun. 59, 327–330 (1986).
[CrossRef]

1985 (4)

1984 (2)

1983 (4)

H. P. Van Albada, A. Lagendijk, “Observation of weak localization of light in a random medium,” Phys. Rev. Lett. 55, 2692–2695 (1983).
[CrossRef]

N. Garcia, A. A. Maradudin, “Exact calculations of the diffraction of s-polarized electromagnetic radiation from large-amplitude dielectric gratings,” Opt. Commun. 45, 301–306 (1983).
[CrossRef]

N. Garcia, “Exact calculations of p-polarized electromagnetic fields incident on grating surfaces: Surface polaritons resonances,” Opt. Commun. 45, 307–310 (1983), and references therein.
[CrossRef]

D. Maystre, “Electromagnetic scattering from perfectly conducting rough surfaces in the resonance region,”IEEE Trans. Antennas. Propag. AP-31, 885–895 (1983).
[CrossRef]

1982 (2)

1981 (1)

M. Nieto-Vesperinas, N. Garcia, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
[CrossRef]

1980 (1)

J. Shen, A. Maradudin, “Multiple scattering of waves from random rough surfaces,” Phys. Rev. B 22, 4234–4240 (1980).
[CrossRef]

1978 (3)

R. M. Axline, A. K. Fung, “Numerical computation of scattering from a perfectly conducting random surface,”IEEE Trans. Antennas Propag. AP-16, 482–488 (1978); H. L. Chan, A. K. Fung, “A numerical study of the Kirchhoff approximation in horizontally polarized backscattering from a random surface,” Radio Sci. 13, 811–818 (1978); A. K. Fung, M. F. Chen, “Numerical simulation of scattering from simple and composite random surfaces,” J. Opt. Soc. Am. A 2, 2274–2284 (1985).
[CrossRef]

N. Garcia, N. Cabrera, “New method for solving the scattering of waves from a periodic hard surface,” Phys. Rev. B 18, 576–589 (1978).
[CrossRef]

N. R. Hill, V. Celli, “Multiple hits in atom-surface diffraction,” Surf. Sci. 75, 577–591 (1978).
[CrossRef]

1977 (2)

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. 15, 5618–5626 (1977).
[CrossRef]

G. S. Agarwal, “Interaction of electromagnetic waves at rough dielectric surfaces,” Phys. Rev. B 15, 2371–2383 (1977).
[CrossRef]

1975 (2)

A. Hessel, J. Schmoys, D. Y. Tseng, “Bragg-angle blazing of diffraction gratings,”J. Opt. Soc. Am. 65, 380–384 (1975).
[CrossRef]

V. Garibaldi, A. C. Levi, R. Spadacini, G. E. Tommei, “Quantum theory of atom-surface scattering: diffraction and rainbow,” Surf. Sci. 48, 649–675 (1975).
[CrossRef]

1972 (2)

D. N. Pattanayak, E. Wolf, “General form and a new interpretation of the Ewald–Oseen extinction theorem,” Opt. Commun. 6, 217–220 (1972).
[CrossRef]

G. R. Valenzuela, J. W. Wright, J. C. Leader, “Comments on the relationship between the Kirchhoff approach and small perturbation analysis in rough surface scattering theory,”IEEE Trans. Antennas Propag. AP-20, 536–539 (1972).
[CrossRef]

1971 (2)

J. C. Leader, “Bidirectional scattering of electromagnetic waves from rough surfaces,” J. Appl. Phys. 42, 4808–4816 (1971).
[CrossRef]

J. C. Leader, “The relationship between the Kirchhoff approach and small perturbation analysis in rough surface scattering theory,”IEEE Trans. Antennas Propag. AP-19, 786–788 (1971).
[CrossRef]

1969 (1)

D. Y. Tseng, A. Hessel, A. A. Oliner, “Scattering by a multimode corrugated structure with application to p type Wood anomalies,” Alta Freq. 38 (special issue: URSI Symposium on EM Waves), 82–88 (1969).

1967 (1)

G. R. Valenzuela, “Depolarization of EM waves by slightly rough surfaces,”IEEE Trans. Antennas Propag. AP-15, 522–557 (1967).

1965 (2)

A. Hessel, A. A. Oliner, “A new theory of Wood’s anomalies on optical gratings,” Appl. Opt. 4, 1275–1279 (1965).
[CrossRef]

J. Renau, J. A. Collinson, “Measurements of electromagnetic backscattering from known, rough surfaces,” Bell Syst. Tech. J. 44, 2203–2226 (1965).

1964 (1)

M. G. Andreasen, “Scattering from parallel metallic cylinders with arbitrary cross sections,”IEEE Trans. Antennas Propag. AP-12, 746–754 (1964).
[CrossRef]

1951 (1)

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 364.

Agarwal, G. S.

G. S. Agarwal, “Interaction of electromagnetic waves at rough dielectric surfaces,” Phys. Rev. B 15, 2371–2383 (1977).
[CrossRef]

Andreasen, M. G.

M. G. Andreasen, “Scattering from parallel metallic cylinders with arbitrary cross sections,”IEEE Trans. Antennas Propag. AP-12, 746–754 (1964).
[CrossRef]

Axline, R. M.

