Abstract

A rigorous method for analyzing plane-wave scattering from perfectly conducting periodic surfaces is examined and applied to trapezoidal profiles. Both TE and TM polarizations of the incident plane wave are considered. An integral equation for the unknown current distribution in the scatter surface is formulated by invoking the extended boundary condition. Upon expressing the current density in terms of its physical optics approximation multiplied by a Fourier series, the integral equation reduces to a linear system of equations. For the case of a piecewise linear surface profile, the coefficient matrix of this system is amenable to efficient computer evaluation, which furnishes the Fourier coefficients of the current distribution. The method is applied to trapezoidal scatterers for which little data is available in the literature, and, by using appropriate limiting procedures, to triangular and rectangular profiles. Scatter fields and surface current densities are calculated. The accuracy of the method, its range, and its limitations, are investigated and comparisons are made with the results of others. The method has given accurate results for surface groove depths of less than half a wavelength and for surface periods of greater than a wavelength at minimal computational cost.

© 1980 Optical Society of America

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  1. K. A. Zaki and A. R. Neureuther, “Scattering from a perfectly conducting surface with sinusoidal height profile, TE polarization,” IEEE Trans. Antennas Propag. AP-19, 208–214 (1971).
    [Crossref]
  2. K. A. Zaki and A. R. Neureuther, “Scattering from a perfectly conducting surface with sinusoidal height profile, TM polarization, “IEEE Trans. Antennas Propag. AP-19, 747–751 (1971).
    [Crossref]
  3. R. B. Green, “Diffraction efficiencies for infinite perfectly conducting gratings of arbitrary profile,” IEEE Trans. Microwave Theory Tech. MTT-18, 313–318 (1970).
    [Crossref]
  4. T. C. H. Tong and T. B. A. Senior, “Scattering of electromagnetic waves by a periodic surface with arbitrary profile,” University of Michigan, Dept. of Electrical and Computing Engineering, Radiation Laboratory, Scientific Report No. 13, AFCRL-72-0258, 1972 (unpublished).
  5. A. Hessel and J. Shmoys, “Computer analysis of propagation reflection phenomena,” Polytechnic Institute of Brookly Scinetific Report, Contract number DAAB07-73-M-2716, 1973 (unpublished).
  6. A. Hessel, J. Shmoys, and D. Y. Tseng, e blazing of diffraction graftings,” J. Opt. Soc. Am. 65, 380–384 (1975).
    [Crossref]
  7. H. Ikuno and K. Yasuura, “Improved point-matching method with application to scattering from a periodic surface,” IEEE Trans. Antennas Propag. AP-21, 657–662 (1973).
    [Crossref]
  8. P. C. Waterman, “Scattering by periodic surfaces,” J. Acoust. Soc. Am. 57, 791–802 (1975).
    [Crossref]
  9. G. Whitman and F. Schwering, “Scattering by periodic metal surfaces with sinusoidal height profiles-a theoretical approach,” IEEE Trans. Antennas Propag. AP-25, 869–876 (1977).
    [Crossref]
  10. H. A. Kalhor and A. R. Neureuther, “Numerical Method for the Analysis of Diffraction Graftings,” J. Opt. Soc. Am. 61, 43–48 (1971).
    [Crossref]
  11. R. Petit, “Diffraction d’une onde plane par un reseau metallique, “Rev. d’Opt. 45, 353–369 (1966).
  12. J. A. DeSanto, “Scattering from a sinusoid; derivationof derivation of linear equations for the field amplitudes,” J. Acoust. Soc. Am. 57, 1195–1197 (1975).
    [Crossref]
  13. Exceptions to the integral equation approach to periodic surface scattering problems include the methods of Hessel and Shmoys5 and of Maystre and Petit37 which use a mode expansion technique (for rectangular surface profiles), and the method by Neviere et al.19,20 which reduces the problem to numerical treatment of a system of coupled differential equations.
  14. G. M. Whitman and F. Schwering, “TE- and TM-polarized plane wave scattering by periodic metal surfaces with sinusoidal height profile-A theoretical approach-Part I: Theory, Part II: Numerical results,” Technical Reports CORAD COM-75-4 and 79-5, 1979 (unpublished).
  15. R. F. Millar, “The Rayleigh hypothesis and a related least-square solution to scattering problems for periodic surfaces and other scatterers,” Rad. Sci. 8, 785–796 (1973).
