Abstract

The rigorous coupled-wave analysis of diffraction by grating(s) formed in general anisotropic media is reviewed and extended. The method is first applied to a single slanted phase and/or amplitude grating with general three-dimensional incidence of a plane wave. The regions external to the grating can be isotropic, uniaxial, or biaxial anisotropic. The cases of gratings in isotropic media and of the grating vector lying in the plane of incidence (scalar analysis) are obtained as limiting cases of this general analysis. Coupling between the two orthogonal polarizations vanishes in these limiting cases. The Bragg conditions for various combinations of ordinary (for isotropic and uniaxial) and extraordinary (for uniaxial) polarized waves are quantified. The analysis is then extended to multiple cascaded gratings and to volume-superposed gratings. Sample calculations are presented for single anisotropic gratings (a lithium niobate photorefractive hologram in air and an interdigitated-electrode-induced grating in an electro-optic crystal), for multiple cascaded gratings (a lithium niobate hologram with grating strength varying with thickness), and for superposed gratings (multiplexed hologram storage). Applications for this analysis include optical storage, switching, modulation, deflection, optical interconnects, beam splitting, beam combining, and data processing.

© 1990 Optical Society of America

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  47. A. Knoesen, T. K. Gaylord, M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar waveguides,” IEEE J. Lightwave Technol. LT-6, 1083–1104 (1988).
    [CrossRef]
  48. A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic propagation at interfaces and in waveguides in uniaxial crystals: surface impedance/admittance approach,” Appl. Phys. B 38, 171–178 (1985).
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  51. M. G. Moharam, T. K. Gaylord, “Chain-matrix analysis of arbitrary-thickness dielectric reflection gratings,” J. Opt. Soc. Am. 72, 187–190 (1982).
    [CrossRef]
  52. R. S. Weis, T. K. Gaylord, “Electromagnetic transmission and reflection characteristics of anisotropic multilayered structures,” J. Opt. Soc. Am. A 3, 1720–1740 (1987).
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    [CrossRef]
  59. M. G. Moharam, T. K. Gaylord, R. Magnusson, “Criteria for Bragg regime diffraction by phase gratings,” Opt. Commun. 32, 14–18 (1980).
    [CrossRef]
  60. E. N. Glytsis, T. K. Gaylord, M. G. Moharam, “Electric field, permittivity, and strain distributions induced by interdigitated electrodes on electro-optic waveguides,” IEEE J. Lightwave Technol. LT-5, 668–683 (1987).
    [CrossRef]

1989 (1)

1988 (3)

H. Lee, “Cross-talk effects in multiplexed volume holograms,” Opt. Lett. 13, 874–876 (1988).
[CrossRef] [PubMed]

E. N. Glytsis, T. K. Gaylord, “Anisotropic guided-wave diffraction by interdigitated-electrode-induced phase gratings,” Appl. Opt. 27, 5035–5050 (1988).
[CrossRef]

A. Knoesen, T. K. Gaylord, M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar waveguides,” IEEE J. Lightwave Technol. LT-6, 1083–1104 (1988).
[CrossRef]

1987 (3)

R. S. Weis, T. K. Gaylord, “Electromagnetic transmission and reflection characteristics of anisotropic multilayered structures,” J. Opt. Soc. Am. A 3, 1720–1740 (1987).
[CrossRef]

E. N. Glytsis, T. K. Gaylord, M. G. Moharam, “Electric field, permittivity, and strain distributions induced by interdigitated electrodes on electro-optic waveguides,” IEEE J. Lightwave Technol. LT-5, 668–683 (1987).
[CrossRef]

E. N. Glytsis, T. K. Gaylord, “Rigorous three-dimensional coupled-wave diffraction analysis of single and cascaded anisotropic gratings,” J. Opt. Soc. Am. A 4, 2061–2080 (1987).
[CrossRef]

1986 (2)

R. V. Johnson, A. R. Tanguay, “Optical beam propagation method for birefringent phase grating diffraction,” Opt. Eng. 25, 235–249 (1986).
[CrossRef]

C. W. Slinger, L. Solymar, “Grating interactions in holograms recorded with two object waves,” Appl. Opt. 25, 3283–3287 (1986).
[CrossRef] [PubMed]

1985 (3)

