Abstract

An efficient and stable eigenmode method is presented for the analysis of dielectric gratings of arbitrary profiles and thicknesses. The solution is expanded in terms of multiple sets of modes. The method differs from the usual eigenmode methods in that mode couplings are achieved through an iterative one-way wave multiple reflection series requiring only matrix–vector multiplications. The solution is computationally simple, has a physical interpretation, and remains stable for gratings of any thickness. Numerical results agree well with those of previous methods.

© 1991 Optical Society of America

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References

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  1. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  2. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [CrossRef]
  3. S. T. Peng, H. L. Bertoni, T. Tamir, “Analysis of periodic thin-film structures with rectangular profiles,” Opt. Commun. 10, 91–94 (1974).
    [CrossRef]
  4. S. T. Peng, T. Tamir, “Directional blazing of waves guided by asymmetrical dielectric gratings,” Opt. Commun. 11, 405–409 (1974).
    [CrossRef]
  5. S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
    [CrossRef]
  6. D. Marcuse, “Exact theory of TE-wave scattering from blazed dielectric gratings,” Bell Syst. Tech. J. 55, 1295–1317 (1976).
  7. W. Streifer, D. R. Scifres, R. D. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and wavelengths,” IEEE J. Quantum Electron. QE-12, 422–428 (1976).
    [CrossRef]
  8. T. Tamir, S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235–254 (1977).
    [CrossRef]
  9. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,”J. Opt. Soc. Am. 68, 1206–1210 (1978).
    [CrossRef]
  10. K. C. Chang, T. Tamir, “Simplified approach to surface scattering by blazed dielectric gratings,” Appl. Opt. 19, 282–288 (1980).
    [CrossRef] [PubMed]
  11. A. R. Neureuther, K. Zaki, “Numerical methods for the analysis of scattering from nonplanar and periodic structures,” Alta Freq. 38, 282–285 (1969).
  12. A. Wirgin, “A new theoretical approach to scattering from a periodic interface,” Opt. Commun. 27, 189–194 (1978).
    [CrossRef]
  13. D. Maystre, “A new general integral theory for dielectric coated gratings,”J. Opt. Soc. Am. 68, 490–495 (1978).
    [CrossRef]
  14. N. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
    [CrossRef]
  15. N. Nevière, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
    [CrossRef]
  16. D. E. Tremain, K. K. Mei, “Application of the unimoment method to scattering from periodic dielectric structures,”J. Opt. Soc. Am. 68, 775–783 (1978).
    [CrossRef]
  17. W.-H. Lee, W. Streifer, “Radiation loss calculation for corrugated dielectric waveguides,”J. Opt. Soc. Am. 68, 1701–1707 (1978); “Radiation loss calculation for corrugated dielectric waveguides. II. TM polarization,” J. Opt. Soc. Am. 69, 1671–1676 (1979).
    [CrossRef]
  18. K. C. Chang, V. Shah, T. Tamir, “Scattering and guiding of waves by dielectric gratings with arbitrary profiles,”J. Opt. Soc. Am. 70, 804–813 (1980).
    [CrossRef]
  19. R. Magnusson, T. K. Gaylord, “Equivalence of multiwave coupled-wave theory and modal theory of periodic-media diffraction,”J. Opt. Soc. Am. 68, 1777–1779 (1978).
    [CrossRef]
  20. T. Tamir, H. C. Wang, A. A. Oliner, “Wave propagation in sinusoidally stratified dielectric media,” IEEE Trans. Microwave Theory Tech. MTT-12, 323–335 (1964).
    [CrossRef]
  21. T. Tamir, H. C. Wang, “Scattering of electromagnetic waves by a sinusoidally stratified half-space: I. Formal solution and analysis approximations,” Can. J. Phys. 44, 2073–2094 (1966).
    [CrossRef]
  22. T. Tamir, “Scattering of electromagnetic waves by a sinusoidally stratified half-space: II. Diffraction aspects of the Rayleigh and Bragg wavelengths,” Can. J. Phys. 44, 2461–2494 (1966).
    [CrossRef]
  23. C. B. Burckhardt, “Diffraction of a plane wave at a sinusoidally stratified dielectric grating,”J. Opt. Soc. Am. 56, 1502–1509 (1966).
    [CrossRef]
  24. L. Bergstein, D. Kermisch, “Image storage and reconstruction in volume holography,” Proc. Symp. Mod. Opt. 17, 655–680 (1967).
  25. R. S. Chu, T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Tech. MTT-18, 486–504 (1970).
  26. R. S. Chu, T. Tamir, “Wave propagation and dispersion in space-time periodic media,” Proc. IEEE 19, 797–806 (1972).
  27. F. G. Kaspar, “Diffraction by thick periodically stratified gratings with complex dielectric constant,”J. Opt. Soc. Am. 63, 37–45 (1973).
    [CrossRef]
  28. R. S. Chu, J. A. Kong, “Modal theory of spatially periodic media,” IEEE Trans. Microwave Theory Tech. MTT-25, 18–24 (1977).
  29. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar grating diffraction,”J. Opt. Soc. Am. 71, 811–818 (1981).
    [CrossRef]
  30. K. Yasuura, Y. Okuno, “Numerical analysis of diffraction from a grating by the mode-matching method with a smoothing procedure,”J. Opt. Soc. Am. 72, 847–852 (1982).
    [CrossRef]
  31. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,”J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  32. K. A. Awada, “Analysis of dielectric gratings of arbitrary profiles,” Master’s thesis (University of Houston, Houston, Texas, 1990).