    [Crossref]
  16. TM (TE) polarization is characterized by a magnetic (electric) field strength directed parallel to the surface grooves.
  17. P. S. Demko, “Polarization/multipath study,” VL-5-72, Avionics Laboratory, U.S. Army Electronics Command, Fort Monmouth, N.J., 1972 (unpublished).
  18. D. A. Schnidman, “Airport Survey of MLS Multipath Issues,” Project Report FAA-RD-195, Lincoln Laboratory, Massachusetts Institute of Technology, Lexington, Mass., 1975 (unpublished).
  19. M. Nevière, M. Cadilhac, and R. Petit, “Applications of conformal mappings to the diffraction of electromagnetic waves by a grating,” IEEE Trans. Antennas Propag. AP-21, 37–46 (1973).
    [Crossref]
  20. E. G. Loewen, M. Neviére, and D. Maystre, “Review of grating efficiency theory as it applies to blazed and holographic gratings,” Appl. Opt. 16, 2711–2721 (1977).
    [Crossref] [PubMed]
  21. D. Maystre, “A new general integral theory for dielectric coated gratings,” J. Opt. Soc. Am 68, 490–495 (1978).
    [Crossref]
  22. D. Maystre and R. Petit, “Some recent theoretical results for gratings; application to their use in the very far ultraviolet,” Nouv. Rev. Opt. 3, 165–180 (1976).
    [Crossref]
  23. R. Petit and D. Maystre, “Application des lois de l’electromagnétisme a 1’etude des reseaux,” Rev. Phys. Appl. 7, 427–441 (1972).
    [Crossref]
  24. R. Petit, “Electromagnetic grating theories: limitations and successes,” Nouv. Rev. Opt. 3, 129–135 (1975).
    [Crossref]
  25. J. A. DeSanto (Editor), Ocean Acoustics (Springer-Verlag, Berlin, Heidleberg, New York, 1978).
  26. G. M. Whitman, D. M. Leskiw, and F. Schwering, “Scattering by height profile TE- and TM-polarizations,” Technical Report CORADCOM-73-3, 1979 (unpublished).
  27. In the case of hangars with surface corrugations running in the vertical direction, this indicates the superiority of horizontal over vertical polarization in suppressing undesirable specular reflection.
  28. G. M. Whitman and F. Schwering, “Reciprocity identity for periodic surface scattering,” IEEE Trans. Antennas Propag. AP-27, 252–254 (1979).
    [Crossref]
  29. J. L. Uretsky, “The scattering of plane waves from periodic surfaces,” Ann. Phys. 33, 400–427 (1965).
    [Crossref]
  30. D. Maystre and R. C. McPhedran, “Le Théorème de réciptocité pour les Réseaux de Conductivité Finie: Demonstration et Applications,” Opt. Commun. 12, 164–167 (1974).
    [Crossref]
  31. D. N. Zuckerman and P. Diament, “Rank reduction of ill-conditioned matrices in waveguide junction problems” IEEE Trans. Microwave Theory Tech. MTT-25, 613–619 (1977).
    [Crossref]
  32. E. G. Loewen, M. Nevière, and D. Maystre, “On an asymptotic theory of diffraction gratings used in the scalar domain,” J. Opt. Soc. Am. 68, 496–501 (1978).
    [Crossref]
  33. Comparison with the power spectrum for θ = 0.1° in Table III indicated that polarization sensitivity near an anomaly is stronger for current than for power, an observation previously noted for sinusoidal profiles.
  34. E. V. Jull, J. W. Heath, and G. R. Ebbeson, “Graftings that diffract all incident energy, J. Opt. Soc. Am 67, 557–560 (1977).
    [Crossref]
  35. A. Wirgin, Selected paper from the URSI Symposium, Alta Freq.38, 332 (1969).
  36. J. Pavageau and J. Bousquet, “Diffraction par un Réseau Conducteur,” Opt. Acta,  17, 469–478 (1970).
    [Crossref]
  37. D. Maystre and R. Petit, “Diffraction par un Réseau Lamellaire Infinitment Conducteur,” Opt. Commun. 5, 90–93 (1972).
    [Crossref]
  38. D. Maystre and R. Petit, “Essai de Determination Theorique du Profil Optimal d’un Réseau Holographique,” Opt. Commun. 4, 25–28 (1971).
    [Crossref]

1979 (1)

G. M. Whitman and F. Schwering, “Reciprocity identity for periodic surface scattering,” IEEE Trans. Antennas Propag. AP-27, 252–254 (1979).
[Crossref]

1978 (2)

1977 (4)

E. V. Jull, J. W. Heath, and G. R. Ebbeson, “Graftings that diffract all incident energy, J. Opt. Soc. Am 67, 557–560 (1977).
[Crossref]

D. N. Zuckerman and P. Diament, “Rank reduction of ill-conditioned matrices in waveguide junction problems” IEEE Trans. Microwave Theory Tech. MTT-25, 613–619 (1977).
[Crossref]

E. G. Loewen, M. Neviére, and D. Maystre, “Review of grating efficiency theory as it applies to blazed and holographic gratings,” Appl. Opt. 16, 2711–2721 (1977).
[Crossref] [PubMed]

