Abstract

Diffraction by an arbitrarily oriented planar grating with slanted fringes is analyzed using rigorous three-dimensional vector coupled-wave analysis. The method applies to any sinusoidal or nonsinusoidal amplitude and/or phase grating, any plane-wave angle of incidence, and any linear polarization. In the resulting (conical) diffraction, it is shown that coupling exists between all space-harmonic vector fields inside the grating (corresponding to diffracted orders outside the grating). Therefore the TE and TM components of an incident wave are each coupled to all the TE and TM components of all the forward- and backward-diffracted waves. For a general Bragg angle of incidence, it is shown that the diffraction efficiency can approach 100% for a lossless grating if either the incident electric field or the magnetic field is perpendicular to the grating vector. Maximum coupling between incident and diffracted waves is shown to occur when the incident electric field is perpendicular to the grating vector. In general, the diffracted waves are shown to be elliptically polarized. The three-dimensional vector coupled-wave analysis presented is shown to reduce to ordinary rigorous coupled-wave theory when the grating vector lies in the plane of incidence.

© 1983 Optical Society of America

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  1. R. R. Aggrawal, "Diffraction of light by ultrasonic waves," Proc. Indian Acad. Sci. 31, 417–426 (1950).
  2. P. Phariseau, "On the diffraction of light by progressive supersonic waves," Proc. Indian Acad. Sci. Sect. A 44, 165–170 (1965).
  3. W. R. Klein and B. D. Cook, "Unified approach to ultrasonic light diffraction," IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).
  4. H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909–2947 (1969).
  5. G. L. Fillmore and R. F. Tynan, "Sensitometric characteristics of hardened dichromated-gelatin films," J. Opt. Soc. Am. 61, 199–203 (1971).
  6. J. A. Kong, "Second-order coupled-mode equations for spatially periodic media," J. Opt. Soc. Am. 67, 825–829 (1977).
  7. R. Magnusson and T. K. Gaylord, "Analysis of multiwave diffraction by thick gratings," J. Opt. Soc. Am. 67, 1165–1170 (1977).
  8. M. G. Moharam and T. K. Gaylord, "Rigorous coupled-wave analysis of planar-grating diffraction," J. Opt. Soc. Am. 71, 811–818 (1981).
  9. M. G. Moharam and T. K. Gaylord, "Rigorous coupled-wave analysis of grating diffraction—E-mode polarization and losses," J. Opt. Soc. Am. 73, 451–455 (1983).
  10. T. Tamir, H. C. Wang, and A. A. Oliner, "Wave propagation in sinusoidally stratified dielectric media," IEEE Trans. Microwave Theory Tech. MTT-12, 323–335 (1964).
  11. T. Tamir and 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).
  12. T. Tamir, "Scattering of electromagnetic waves by a sinusoidally stratified half-space: II. Diffraction aspects at the Rayleigh and Bragg wavelengths," Can. J. Phys. 44, 2461–2494 (1966).
  13. C. B. Burckhardt, "Diffraction of a plane wave at a sinusoidally stratified dielectric grating," J. Opt. Soc. Am. 56, 1502–1509 (1966).
  14. L. Bergstein and D. Kermisch, "Image storage and reconstruction in volume holography," Proc. Symp. Modern Opt. 17, 655–680 (1967).
  15. R. S. Chu and T. Tamir, "Guided wave theory of light diffraction by acoustic microwaves," IEEE Trans. Microwave Theory Tech. MTT-18, 486–504 (1970).
  16. R. S. Chu and T. Tamir, "Wave propagation and dispersion in space—time periodic media," Proc. IEE 119, 797–806 (1972).
  17. F. G. Kaspar, "Diffraction by thick periodically stratified gratings with complex dielectric constant," J. Opt. Soc. Am. 63, 37–45 (1973).
  18. S. T. Peng, T. Tamir, and H. L. Bertoni, "Theory of periodic dielectric wavelengths," IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
  19. R. S. Chu and J. A. Kong, "Modal theory of spatially periodic media," IEEE Trans. Microwave Theory Tech. MTT-25, 18–24 (1977).
  20. M. Neviere, R. Petit, and M. Cadilhac, "About the theory of optical grating coupler-waveguide systems," Opt. Commun. 8, 113–117 (1973).
  21. M. Neviere, P. Vincent, R. Petit, and M. Cadilhac, "Systematic study of resonances of holographic thin film couplers," Opt. Commun. 9, 48–53 (1973).
  22. M. Neviere, P. Vincent, and R. Petit, "Sur la théorie du réseau conducteur et ses applications a l'optique," Nouv. Rev. Opt. 5, 65–77 (1974).
  23. 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).
  24. K. Knop, "Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves," J. Opt. Soc. Am. 68, 1206–1210 (1978).
  25. D. Maystre, "Integral methods" in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980).
  26. 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).
  27. M. G. Moharam and T. K. Gaylord, "Chain-matrix analysis of arbitrary-thickness dielectric reflection gratings," J. Opt. Soc. Am. 72, 187–190 (1982).
  28. T. K. Gaylord and M. G. Moharam, "Planar dielectric grating diffraction theories," Appl. Phys. B 28, 1–14 (1982).
  29. E.g., program EIGRF from the International Mathematics and Statistics Library, Houston, Texas.
  30. E.g., B. Carnahan, H. A. Luther, and J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).
  31. M. G. Moharam and T. K. Gaylord, "Diffraction analysis of dielectric surface-relief gratings," J. Opt. Soc. Am. 72, 1385–1392 (1982).

1983 (1)

1982 (4)

M. G. Moharam and T. K. Gaylord, "Chain-matrix analysis of arbitrary-thickness dielectric reflection gratings," J. Opt. Soc. Am. 72, 187–190 (1982).