1985

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

1982

1981

1980

1978

1977

T. Tamir, S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235–254 (1977).
[CrossRef]

R. S. Chu, J. A. Kong, “Modal theory of spatially periodic media,” IEEE Trans. Microwave Theory Tech. MTT-25, 18–24 (1977).

1976

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

W. Streifer, D. R. Scifres, R. D. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and wavelengths,” IEEE J. Quantum Electron. QE-12, 422–428 (1976).
[CrossRef]

1975

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

1974

S. T. Peng, H. L. Bertoni, T. Tamir, “Analysis of periodic thin-film structures with rectangular profiles,” Opt. Commun. 10, 91–94 (1974).
[CrossRef]

S. T. Peng, T. Tamir, “Directional blazing of waves guided by asymmetrical dielectric gratings,” Opt. Commun. 11, 405–409 (1974).
[CrossRef]

1973

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

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

F. G. Kaspar, “Diffraction by thick periodically stratified gratings with complex dielectric constant,”J. Opt. Soc. Am. 63, 37–45 (1973).
[CrossRef]

1972

R. S. Chu, T. Tamir, “Wave propagation and dispersion in space-time periodic media,” Proc. IEEE 19, 797–806 (1972).

1970

R. S. Chu, T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Tech. MTT-18, 486–504 (1970).

1969

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

1967

L. Bergstein, D. Kermisch, “Image storage and reconstruction in volume holography,” Proc. Symp. Mod. Opt. 17, 655–680 (1967).

1966

T. Tamir, H. C. Wang, “Scattering of electromagnetic waves by a sinusoidally stratified half-space: I. Formal solution and analysis approximations,” Can. J. Phys. 44, 2073–2094 (1966).
[CrossRef]

T. Tamir, “Scattering of electromagnetic waves by a sinusoidally stratified half-space: II. Diffraction aspects of the Rayleigh and Bragg wavelengths,” Can. J. Phys. 44, 2461–2494 (1966).
[CrossRef]

C. B. Burckhardt, “Diffraction of a plane wave at a sinusoidally stratified dielectric grating,”J. Opt. Soc. Am. 56, 1502–1509 (1966).
[CrossRef]

1964

T. Tamir, H. C. Wang, A. A. Oliner, “Wave propagation in sinusoidally stratified dielectric media,” IEEE Trans. Microwave Theory Tech. MTT-12, 323–335 (1964).
[CrossRef]

Awada, K. A.

K. A. Awada, “Analysis of dielectric gratings of arbitrary profiles,” Master’s thesis (University of Houston, Houston, Texas, 1990).

Bergstein, L.

L. Bergstein, D. Kermisch, “Image storage and reconstruction in volume holography,” Proc. Symp. Mod. Opt. 17, 655–680 (1967).

Bertoni, H. L.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

S. T. Peng, H. L. Bertoni, T. Tamir, “Analysis of periodic thin-film structures with rectangular profiles,” Opt. Commun. 10, 91–94 (1974).
[CrossRef]

Burckhardt, C. B.

Burnham, R. D.