G. Whitman and F. Schwering, “Scattering by periodic metal surfaces with sinusoidal height profiles-a theoretical approach,” IEEE Trans. Antennas Propag. AP-25, 869–876 (1977).
[Crossref]

1976 (1)

D. Maystre and R. Petit, “Some recent theoretical results for gratings; application to their use in the very far ultraviolet,” Nouv. Rev. Opt. 3, 165–180 (1976).
[Crossref]

1975 (4)

P. C. Waterman, “Scattering by periodic surfaces,” J. Acoust. Soc. Am. 57, 791–802 (1975).
[Crossref]

R. Petit, “Electromagnetic grating theories: limitations and successes,” Nouv. Rev. Opt. 3, 129–135 (1975).
[Crossref]

J. A. DeSanto, “Scattering from a sinusoid; derivationof derivation of linear equations for the field amplitudes,” J. Acoust. Soc. Am. 57, 1195–1197 (1975).
[Crossref]

A. Hessel, J. Shmoys, and D. Y. Tseng, e blazing of diffraction graftings,” J. Opt. Soc. Am. 65, 380–384 (1975).
[Crossref]

1974 (1)

D. Maystre and R. C. McPhedran, “Le Théorème de réciptocité pour les Réseaux de Conductivité Finie: Demonstration et Applications,” Opt. Commun. 12, 164–167 (1974).
[Crossref]

1973 (3)

H. Ikuno and K. Yasuura, “Improved point-matching method with application to scattering from a periodic surface,” IEEE Trans. Antennas Propag. AP-21, 657–662 (1973).
[Crossref]

R. F. Millar, “The Rayleigh hypothesis and a related least-square solution to scattering problems for periodic surfaces and other scatterers,” Rad. Sci. 8, 785–796 (1973).
[Crossref]

M. Nevière, M. Cadilhac, and R. Petit, “Applications of conformal mappings to the diffraction of electromagnetic waves by a grating,” IEEE Trans. Antennas Propag. AP-21, 37–46 (1973).
[Crossref]

1972 (2)

R. Petit and D. Maystre, “Application des lois de l’electromagnétisme a 1’etude des reseaux,” Rev. Phys. Appl. 7, 427–441 (1972).
[Crossref]

D. Maystre and R. Petit, “Diffraction par un Réseau Lamellaire Infinitment Conducteur,” Opt. Commun. 5, 90–93 (1972).
[Crossref]

1971 (4)

D. Maystre and R. Petit, “Essai de Determination Theorique du Profil Optimal d’un Réseau Holographique,” Opt. Commun. 4, 25–28 (1971).
[Crossref]

H. A. Kalhor and A. R. Neureuther, “Numerical Method for the Analysis of Diffraction Graftings,” J. Opt. Soc. Am. 61, 43–48 (1971).
[Crossref]

K. A. Zaki and A. R. Neureuther, “Scattering from a perfectly conducting surface with sinusoidal height profile, TE polarization,” IEEE Trans. Antennas Propag. AP-19, 208–214 (1971).
[Crossref]

K. A. Zaki and A. R. Neureuther, “Scattering from a perfectly conducting surface with sinusoidal height profile, TM polarization, “IEEE Trans. Antennas Propag. AP-19, 747–751 (1971).
[Crossref]

1970 (2)

R. B. Green, “Diffraction efficiencies for infinite perfectly conducting gratings of arbitrary profile,” IEEE Trans. Microwave Theory Tech. MTT-18, 313–318 (1970).
[Crossref]

J. Pavageau and J. Bousquet, “Diffraction par un Réseau Conducteur,” Opt. Acta,  17, 469–478 (1970).
[Crossref]

1966 (1)

R. Petit, “Diffraction d’une onde plane par un reseau metallique, “Rev. d’Opt. 45, 353–369 (1966).

1965 (1)

J. L. Uretsky, “The scattering of plane waves from periodic surfaces,” Ann. Phys. 33, 400–427 (1965).
[Crossref]

Bousquet, J.

J. Pavageau and J. Bousquet, “Diffraction par un Réseau Conducteur,” Opt. Acta,  17, 469–478 (1970).
[Crossref]

Cadilhac, M.

M. Nevière, M. Cadilhac, and R. Petit, “Applications of conformal mappings to the diffraction of electromagnetic waves by a grating,” IEEE Trans. Antennas Propag. AP-21, 37–46 (1973).
[Crossref]

Demko, P. S.

P. S. Demko, “Polarization/multipath study,” VL-5-72, Avionics Laboratory, U.S. Army Electronics Command, Fort Monmouth, N.J., 1972 (unpublished).

DeSanto, J. A.

J. A. DeSanto, “Scattering from a sinusoid; derivationof derivation of linear equations for the field amplitudes,” J. Acoust. Soc. Am. 57, 1195–1197 (1975).
[Crossref]

Diament, P.