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

R. S. Weis, T. K. Gaylord, “Lithium niobate: summary of physical properties and crystal structure,” Appl. Phys. A 37, 191–203 (1985).
[CrossRef]

A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic propagation at interfaces and in waveguides in uniaxial crystals: surface impedance/admittance approach,” Appl. Phys. B 38, 171–178 (1985).
[CrossRef]

1984 (2)

P. J. Lin-Chung, S. Teitler, “4 × 4 matrix formalisms for optics in stratified anisotropic media,” J. Opt. Soc. Am. A 1, 703–705 (1984).
[CrossRef]

C. M. Verber, “Integrated-optical approaches to numerical optical computing,” Proc. IEEE 72, 942–953 (1984).
[CrossRef]

1983 (5)

1982 (4)

1981 (1)

1980 (2)

P. D. Bloch, L. Solymar, “Analysis of a 4-port Bragg device,” Proc. Inst. Electr. Eng. Part H 127, 133–137 (1980).

M. G. Moharam, T. K. Gaylord, R. Magnusson, “Criteria for Bragg regime diffraction by phase gratings,” Opt. Commun. 32, 14–18 (1980).
[CrossRef]

1979 (4)

1978 (4)

R. Kowarschik, “Diffraction efficiency of sequentially stored gratings in transmission volume holograms,” Opt. Acta 25, 67–81 (1978).
[CrossRef]

R. Kowarschik, “Diffraction efficiency of sequentially stored gratings in reflection volume holograms,” Opt. Quantum Electron. 10, 171–178 (1978).
[CrossRef]

M. D. Feit, J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990–3998 (1978).
[CrossRef] [PubMed]

V. V. Kazankova, V. I. Protasevich, Y. A. Pryakhin, “Superposition of holograms taking into account the limits of the dynamic range of the photographic layer,” Opt. Spectrosc. (USSR) 44, 324–326 (1978).

1977 (4)

W. J. Burke, P. Sheng, “Crosstalk noise from multiple thick-phase holograms,” J. Appl. Phys. 48, 681–685 (1977).
[CrossRef]

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

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

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

1976 (1)

R. A. Bartolini, A. Bloom, J. S. Escher, “Multiple storage of holograms in an organic medium,” Appl. Phys. Lett. 28, 506–507 (1976).
[CrossRef]

1975 (4)

R. P. Kenan, “Theory of diffraction of guided optical waves by thick holograms,” J. Appl. Phys. 46, 4545–4551 (1975).
[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]

R. Alferness, S. K. Case, “Coupling in doubly exposed, thick holographic gratings,” J. Opt. Soc. Am. 65, 730–739 (1975).
[CrossRef]

S. K. Case, “Coupled-wave theory for multiply exposed thick holographic gratings,” J. Opt. Soc. Am. 65, 724–729 (1975).
[CrossRef]

1974 (1)

1973 (1)

1972 (2)

R. S. Chu, T. Tamir, “Wave propagation and dispersion in space time periodic media,” Proc. Inst. Electr. Eng. 119, 797–806 (1972).
[CrossRef]

D. W. Berreman, “Optics in stratified and anisotropic media,” J. Opt. Soc. Am. 62, 502–510 (1972).
[CrossRef]

1970 (1)

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 (1)

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

1967 (3)

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

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

R. W. Dixon, “Acoustic diffraction of light in anisotropic media,” IEEE J. Quantum Electron. QE-3, 85–93 (1967).
[CrossRef]

1966 (3)

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 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, 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. A 31, 417–426 (1950).

Aggrawal, R. R.

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

Alferness, R.

Bartolini, R. A.

R. A. Bartolini, A. Bloom, J. S. Escher, “Multiple storage of holograms in an organic medium,” Appl. Phys. Lett. 28, 506–507 (1976).
[CrossRef]

Benlarbi, B.

B. Benlarbi, L. Solymar, “The effect of the relative intensity of the reference beam on the reconstructing properties of volume phase holograms,” Opt. Acta 26, 271–278 (1979).
[CrossRef]

Bergstein, L.

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

Berreman, D. W.

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]

Bloch, P. D.

P. D. Bloch, L. Solymar, “Analysis of a 4-port Bragg device,” Proc. Inst. Electr. Eng. Part H 127, 133–137 (1980).

Bloom, A.

