Abstract

Stratified media with periodic surfaces have been used in waveguide-input and waveguide-output couplers and distributed-feedback lasers. We apply the extended-boundary-condition approach to treat the problem of scattering and guidance by stratified media with period surfaces for both the TE and TM polarizations in a unified manner with the use of a transition matrix. Theoretical results are shown to compare well with the experimental data for metallic gratings with surface-plasmon excitations. The anomalous behavior of the diffraction efficiencies is explained with the evanescent Floquet waves, which are excited with incidence angles corresponding to the locations of the dips in the total reflected efficiency plots. Diffraction efficiencies are plotted for wave scattering from corrugated thin-film waveguides with an angle of incidence near that for the waveguide-mode excitation. The guiding constants are also calculated and compared with the results obtained by other numerical methods. The origins of the imaginary parts in the guiding constants are discussed in light of the nonevanscent waves excited in the substrate.

© 1983 Optical Society of America

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    [CrossRef]
  4. D. Maystre, “A new theory for multiprofile, buried gratings,” Opt. Commun. 26, 127–132 (1978).
    [CrossRef]
  5. M. Neviere, R. Petit, and M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
    [CrossRef]
  6. M. Neviere, P. Vincent, R. Petit, and M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
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  7. M. Neviere, P. Vincent, R. Petit, and M. Cadilhac, “Determination of the coupling coefficient of a holographic thin film coupler,” Opt. Commun. 9, 240–244 (1973).
    [CrossRef]
  8. K. C. Chang, V. Shah, and T. Tamir, “Scattering and guiding of waves by dielectric gratings with arbitrary profiles,” J. Opt. Soc. Am. 70, 804–813 (1980).
    [CrossRef]
  9. P. C. Waterman, “Scattering by periodic surfaces,” J. Acoust. Soc. Am. 57, 791–802 (1975).
    [CrossRef]
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    [CrossRef]
  11. S. L. Chuang and J. A. Kong, “Wave scattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
    [CrossRef]
  12. S. L. Chuang and R. K. Johnson, “Acoustic wave scattering from a fluid/solid periodic rough surface,” J. Acoust. Soc. Am. 71, 1368–1376 (1982).
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  16. D. N. Pattanayak and E. Wolf, “General form and a new interpretation of the Ewald-Oseen extinction theorem,” Opt. Commun. 6, 217–220 (1972).
    [CrossRef]
  17. J. A. Kong, Theory of Electromagnetic Waves (Wiley, New York, 1975), Chap. 4, pp. 112–121.
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  19. T. Tamir, “Beam and waveguide couplers,” in Integrated Optics, T. Tamir, ed. (Springer-Verlag, New York, 1979).
  20. W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron. QE-13, 134–141 (1977).
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  24. M. C. Hutley, J. P. Verrill, R. C. McPhedran, M. Neviere, and P. Vincent, “Presentation and verification of a differential formulation for the diffraction by conducting gratings,” Nouv. Rev. Optique 6, 87–95 (1975).
    [CrossRef]
  25. R. C. McPhedran and D. Maystre, “A detailed theoretical study of the anomalies of a sinusoidal diffraction grating,” Opt. Acta 21, 413–421 (1974).
    [CrossRef]
  26. G. Armand and J. R. Manson, “Scattering by a hard corrugated wall: an exact solution,” Phys. Rev. B 18, 6510–6518 (1978).
    [CrossRef]
  27. M. C. Hutley, “An experimental study of the anomalies of sinusoidal diffraction gratings,” Opt. Acta 20, 607–624 (1973).
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  28. M. C. Hutley and V. M. Bird, “A detailed experimental study of the anomalies of a sinusoidal diffraction grating,” Opt. Acta 20, 771–782 (1973).
    [CrossRef]
  29. Y. Y. Teng and E. A. Stern, “Plasma radiation from metal grating surfaces,” Phys. Rev. Lett. 19, 511–514 (1967).
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  30. P. Halevi, “Polariton modes at the interface between the conducting or dielectric media,” Surf. Sci. 76, 64–90 (1978).
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    [CrossRef]
  35. M. L. Dakss, L. Kuhn, P. F. Heidrich, and B. A. Scott, “Grating coupler for efficient excitation of optical guided waves in thin films,” Appl. Phys. Lett. 16, 523–525 (1970).
    [CrossRef]
  36. A. Hessel and A. A. Oliner, “A new theory of Wood’s anomalies on optical gratings,” Appl. Opt. 4, 1275–1297 (1965).
    [CrossRef]
  37. K. C. Chang and T. Tamir, “Bragg-reflection approach for blazed dielectric gratings,” Opt. Commun. 26, 327–330 (1978).
    [CrossRef]
  38. K. C. Chang and T. Tamir, “Simplified approach to surface wave scattering by blazed dielectric gratings,” Appl. Opt. 19, 282–288 (1980).
    [CrossRef] [PubMed]
  39. S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Direct Measurement (Elsevier, New York, 1977).