D. N. Zuckerman and P. Diament, “Rank reduction of ill-conditioned matrices in waveguide junction problems” IEEE Trans. Microwave Theory Tech. MTT-25, 613–619 (1977).
[Crossref]

Ebbeson, G. R.

E. V. Jull, J. W. Heath, and G. R. Ebbeson, “Graftings that diffract all incident energy, J. Opt. Soc. Am 67, 557–560 (1977).
[Crossref]

Green, R. B.

R. B. Green, “Diffraction efficiencies for infinite perfectly conducting gratings of arbitrary profile,” IEEE Trans. Microwave Theory Tech. MTT-18, 313–318 (1970).
[Crossref]

Heath, J. W.

E. V. Jull, J. W. Heath, and G. R. Ebbeson, “Graftings that diffract all incident energy, J. Opt. Soc. Am 67, 557–560 (1977).
[Crossref]

Hessel, A.

A. Hessel, J. Shmoys, and D. Y. Tseng, e blazing of diffraction graftings,” J. Opt. Soc. Am. 65, 380–384 (1975).
[Crossref]

A. Hessel and J. Shmoys, “Computer analysis of propagation reflection phenomena,” Polytechnic Institute of Brookly Scinetific Report, Contract number DAAB07-73-M-2716, 1973 (unpublished).

Ikuno, H.

H. Ikuno and K. Yasuura, “Improved point-matching method with application to scattering from a periodic surface,” IEEE Trans. Antennas Propag. AP-21, 657–662 (1973).
[Crossref]

Jull, E. V.

E. V. Jull, J. W. Heath, and G. R. Ebbeson, “Graftings that diffract all incident energy, J. Opt. Soc. Am 67, 557–560 (1977).
[Crossref]

Kalhor, H. A.

Leskiw, D. M.

G. M. Whitman, D. M. Leskiw, and F. Schwering, “Scattering by height profile TE- and TM-polarizations,” Technical Report CORADCOM-73-3, 1979 (unpublished).

Loewen, E. G.

Maystre, D.

E. G. Loewen, M. Nevière, and D. Maystre, “On an asymptotic theory of diffraction gratings used in the scalar domain,” J. Opt. Soc. Am. 68, 496–501 (1978).
[Crossref]

D. Maystre, “A new general integral theory for dielectric coated gratings,” J. Opt. Soc. Am 68, 490–495 (1978).
[Crossref]

E. G. Loewen, M. Neviére, and D. Maystre, “Review of grating efficiency theory as it applies to blazed and holographic gratings,” Appl. Opt. 16, 2711–2721 (1977).
[Crossref] [PubMed]

D. Maystre and R. Petit, “Some recent theoretical results for gratings; application to their use in the very far ultraviolet,” Nouv. Rev. Opt. 3, 165–180 (1976).
[Crossref]

D. Maystre and R. C. McPhedran, “Le Théorème de réciptocité pour les Réseaux de Conductivité Finie: Demonstration et Applications,” Opt. Commun. 12, 164–167 (1974).
[Crossref]

R. Petit and D. Maystre, “Application des lois de l’electromagnétisme a 1’etude des reseaux,” Rev. Phys. Appl. 7, 427–441 (1972).
[Crossref]

D. Maystre and R. Petit, “Diffraction par un Réseau Lamellaire Infinitment Conducteur,” Opt. Commun. 5, 90–93 (1972).
[Crossref]

D. Maystre and R. Petit, “Essai de Determination Theorique du Profil Optimal d’un Réseau Holographique,” Opt. Commun. 4, 25–28 (1971).
[Crossref]

McPhedran, R. C.

D. Maystre and R. C. McPhedran, “Le Théorème de réciptocité pour les Réseaux de Conductivité Finie: Demonstration et Applications,” Opt. Commun. 12, 164–167 (1974).
[Crossref]

Millar, R. F.

R. F. Millar, “The Rayleigh hypothesis and a related least-square solution to scattering problems for periodic surfaces and other scatterers,” Rad. Sci. 8, 785–796 (1973).
[Crossref]

Neureuther, A. R.

K. A. Zaki and A. R. Neureuther, “Scattering from a perfectly conducting surface with sinusoidal height profile, TE polarization,” IEEE Trans. Antennas Propag. AP-19, 208–214 (1971).
[Crossref]

K. A. Zaki and A. R. Neureuther, “Scattering from a perfectly conducting surface with sinusoidal height profile, TM polarization, “IEEE Trans. Antennas Propag. AP-19, 747–751 (1971).
[Crossref]

H. A. Kalhor and A. R. Neureuther, “Numerical Method for the Analysis of Diffraction Graftings,” J. Opt. Soc. Am. 61, 43–48 (1971).
[Crossref]

Neviére, M.

Nevière, M.

