Abstract

Based on an exact solution of the pertinent boundary-value problem, a method is presented for finding the electromagnetic fields scattered or guided by lossy dielectric gratings having arbitrary profiles. This method unifies the treatment of both perpendicular (TE) and parallel (TM) polarizations by expressing the fields in terms of two coupled first-order differential equations. Their solution is obtained by resorting to difference equations in conjunction with the algorithm of Adams—Moulton, which easily leads to accurate results for a large variety of practical problems. To illustrate the application of this approach, quantitative results are presented for the scattering of plane waves by lossy corrugated structures and for the guiding of (leaky) surface waves by triangular gratings with symmetric or asymmetric profiles.

© 1980 Optical Society of America

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  1. A. R. Neureuther and K. Zaki, "Numerical methods for the analysis of scattering from nonplanar and periodic structures," Alta Freq. 38, 282–285 (1969).
  2. P. M. van den Berg, "Rigorous diffraction theory of optical reflection and transmission gratings," Thesis, Delft University of Technolgy, The Netherlands, Report No. 1971-16, 1971 (unpublished).
  3. N. Nevière, R. Petit, and M. Cadilhac, "About the theory of optical grating coupler-waveguide systems." Opt. Commun. 8, 113–117 (1973).
  4. M. Nevière, P. Vincent, R. Petit, and M. Cadilhac, "Systematic study of resonances of holographic thin film couplers," Opt. Commun. 9, 48–53 (1973).
  5. M. Nevière, P. Vincent, and R. Petit, "Sur la théorie du réseau conducteur et ses applications à l'optique," Nouv. Rev. Opt. 5, 65–77 (1974).
  6. S. T. Peng, T. Tamir, and H. L. Bertoni, "Theory of periodic dielectric waveguides," IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
  7. D. Marcuse, "Exact theory of TE-wave scattering from blazed dielectric gratings," Bell Syst. Tech. J. 55, 1295–1317 (1976).
  8. D. E. Tremain and K. K. Mei, "Application of the unimoment method to scattering from periodic dielectric structures," J. Opt. Soc. Am. 68, 775–783 (1978).
  9. A. Wirgin, "A new theoretical approach to scattering from a periodic interface," Opt. Commun. 27, 189–194 (1978).
  10. D. Maystre, "A new general integral theory for dielectric coated gratings," J. Opt. Soc. Am. 68, 490–495 (1978).
  11. K. C. Chang, "Surface-wave scattering by dielectric gratings with arbitrary profiles," Ph.D. thesis, Polytechnic Institute of New York, Brooklyn, N.Y., 1979 (unpublished).
  12. W.-H. Lee and W. Streifer, "Radiation loss calculation for corrugated dielectric waveguides," J. Opt. Soc. Am. 68, 1701–1707 (1978); 69, 1671–1676 (1979).
  13. K. C. Chang and T. Tamir, "Simplified approach to surface-wave scattering by blazed dielectric gratings," Appl. Opt. 19, 282–288 (1980).
  14. For surface waves, the sign is chosen as in the case of the scattering problem. For leaky waves, the same rule holds except that if n < 0 and βn β 0, the sign must usually be chosen so that the imaginary part of k(u)znis negative. The reader should refer for a detailed discussion of this aspect to Antenna Theory, edited by R. E. Collin and F. J. Zucker (McGraw-Hill, New York, 1969), Sec. 19.10, pp. 203–208.
  15. C. F. Gerald, Applied Numerical Analysis (Addison-Wesley, Reading, Mass., 1973), p. 124.
  16. Reference 15, p. 116.
  17. M. Nevière (personal communication).
  18. D. Maystre and R. Petit, "Brewster incidence for metallic gratings," Opt. Commun. 17, 196–200 (1976).
  19. E. G. Loewen and M. Nevière, "Dielectric coated gratings: A curious property," Appl. Opt. 16, 3009–3011 (1977).
  20. V. Shah and T. Tamir, "Brewster phenomena in lossy structures," Opt. Commun. 23, 113–117 (1977).
  21. T. Tamir and S. T. Peng, "Analysis and design of grating couplers," Appl. Phys. 14, 235–254 (1977).