1982 (2)

S. L. Chuang and J. A. Kong, “Wave scattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
[CrossRef]

S. L. Chuang and R. K. Johnson, “Acoustic wave scattering from a fluid/solid periodic rough surface,” J. Acoust. Soc. Am. 71, 1368–1376 (1982).
[CrossRef]

1981 (1)

S. L. Chuang and J. A. Kong, “Scattering of waves from periodic surfaces,” Proc. IEEE 69, 1132–1144 (1981).
[CrossRef]

1980 (2)

1978 (5)

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

K. C. Chang and T. Tamir, “Bragg-reflection approach for blazed dielectric gratings,” Opt. Commun. 26, 327–330 (1978).
[CrossRef]

G. Armand and J. R. Manson, “Scattering by a hard corrugated wall: an exact solution,” Phys. Rev. B 18, 6510–6518 (1978).
[CrossRef]

P. Halevi, “Polariton modes at the interface between the conducting or dielectric media,” Surf. Sci. 76, 64–90 (1978).
[CrossRef]

D. Maystre, “A new theory for multiprofile, buried gratings,” Opt. Commun. 26, 127–132 (1978).
[CrossRef]

1977 (1)

W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron. QE-13, 134–141 (1977).
[CrossRef]

1976 (2)

G. S. Agarwal, “Relation between Waterman’s extended boundary condition and the generalized extinction theorem,” Phys. Rev. D 14, 1168–1171 (1976).
[CrossRef]

M. C. Hutley and D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19, 431–436 (1976).
[CrossRef]

1975 (2)

M. C. Hutley, J. P. Verrill, R. C. McPhedran, M. Neviere, and P. Vincent, “Presentation and verification of a differential formulation for the diffraction by conducting gratings,” Nouv. Rev. Optique 6, 87–95 (1975).
[CrossRef]

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

1974 (1)

R. C. McPhedran and D. Maystre, “A detailed theoretical study of the anomalies of a sinusoidal diffraction grating,” Opt. Acta 21, 413–421 (1974).
[CrossRef]

1973 (6)

T. Tamir, “Inhomogeneous wave types at planar interfaces: III. Leaky waves,” Optik 38, 269–297 (1973).

M. Neviere, R. Petit, and M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

M. Neviere, P. Vincent, R. Petit, and M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

M. Neviere, P. Vincent, R. Petit, and M. Cadilhac, “Determination of the coupling coefficient of a holographic thin film coupler,” Opt. Commun. 9, 240–244 (1973).
[CrossRef]

M. C. Hutley, “An experimental study of the anomalies of sinusoidal diffraction gratings,” Opt. Acta 20, 607–624 (1973).
[CrossRef]

M. C. Hutley and V. M. Bird, “A detailed experimental study of the anomalies of a sinusoidal diffraction grating,” Opt. Acta 20, 771–782 (1973).
[CrossRef]

1972 (2)

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

L. L. Hope, “Theory of optical grating couplers,” Opt. Commun. 5, 179–182 (1972).
[CrossRef]

1971 (1)

P. M. Van den Berg, “Diffraction theory of a reflection grating,” Appl. Sci. Res. 24, 261–293 (1971).

1970 (1)

M. L. Dakss, L. Kuhn, P. F. Heidrich, and B. A. Scott, “Grating coupler for efficient excitation of optical guided waves in thin films,” Appl. Phys. Lett. 16, 523–525 (1970).
[CrossRef]

1967 (1)

Y. Y. Teng and E. A. Stern, “Plasma radiation from metal grating surfaces,” Phys. Rev. Lett. 19, 511–514 (1967).
[CrossRef]

1965 (1)

1963 (1)

J. L. Uretsky, “Reflection of a plane sound wave from a sinusoidal surface,” J. Acoust. Soc. Am. 35, 1293–1294 (1963).
[CrossRef]

1957 (1)

J. M. Proud and P. Tamarkin, “Reflection of sound from a surface of saw-tooth profile,” J. Appl. Phys. 28, 1298–1301 (1957).
[CrossRef]

1954 (2)

Agarwal, G. S.

G. S. Agarwal, “Relation between Waterman’s extended boundary condition and the generalized extinction theorem,” Phys. Rev. D 14, 1168–1171 (1976).
[CrossRef]

Armand, G.

G. Armand and J. R. Manson, “Scattering by a hard corrugated wall: an exact solution,” Phys. Rev. B 18, 6510–6518 (1978).
[CrossRef]

Bird, V. M.