E. G. Loewen, M. Nevière, and D. Maystre, “On an asymptotic theory of diffraction gratings used in the scalar domain,” J. Opt. Soc. Am. 68, 496–501 (1978).
[Crossref]

M. Nevière, M. Cadilhac, and R. Petit, “Applications of conformal mappings to the diffraction of electromagnetic waves by a grating,” IEEE Trans. Antennas Propag. AP-21, 37–46 (1973).
[Crossref]

Pavageau, J.

J. Pavageau and J. Bousquet, “Diffraction par un Réseau Conducteur,” Opt. Acta,  17, 469–478 (1970).
[Crossref]

Petit, R.

D. Maystre and R. Petit, “Some recent theoretical results for gratings; application to their use in the very far ultraviolet,” Nouv. Rev. Opt. 3, 165–180 (1976).
[Crossref]

R. Petit, “Electromagnetic grating theories: limitations and successes,” Nouv. Rev. Opt. 3, 129–135 (1975).
[Crossref]

M. Nevière, M. Cadilhac, and R. Petit, “Applications of conformal mappings to the diffraction of electromagnetic waves by a grating,” IEEE Trans. Antennas Propag. AP-21, 37–46 (1973).
[Crossref]

R. Petit and D. Maystre, “Application des lois de l’electromagnétisme a 1’etude des reseaux,” Rev. Phys. Appl. 7, 427–441 (1972).
[Crossref]

D. Maystre and R. Petit, “Diffraction par un Réseau Lamellaire Infinitment Conducteur,” Opt. Commun. 5, 90–93 (1972).
[Crossref]

D. Maystre and R. Petit, “Essai de Determination Theorique du Profil Optimal d’un Réseau Holographique,” Opt. Commun. 4, 25–28 (1971).
[Crossref]

R. Petit, “Diffraction d’une onde plane par un reseau metallique, “Rev. d’Opt. 45, 353–369 (1966).

Schnidman, D. A.

D. A. Schnidman, “Airport Survey of MLS Multipath Issues,” Project Report FAA-RD-195, Lincoln Laboratory, Massachusetts Institute of Technology, Lexington, Mass., 1975 (unpublished).

Schwering, F.

G. M. Whitman and F. Schwering, “Reciprocity identity for periodic surface scattering,” IEEE Trans. Antennas Propag. AP-27, 252–254 (1979).
[Crossref]

G. Whitman and F. Schwering, “Scattering by periodic metal surfaces with sinusoidal height profiles-a theoretical approach,” IEEE Trans. Antennas Propag. AP-25, 869–876 (1977).
[Crossref]

G. M. Whitman, D. M. Leskiw, and F. Schwering, “Scattering by height profile TE- and TM-polarizations,” Technical Report CORADCOM-73-3, 1979 (unpublished).

G. M. Whitman and F. Schwering, “TE- and TM-polarized plane wave scattering by periodic metal surfaces with sinusoidal height profile-A theoretical approach-Part I: Theory, Part II: Numerical results,” Technical Reports CORAD COM-75-4 and 79-5, 1979 (unpublished).

Senior, T. B. A.

T. C. H. Tong and T. B. A. Senior, “Scattering of electromagnetic waves by a periodic surface with arbitrary profile,” University of Michigan, Dept. of Electrical and Computing Engineering, Radiation Laboratory, Scientific Report No. 13, AFCRL-72-0258, 1972 (unpublished).

Shmoys, J.

A. Hessel, J. Shmoys, and D. Y. Tseng, e blazing of diffraction graftings,” J. Opt. Soc. Am. 65, 380–384 (1975).
[Crossref]

A. Hessel and J. Shmoys, “Computer analysis of propagation reflection phenomena,” Polytechnic Institute of Brookly Scinetific Report, Contract number DAAB07-73-M-2716, 1973 (unpublished).

Tong, T. C. H.

T. C. H. Tong and T. B. A. Senior, “Scattering of electromagnetic waves by a periodic surface with arbitrary profile,” University of Michigan, Dept. of Electrical and Computing Engineering, Radiation Laboratory, Scientific Report No. 13, AFCRL-72-0258, 1972 (unpublished).

Tseng, D. Y.

Uretsky, J. L.

J. L. Uretsky, “The scattering of plane waves from periodic surfaces,” Ann. Phys. 33, 400–427 (1965).
[Crossref]

Waterman, P. C.

P. C. Waterman, “Scattering by periodic surfaces,” J. Acoust. Soc. Am. 57, 791–802 (1975).
[Crossref]

Whitman, G.

G. Whitman and F. Schwering, “Scattering by periodic metal surfaces with sinusoidal height profiles-a theoretical approach,” IEEE Trans. Antennas Propag. AP-25, 869–876 (1977).
[Crossref]

Whitman, G. M.