1980

1978

1977

V. Shah and T. Tamir, "Brewster phenomena in lossy structures," Opt. Commun. 23, 113–117 (1977).

T. Tamir and S. T. Peng, "Analysis and design of grating couplers," Appl. Phys. 14, 235–254 (1977).

E. G. Loewen and M. Nevière, "Dielectric coated gratings: A curious property," Appl. Opt. 16, 3009–3011 (1977).

1976

D. Maystre and R. Petit, "Brewster incidence for metallic gratings," Opt. Commun. 17, 196–200 (1976).

D. Marcuse, "Exact theory of TE-wave scattering from blazed dielectric gratings," Bell Syst. Tech. J. 55, 1295–1317 (1976).

1975

S. T. Peng, T. Tamir, and H. L. Bertoni, "Theory of periodic dielectric waveguides," IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).

1974

M. Nevière, P. Vincent, and R. Petit, "Sur la théorie du réseau conducteur et ses applications à l'optique," Nouv. Rev. Opt. 5, 65–77 (1974).

1973

N. Nevière, R. Petit, and M. Cadilhac, "About the theory of optical grating coupler-waveguide systems." Opt. Commun. 8, 113–117 (1973).

M. Nevière, P. Vincent, R. Petit, and M. Cadilhac, "Systematic study of resonances of holographic thin film couplers," Opt. Commun. 9, 48–53 (1973).

1969

A. R. Neureuther and K. Zaki, "Numerical methods for the analysis of scattering from nonplanar and periodic structures," Alta Freq. 38, 282–285 (1969).

Bertoni, H. L.

S. T. Peng, T. Tamir, and H. L. Bertoni, "Theory of periodic dielectric waveguides," IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).

Cadilhac, M.

N. Nevière, R. Petit, and M. Cadilhac, "About the theory of optical grating coupler-waveguide systems." Opt. Commun. 8, 113–117 (1973).

M. Nevière, P. Vincent, R. Petit, and M. Cadilhac, "Systematic study of resonances of holographic thin film couplers," Opt. Commun. 9, 48–53 (1973).

Chang, K. C.

K. C. Chang and T. Tamir, "Simplified approach to surface-wave scattering by blazed dielectric gratings," Appl. Opt. 19, 282–288 (1980).

K. C. Chang, "Surface-wave scattering by dielectric gratings with arbitrary profiles," Ph.D. thesis, Polytechnic Institute of New York, Brooklyn, N.Y., 1979 (unpublished).

Gerald, C. F.

C. F. Gerald, Applied Numerical Analysis (Addison-Wesley, Reading, Mass., 1973), p. 124.

Lee, W.-H.

Loewen, E. G.

Marcuse, D.

D. Marcuse, "Exact theory of TE-wave scattering from blazed dielectric gratings," Bell Syst. Tech. J. 55, 1295–1317 (1976).

Maystre, D.

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

D. Maystre and R. Petit, "Brewster incidence for metallic gratings," Opt. Commun. 17, 196–200 (1976).

Mei, K. K.

Neureuther, A. R.

A. R. Neureuther and K. Zaki, "Numerical methods for the analysis of scattering from nonplanar and periodic structures," Alta Freq. 38, 282–285 (1969).

Nevière, M.

E. G. Loewen and M. Nevière, "Dielectric coated gratings: A curious property," Appl. Opt. 16, 3009–3011 (1977).

M. Nevière, P. Vincent, and R. Petit, "Sur la théorie du réseau conducteur et ses applications à l'optique," Nouv. Rev. Opt. 5, 65–77 (1974).

M. Nevière, P. Vincent, R. Petit, and M. Cadilhac, "Systematic study of resonances of holographic thin film couplers," Opt. Commun. 9, 48–53 (1973).

M. Nevière (personal communication).

Nevière, N.

N. Nevière, R. Petit, and M. Cadilhac, "About the theory of optical grating coupler-waveguide systems." Opt. Commun. 8, 113–117 (1973).

Peng, S. T.

T. Tamir and S. T. Peng, "Analysis and design of grating couplers," Appl. Phys. 14, 235–254 (1977).

S. T. Peng, T. Tamir, and H. L. Bertoni, "Theory of periodic dielectric waveguides," IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).

Petit, R.

D. Maystre and R. Petit, "Brewster incidence for metallic gratings," Opt. Commun. 17, 196–200 (1976).

M. Nevière, P. Vincent, and R. Petit, "Sur la théorie du réseau conducteur et ses applications à l'optique," Nouv. Rev. Opt. 5, 65–77 (1974).

M. Nevière, P. Vincent, R. Petit, and M. Cadilhac, "Systematic study of resonances of holographic thin film couplers," Opt. Commun. 9, 48–53 (1973).

N. Nevière, R. Petit, and M. Cadilhac, "About the theory of optical grating coupler-waveguide systems." Opt. Commun. 8, 113–117 (1973).