R. M. Axline, A. K. Fung, “Numerical computation of scattering from a perfectly conducting random surface,”IEEE Trans. Antennas Propag. AP-16, 482–488 (1978); H. L. Chan, A. K. Fung, “A numerical study of the Kirchhoff approximation in horizontally polarized backscattering from a random surface,” Radio Sci. 13, 811–818 (1978); A. K. Fung, M. F. Chen, “Numerical simulation of scattering from simple and composite random surfaces,” J. Opt. Soc. Am. A 2, 2274–2284 (1985).
[CrossRef]

Bahar, E.

E. Bahar, M. A. Fitzwater, “Full wave theory and controlled optical experiments for enhanced scatter and depolarization by random rough surfaces,” Opt. Commun. 63, 355–360 (1987).
[CrossRef]

Bass, F. G.

F. G. Bass, I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, Oxford, 1979).

Beckmann, P.

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

P. Beckmann, “Scattering of light by rough surfaces,” in Progress in Optics VI, E. Wolf, ed. (North Holland, Amsterdam, 1961), pp. 55–69.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978), p. 214.

Brown, G. S.

G. S. Brown, “A comparison of approximate theories for scattering from rough surfaces,” Wave Motion 7, 195–205 (1985).
[CrossRef]

J. A. de Santo, G. S. Brown, “Analytical techniques for multiple scattering from rough surfaces,” in Progress in Optics XXIII, E. Wolf, ed. (North-Holland, Amsterdam, 1986), pp. 3–62.

Cabrera, N.

N. Garcia, N. Cabrera, “New method for solving the scattering of waves from a periodic hard surface,” Phys. Rev. B 18, 576–589 (1978).
[CrossRef]

Celli, V.

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); see also 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]

N. R. Hill, V. Celli, “Multiple hits in atom-surface diffraction,” Surf. Sci. 75, 577–591 (1978).
[CrossRef]

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. 15, 5618–5626 (1977).
[CrossRef]

Collinson, J. A.

J. Renau, J. A. Collinson, “Measurements of electromagnetic backscattering from known, rough surfaces,” Bell Syst. Tech. J. 44, 2203–2226 (1965).

Dainty, J. C.

J. C. Dainty, M. J. Kim, Imperial College, London, UK (personal communication, 1988).

de Santo, J. A.

J. A. de Santo, G. S. Brown, “Analytical techniques for multiple scattering from rough surfaces,” in Progress in Optics XXIII, E. Wolf, ed. (North-Holland, Amsterdam, 1986), pp. 3–62.

Edrei, M.

M. Kaveh, M. Rosenbluth, M. Edrei, I. Freund, “Weak localization and light scattering from disordered solids,” Phys. Rev. Lett. 57, 2049–2052 (1986).
[CrossRef] [PubMed]

Fitzwater, M. A.

E. Bahar, M. A. Fitzwater, “Full wave theory and controlled optical experiments for enhanced scatter and depolarization by random rough surfaces,” Opt. Commun. 63, 355–360 (1987).
[CrossRef]

Freund, I.

M. Kaveh, M. Rosenbluth, M. Edrei, I. Freund, “Weak localization and light scattering from disordered solids,” Phys. Rev. Lett. 57, 2049–2052 (1986).
[CrossRef] [PubMed]

Fuks, I. M.

F. G. Bass, I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, Oxford, 1979).

Fung, A. K.

R. M. Axline, A. K. Fung, “Numerical computation of scattering from a perfectly conducting random surface,”IEEE Trans. Antennas Propag. AP-16, 482–488 (1978); H. L. Chan, A. K. Fung, “A numerical study of the Kirchhoff approximation in horizontally polarized backscattering from a random surface,” Radio Sci. 13, 811–818 (1978); A. K. Fung, M. F. Chen, “Numerical simulation of scattering from simple and composite random surfaces,” J. Opt. Soc. Am. A 2, 2274–2284 (1985).
[CrossRef]

Garcia, N.

N. Garcia, “Exact calculations of p-polarized electromagnetic fields incident on grating surfaces: Surface polaritons resonances,” Opt. Commun. 45, 307–310 (1983), and references therein.
[CrossRef]

N. Garcia, A. A. Maradudin, “Exact calculations of the diffraction of s-polarized electromagnetic radiation from large-amplitude dielectric gratings,” Opt. Commun. 45, 301–306 (1983).
[CrossRef]

M. Nieto-Vesperinas, N. Garcia, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
[CrossRef]

N. Garcia, N. Cabrera, “New method for solving the scattering of waves from a periodic hard surface,” Phys. Rev. B 18, 576–589 (1978).
[CrossRef]

Garibaldi, V.

V. Garibaldi, A. C. Levi, R. Spadacini, G. E. Tommei, “Quantum theory of atom-surface scattering: diffraction and rainbow,” Surf. Sci. 48, 649–675 (1975).
[CrossRef]

Hessel, A.

