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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References

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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).
    [Crossref]
  4. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [Crossref]
  5. G. L. Fillmore and R. F. Tynan, “Sensitometric characteristics of hardened dichromated-gelatin films,” J. Opt. Soc. Am. 61, 199–203 (1971).
    [Crossref]
  6. J. A. Kong, “Second-order coupled-mode equations for spatially periodic media,” J. Opt. Soc. Am. 67, 825–829 (1977).
    [Crossref]
  7. R. Magnusson and T. K. Gaylord, “Analysis of multiwave diffraction by thick gratings,” J. Opt. Soc. Am. 67, 1165–1170 (1977).
    [Crossref]
  8. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [Crossref]
  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).
    [Crossref]
  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).
    [Crossref]
  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).
    [Crossref]
  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).
    [Crossref]
  13. C. B. Burckhardt, “Diffraction of a plane wave at a sinusoidally stratified dielectric grating,” J. Opt. Soc. Am. 56, 1502–1509 (1966).
    [Crossref]
  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).
    [Crossref]
  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).
    [Crossref]
  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).
    [Crossref]
  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).
    [Crossref]
  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).
    [Crossref]
  24. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68, 1206–1210 (1978).
    [Crossref]
  25. D. Maystre, “Integral methods” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980).
    [Crossref]
  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).
    [Crossref]
  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).
    [Crossref]
  28. T. K. Gaylord and M. G. Moharam, “Planar dielectric grating diffraction theories,” Appl. Phys. B 28, 1–14 (1982).
    [Crossref]
  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).
    [Crossref]

1983 (1)

1982 (4)

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]

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

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

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

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).
[Crossref]

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).
[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]

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

1972 (1)

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

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).
[Crossref]

1967 (2)

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

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

1966 (3)

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).
[Crossref]

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).
[Crossref]

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

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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

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, “Wave propagation and dispersion in space–time periodic media,” Proc. IEE 119, 797–806 (1972).

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).

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]

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).
[Crossref]

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).
[Crossref]

Knop, K.

Kogelnik, H.

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

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

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).
[Crossref]

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).
[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.

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).
[Crossref]

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).
[Crossref]

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, 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]

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).
[Crossref]

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

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

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, “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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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

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

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

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).
[Crossref]

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]

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

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

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

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).
[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]

Proc. IEE (1)

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

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).
[Crossref]

Other (3)

D. Maystre, “Integral methods” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980).
[Crossref]

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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Figures (7)

Fig. 1
Fig. 1

Geometry of a slanted-fringe planar grating with a plane wave of wave vector k1 incident at an arbitrary angle and with arbitrary linear polarization.

Fig. 2
Fig. 2

Geometry associated with the ith forward-diffracted wave.

Fig. 3
Fig. 3

Geometry of forward-diffracted wave vectors showing conical nature of diffraction. All forward-diffracted waves (i = −1 to i = +2) have wave vectors that are equal in magnitude and have the same y component.

Fig. 4
Fig. 4

Characteristics of first-order (i = +1) forward-diffracted wave for an unslanted-fringe grating (ϕ = 90°) with a plane wave of wavelength λ = 0.500 μm incident at the first Bragg angle (α = 30°) with the grating vector lying on the plane of incidence (δ = 0°). The grating is lossless and has an average relative permittivity of ɛ0 = 2.25 and a relative permittivity modulation of ɛ1 = 0.01. The relative permittivity outside the grating region is the same as the average relative permittivity of the grating. (a) TE component of the normalized diffracted electric-field amplitude for various incident linear polarizations. (b) TM component of the normalized diffracted electric field. (c) Diffraction efficiency of diffracted wave. (d) Angular selectivity for a grating with a thickness of d = 50 μm.

Fig. 5
Fig. 5

Characteristics of first-order (i = +1) forward-diffracted wave for the same unslanted-fringe grating (ϕ = 90°) as in Fig. 4 with a plane wave of wavelength of λ = 0.500 μm incident at the first Bragg angle (α = 35.26°) with the plane of incidence inclined at an angle of δ = 30°. The grating vector is therefore not in the plane of incidence. (a) TE component of the normalized diffracted electric field for various incident linear polarizations. (b) TM component of the normalized diffracted electric field. (c) Diffraction efficiency of diffracted wave. (d) Angular selectivity for a grating with a thickness of d = 50 μm.

Fig. 6
Fig. 6

Normalized diffracted electric-field amplitude for the same grating as in Figs. 4 and 5 for a grating vector both in plane of incidence (δ = 0°) and out of plane of incidence (δ = 30°) for (a) incident electric field perpendicular to the grating vector and (b) incident magnetic field perpendicular to grating vector. The solid and dashed lines are the normalized electric fields. The dotted line is the phase.

Fig. 7
Fig. 7

Diffraction efficiency of first-order (i = +1) backward-diffracted wave for a slanted-fringe (ϕ = 10°) grating with a plane wave of wavelength of λ = 0.500 μm incident at the first Bragg angle (α = 25.87°) with the plane of incidence inclined at an angle of δ = 45°. The grating vector is therefore not in the plane of incidence. The grating is lossless and has an average relative permittivity of ɛ0 = 2.25 and a relative permittivity modulation of 0.01.