G. M. Whitman and F. Schwering, “Reciprocity identity for periodic surface scattering,” IEEE Trans. Antennas Propag. AP-27, 252–254 (1979).
[Crossref]

G. M. Whitman, D. M. Leskiw, and F. Schwering, “Scattering by height profile TE- and TM-polarizations,” Technical Report CORADCOM-73-3, 1979 (unpublished).

G. M. Whitman and F. Schwering, “TE- and TM-polarized plane wave scattering by periodic metal surfaces with sinusoidal height profile-A theoretical approach-Part I: Theory, Part II: Numerical results,” Technical Reports CORAD COM-75-4 and 79-5, 1979 (unpublished).

Wirgin, A.

A. Wirgin, Selected paper from the URSI Symposium, Alta Freq.38, 332 (1969).

Yasuura, K.

H. Ikuno and K. Yasuura, “Improved point-matching method with application to scattering from a periodic surface,” IEEE Trans. Antennas Propag. AP-21, 657–662 (1973).
[Crossref]

Zaki, K. A.

K. A. Zaki and A. R. Neureuther, “Scattering from a perfectly conducting surface with sinusoidal height profile, TE polarization,” IEEE Trans. Antennas Propag. AP-19, 208–214 (1971).
[Crossref]

K. A. Zaki and A. R. Neureuther, “Scattering from a perfectly conducting surface with sinusoidal height profile, TM polarization, “IEEE Trans. Antennas Propag. AP-19, 747–751 (1971).
[Crossref]

Zuckerman, D. N.

D. N. Zuckerman and P. Diament, “Rank reduction of ill-conditioned matrices in waveguide junction problems” IEEE Trans. Microwave Theory Tech. MTT-25, 613–619 (1977).
[Crossref]

Ann. Phys. (1)

J. L. Uretsky, “The scattering of plane waves from periodic surfaces,” Ann. Phys. 33, 400–427 (1965).
[Crossref]

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (6)

K. A. Zaki and A. R. Neureuther, “Scattering from a perfectly conducting surface with sinusoidal height profile, TE polarization,” IEEE Trans. Antennas Propag. AP-19, 208–214 (1971).
[Crossref]

K. A. Zaki and A. R. Neureuther, “Scattering from a perfectly conducting surface with sinusoidal height profile, TM polarization, “IEEE Trans. Antennas Propag. AP-19, 747–751 (1971).
[Crossref]

H. Ikuno and K. Yasuura, “Improved point-matching method with application to scattering from a periodic surface,” IEEE Trans. Antennas Propag. AP-21, 657–662 (1973).
[Crossref]

G. Whitman and F. Schwering, “Scattering by periodic metal surfaces with sinusoidal height profiles-a theoretical approach,” IEEE Trans. Antennas Propag. AP-25, 869–876 (1977).
[Crossref]

M. Nevière, M. Cadilhac, and R. Petit, “Applications of conformal mappings to the diffraction of electromagnetic waves by a grating,” IEEE Trans. Antennas Propag. AP-21, 37–46 (1973).
[Crossref]

G. M. Whitman and F. Schwering, “Reciprocity identity for periodic surface scattering,” IEEE Trans. Antennas Propag. AP-27, 252–254 (1979).
[Crossref]

IEEE Trans. Microwave Theory Tech. (2)

R. B. Green, “Diffraction efficiencies for infinite perfectly conducting gratings of arbitrary profile,” IEEE Trans. Microwave Theory Tech. MTT-18, 313–318 (1970).
[Crossref]

D. N. Zuckerman and P. Diament, “Rank reduction of ill-conditioned matrices in waveguide junction problems” IEEE Trans. Microwave Theory Tech. MTT-25, 613–619 (1977).
[Crossref]

J. Acoust. Soc. Am. (2)

P. C. Waterman, “Scattering by periodic surfaces,” J. Acoust. Soc. Am. 57, 791–802 (1975).
[Crossref]

J. A. DeSanto, “Scattering from a sinusoid; derivationof derivation of linear equations for the field amplitudes,” J. Acoust. Soc. Am. 57, 1195–1197 (1975).
[Crossref]

J. Opt. Soc. Am (2)

E. V. Jull, J. W. Heath, and G. R. Ebbeson, “Graftings that diffract all incident energy, J. Opt. Soc. Am 67, 557–560 (1977).
[Crossref]

D. Maystre, “A new general integral theory for dielectric coated gratings,” J. Opt. Soc. Am 68, 490–495 (1978).
[Crossref]

J. Opt. Soc. Am. (3)

Nouv. Rev. Opt. (2)

D. Maystre and R. Petit, “Some recent theoretical results for gratings; application to their use in the very far ultraviolet,” Nouv. Rev. Opt. 3, 165–180 (1976).
[Crossref]