A. Hessel, A. A. Oliner, “Wood’s anomaly effects on gratings of large amplitude,” Opt. Commun. 59, 327–330 (1986).
[CrossRef]

A. Hessel, J. Schmoys, D. Y. Tseng, “Bragg-angle blazing of diffraction gratings,”J. Opt. Soc. Am. 65, 380–384 (1975).
[CrossRef]

D. Y. Tseng, A. Hessel, A. A. Oliner, “Scattering by a multimode corrugated structure with application to p type Wood anomalies,” Alta Freq. 38 (special issue: URSI Symposium on EM Waves), 82–88 (1969).

A. Hessel, A. A. Oliner, “A new theory of Wood’s anomalies on optical gratings,” Appl. Opt. 4, 1275–1279 (1965).
[CrossRef]

Hill, N. R.

N. R. Hill, V. Celli, “Multiple hits in atom-surface diffraction,” Surf. Sci. 75, 577–591 (1978).
[CrossRef]

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. 15, 5618–5626 (1977).
[CrossRef]

Ishimaru, A.

Kaveh, M.

M. Kaveh, M. Rosenbluth, M. Edrei, I. Freund, “Weak localization and light scattering from disordered solids,” Phys. Rev. Lett. 57, 2049–2052 (1986).
[CrossRef] [PubMed]

Kim, M. J.

J. C. Dainty, M. J. Kim, Imperial College, London, UK (personal communication, 1988).

Kuga, Y.

Lagendijk, A.

H. P. Van Albada, A. Lagendijk, “Observation of weak localization of light in a random medium,” Phys. Rev. Lett. 55, 2692–2695 (1983).
[CrossRef]

Leader, J. C.

G. R. Valenzuela, J. W. Wright, J. C. Leader, “Comments on the relationship between the Kirchhoff approach and small perturbation analysis in rough surface scattering theory,”IEEE Trans. Antennas Propag. AP-20, 536–539 (1972).
[CrossRef]

J. C. Leader, “Bidirectional scattering of electromagnetic waves from rough surfaces,” J. Appl. Phys. 42, 4808–4816 (1971).
[CrossRef]

J. C. Leader, “The relationship between the Kirchhoff approach and small perturbation analysis in rough surface scattering theory,”IEEE Trans. Antennas Propag. AP-19, 786–788 (1971).
[CrossRef]

Levi, A. C.

V. Garibaldi, A. C. Levi, R. Spadacini, G. E. Tommei, “Quantum theory of atom-surface scattering: diffraction and rainbow,” Surf. Sci. 48, 649–675 (1975).
[CrossRef]

Maradudin, A.

J. Shen, A. Maradudin, “Multiple scattering of waves from random rough surfaces,” Phys. Rev. B 22, 4234–4240 (1980).
[CrossRef]

Maradudin, A. A.

Maret, G.

P. E. Wolf, G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[CrossRef] [PubMed]

Marvin, A.

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. 15, 5618–5626 (1977).
[CrossRef]

Marvin, A. M.

Maystre, D.

D. Maystre, J. P. Rossi, “Implementation of a rigorous vector theory of speckle for two-dimensional microrough surfaces,” J. Opt. Soc. Am. A 3, 1276–1282 (1986).
[CrossRef]

D. Maystre, “Electromagnetic scattering from perfectly conducting rough surfaces in the resonance region,”IEEE Trans. Antennas. Propag. AP-31, 885–895 (1983).
[CrossRef]

D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics XXI, E. Wolf, ed. (North-Holland, Amsterdam, 1984).
[CrossRef]

McGurn, A. R.

Mendez, E. R.

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

E. R. Mendez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 91–95 (1987).
[CrossRef]

Nieto-Vesperinas, M.

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

M. Nieto-Vesperinas, J. M. Soto-Crespo, “Monte Carlo simulations for scattering of electromagnetic waves from perfectly conductive random rough surfaces,” Opt. Lett.979–981 (1987).
[CrossRef] [PubMed]

M. Nieto-Vesperinas, “Radiometry of rough surfaces,” Opt. Acta 29, 961–971 (1982).
[CrossRef]

M. Nieto-Vesperinas, “Depolarization of EM waves scattered from slightly rough random surfaces: a study by means of the extinction theorem,”J. Opt. Soc. Am. 72, 539–547 (1982).
[CrossRef]

M. Nieto-Vesperinas, N. Garcia, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
[CrossRef]

J. M. Soto-Crespo, M. Nieto-Vesperinas, “Enhancement of all antispecular orders from deep gratings,” Opt. Commun. (to be published).

O’Donnell, K. A.

E. R. Mendez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 91–95 (1987).
[CrossRef]

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

Oliner, A. A.

A. Hessel, A. A. Oliner, “Wood’s anomaly effects on gratings of large amplitude,” Opt. Commun. 59, 327–330 (1986).
[CrossRef]

D. Y. Tseng, A. Hessel, A. A. Oliner, “Scattering by a multimode corrugated structure with application to p type Wood anomalies,” Alta Freq. 38 (special issue: URSI Symposium on EM Waves), 82–88 (1969).

A. Hessel, A. A. Oliner, “A new theory of Wood’s anomalies on optical gratings,” Appl. Opt. 4, 1275–1279 (1965).
[CrossRef]

Pattanayak, D. N.