M. C. Hutley and V. M. Bird, “A detailed experimental study of the anomalies of a sinusoidal diffraction grating,” Opt. Acta 20, 771–782 (1973).
[CrossRef]

Burnham, R. D.

W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron. QE-13, 134–141 (1977).
[CrossRef]

Cadilhac, M.

M. Neviere, R. Petit, and M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

M. Neviere, P. Vincent, R. Petit, and M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

M. Neviere, P. Vincent, R. Petit, and M. Cadilhac, “Determination of the coupling coefficient of a holographic thin film coupler,” Opt. Commun. 9, 240–244 (1973).
[CrossRef]

Chang, K. C.

Chuang, S. L.

S. L. Chuang and J. A. Kong, “Wave scattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
[CrossRef]

S. L. Chuang and R. K. Johnson, “Acoustic wave scattering from a fluid/solid periodic rough surface,” J. Acoust. Soc. Am. 71, 1368–1376 (1982).
[CrossRef]

S. L. Chuang and J. A. Kong, “Scattering of waves from periodic surfaces,” Proc. IEEE 69, 1132–1144 (1981).
[CrossRef]

Collin, R. E.

R. E. Collin and E. J. Zucker, Antenna Theory, Part II (McGraw-Hill, New York, 1969), Sec. 19.10, pp. 203–208.

Dakss, M. L.

M. L. Dakss, L. Kuhn, P. F. Heidrich, and B. A. Scott, “Grating coupler for efficient excitation of optical guided waves in thin films,” Appl. Phys. Lett. 16, 523–525 (1970).
[CrossRef]

Halevi, P.

P. Halevi, “Polariton modes at the interface between the conducting or dielectric media,” Surf. Sci. 76, 64–90 (1978).
[CrossRef]

Heidrich, P. F.

M. L. Dakss, L. Kuhn, P. F. Heidrich, and B. A. Scott, “Grating coupler for efficient excitation of optical guided waves in thin films,” Appl. Phys. Lett. 16, 523–525 (1970).
[CrossRef]

Hessel, A.

Hope, L. L.

L. L. Hope, “Theory of optical grating couplers,” Opt. Commun. 5, 179–182 (1972).
[CrossRef]

Hutley, M. C.

M. C. Hutley and D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19, 431–436 (1976).
[CrossRef]

M. C. Hutley, J. P. Verrill, R. C. McPhedran, M. Neviere, and P. Vincent, “Presentation and verification of a differential formulation for the diffraction by conducting gratings,” Nouv. Rev. Optique 6, 87–95 (1975).
[CrossRef]

M. C. Hutley, “An experimental study of the anomalies of sinusoidal diffraction gratings,” Opt. Acta 20, 607–624 (1973).
[CrossRef]

M. C. Hutley and V. M. Bird, “A detailed experimental study of the anomalies of a sinusoidal diffraction grating,” Opt. Acta 20, 771–782 (1973).
[CrossRef]

Johnson, R. K.

S. L. Chuang and R. K. Johnson, “Acoustic wave scattering from a fluid/solid periodic rough surface,” J. Acoust. Soc. Am. 71, 1368–1376 (1982).
[CrossRef]

Kong, J. A.

S. L. Chuang and J. A. Kong, “Wave scattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
[CrossRef]

S. L. Chuang and J. A. Kong, “Scattering of waves from periodic surfaces,” Proc. IEEE 69, 1132–1144 (1981).
[CrossRef]

J. A. Kong, Theory of Electromagnetic Waves (Wiley, New York, 1975), Chap. 4, pp. 112–121.

Kuhn, L.

M. L. Dakss, L. Kuhn, P. F. Heidrich, and B. A. Scott, “Grating coupler for efficient excitation of optical guided waves in thin films,” Appl. Phys. Lett. 16, 523–525 (1970).
[CrossRef]

Manson, J. R.

G. Armand and J. R. Manson, “Scattering by a hard corrugated wall: an exact solution,” Phys. Rev. B 18, 6510–6518 (1978).
[CrossRef]

Maystre, D.

D. Maystre, “A new theory for multiprofile, buried gratings,” Opt. Commun. 26, 127–132 (1978).
[CrossRef]

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

M. C. Hutley and D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19, 431–436 (1976).
[CrossRef]

R. C. McPhedran and D. Maystre, “A detailed theoretical study of the anomalies of a sinusoidal diffraction grating,” Opt. Acta 21, 413–421 (1974).
[CrossRef]

McPhedran, R. C.

M. C. Hutley, J. P. Verrill, R. C. McPhedran, M. Neviere, and P. Vincent, “Presentation and verification of a differential formulation for the diffraction by conducting gratings,” Nouv. Rev. Optique 6, 87–95 (1975).
[CrossRef]

R. C. McPhedran and D. Maystre, “A detailed theoretical study of the anomalies of a sinusoidal diffraction grating,” Opt. Acta 21, 413–421 (1974).
[CrossRef]

Neviere, M.