Shah, V.

V. Shah and T. Tamir, "Brewster phenomena in lossy structures," Opt. Commun. 23, 113–117 (1977).

Streifer, W.

Tamir, T.

K. C. Chang and T. Tamir, "Simplified approach to surface-wave scattering by blazed dielectric gratings," Appl. Opt. 19, 282–288 (1980).

T. Tamir and S. T. Peng, "Analysis and design of grating couplers," Appl. Phys. 14, 235–254 (1977).

V. Shah and T. Tamir, "Brewster phenomena in lossy structures," Opt. Commun. 23, 113–117 (1977).

S. T. Peng, T. Tamir, and H. L. Bertoni, "Theory of periodic dielectric waveguides," IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).

Tremain, D. E.

van den Berg, P. M.

P. M. van den Berg, "Rigorous diffraction theory of optical reflection and transmission gratings," Thesis, Delft University of Technolgy, The Netherlands, Report No. 1971-16, 1971 (unpublished).

Vincent, P.

M. Nevière, P. Vincent, and R. Petit, "Sur la théorie du réseau conducteur et ses applications à l'optique," Nouv. Rev. Opt. 5, 65–77 (1974).

M. Nevière, P. Vincent, R. Petit, and M. Cadilhac, "Systematic study of resonances of holographic thin film couplers," Opt. Commun. 9, 48–53 (1973).

Wirgin, A.

A. Wirgin, "A new theoretical approach to scattering from a periodic interface," Opt. Commun. 27, 189–194 (1978).

Zaki, K.

A. R. Neureuther and K. Zaki, "Numerical methods for the analysis of scattering from nonplanar and periodic structures," Alta Freq. 38, 282–285 (1969).

Alta Freq.

A. R. Neureuther and K. Zaki, "Numerical methods for the analysis of scattering from nonplanar and periodic structures," Alta Freq. 38, 282–285 (1969).

Appl. Opt.

Appl. Phys.

T. Tamir and S. T. Peng, "Analysis and design of grating couplers," Appl. Phys. 14, 235–254 (1977).

Bell Syst. Tech. J.

D. Marcuse, "Exact theory of TE-wave scattering from blazed dielectric gratings," Bell Syst. Tech. J. 55, 1295–1317 (1976).

IEEE Trans. Microwave Theory Tech.

S. T. Peng, T. Tamir, and H. L. Bertoni, "Theory of periodic dielectric waveguides," IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).

J. Opt. Soc. Am.

Nouv. Rev. Opt.

M. Nevière, P. Vincent, and R. Petit, "Sur la théorie du réseau conducteur et ses applications à l'optique," Nouv. Rev. Opt. 5, 65–77 (1974).

Opt. Commun.

D. Maystre and R. Petit, "Brewster incidence for metallic gratings," Opt. Commun. 17, 196–200 (1976).

V. Shah and T. Tamir, "Brewster phenomena in lossy structures," Opt. Commun. 23, 113–117 (1977).

A. Wirgin, "A new theoretical approach to scattering from a periodic interface," Opt. Commun. 27, 189–194 (1978).

N. Nevière, R. Petit, and M. Cadilhac, "About the theory of optical grating coupler-waveguide systems." Opt. Commun. 8, 113–117 (1973).

M. Nevière, P. Vincent, R. Petit, and M. Cadilhac, "Systematic study of resonances of holographic thin film couplers," Opt. Commun. 9, 48–53 (1973).

Other

P. M. van den Berg, "Rigorous diffraction theory of optical reflection and transmission gratings," Thesis, Delft University of Technolgy, The Netherlands, Report No. 1971-16, 1971 (unpublished).

K. C. Chang, "Surface-wave scattering by dielectric gratings with arbitrary profiles," Ph.D. thesis, Polytechnic Institute of New York, Brooklyn, N.Y., 1979 (unpublished).

For surface waves, the sign is chosen as in the case of the scattering problem. For leaky waves, the same rule holds except that if n < 0 and βn β 0, the sign must usually be chosen so that the imaginary part of k(u)znis negative. The reader should refer for a detailed discussion of this aspect to Antenna Theory, edited by R. E. Collin and F. J. Zucker (McGraw-Hill, New York, 1969), Sec. 19.10, pp. 203–208.

C. F. Gerald, Applied Numerical Analysis (Addison-Wesley, Reading, Mass., 1973), p. 124.

Reference 15, p. 116.

M. Nevière (personal communication).

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