D. N. Pattanayak, E. Wolf, “General form and a new interpretation of the Ewald–Oseen extinction theorem,” Opt. Commun. 6, 217–220 (1972).
[CrossRef]

Renau, J.

J. Renau, J. A. Collinson, “Measurements of electromagnetic backscattering from known, rough surfaces,” Bell Syst. Tech. J. 44, 2203–2226 (1965).

Rice, S. O.

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
[CrossRef]

Rosenbluth, M.

M. Kaveh, M. Rosenbluth, M. Edrei, I. Freund, “Weak localization and light scattering from disordered solids,” Phys. Rev. Lett. 57, 2049–2052 (1986).
[CrossRef] [PubMed]

Rossi, J. P.

Schmoys, J.

Shen, J.

J. Shen, A. Maradudin, “Multiple scattering of waves from random rough surfaces,” Phys. Rev. B 22, 4234–4240 (1980).
[CrossRef]

Soto-Crespo, J. M.

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

M. Nieto-Vesperinas, J. M. Soto-Crespo, “Monte Carlo simulations for scattering of electromagnetic waves from perfectly conductive random rough surfaces,” Opt. Lett.979–981 (1987).
[CrossRef] [PubMed]

J. M. Soto-Crespo, M. Nieto-Vesperinas, “Enhancement of all antispecular orders from deep gratings,” Opt. Commun. (to be published).

Spadacini, R.

V. Garibaldi, A. C. Levi, R. Spadacini, G. E. Tommei, “Quantum theory of atom-surface scattering: diffraction and rainbow,” Surf. Sci. 48, 649–675 (1975).
[CrossRef]

Spizzichino, A.

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

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 364.

Thorsos, E. I.

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

Toigo, F.

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. 15, 5618–5626 (1977).
[CrossRef]

Tommei, G. E.

V. Garibaldi, A. C. Levi, R. Spadacini, G. E. Tommei, “Quantum theory of atom-surface scattering: diffraction and rainbow,” Surf. Sci. 48, 649–675 (1975).
[CrossRef]

Tsang, L.

Tseng, D. Y.

A. Hessel, J. Schmoys, D. Y. Tseng, “Bragg-angle blazing of diffraction gratings,”J. Opt. Soc. Am. 65, 380–384 (1975).
[CrossRef]

D. Y. Tseng, A. Hessel, A. A. Oliner, “Scattering by a multimode corrugated structure with application to p type Wood anomalies,” Alta Freq. 38 (special issue: URSI Symposium on EM Waves), 82–88 (1969).

Valenzuela, G. R.

G. R. Valenzuela, J. W. Wright, J. C. Leader, “Comments on the relationship between the Kirchhoff approach and small perturbation analysis in rough surface scattering theory,”IEEE Trans. Antennas Propag. AP-20, 536–539 (1972).
[CrossRef]

G. R. Valenzuela, “Depolarization of EM waves by slightly rough surfaces,”IEEE Trans. Antennas Propag. AP-15, 522–557 (1967).

Van Albada, H. P.

H. P. Van Albada, A. Lagendijk, “Observation of weak localization of light in a random medium,” Phys. Rev. Lett. 55, 2692–2695 (1983).
[CrossRef]

Winebrenner, D. P.

Wolf, E.

D. N. Pattanayak, E. Wolf, “General form and a new interpretation of the Ewald–Oseen extinction theorem,” Opt. Commun. 6, 217–220 (1972).
[CrossRef]

Wolf, P. E.

P. E. Wolf, G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[CrossRef] [PubMed]

Wright, J. W.

G. R. Valenzuela, J. W. Wright, J. C. Leader, “Comments on the relationship between the Kirchhoff approach and small perturbation analysis in rough surface scattering theory,”IEEE Trans. Antennas Propag. AP-20, 536–539 (1972).
[CrossRef]

Alta Freq. (1)

D. Y. Tseng, A. Hessel, A. A. Oliner, “Scattering by a multimode corrugated structure with application to p type Wood anomalies,” Alta Freq. 38 (special issue: URSI Symposium on EM Waves), 82–88 (1969).

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

J. Renau, J. A. Collinson, “Measurements of electromagnetic backscattering from known, rough surfaces,” Bell Syst. Tech. J. 44, 2203–2226 (1965).

Commun. Pure Appl. Math. (1)

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
[CrossRef]

IEEE Trans. Antennas Propag. (5)

J. C. Leader, “The relationship between the Kirchhoff approach and small perturbation analysis in rough surface scattering theory,”IEEE Trans. Antennas Propag. AP-19, 786–788 (1971).
[CrossRef]

G. R. Valenzuela, J. W. Wright, J. C. Leader, “Comments on the relationship between the Kirchhoff approach and small perturbation analysis in rough surface scattering theory,”IEEE Trans. Antennas Propag. AP-20, 536–539 (1972).
[CrossRef]

G. R. Valenzuela, “Depolarization of EM waves by slightly rough surfaces,”IEEE Trans. Antennas Propag. AP-15, 522–557 (1967).