M. C. Hutley, J. P. Verrill, R. C. McPhedran, M. Neviere, and P. Vincent, “Presentation and verification of a differential formulation for the diffraction by conducting gratings,” Nouv. Rev. Optique 6, 87–95 (1975).
[CrossRef]

M. Neviere, P. Vincent, R. Petit, and M. Cadilhac, “Determination of the coupling coefficient of a holographic thin film coupler,” Opt. Commun. 9, 240–244 (1973).
[CrossRef]

M. Neviere, R. Petit, and M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

M. Neviere, P. Vincent, R. Petit, and M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

Oliner, A. A.

Pattanayak, D. N.

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

Petit, R.

M. Neviere, P. Vincent, R. Petit, and M. Cadilhac, “Determination of the coupling coefficient of a holographic thin film coupler,” Opt. Commun. 9, 240–244 (1973).
[CrossRef]

M. Neviere, P. Vincent, R. Petit, and M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

M. Neviere, R. Petit, and M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

Proud, J. M.

J. M. Proud and P. Tamarkin, “Reflection of sound from a surface of saw-tooth profile,” J. Appl. Phys. 28, 1298–1301 (1957).
[CrossRef]

Schultz, L. G.

Schulz, L. G.

Scifres, D. R.

W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron. QE-13, 134–141 (1977).
[CrossRef]

Scott, B. A.

M. L. Dakss, L. Kuhn, P. F. Heidrich, and B. A. Scott, “Grating coupler for efficient excitation of optical guided waves in thin films,” Appl. Phys. Lett. 16, 523–525 (1970).
[CrossRef]

Shah, V.

Stern, E. A.

Y. Y. Teng and E. A. Stern, “Plasma radiation from metal grating surfaces,” Phys. Rev. Lett. 19, 511–514 (1967).
[CrossRef]

Streifer, W.

W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron. QE-13, 134–141 (1977).
[CrossRef]

Tamarkin, P.

J. M. Proud and P. Tamarkin, “Reflection of sound from a surface of saw-tooth profile,” J. Appl. Phys. 28, 1298–1301 (1957).
[CrossRef]

Tamir, T.

K. C. Chang, V. Shah, and T. Tamir, “Scattering and guiding of waves by dielectric gratings with arbitrary profiles,” J. Opt. Soc. Am. 70, 804–813 (1980).
[CrossRef]

K. C. Chang and T. Tamir, “Simplified approach to surface wave scattering by blazed dielectric gratings,” Appl. Opt. 19, 282–288 (1980).
[CrossRef] [PubMed]

K. C. Chang and T. Tamir, “Bragg-reflection approach for blazed dielectric gratings,” Opt. Commun. 26, 327–330 (1978).
[CrossRef]

T. Tamir, “Inhomogeneous wave types at planar interfaces: III. Leaky waves,” Optik 38, 269–297 (1973).

T. Tamir, “Beam and waveguide couplers,” in Integrated Optics, T. Tamir, ed. (Springer-Verlag, New York, 1979).

Tangherlini, F. R.

Teng, Y. Y.

Y. Y. Teng and E. A. Stern, “Plasma radiation from metal grating surfaces,” Phys. Rev. Lett. 19, 511–514 (1967).
[CrossRef]

Twomey, S.

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Direct Measurement (Elsevier, New York, 1977).

Uretsky, J. L.

J. L. Uretsky, “Reflection of a plane sound wave from a sinusoidal surface,” J. Acoust. Soc. Am. 35, 1293–1294 (1963).
[CrossRef]

Van den Berg, P. M.

P. M. Van den Berg, “Diffraction theory of a reflection grating,” Appl. Sci. Res. 24, 261–293 (1971).

Varadan, V. K.

V. K. Varadan and V. V. Varadan, Acoustic, Electromagnetic and Elastic Wave Scattering—Focus on the T-Matrix Approach (Pergamon, New York, 1979).

Varadan, V. V.

V. K. Varadan and V. V. Varadan, Acoustic, Electromagnetic and Elastic Wave Scattering—Focus on the T-Matrix Approach (Pergamon, New York, 1979).

Verrill, J. P.

M. C. Hutley, J. P. Verrill, R. C. McPhedran, M. Neviere, and P. Vincent, “Presentation and verification of a differential formulation for the diffraction by conducting gratings,” Nouv. Rev. Optique 6, 87–95 (1975).
[CrossRef]

Vincent, P.