W. Streifer, D. R. Scifres, R. D. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and wavelengths,” IEEE J. Quantum Electron. QE-12, 422–428 (1976).
[CrossRef]

Cadilhac, M.

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

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

Chang, K. C.

Chu, R. S.

R. S. Chu, J. A. Kong, “Modal theory of spatially periodic media,” IEEE Trans. Microwave Theory Tech. MTT-25, 18–24 (1977).

R. S. Chu, T. Tamir, “Wave propagation and dispersion in space-time periodic media,” Proc. IEEE 19, 797–806 (1972).

R. S. Chu, T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Tech. MTT-18, 486–504 (1970).

Gaylord, T. K.

Kaspar, F. G.

Kermisch, D.

L. Bergstein, D. Kermisch, “Image storage and reconstruction in volume holography,” Proc. Symp. Mod. Opt. 17, 655–680 (1967).

Knop, K.

Kong, J. A.

R. S. Chu, J. A. Kong, “Modal theory of spatially periodic media,” IEEE Trans. Microwave Theory Tech. MTT-25, 18–24 (1977).

Lee, W.-H.

Magnusson, R.

Marcuse, D.

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

Maystre, D.

Mei, K. K.

Moharam, M. G.

Neureuther, A. R.

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

Nevière, N.

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

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

Okuno, Y.

Oliner, A. A.

T. Tamir, H. C. Wang, A. A. Oliner, “Wave propagation in sinusoidally stratified dielectric media,” IEEE Trans. Microwave Theory Tech. MTT-12, 323–335 (1964).
[CrossRef]

Peng, S. T.

T. Tamir, S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235–254 (1977).
[CrossRef]

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

S. T. Peng, T. Tamir, “Directional blazing of waves guided by asymmetrical dielectric gratings,” Opt. Commun. 11, 405–409 (1974).
[CrossRef]

S. T. Peng, H. L. Bertoni, T. Tamir, “Analysis of periodic thin-film structures with rectangular profiles,” Opt. Commun. 10, 91–94 (1974).
[CrossRef]

Petit, R.

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

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

Scifres, D. R.

W. Streifer, D. R. Scifres, R. D. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and wavelengths,” IEEE J. Quantum Electron. QE-12, 422–428 (1976).
[CrossRef]

Shah, V.

Streifer, W.

Tamir, T.

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

T. Tamir, S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235–254 (1977).
[CrossRef]

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

S. T. Peng, T. Tamir, “Directional blazing of waves guided by asymmetrical dielectric gratings,” Opt. Commun. 11, 405–409 (1974).
[CrossRef]

S. T. Peng, H. L. Bertoni, T. Tamir, “Analysis of periodic thin-film structures with rectangular profiles,” Opt. Commun. 10, 91–94 (1974).
[CrossRef]

R. S. Chu, T. Tamir, “Wave propagation and dispersion in space-time periodic media,” Proc. IEEE 19, 797–806 (1972).

R. S. Chu, T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Tech. MTT-18, 486–504 (1970).

T. Tamir, H. C. Wang, “Scattering of electromagnetic waves by a sinusoidally stratified half-space: I. Formal solution and analysis approximations,” Can. J. Phys. 44, 2073–2094 (1966).
[CrossRef]

T. Tamir, “Scattering of electromagnetic waves by a sinusoidally stratified half-space: II. Diffraction aspects of the Rayleigh and Bragg wavelengths,” Can. J. Phys. 44, 2461–2494 (1966).
[CrossRef]

T. Tamir, H. C. Wang, A. A. Oliner, “Wave propagation in sinusoidally stratified dielectric media,” IEEE Trans. Microwave Theory Tech. MTT-12, 323–335 (1964).
[CrossRef]

Tremain, D. E.

Vincent, P.

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

Wang, H. C.

T. Tamir, H. C. Wang, “Scattering of electromagnetic waves by a sinusoidally stratified half-space: I. Formal solution and analysis approximations,” Can. J. Phys. 44, 2073–2094 (1966).
[CrossRef]

T. Tamir, H. C. Wang, A. A. Oliner, “Wave propagation in sinusoidally stratified dielectric media,” IEEE Trans. Microwave Theory Tech. MTT-12, 323–335 (1964).
[CrossRef]

Wirgin, A.

A. Wirgin, “A new theoretical approach to scattering from a periodic interface,” Opt. Commun. 27, 189–194 (1978).
[CrossRef]

Yasuura, K.