R. Petit, “Electromagnetic grating theories: limitations and successes,” Nouv. Rev. Opt. 3, 129–135 (1975).
[Crossref]

Opt. Acta (1)

J. Pavageau and J. Bousquet, “Diffraction par un Réseau Conducteur,” Opt. Acta,  17, 469–478 (1970).
[Crossref]

Opt. Commun. (3)

D. Maystre and R. Petit, “Diffraction par un Réseau Lamellaire Infinitment Conducteur,” Opt. Commun. 5, 90–93 (1972).
[Crossref]

D. Maystre and R. Petit, “Essai de Determination Theorique du Profil Optimal d’un Réseau Holographique,” Opt. Commun. 4, 25–28 (1971).
[Crossref]

D. Maystre and R. C. McPhedran, “Le Théorème de réciptocité pour les Réseaux de Conductivité Finie: Demonstration et Applications,” Opt. Commun. 12, 164–167 (1974).
[Crossref]

Rad. Sci. (1)

R. F. Millar, “The Rayleigh hypothesis and a related least-square solution to scattering problems for periodic surfaces and other scatterers,” Rad. Sci. 8, 785–796 (1973).
[Crossref]

Rev. d’Opt. (1)

R. Petit, “Diffraction d’une onde plane par un reseau metallique, “Rev. d’Opt. 45, 353–369 (1966).

Rev. Phys. Appl. (1)

R. Petit and D. Maystre, “Application des lois de l’electromagnétisme a 1’etude des reseaux,” Rev. Phys. Appl. 7, 427–441 (1972).
[Crossref]

Other (12)

J. A. DeSanto (Editor), Ocean Acoustics (Springer-Verlag, Berlin, Heidleberg, New York, 1978).

G. M. Whitman, D. M. Leskiw, and F. Schwering, “Scattering by height profile TE- and TM-polarizations,” Technical Report CORADCOM-73-3, 1979 (unpublished).

In the case of hangars with surface corrugations running in the vertical direction, this indicates the superiority of horizontal over vertical polarization in suppressing undesirable specular reflection.

Comparison with the power spectrum for θ = 0.1° in Table III indicated that polarization sensitivity near an anomaly is stronger for current than for power, an observation previously noted for sinusoidal profiles.

A. Wirgin, Selected paper from the URSI Symposium, Alta Freq.38, 332 (1969).

T. C. H. Tong and T. B. A. Senior, “Scattering of electromagnetic waves by a periodic surface with arbitrary profile,” University of Michigan, Dept. of Electrical and Computing Engineering, Radiation Laboratory, Scientific Report No. 13, AFCRL-72-0258, 1972 (unpublished).

A. Hessel and J. Shmoys, “Computer analysis of propagation reflection phenomena,” Polytechnic Institute of Brookly Scinetific Report, Contract number DAAB07-73-M-2716, 1973 (unpublished).

TM (TE) polarization is characterized by a magnetic (electric) field strength directed parallel to the surface grooves.

P. S. Demko, “Polarization/multipath study,” VL-5-72, Avionics Laboratory, U.S. Army Electronics Command, Fort Monmouth, N.J., 1972 (unpublished).

D. A. Schnidman, “Airport Survey of MLS Multipath Issues,” Project Report FAA-RD-195, Lincoln Laboratory, Massachusetts Institute of Technology, Lexington, Mass., 1975 (unpublished).

Exceptions to the integral equation approach to periodic surface scattering problems include the methods of Hessel and Shmoys5 and of Maystre and Petit37 which use a mode expansion technique (for rectangular surface profiles), and the method by Neviere et al.19,20 which reduces the problem to numerical treatment of a system of coupled differential equations.

G. M. Whitman and F. Schwering, “TE- and TM-polarized plane wave scattering by periodic metal surfaces with sinusoidal height profile-A theoretical approach-Part I: Theory, Part II: Numerical results,” Technical Reports CORAD COM-75-4 and 79-5, 1979 (unpublished).

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Figures (9)

FIG. 1
FIG. 1

Trapezoidal metal surface illuminated by plane wave; coordinates and geometry of scatter problem.

FIG. 2
FIG. 2

Modulus of current distribution versus x/d for trapezoidal surfaces d/λ = 3.0, a/b = 0.5, h/λ = 0.1 (- -), 0.2 (—), 0.3 (-·-) illuminated by a normally incident plane wave. The arrows (→) indicate comers of the trapezoid, (a) TE polarization (b) TM polarization

FIG. 3
FIG. 3

Powers Pm of propagating space harmonics versus incidence angle θ of primary wave for trapezoidal surface h/λ = 0.3, d/λ = 3.0, a/b = 0.5; (→) indicates Rayleigh-Wood anomalies, (a) TE polarization (b) TM polarization

FIG. 4
FIG. 4

Powers P0 of specular space harmonics versus incidence angle θ of primary wave for trapezoidal surfaces d/λ = 3.0, a/b = 0.5, h/λ = 0.1 (- -), 0.2 (-·-), 0.3 (—) for both TE and TM polarizations. The arrows (→) indicate Rayleigh-Wood anomalies.