M. C. Hutley, J. P. Verrill, R. C. McPhedran, M. Neviere, and P. Vincent, “Presentation and verification of a differential formulation for the diffraction by conducting gratings,” Nouv. Rev. Optique 6, 87–95 (1975).
[CrossRef]

M. Neviere, P. Vincent, R. Petit, and M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

M. Neviere, P. Vincent, R. Petit, and M. Cadilhac, “Determination of the coupling coefficient of a holographic thin film coupler,” Opt. Commun. 9, 240–244 (1973).
[CrossRef]

Waterman, P. C.

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

Wolf, E.

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

Zucker, E. J.

R. E. Collin and E. J. Zucker, Antenna Theory, Part II (McGraw-Hill, New York, 1969), Sec. 19.10, pp. 203–208.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

M. L. Dakss, L. Kuhn, P. F. Heidrich, and B. A. Scott, “Grating coupler for efficient excitation of optical guided waves in thin films,” Appl. Phys. Lett. 16, 523–525 (1970).
[CrossRef]

Appl. Sci. Res. (1)

P. M. Van den Berg, “Diffraction theory of a reflection grating,” Appl. Sci. Res. 24, 261–293 (1971).

IEEE J. Quantum Electron. (1)

W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron. QE-13, 134–141 (1977).
[CrossRef]

J. Acoust. Soc. Am. (3)

J. L. Uretsky, “Reflection of a plane sound wave from a sinusoidal surface,” J. Acoust. Soc. Am. 35, 1293–1294 (1963).
[CrossRef]

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

S. L. Chuang and R. K. Johnson, “Acoustic wave scattering from a fluid/solid periodic rough surface,” J. Acoust. Soc. Am. 71, 1368–1376 (1982).
[CrossRef]

J. Appl. Phys. (1)

J. M. Proud and P. Tamarkin, “Reflection of sound from a surface of saw-tooth profile,” J. Appl. Phys. 28, 1298–1301 (1957).
[CrossRef]

J. Opt. Soc. Am. (4)

Nouv. Rev. Optique (1)

M. C. Hutley, J. P. Verrill, R. C. McPhedran, M. Neviere, and P. Vincent, “Presentation and verification of a differential formulation for the diffraction by conducting gratings,” Nouv. Rev. Optique 6, 87–95 (1975).
[CrossRef]

Opt. Acta (3)

R. C. McPhedran and D. Maystre, “A detailed theoretical study of the anomalies of a sinusoidal diffraction grating,” Opt. Acta 21, 413–421 (1974).
[CrossRef]

M. C. Hutley, “An experimental study of the anomalies of sinusoidal diffraction gratings,” Opt. Acta 20, 607–624 (1973).
[CrossRef]

M. C. Hutley and V. M. Bird, “A detailed experimental study of the anomalies of a sinusoidal diffraction grating,” Opt. Acta 20, 771–782 (1973).
[CrossRef]

Opt. Commun. (8)

M. C. Hutley and D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19, 431–436 (1976).
[CrossRef]

K. C. Chang and T. Tamir, “Bragg-reflection approach for blazed dielectric gratings,” Opt. Commun. 26, 327–330 (1978).
[CrossRef]

D. Maystre, “A new theory for multiprofile, buried gratings,” Opt. Commun. 26, 127–132 (1978).
[CrossRef]

M. Neviere, R. Petit, and M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

M. Neviere, P. Vincent, R. Petit, and M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

M. Neviere, P. Vincent, R. Petit, and M. Cadilhac, “Determination of the coupling coefficient of a holographic thin film coupler,” Opt. Commun. 9, 240–244 (1973).
[CrossRef]

L. L. Hope, “Theory of optical grating couplers,” Opt. Commun. 5, 179–182 (1972).
[CrossRef]

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

Optik (1)

T. Tamir, “Inhomogeneous wave types at planar interfaces: III. Leaky waves,” Optik 38, 269–297 (1973).

Phys. Rev. B (1)

G. Armand and J. R. Manson, “Scattering by a hard corrugated wall: an exact solution,” Phys. Rev. B 18, 6510–6518 (1978).
[CrossRef]

Phys. Rev. D (1)

G. S. Agarwal, “Relation between Waterman’s extended boundary condition and the generalized extinction theorem,” Phys. Rev. D 14, 1168–1171 (1976).
[CrossRef]

Phys. Rev. Lett. (1)

Y. Y. Teng and E. A. Stern, “Plasma radiation from metal grating surfaces,” Phys. Rev. Lett. 19, 511–514 (1967).
[CrossRef]

Proc. IEEE (1)

S. L. Chuang and J. A. Kong, “Scattering of waves from periodic surfaces,” Proc. IEEE 69, 1132–1144 (1981).
[CrossRef]

Radio Sci. (1)

S. L. Chuang and J. A. Kong, “Wave scattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
[CrossRef]

Surf. Sci. (1)

P. Halevi, “Polariton modes at the interface between the conducting or dielectric media,” Surf. Sci. 76, 64–90 (1978).
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[CrossRef]

T. Tamir, “Beam and waveguide couplers,” in Integrated Optics, T. Tamir, ed. (Springer-Verlag, New York, 1979).

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S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Direct Measurement (Elsevier, New York, 1977).