Zaki, K.

A. R. Neureuther, 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, 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, S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235–254 (1977).
[CrossRef]

Bell Syst. Tech. J.

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

Can. J. Phys.

T. Tamir, H. C. Wang, “Scattering of electromagnetic waves by a sinusoidally stratified half-space: I. Formal solution and analysis approximations,” Can. J. Phys. 44, 2073–2094 (1966).
[CrossRef]

T. Tamir, “Scattering of electromagnetic waves by a sinusoidally stratified half-space: II. Diffraction aspects of the Rayleigh and Bragg wavelengths,” Can. J. Phys. 44, 2461–2494 (1966).
[CrossRef]

IEEE J. Quantum Electron.

W. Streifer, D. R. Scifres, R. D. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and wavelengths,” IEEE J. Quantum Electron. QE-12, 422–428 (1976).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

R. S. Chu, T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Tech. MTT-18, 486–504 (1970).

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

T. Tamir, H. C. Wang, A. A. Oliner, “Wave propagation in sinusoidally stratified dielectric media,” IEEE Trans. Microwave Theory Tech. MTT-12, 323–335 (1964).
[CrossRef]

R. S. Chu, J. A. Kong, “Modal theory of spatially periodic media,” IEEE Trans. Microwave Theory Tech. MTT-25, 18–24 (1977).

J. Opt. Soc. Am.

C. B. Burckhardt, “Diffraction of a plane wave at a sinusoidally stratified dielectric grating,”J. Opt. Soc. Am. 56, 1502–1509 (1966).
[CrossRef]

F. G. Kaspar, “Diffraction by thick periodically stratified gratings with complex dielectric constant,”J. Opt. Soc. Am. 63, 37–45 (1973).
[CrossRef]

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

D. E. Tremain, K. K. Mei, “Application of the unimoment method to scattering from periodic dielectric structures,”J. Opt. Soc. Am. 68, 775–783 (1978).
[CrossRef]

K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,”J. Opt. Soc. Am. 68, 1206–1210 (1978).
[CrossRef]

W.-H. Lee, W. Streifer, “Radiation loss calculation for corrugated dielectric waveguides,”J. Opt. Soc. Am. 68, 1701–1707 (1978); “Radiation loss calculation for corrugated dielectric waveguides. II. TM polarization,” J. Opt. Soc. Am. 69, 1671–1676 (1979).
[CrossRef]

R. Magnusson, T. K. Gaylord, “Equivalence of multiwave coupled-wave theory and modal theory of periodic-media diffraction,”J. Opt. Soc. Am. 68, 1777–1779 (1978).
[CrossRef]

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

M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar grating diffraction,”J. Opt. Soc. Am. 71, 811–818 (1981).
[CrossRef]

K. Yasuura, Y. Okuno, “Numerical analysis of diffraction from a grating by the mode-matching method with a smoothing procedure,”J. Opt. Soc. Am. 72, 847–852 (1982).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,”J. Opt. Soc. Am. 72, 1385–1392 (1982).
[CrossRef]

Opt. Commun.

A. Wirgin, “A new theoretical approach to scattering from a periodic interface,” Opt. Commun. 27, 189–194 (1978).
[CrossRef]

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

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

S. T. Peng, H. L. Bertoni, T. Tamir, “Analysis of periodic thin-film structures with rectangular profiles,” Opt. Commun. 10, 91–94 (1974).
[CrossRef]

S. T. Peng, T. Tamir, “Directional blazing of waves guided by asymmetrical dielectric gratings,” Opt. Commun. 11, 405–409 (1974).
[CrossRef]

Proc. IEEE

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

R. S. Chu, T. Tamir, “Wave propagation and dispersion in space-time periodic media,” Proc. IEEE 19, 797–806 (1972).

Proc. Symp. Mod. Opt.

L. Bergstein, D. Kermisch, “Image storage and reconstruction in volume holography,” Proc. Symp. Mod. Opt. 17, 655–680 (1967).

Other

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
[CrossRef]

K. A. Awada, “Analysis of dielectric gratings of arbitrary profiles,” Master’s thesis (University of Houston, Houston, Texas, 1990).

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Figures (10)

Fig. 1
Fig. 1

Rectangular grating and arbitrary-profile grating.

Fig. 2
Fig. 2

Transmission and reflection matrices at a layer interface.