FIG. 5
FIG. 5

Powers P0 of specular space harmonics versus incidence angle θ of primary wave for trapezoidal surfaces h/λ = 0.2, a/b = 0.5, d/λ = 2.0 (—), 3.0 (-·-), 4.0 (- -) for both TE and TM polarizations. The arrows (→) indicate Rayleigh-Wood anomalies.

FIG. 6
FIG. 6

Relative power error versus incidence angle θ for trapezoidal surface h/λ = 0.3, d/λ = 3.0, a/b = 0.5 for TE (---) and TM (—) polarizations.

FIG. 7
FIG. 7

Percent error in conservation of power TE versus the number of Fourier coefficients for h/λ = 0.15, d/λ = 3.0, a/b = 0.5, θ = 5°.

FIG. 8
FIG. 8

Percent error in conservation of power TM versus the number of Fourier coefficients for h/λ = 0.15, d/λ = 3.0, a/b = 0.5, θ = 5°.

FIG. 9
FIG. 9

Induced surface currents versus x′/d for trapezoidal surface h/λ = 0.2, d/λ = 3.0, a/b = 0.27815 (—) and sinusoidal surface hs/λ = 0.2254, d/λ = 3.0 (---) for both TE and TM polarizations and incidence angle θ = 0.1°. The arrows (→) indicate corners of the trapezoid.

Tables (3)

Tables Icon

TABLE I Comparison of power spectrum and relative power errors with method of Hessel and Shmoys—trapezoidal versus rectangular profiles.

Tables Icon

TABLE II Comparison of magnitudes of space-harmonic amplitudes and total power with the method of Kalhor and Neureuther (KN) and of Petit (P) for triangular profiles. d/λ = 2.29, a/b = 0.0, θ = 0.01°

Tables Icon

TABLE III Comparison of power spectrum and relative power errors—trapezoidal versus sinusoidal profiles.

Equations (21)

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z = f ( x ) = { h 2 h / ( b a ) ( | x | b / 2 ) 0 , 0 | x | a / 2 , a / 2 | x | b / 2 , b / 2 | x | d / 2 f ( x + d ) = f ( x ) , d = a + b .
U y i = exp [ j k ( x sin θ z cos θ ) ] , U y i = { E y i for TE H y i for TM .
K ¯ = { K y ( x ) e ̂ y = K p y ( x ) F y ( x ) e ̂ y for TE K t ( x ) e ̂ t = K p t ( x ) F t ( x ) e ̂ t for TM
K p y ( x ) = 2 cos θ η 1 + ( d f / d x ) 2 exp [ j k ( x sin θ f ( x ) cos θ ] K p t ( x ) = 2 exp [ j k ( x sin θ f ( x ) cos θ ) ]
F y ( x ) = n = C n TE exp [ j 2 π n x / d ] , F t ( x ) = n = C n TM exp [ j 2 π n x / d ] .
U y + = m = U m + exp [ j k ( x sin θ m + z cos θ m ) ] , for z > max ( f ( x ) ) = 0
U y = m = U m exp [ j k ( x sin θ m z cos θ m ) ] , for z < ( f ( x ) ) = h
sin θ m = sin θ + m λ / d ,
Re ( cos θ m ) > 0 , Im ( cos θ m ) < 0 .
U m ± = n = E m , n ± C n TE with E m , n ± = cos θ cos θ m Q m , n , ±
U m ± = m = + H m , n ± C n TM with H m , n ± = [ 1 ± ( 2 β m n h / α m ± ) × tan θ m ] Q m , n , ±
Q m , n ± = a d [ 1 + exp ( j α m ± ] j b a d [ 1 exp ( j α m ± ) ] 1 α m ± for m = n = α m ± β m n 2 ( b a ) 2 ( α m ± ) 2 × { α m ± β m n d [ sin ( β m n b ) sin ( β m n a ) exp ( j α m ± ) ] + j b a d [ cos ( β m n d ) cos ( β m n a ) exp ( j α m ± ) ] } for m n
β m n = ( m + n ) π / d , α m ± = k h ( cos θ ± cos θ m ) .
U y i + U y = 0 for z < h , x ,
U m + δ m 0 = × { n = + E m , n C n TE + δ m 0 = 0 for TE n = + H m , n C n TM + δ m 0 = 0 for TM
δ m 0 = 0 for m 0 , δ 00 = 1 .
P m = | U m + | 2 cos θ m / cos θ .
TE , TM = | m P m 1 | × 100 ,
w = P m t = P m t ( | P m t P m r | / P m r ) × 100 ,
w TE
w TM