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Figures (10)

Fig. 1
Fig. 1

Scattering by one periodic rough interface between two media.

Fig. 2
Fig. 2

Scattering by a periodic surface above some stratifications.

Fig. 3
Fig. 3

Phase-matching diagram: a geometrical construction of the wave vectors of the Floquet waves. A−2A−1 = A−1A0 = A0A1 = 2π/P; B−2B−1 = B−1B0 = B0B1 = 2π/P; OB0 = OA0 = k sin θi. The wave vectors kn+, kn, k1n+, and k1n are constructed in this figure.

Fig. 4
Fig. 4

(a), (b) Comparison between theory (solid lines) and the experimental data (dashed lines) for optical gratings with a sinusoidal profile coated with silver or aluminum where surface plasmons are excited. The wavelength is 476 nm. The total reflected powers show clear dips when the surface plasmons are excited. The period is P = 1205 nm; the height is h = 100 nm. At this wavelength, the permittivity for silver is 1 ≅ (−7.0410 + i0.2781)0, and, for aluminum, it is 1 = (−20.5703 + i5.096)0. (c), (d) Comparison between the theory (solid lines) and the experimental data (dashed lines) for a gold grating with the same profile as in (a) and (b). At λ = 476 nm, 1 ≅ (−2.2272 + i4.123)0, and at λ = 647 nm, 1 ≅ (−11.9914 + i1.3346) 0.

Fig. 5
Fig. 5

The amplitudes of the evanescent waves |bn|2 and the total diffraction efficiency plotted versus the angle of incidence. When the surface plasmons are excited, the evanescent waves are strong, showing peaks in the field amplitudes, and the total efficiency exhibits energy loss into the grating. The grating is made with aluminum, and λ = 476 nm.

Fig. 6
Fig. 6

Same as Fig. 5 except that the grating is silver and the wavelength of the incident wave is at 647 nm.

Fig. 7
Fig. 7

The guiding constants for thin-film waveguides with a flattop surface (2h = 0, solid lines) or a periodic rough surface (2h/d1 = 0.2). The other parameters are d1 = 0.76 μm, P = 0.665 μm; the refractive indices are nf = 1.73 and ns = 1.515. We plot both the real part and the imaginary parts of the guiding constant (β0 + for the periodic surface case) normalized to k for both TE0 and TE1 modes.

Fig. 8
Fig. 8

Same as Fig. 7 except that we plot the guiding constants (β0 + normalized to k) for the TM0 and TM1 modes.

Fig. 9
Fig. 9

Dispersion diagram with Floquet wave β0k relationship used to determine the leaky behavior of the periodic structure. The intersection of the straight line β0 = (2π/P) − kns and the dispersion curve for the TE0 mode gives the wave number k ≃ 3.0 (d1/λ ≃ 0.36) at which the attenuation constant α of the structure starts to appear, as is given in Fig. 7.

Fig. 10
Fig. 10

Diffracted powers for the reflected modes, transmitted modes, and the total diffracted power for one of the gratings in Table 2 for the TE incident wave at the angle of incidence 29–30°. P = 0.606 μm, h = 0.1 μm, d1 = 0.48 μm, d2 = 0.8 μm, λ = 0.6328 μm; 1 = (1.62 + i10−3)20, 2 = 2.4650, 3 = 2.2950.

Tables (2)

Tables Icon

Table 1 Comparison between Theory and Experiment for the Angles of Incidence for Surface-Plasmon Excitations in a Sinusoidal Gratinga

Tables Icon

Table 2 Comparison of the Numerical Results for the Guiding Constants β0 + of Sinusoidal Gratingsa

Equations (67)