R. M. Axline, A. K. Fung, “Numerical computation of scattering from a perfectly conducting random surface,”IEEE Trans. Antennas Propag. AP-16, 482–488 (1978); H. L. Chan, A. K. Fung, “A numerical study of the Kirchhoff approximation in horizontally polarized backscattering from a random surface,” Radio Sci. 13, 811–818 (1978); A. K. Fung, M. F. Chen, “Numerical simulation of scattering from simple and composite random surfaces,” J. Opt. Soc. Am. A 2, 2274–2284 (1985).
[CrossRef]

M. G. Andreasen, “Scattering from parallel metallic cylinders with arbitrary cross sections,”IEEE Trans. Antennas Propag. AP-12, 746–754 (1964).
[CrossRef]

IEEE Trans. Antennas. Propag. (1)

D. Maystre, “Electromagnetic scattering from perfectly conducting rough surfaces in the resonance region,”IEEE Trans. Antennas. Propag. AP-31, 885–895 (1983).
[CrossRef]

J. Acoust. Soc. Am. (1)

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

J. Appl. Phys. (1)

J. C. Leader, “Bidirectional scattering of electromagnetic waves from rough surfaces,” J. Appl. Phys. 42, 4808–4816 (1971).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (6)

Opt. Acta (2)

M. Nieto-Vesperinas, N. Garcia, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
[CrossRef]

M. Nieto-Vesperinas, “Radiometry of rough surfaces,” Opt. Acta 29, 961–971 (1982).
[CrossRef]

Opt. Commun. (6)

N. Garcia, “Exact calculations of p-polarized electromagnetic fields incident on grating surfaces: Surface polaritons resonances,” Opt. Commun. 45, 307–310 (1983), and references therein.
[CrossRef]

N. Garcia, A. A. Maradudin, “Exact calculations of the diffraction of s-polarized electromagnetic radiation from large-amplitude dielectric gratings,” Opt. Commun. 45, 301–306 (1983).
[CrossRef]

A. Hessel, A. A. Oliner, “Wood’s anomaly effects on gratings of large amplitude,” Opt. Commun. 59, 327–330 (1986).
[CrossRef]

E. Bahar, M. A. Fitzwater, “Full wave theory and controlled optical experiments for enhanced scatter and depolarization by random rough surfaces,” Opt. Commun. 63, 355–360 (1987).
[CrossRef]

D. N. Pattanayak, E. Wolf, “General form and a new interpretation of the Ewald–Oseen extinction theorem,” Opt. Commun. 6, 217–220 (1972).
[CrossRef]

E. R. Mendez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 91–95 (1987).
[CrossRef]

Opt. Lett. (1)

M. Nieto-Vesperinas, J. M. Soto-Crespo, “Monte Carlo simulations for scattering of electromagnetic waves from perfectly conductive random rough surfaces,” Opt. Lett.979–981 (1987).
[CrossRef] [PubMed]

Phys. Rev. (1)

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. 15, 5618–5626 (1977).
[CrossRef]

Phys. Rev. B (4)

G. S. Agarwal, “Interaction of electromagnetic waves at rough dielectric surfaces,” Phys. Rev. B 15, 2371–2383 (1977).
[CrossRef]

J. Shen, A. Maradudin, “Multiple scattering of waves from random rough surfaces,” Phys. Rev. B 22, 4234–4240 (1980).
[CrossRef]

N. Garcia, N. Cabrera, “New method for solving the scattering of waves from a periodic hard surface,” Phys. Rev. B 18, 576–589 (1978).
[CrossRef]

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

Phys. Rev. Lett. (3)

H. P. Van Albada, A. Lagendijk, “Observation of weak localization of light in a random medium,” Phys. Rev. Lett. 55, 2692–2695 (1983).
[CrossRef]

P. E. Wolf, G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[CrossRef] [PubMed]

M. Kaveh, M. Rosenbluth, M. Edrei, I. Freund, “Weak localization and light scattering from disordered solids,” Phys. Rev. Lett. 57, 2049–2052 (1986).
[CrossRef] [PubMed]

Surf. Sci. (2)

V. Garibaldi, A. C. Levi, R. Spadacini, G. E. Tommei, “Quantum theory of atom-surface scattering: diffraction and rainbow,” Surf. Sci. 48, 649–675 (1975).
[CrossRef]

N. R. Hill, V. Celli, “Multiple hits in atom-surface diffraction,” Surf. Sci. 75, 577–591 (1978).
[CrossRef]

Wave Motion (1)

G. S. Brown, “A comparison of approximate theories for scattering from rough surfaces,” Wave Motion 7, 195–205 (1985).
[CrossRef]

Other (12)

J. A. de Santo, G. S. Brown, “Analytical techniques for multiple scattering from rough surfaces,” in Progress in Optics XXIII, E. Wolf, ed. (North-Holland, Amsterdam, 1986), pp. 3–62.

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

P. Beckmann, “Scattering of light by rough surfaces,” in Progress in Optics VI, E. Wolf, ed. (North Holland, Amsterdam, 1961), pp. 55–69.

F. G. Bass, I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, Oxford, 1979).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 364.

J. C. Dainty, M. J. Kim, Imperial College, London, UK (personal communication, 1988).

J. M. Soto-Crespo, M. Nieto-Vesperinas, “Enhancement of all antispecular orders from deep gratings,” Opt. Commun. (to be published).