Equations (48)

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

ɛ ( x , z ) = h ɛ ˆ h exp ( j h K · r ) .
K = K x x ˆ + K z z ˆ = K sin ϕ x ˆ + K cos ϕ z ˆ ,
E inc = u ˆ exp ( - j k 1 · r ) ,
k 1 = k 1 ( sin α cos δ x ˆ + sin α sin δ y ˆ + cos α z ˆ ) ,
u ˆ = u x x ˆ + u y y ˆ + u z z ˆ = ( cos ψ cos α cos δ - sin ψ sin δ ) x ˆ + ( cos ψ cos α sin δ + sin ψ cos δ ) y ˆ - cos ψ sin α z ˆ ,
E 1 = E inc + i R i exp ( - j k 1 i · r ) ,
E 3 = i T i exp [ - j k 3 i · ( r - d ) ] ,
k l i = [ ( k 1 - i K ) · x ˆ ] x ˆ + [ ( k 1 - i K ) · y ˆ ] y ˆ + k z l i z ˆ = k x i x ˆ + k y y ˆ + k z l i z ˆ ,
k x i = k 1 sin α cos δ - i K sin ϕ ,
k y = k 1 sin α sin δ ,
k z l i = ( k l 2 - k x i 2 - k y 2 ) 1 / 2
tan β l i = ( k x i 2 + k y 2 ) 1 / 2 / k z l i ,
tan δ i = k y / k x i .
H = ( j / ω μ 0 ) × E ,
E 2 = i [ S x i ( z ) x ˆ + S y i ( z ) y ˆ + S z i ( z ) z ˆ ] exp ( - j σ i · r ) ,
H 2 = ( 0 / μ 0 ) 1 / 2 i [ U x i ( z ) x ˆ + U y i ( z ) y ˆ + U z i ( z ) z ˆ ] exp ( - j σ i · r ) ,
σ i = k x i x ˆ + k y y ˆ - i K z z ˆ .
× E 2 = - j ω μ 0 H 2 ,
× H 2 = j ω 0 ɛ ( x , z ) E 2
d S x i ( z ) d z = - j { i K z S x i ( z ) + ( k x i / k ) p a i - p [ k y U x p ( z ) - k x p U y p ( z ) ] + k U y i ( z ) } ,
d S y i ( z ) d z = - j { i K z S y i ( z ) - k U x i ( z ) + ( k y / k ) p a i - p [ k y U x p ( z ) - k x p U y p ( z ) ] } ,
d U x i ( z ) d z = j { ( k x i / k ) [ k y S x i ( z ) - k x i S y i ( z ) ] + k p ɛ ˆ i - p S y p ( z ) - i K z U x i ( z ) } ,
d U y i ( z ) d z = - j { k p ɛ ˆ i - p S x p ( z ) - ( k y / k ) [ k y S x i ( z ) - k x i S y i ( z ) ] + i K z U y i ( z ) } ,
ɛ - 1 ( x , z ) = h a h exp ( j h K · r ) .
[ S ˙ x i S ˙ y i U ˙ x i U ˙ y i ] = [ a 0 c d 0 f g h i j k 0 m n 0 p ] [ S x i S y i U x i U y i ]
V ˙ = AV ,
S x i ( z ) = m C m w 1 , i m exp ( λ m z ) ,
S y i ( z ) = m C m w 2 , i m exp ( λ m z ) ,
U x i ( z ) = m C m w 3 , i m exp ( λ m z ) ,
U y i ( z ) = m C m w 4 , i m exp ( λ m z ) ,
u x δ i 0 + R x i = S x i ( 0 ) ,
u y δ i 0 + R y i = S y i ( 0 ) ,
δ i 0 ( k y u z - k 1 cos α u y ) - k z 1 i R y i + k y R z i = k U x i ( 0 ) ,
δ i 0 ( k 1 cos α u x - k x 0 u z ) + k z 1 i R x i - k x i R z i = k U y i ( 0 ) .
T x i = S x i ( d ) exp ( j i K z d ) ,
T y i = S y i ( d ) exp ( j i K z d ) ,
- k z 3 i T y i + k y T z i = k U x i ( d ) exp ( j i K z d ) ,
k z 3 i T x i - k x i T z i = k U y i ( d ) exp ( j i K z d ) .
k x i R x i + k y R y i + k z 1 i R z i = 0 ,
k x i T x i + k y T y i + k z 3 i T z i = 0.
DE 1 i = - Re ( k z 1 i / k 1 cos ) R i 2 ,
DE 3 i = Re ( k z 3 i / k 1 cos ) T i 2 ,
i ( DE 1 i + DE 3 i ) = 1.
R i ( ψ ) = [ sin ( ψ 2 - ψ ) R i ( ψ 1 ) - sin ( ψ 1 - ψ ) R i ( ψ 2 ) ] / sin ( ψ 2 - ψ 1 ) ,
T i ( ψ ) = [ sin ( ψ 2 - ψ ) T i ( ψ 1 ) - sin ( ψ 1 - ψ ) T i ( ψ 2 ) ] / sin ( ψ 2 - ψ 1 ) .
m ( 2 / K 2 ) [ k x 0 K x + Re ( k 2 ɛ ˆ 0 - k x 0 2 - k y 2 ) 1 / 2 K z ]
m [ 2 ( ɛ 1 ) 1 / 2 Λ / λ ] [ sin α sin ϕ cos δ + ( ɛ 0 / ɛ I - sin 2 α ) 1 / 2 cos ϕ ] .
tan ψ = cos α cot δ - sin α csc δ cot ϕ .