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ψ r ( r ) = n = - b n exp ( i k x n x + i k z n z ) ( k z n ) 1 / 2 ,             z > max z ( x )
ψ t ( r ) = n = - A n exp ( i k 1 x n x - i k 1 z n z ) ( ν 1 k 1 z n ) 1 / 2 ,             z < min z ( x ) ,
ν 1 = { 0 / 1 TM polarization μ 0 / μ 1 TE polarization .
k n ± = x ˆ k x n ± z ˆ k z n ,
k 1 n ± = x ˆ k 1 x n ± z ˆ k 1 z n
ψ i ( r ) = n a n exp ( i k n - · r ) ( k z n ) 1 / 2
ψ i , 1 ( r ) = n B n exp ( i k 1 n + · r ) ( ν 1 k 1 z n ) 1 / 2 .
z > z ( x ) , ψ ( r ) z < z ( x ) , 0 } = ψ i ( r ) - d σ · [ G ( r , r ) ψ ( r ) - ψ ( r ) G ( r , r ) ]
z > z ( x ) 0 z < z ( x ) ψ 1 ( r ) } = ψ i , 1 ( r ) + d σ · [ G 1 ( r , r ) ψ 1 ( r ) - ψ 1 ( r ) G 1 ( r , r ) ] ,
d σ = d x [ z ˆ - x ˆ d z ( x ) d x ) ]
G ( r , r ) = n - 1 2 i k z n P exp [ i k x n ( x - x ) + i k z n z - z ] ,
G 1 ( r , r ) = n - 1 2 i k 1 z n P exp [ i k 1 x n ( x - x ) + i k 1 z n z - z ] .
z max z ( x ) , ψ ( r ) z min z ( x ) , 0 } = ψ i ( r ) + { n b n exp ( i k n + · r ) ( k z n ) 1 / 2 , - n a n exp ( i k n - · r ) ( k z n ) 1 / 2 ,
z max z ( x ) , 0 z min z ( x ) , ψ 1 ( r ) } = ψ i , 1 ( r ) + { - n B n exp ( i k 1 n + · r ) ( ν 1 k 1 z n ) 1 / 2 , n A n exp ( i k 1 n - · r ) ( ν 1 k 1 z n ) 1 / 2 ,
b n = 1 2 i P d σ · { exp [ - i k n + · r ( x ) ] ( k z n ) 1 / 2 ψ ( r ) - ψ ( r ) exp [ - i k n + · r ( x ) ] ( k z n ) 1 / 2 } ,
a n = - 1 2 i P d σ · { exp [ - i k n - · r ( x ) ] ( k z n ) 1 / 2 ψ ( r ) - ψ ( r ) exp [ - i k n - · r ( x ) ] ( k z n ) 1 / 2 } ,
B n = ν 1 2 i P d σ · { exp [ - i k 1 n + · r ( x ) ] ( ν 1 k 1 z n ) 1 / 2 ψ 1 ( r ) - ψ 1 ( r ) exp [ - i k 1 n + · r ( x ) ] ( ν 1 k 1 z n ) 1 / 2 } ,
A n = - ν 1 2 i P d σ · { exp [ - i k 1 n - · r ( x ) ] ( ν 1 k 1 z n ) 1 / 2 ψ 1 ( r ) - ψ 1 ( r ) exp [ - i k 1 n - · r ( x ) ] ( ν 1 k 1 z n ) 1 / 2 } .
ψ 1 ( r ) = ψ ( r ) ,
n ˆ · ψ 1 ( r ) = 1 ν 1 n ˆ · ψ ( r ) ,
ψ ( r ) = n 2 α n s exp ( i k x n x ) ,
d σ · ψ ( r ) = - i d x n 2 β n s exp ( i k x n x ) .
[ a B ] = - [ Q D - Q N - Q D 1 + Q N 1 + ] [ β s α s ] ,
[ b A ] = - [ Q D + Q N + Q D 1 - Q N 1 - ] [ β s α s ] ,
[ Q D + ] m n = ± 1 P d x exp [ - i k m ± · r ( x ) ] ( k z m ) 1 / 2 exp ( i k x n x ) ,
[ Q N + ] m n = ± 1 i P { d σ · exp [ - i k m ± · r ( x ) ] ( k z m ) 1 / 2 } exp ( i k x n x ) ,
[ Q D 1 ± ] m n = ± 1 P d x exp [ - i k 1 m ± · r ( x ) ] ( ν 1 k 1 z m ) 1 / 2 exp ( i k x n x ) ,
[ Q N 1 ± ] m n = ± ν 1 i P { d σ · exp [ - i k 1 m ± · r ( x ) ] ( ν 1 k 1 z m ) 1 / 2 } exp ( i k x n x ) ,
[ b A ] = T · [ a B ] ,
T [ Q D + Q N + Q D 1 - Q N 1 - ] [ Q D - Q N - Q D 1 + Q N 1 + ] - 1 .