See Ref. 29, pp. 159 – 225 .

R. Petit, ed., “Electromagnetic theory of gratings,” Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980).
[CrossRef]

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978), p. 214.

D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics XXI, E. Wolf, ed. (North-Holland, Amsterdam, 1984).
[CrossRef]

See Ref. 38, p. 823 .

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Figures (23)

Fig. 1
Fig. 1

Illustration of the scattering geometry.

Fig. 2
Fig. 2

Mean scattered intensity from a random surface with T = 1.8λ and σ = 1.5λ for s polarization (solid curves) and for p polarization (dotted curves): (a) θ0 = 0°, (b) θ0 = 10°, (c) θ0 = 20°, (d) θ0 = 40°. The two upper peaks mark the backscattering (left) and specular (right) directions. Unitarity of these results is kept within 4% error.

Fig. 3
Fig. 3

Same as Fig. 2 for σ = 0.1λ, T = 0.25λ, and θ0 = 60°.

Fig. 4
Fig. 4

Same as Fig. 2 for T = 0.2λ and σ = 0.2λ. (a) θ0 = 10°, (b) θ0 = 5°.

Fig. 5
Fig. 5

Same as Fig. 2 for T = 2.5λ and σ = 1.7λ. (a) θ0 = 0°, (b) θ0 = 10°, (c) θ0 = 20°.

Fig. 6
Fig. 6

Same as Fig. 2 for T = 0.5λ and σ = 0.5λ. (a) θ0 = 0°, (b) θ0 = 10°, (c) θ0 = 20°, (d) θ0 = 40°.

Fig. 7
Fig. 7

Same as Fig. 2 for T = λ and σ = λ. (a) θ0 = 0°, (b) θ0 = 10°, (c) θ0 = 20°, (d) θ0 = 40°.

Fig. 8
Fig. 8

Same as Fig. 2 for T = 0.5λ and σ = 1.8λ. (a) θ0 = 0°, (b) θ0 = 10°, (c) θ0 = 20°.

Fig. 9
Fig. 9

Same as Fig. 2 for T = 1.8λ and σ = λ. (a) θ0 =0°, (b) θ0 = 10°, (c) θ0 = 20°.

Fig. 10
Fig. 10

Same as Fig. 2 for T = 4.8λ and σ = 2λ. (a) θ0 = 0°, (b) θ0 = 10°, (c) θ0 = 20°.

Fig. 11
Fig. 11

Plots of σ/T cos(θ0) versus T/λ at three different incidence angles: θ0 = 0°, 35°, 70°, marking the zones of validity of the Kirchhoff approximation.

Fig. 12
Fig. 12

(a) Same as Fig. 2 for T = 4.8λ, σ = λ, and θ0 = 0°. Solid curves, ET result, s polarization; dotted curves, ET result, p polarization; short-dashed curves, KA averaging over 200 samples; long-dashed curves, KA from Eq. (12). (b) Same as (a) for T = λ, σ = 0.2λ, and θ0 = 20°.

Fig. 13
Fig. 13

(a) Specular and (b) antispecular intensities for s waves versus h/λ from z = h cos(2πx/a), with a = 4.44λ and θ0 = 13°. Shown are exact results (solid curves) and results of the approximate calculation (dashed curves) as described in text.

Fig. 14
Fig. 14

Diffracted intensities of s waves from sinusoidal grating with a = 6λ, h = 1.98λ, and θ0 = 56.4° showing approximately 60% enhancement at n = −10.

Fig. 15
Fig. 15

Diffracted intensities of s waves n = −15 versus σ/λ for the profile at the top: a = 8λ, θ0 = 70°. The random series from which this profile has been simulated has T = 0.7λ.

Fig. 16
Fig. 16

Diffracted intensities of s waves from sinusoidal gratings versus h/λ. Curve (a), a = 1.78λ, θ0 = 57.6°, n = −3; curve (b), a = 3.2λ, θ0 = 70°, n = −6; curve (c), a = 4.44λ, θ0 = 64°, n = −8; curve (d), a = 8λ, θ0 = 70°, n = −15.

Fig. 17
Fig. 17

Diffracted intensities of s waves from a sinusoidal grating with a = 21.33λ and θ0 = 70°. (a) h = 0.71λ, (b) h = 5.66λ, (c) h = 11.31λ.

Fig. 18
Fig. 18

Diffracted intensities of s waves for θ0 = 34° versus σ/λ for two gratings (a) and (b) with a period a = 1.78λ and with a profile shown at the top, extracted from two different portions of the same series with T = 0.24λ. Solid curves, specular order; long-dashed curves, antispecular order (n = −2); short-dashed curves, order n = −1.

Fig. 19
Fig. 19

Same as Fig. 18 for grating profile shown at the top, with θ0 = 27°, a = 4.44λ; and T = 0.7λ. Short-dashed curve, antispecular order (n = −4); solid curves, specular order.

Fig. 20
Fig. 20

Averaged diffracted intensities of s waves from 200 profiles with T = 0.24λ, a = 0.5λ, and a = 1.78λ, for θ0 = 0°, 16°, 46°, 57°.