Fig. 3
Fig. 3

One-way wave multiple reflection series.

Fig. 4
Fig. 4

Convergence of the multiple reflection series, transmitted wave (N = 21).

Fig. 5
Fig. 5

Convergence of the multiple reflection series, reflected wave (N = 21).

Fig. 6
Fig. 6

Diffraction by a rectangular grating (N = 21, NT = 19).

Fig. 7
Fig. 7

Diffraction by a rectangular grating of extreme thickness (N = 33, NT = 21).

Fig. 8
Fig. 8

Diffraction by a sinusoidal grating (N = 21, NT = 25).

Fig. 9
Fig. 9

Diffraction by a triangular grating (N = 21, NT = 25).

Fig. 10
Fig. 10

Convergence with respect to layer thickness (N = 21, NT = 25).

Equations (32)

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

[ 2 x 2 + 2 y 2 + s ( x , z ) ] ψ ( x , z ) = 0 ,
s ( x , z ) = ω 2 μ ( x , z ) .
[ d 2 d z 2 + K + S ( z ) ] Ψ k ( z ) = 0 ,
K = [ - k 2 - ( N - 1 ) / 2 0 0 - k 2 ( N - 1 ) / 2 ] , S ( z ) = [ s 0 s - ( N - 1 ) s N - 1 s 0 ] .
s m ( z ) = 1 L - L / 2 L / 2 s ( x , z ) exp ( - i m Δ k x ) d x .
k n = k in + n Δ k ,
k in = ω ( μ 0 ) 1 / 2 sin θ .
K + S = M - 1 E 2 M ,
M - 1 = [ e 1 e N ] ,             E = [ e 1 0 0 e N ] , M = [ e 1 e N ] .
d 2 d z 2 Ψ ( z ) + E 2 Ψ ( z ) = 0.
Ψ ( z ) = M Ψ k ( z ) .
ψ i ( z 2 ) = u i ( z 1 ) exp [ - i e i ( z 2 - z 1 ) ] + d i ( z 1 ) exp [ i e i ( z 2 - z 1 ) ] ,
u ( z 2 ) = L u ( z 2 , z 1 ) u ( z 1 ) ,
L u ( z 2 , z 1 ) = [ exp [ - i e 1 ( z 2 - z 1 ) ] exp [ - i e N ( z 2 - z 1 ) ] ] .
d ( z 2 ) = L d ( z 2 , z 1 ) d ( z 1 ) ,
L d ( z 2 , z 1 ) = [ exp [ i e 1 ( z 2 - z 1 ) ] exp [ i e N ( z 2 - z 1 ) ] ] .
[ u + d + ] = [ C u u C u d C d u C d d ] [ u - d - ] ,
u j - a j = k = 1 N ( u k + b k + d k + b k ) , - i a j u j - a j = k = 1 N ( - i b k u k + b k + i b k d k + b k ) ,
u i + = u j - ( b i + a j 2 b i ) ( b i a j ) ,             d i + = u j - ( b i - a j 2 b i ) ( b i a j ) .
( C u u ) i j = b i + a j 2 b i ( b i a j ) ,             ( C d u ) i j = b i - a j 2 b i ( b i a j ) .
[ 0 d + ] = [ C u u C u d C d u C d d ] [ u - d - ] .
u - = R u d ( z i ) d - ,
R u d ( z i ) = - C u u - 1 C u d .
d + = T d d ( z i ) d - ,
T d d ( z i ) = - C d u C u u - 1 C u d + C d d .
Ψ ( z ) = d 0 ( z ) + u 1 ( z ) + d 2 ( z ) + ,
d 0 ( z ) = P d ( z , z t ) d in ,
P d ( z , z t ) = L d ( z , z 1 ) T d d ( z 1 ) L d ( z 1 , z t ) ,
u 1 ( z ) = z i P u ( z , z i ) R u d ( z i ) d 0 ( z i ) ,
d 2 ( z ) = z i P d ( z , z i ) R d u ( z i ) u 1 ( z i ) = z i P d ( z , z i ) R d u ( z i ) z j P u ( z i , z j ) R u d ( z j ) d 0 ( z j ) ,
d t ( z b ) = d 0 ( z b ) + d 2 ( z b ) + d 4 ( z b ) + .
u r ( z t ) = u 1 ( z t ) + u 3 ( z t ) + u 5 ( z t ) + .

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