ψ = n a n exp ( i k n - · r ) ( k z n ) 1 / 2 + n b n exp ( i k n + · r ) ( k z n ) 1 / 2 ,             z h ,
ψ 1 = n A n exp ( i k 1 n - · r ) ( ν 1 k 1 z n ) 1 / 2 + n B n exp ( i k 1 n + · r ) ( ν 1 k 1 z n ) 1 / 2 ,             - d 1 z - h ,
ψ 2 = 2 A n ( 2 ) exp ( i k 2 n - · r ) ( ν 2 k 2 z n ) + n B n ( 2 ) exp ( i k 2 n + · r ) ( ν 2 k 2 z n ) 1 / 2 ,             - ( d 1 + d 2 ) z - d 1 ,
ψ 3 = n A n ( 3 ) exp ( i k 3 n - · r ) ( ν 3 k 3 z n ) ,             z - ( d 1 + d 2 ) ,
k 2 n ± = x ˆ k 2 x n ± z ˆ k 2 z n ,
k 3 n - = x ˆ k 3 x n - z ˆ k 3 z n ,
ν 2 = { 0 / 2 μ 0 / μ 2 ν 3 = { 0 / 3 TM polarization μ 0 / μ 3 TE polariaztion .
k 3 x n = k 2 x n = k 1 x n = k x n = k sin θ i + n 2 π P .
k 2 z n = ( ω 2 μ 2 2 - k 2 x n 2 ) 1 / 2 ,             Im k 2 z n 0 ,
k 3 z n = ( ω 2 μ 3 3 - k 3 x n 2 ) 1 / 2 ,             Im k 3 z n 0 ,
[ S a b S a A S B b S B A ] [ b A ] = [ a B ] ,
[ S a b S a A S B b S B A ] T - 1 = [ Q D - Q N - Q D 1 + Q N 1 + ] [ Q D + Q N + Q D 1 - Q N 1 - ] - 1 .
B n = R n exp ( 2 i k 1 z n d 1 ) A n ,
R n = R 12 ( n ) + R 23 ( n ) exp ( i 2 k 2 z n d 2 ) 1 + R 12 ( n ) R 23 ( n ) exp ( i 2 k 2 z n d 2 ) ,
R 12 ( n ) = ν 1 k 1 z n - ν 2 k 2 z n ν 1 k 1 z n + ν 2 k 2 z n ,
R 23 ( n ) = ν 2 k 2 z n - ν 3 k 3 z n ν 2 k 2 z n + ν 3 k 3 z n ,
B = Λ A ,
Λ = diag [ R n exp ( 2 i k 1 z n d 1 ) ] .
[ S a b S a A S B b S B A - Λ ] [ b A ] = [ a 0 ] .
[ b A ] = [ S a b S a A S B b S B A - Λ ] - 1 [ a 0 ] .
A n ( 3 ) = T n ( ν 3 k 3 z n ) 1 / 2 ( ν 1 k 1 z n ) 1 / 2 exp [ i ( k 1 z n - k 3 z n ) d 1 ] A n ,
T n 4 exp [ i ( k 2 z n - k 3 z n ) d 2 ] ( 1 + ν 2 k 2 z n ν 1 k 1 z n ) ( 1 + ν 3 k 3 z n ν 2 k 2 z n ) [ 1 + R 12 ( n ) R 23 ( n ) exp ( i 2 k 2 z n d 2 ) ]
P r = n b n 2 .
P t = n A n ( 3 ) 2
nonevanes b n 2 + nonenanes A n ( 3 ) 2 = 1
det [ S a b S a A S B b S B A - Λ ] = 0
P t = n Re ( ν 1 k 1 z n ) A n 2 ν 1 k 1 z n exp ( - 2 k 1 z n h ) ,
k x n = k 1 x n = k x 0 + n 2 π P .
k x = k ( 1 1 + 0 ) 2
sin θ i + n λ p = ( 1 1 + 0 ) 1 / 2 .
k 1 z d = tan - 1 α z ν 1 k 1 z + tan - 1 ν 2 α 2 z ν 1 k 1 z + n π ,
k 1 z = ( k 2 n f 2 - β 0 2 ) 1 / 2 ,
α z = ( β 0 2 - k 2 ) 1 / 2 ,
α 2 z = ( β 0 2 - k 2 n s 2 ) 1 / 2 ,
( n f 2 - n s 2 ) 1 / 2 k d = tan - 1 [ ( n s 2 - 1 ) 1 / 2 ν 1 ( n f 2 - n s 2 ) 1 / 2 ] + n π .
Z ( x ) = - h cos 2 π P x ,
[ Q D ± ] m n = ± ( ± i ) m - n ( k z m ) 1 / 2 J m - n ( k z m h ) , [ Q N ± ] m n = ( - k 2 + k x m k x n ± k z m ) [ Q D ± ] m n , [ Q D 1 ± ] m n = ± ( ± i ) m - n ( ν 1 k 1 z m ) 1 / 2 J m - n ( k 1 z m h ) , [ Q N 1 ± ] m n = ν 1 ( - k 1 2 + k 1 x m k 1 x n ± k 1 z m ) [ Q D 1 ± ] m n .