Fig. 21
Fig. 21

Mean efficiencies versus θ0 for the same gratings as in Fig. 20.

Fig. 22
Fig. 22

Same as Fig. 20 for T = 0.7λ, σ = 1.2λ, and a = 8λ, θ0 = 4°, 14°, 43°, 54°.

Fig. 23
Fig. 23

Mean efficiencies versus θ0 for the same gratings as in Fig. 22.

Equations (59)

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E ( i ) ( r ) = j ^ E ( i ) exp [ i ( K 0 x - q 0 z ) ] ,
K 0 = k sin ( θ 0 ) ,
q 0 = k cos ( θ 0 )
E ( s ) ( θ ) = ( - π k 2 c π k r > ) exp [ i ( k r > - π / 2 ) ] - J y [ x , D ( x ) ] × exp { - i k [ x sin θ + D ( x ) cos θ ] } [ 1 + ( d D d x ) 2 ] 1 / 2 d x ,
x = r > sin ( θ ) ,
z = r > cos ( θ ) ,
I 0 E ( i ) 2 L cos ( θ 0 ) ;
1 I 0 I s ( s ) ( θ ) = 1 I 0 r > E ( s ) ( θ ) 2 .
1 I 0 I s ( s ) ( θ ) = 2 π k c 2 1 E ( i ) 2 L cos θ 0 × | - d x J y [ x , D ( x ) ] exp { - i k [ x sin θ + D ( x ) cos θ ] } | 2 .
H ( i ) ( r ) = j ^ H ( i ) exp [ i ( K 0 x - q 0 z ) ] ,
H ( s ) ( θ ) = π i k c ( 2 / π k r > ) 1 / 2 exp ( - i 3 π / 4 ) exp ( i k r > ) × - + d x J x [ x , D ( x ) ] × exp { - i k [ x sin θ + D ( x ) cos θ ] } × ( cos θ - d D d x sin θ ) [ 1 + ( d D d x ) 2 ] 1 / 2 .
1 I 0 I p ( s ) ( θ ) = 2 π k c 2 1 H ( i ) 2 L cos θ 0 | - + d x J x [ x , ( D ( x ) ] × exp { - i k [ x sin θ + D ( x ) cos θ ] } × ( cos θ - d D d x sin θ ) [ 1 + ( d D d x ) 2 ] 1 / 2 | 2 .
1 I 0 - π / 2 π / 2 I { s p } ( s ) ( θ ) d θ = 1.
C ( τ ) = D ( x ) D ( x + τ ) = σ 2 exp ( - τ 2 / T 2 ) ,
1 I 0 I K A ( θ ) = k π 1 cos θ 0 [ 1 + cos ( θ 0 + θ ) cos θ 0 + cos θ ] 2 × 0 cos ( v x τ ) exp { - v z 2 [ σ 2 - c ( τ ) ] } d τ ,
v x = k [ sin ( θ 0 ) - sin ( θ ) ] ,
v z = - k [ cos ( θ 0 ) + cos ( θ ) ] ,
E ( i ) ( r < ) + ( 1 + 4 π ) S e ( r < ) = 0 ,
E ( s ) ( r > ) = ( 1 / 4 π ) S e ( r > ) .
S e ( r ) = ( 4 π i k / c ) S J ( r ) G ˜ ( r , r ) d S ,
G ˜ ( r , r ) = ( U ˜ + 1 / k 2 ) G 0 ( r , r ) ,
H ( i ) ( r < ) + ( 1 / 4 π ) S h ( r > ) = 0 ,
H ( s ) ( r > ) = ( 1 / 4 π ) S h ( r > ) ,
S h ( r ) = ( 4 π / c ) S J ( r ) · × G ˜ ( r , r ) d S .
d S = [ 1 + ( d D d x ) 2 ] 1 / 2 d x .
S e ( r ) = - 4 π 2 k c - J ( r ) ( U ˜ + 1 k 2 ) × H 0 ( 1 ) ( k r - r ) [ 1 + ( d D d x ) 2 ] 1 / 2 d x ,
r = [ x , z = D ( x ) ] .
[ S e ( r ) ] x = - 4 π 2 k c - + [ J x ( r ) - J x ( r ) 1 k 2 2 x 2 + J z ( r ) k 2 z x ] H 0 ( 1 ) ( k r - r ) [ 1 + ( d D d x ) 2 ] 1 / 2 d x ,
[ S e ( r ) ] y = - 4 π 2 k c - + J y ( r ) H 0 ( 1 ) ( k r - r ) × [ 1 + ( d D d x ) 2 ] 1 / 2 d x ,
[ S e ( r ) ] y = - 4 π 2 k c - + [ J z ( r ) + J x ( r ) 1 k 2 x z + J z ( r ) k 2 2 z 2 ] H 0 ( 1 ) ( k r - r ) [ 1 + ( d D d x ) 2 ] 1 / 2 d x ,
n = ( - d D d x , 0 , 1 ) / [ 1 + ( d D d x ) 2 ] 1 / 2 ,
J · n = 0
- J x d D / d x + J z = 0.
E ( i ) exp [ ( K 0 x - q 0 z ) ] = π k c - J y ( r ) H 0 ( 1 ) ( k r - r ) [ 1 + ( d D d x ) 2 ] 1 / 2 d x .