M. G. Moharam and T. K. Gaylord, "Diffraction analysis of dielectric surface-relief gratings," J. Opt. Soc. Am. 72, 1385–1392 (1982).

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).

T. K. Gaylord and M. G. Moharam, "Planar dielectric grating diffraction theories," Appl. Phys. B 28, 1–14 (1982).

1981 (1)

1980 (1)

1978 (1)

1977 (3)

1975 (1)

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

1974 (1)

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

1973 (3)

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

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

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

1971 (1)

1970 (1)

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

1969 (1)

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909–2947 (1969).

1967 (2)

W. R. Klein and B. D. Cook, "Unified approach to ultrasonic light diffraction," IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).

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

1966 (3)

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

T. Tamir and 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).

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

1965 (1)

P. Phariseau, "On the diffraction of light by progressive supersonic waves," Proc. Indian Acad. Sci. Sect. A 44, 165–170 (1965).

1964 (1)

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

1950 (1)

R. R. Aggrawal, "Diffraction of light by ultrasonic waves," Proc. Indian Acad. Sci. 31, 417–426 (1950).

Aggrawal, R. R.

R. R. Aggrawal, "Diffraction of light by ultrasonic waves," Proc. Indian Acad. Sci. 31, 417–426 (1950).

Bergstein, L.

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

Bertoni, H. L.

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

Burckhardt, C. B.

Cadilhac, M.

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

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

Carnahan, B.

E.g., B. Carnahan, H. A. Luther, and J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

Chang, K. C.

Chu, R. S.

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

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

R. S. Chu and T. Tamir, "Wave propagation and dispersion in space—time periodic media," Proc. IEE 119, 797–806 (1972).

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).

Cook, B. D.

W. R. Klein and B. D. Cook, "Unified approach to ultrasonic light diffraction," IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).

Fillmore, G. L.

Gaylord, T. K.

Kaspar, F. G.

Kermisch, D.

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

Klein, W. R.

W. R. Klein and B. D. Cook, "Unified approach to ultrasonic light diffraction," IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).

Knop, K.

Kogelnik, H.

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909–2947 (1969).

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).

J. A. Kong, "Second-order coupled-mode equations for spatially periodic media," J. Opt. Soc. Am. 67, 825–829 (1977).

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

Luther, H. A.

E.g., B. Carnahan, H. A. Luther, and J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

Magnusson, R.

Maystre, D.

D. Maystre, "Integral methods" in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980).

Moharam, M. G.

Neviere, M.

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

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

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

Oliner, A. A.

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

Peng, S. T.

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

Petit, R.

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

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

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

Phariseau, P.

P. Phariseau, "On the diffraction of light by progressive supersonic waves," Proc. Indian Acad. Sci. Sect. A 44, 165–170 (1965).

Shah, V.

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).

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

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

T. Tamir and 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).

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

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

R. S. Chu and T. Tamir, "Wave propagation and dispersion in space—time periodic media," Proc. IEE 119, 797–806 (1972).

Tynan, R. F.

Vincent, P.

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

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

Wang, H. C.

T. Tamir and 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).

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

Wilkes, J. O.

E.g., B. Carnahan, H. A. Luther, and J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

Appl. Phys. B (1)

T. K. Gaylord and M. G. Moharam, "Planar dielectric grating diffraction theories," Appl. Phys. B 28, 1–14 (1982).

Bell Syst. Tech. J. (1)

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909–2947 (1969).

Can. J. Phys. (2)

T. Tamir and 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).

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

IEEE Trans. Microwave Theory Tech. (4)

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

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

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

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

IEEE Trans. Sonics Ultrason. (1)

W. R. Klein and B. D. Cook, "Unified approach to ultrasonic light diffraction," IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).

J. Opt. Soc. Am. (11)

G. L. Fillmore and R. F. Tynan, "Sensitometric characteristics of hardened dichromated-gelatin films," J. Opt. Soc. Am. 61, 199–203 (1971).

J. A. Kong, "Second-order coupled-mode equations for spatially periodic media," J. Opt. Soc. Am. 67, 825–829 (1977).

R. Magnusson and T. K. Gaylord, "Analysis of multiwave diffraction by thick gratings," J. Opt. Soc. Am. 67, 1165–1170 (1977).

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

M. G. Moharam and T. K. Gaylord, "Rigorous coupled-wave analysis of grating diffraction—E-mode polarization and losses," J. Opt. Soc. Am. 73, 451–455 (1983).

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

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

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).

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

M. G. Moharam and T. K. Gaylord, "Chain-matrix analysis of arbitrary-thickness dielectric reflection gratings," J. Opt. Soc. Am. 72, 187–190 (1982).

M. G. Moharam and T. K. Gaylord, "Diffraction analysis of dielectric surface-relief gratings," J. Opt. Soc. Am. 72, 1385–1392 (1982).

Nouv. Rev. Opt. (1)

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

Opt. Commun. (2)

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

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

Proc. Indian Acad. Sci. (1)

R. R. Aggrawal, "Diffraction of light by ultrasonic waves," Proc. Indian Acad. Sci. 31, 417–426 (1950).

Proc. Indian Acad. Sci. Sect. A (1)

P. Phariseau, "On the diffraction of light by progressive supersonic waves," Proc. Indian Acad. Sci. Sect. A 44, 165–170 (1965).

Proc. Symp. Modern Opt. (1)

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

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).

Other (4)

D. Maystre, "Integral methods" in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980).

R. S. Chu and T. Tamir, "Wave propagation and dispersion in space—time periodic media," Proc. IEE 119, 797–806 (1972).

E.g., program EIGRF from the International Mathematics and Statistics Library, Houston, Texas.

E.g., B. Carnahan, H. A. Luther, and J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

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