E y ( s ) ( r > ) = π k c - J y ( r ) H 0 ( 1 ) ( k r - r ) [ 1 + ( d D d x ) 2 ] 1 / 2 d x .
[ S h ( r ) ] y = 4 π 2 i k c - J x ( r ) H 1 ( 1 ) ( k r - r ) × ( z - z ) - d D d x ( x - x ) [ ( x - x ) 2 + ( z - z ) 2 ] 1 / 2 [ 1 + ( d D d x ) 2 ] 1 / 2 d x .
H ( i ) exp [ i ( k 0 x - q 0 z ) ] = - π i k c - J x ( r ) H 1 ( 1 ) ( k r - r ) × ( z - z ) - d D d x ( x - x ) [ ( x - x ) 2 + ( z - z ) 2 ] 1 / 2 [ 1 + ( d D d x ) 2 ] 1 / 2 d x .
H ( i ) exp [ i ( K 0 x - q 0 z ) ] = - π i k c { 2 i k ( r - r ) + [ 1 - Δ ( r - r ) ] × - + J x ( r ) H 1 ( 1 ) ( k r - r ) × ( z - z ) - d D d x ( x - x ) [ ( x - x ) 2 + ( z - z ) 2 ] 1 / 2 [ 1 + ( d D d x ) 2 ] 1 / 2 d x } ,
Δ ( r - r ) = { 1 r = r 0 r r .
H y ( s ) ( r > ) = π i k c - + J x ( r ) H 1 ( 1 ) ( k r > - r ) × ( z > - z ) - d D d x ( x > - x ) [ ( x > - x ) 2 + ( z > - z ) 2 ] 1 / 2 [ 1 + ( d D d x ) 2 ] 1 / 2 d x .
J ( x + a ) = J ( x ) exp ( i K 0 a ) ,
E y ( s ) ( r > ) = - π k c - + d x J y ( r ) 1 π - d K exp [ i K ( x - x ) ] × exp ( i q z - z ) q [ 1 + ( d D d x ) 2 ] 1 / 2 .
q = ( k 2 - K 2 ) 1 / 2             if K k ( homogeneous waves )
= i ( K 2 - k 2 ) 1 / 2             if K > k ( evanescent waves ) .
E y ( s ) ( r > ) = - k c 0 a J y ( r ) { - + d K exp [ i K ( x - x ) × exp [ i q z - D ( x ) ] q ] × l = - exp [ i ( K 0 - K ) l a ] [ 1 + ( d D d x ) 2 ] 1 / 2 } d x .
l = - exp [ - i ( K - K 0 ) l a ] = 2 π a n = - δ [ K - ( K 0 + 2 π n a ) ] .
E y ( s ) ( r > ) = 2 π k c a n = - 0 a J y ( r ) 1 q n exp [ i K n ( x - x ) ] × exp [ i q n z - D ( x ) ] [ 1 + ( d D d x ) 2 ] 1 / 2 d x .
K n = K 0 + 2 π n / a ,
q n = ( k 2 - K n ) 1 / 2             if K n k ( homogeneous waves )
= i ( K n - k 2 ) 1 / 2             if K n > k ( evanescent waves ) .
E y ( s ) ( r > ) = - n = - A n exp [ i ( K n x + q n z ) ] ,
A n = 2 π k c a q n 0 a J y [ x , D ( x ) ] exp [ - i ( K n x + q n D ( x ) ] × [ 1 + ( d D d x ) 2 ] 1 / 2 d x
I n s = q n q 0 A n 2 E ( i ) 2 .
E ( i ) exp [ i ( K 0 x - q 0 z ) ] = π k c 0 a J y [ x , D ( x ) ] × [ l = - H 0 ( 1 ) ( k { [ x - ( x + l a ) ] 2 + [ z - D ( x ) ] 2 } 1 / 2 ) exp ( i K 0 l a ) ] × [ 1 + ( d D d x ) 2 ] 1 / 2 d x .
H ( i ) exp [ i ( K 0 x - q 0 z ) ] = - π i k c 0 a J x [ x , D ( x ) ] × [ l = - [ z - D ( x ) ] - d D d x [ x - ( x + l a ) ] { ( x - ( x + l a ) ] 2 + [ z - D ( x ) ] 2 } 1 / 2 × exp ( i K 0 l a ) H 1 ( 1 ) ( k { [ x - ( x + l a ) ] 2 + ( z - D ( x ) ] 2 } 1 / 2 ) ] × [ 1 + ( d D d x ) 2 ] 1 / 2 d x ,
H y ( s ) ( r > ) = n = - B n exp [ i ( K n x + q n z ) ] ,
B n = - 2 π k c a q n 0 a J x [ x , D ( x ) ] exp { - i [ K n x + q n D ( x ) ] } × ( d D d x K n - q n ) [ 1 + ( d D d x ) 2 ] 1 / 2 d x .
I n p = q n q 0 B n 2 E ( i ) 2 .
n I n { s p } = 1 ,

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