R. A. Bartolini, A. Bloom, J. S. Escher, “Multiple storage of holograms in an organic medium,” Appl. Phys. Lett. 28, 506–507 (1976).
[CrossRef]

Burckhardt, C. B.

Burke, W. J.

W. J. Burke, P. Sheng, “Crosstalk noise from multiple thick-phase holograms,” J. Appl. Phys. 48, 681–685 (1977).
[CrossRef]

Case, S. K.

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. Inst. Electr. Eng. 119, 797–806 (1972).
[CrossRef]

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

Cook, B. D.

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

Dixon, R. W.

R. W. Dixon, “Acoustic diffraction of light in anisotropic media,” IEEE J. Quantum Electron. QE-3, 85–93 (1967).
[CrossRef]

Escher, J. S.

R. A. Bartolini, A. Bloom, J. S. Escher, “Multiple storage of holograms in an organic medium,” Appl. Phys. Lett. 28, 506–507 (1976).
[CrossRef]

Feit, M. D.

Fillmore, G. L.

Fleck, J. A.

Gaylord, T. K.

E. N. Glytsis, T. K. Gaylord, “Rigorous 3-D coupled wave diffraction analysis of multiple superposed gratings in anisotropic media,” Appl. Opt. 28, 2401–2421 (1989).
[CrossRef] [PubMed]

E. N. Glytsis, T. K. Gaylord, “Anisotropic guided-wave diffraction by interdigitated-electrode-induced phase gratings,” Appl. Opt. 27, 5035–5050 (1988).
[CrossRef]

A. Knoesen, T. K. Gaylord, M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar waveguides,” IEEE J. Lightwave Technol. LT-6, 1083–1104 (1988).
[CrossRef]

E. N. Glytsis, T. K. Gaylord, “Rigorous three-dimensional coupled-wave diffraction analysis of single and cascaded anisotropic gratings,” J. Opt. Soc. Am. A 4, 2061–2080 (1987).
[CrossRef]

E. N. Glytsis, T. K. Gaylord, M. G. Moharam, “Electric field, permittivity, and strain distributions induced by interdigitated electrodes on electro-optic waveguides,” IEEE J. Lightwave Technol. LT-5, 668–683 (1987).
[CrossRef]

R. S. Weis, T. K. Gaylord, “Electromagnetic transmission and reflection characteristics of anisotropic multilayered structures,” J. Opt. Soc. Am. A 3, 1720–1740 (1987).
[CrossRef]

R. S. Weis, T. K. Gaylord, “Lithium niobate: summary of physical properties and crystal structure,” Appl. Phys. A 37, 191–203 (1985).
[CrossRef]

A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic propagation at interfaces and in waveguides in uniaxial crystals: surface impedance/admittance approach,” Appl. Phys. B 38, 171–178 (1985).
[CrossRef]

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

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

M. G. Moharam, T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 73, 1105–1112 (1983).
[CrossRef]

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

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

M. G. Moharam, T. K. Gaylord, R. Magnusson, “Criteria for Bragg regime diffraction by phase gratings,” Opt. Commun. 32, 14–18 (1980).
[CrossRef]

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

Glytsis, E. N.

E. N. Glytsis, T. K. Gaylord, “Rigorous 3-D coupled wave diffraction analysis of multiple superposed gratings in anisotropic media,” Appl. Opt. 28, 2401–2421 (1989).
[CrossRef] [PubMed]

E. N. Glytsis, T. K. Gaylord, “Anisotropic guided-wave diffraction by interdigitated-electrode-induced phase gratings,” Appl. Opt. 27, 5035–5050 (1988).
[CrossRef]

E. N. Glytsis, T. K. Gaylord, M. G. Moharam, “Electric field, permittivity, and strain distributions induced by interdigitated electrodes on electro-optic waveguides,” IEEE J. Lightwave Technol. LT-5, 668–683 (1987).
[CrossRef]

E. N. Glytsis, T. K. Gaylord, “Rigorous three-dimensional coupled-wave diffraction analysis of single and cascaded anisotropic gratings,” J. Opt. Soc. Am. A 4, 2061–2080 (1987).
[CrossRef]

Hermansson, B.

D. Yevick, B. Hermansson, “Soliton analysis with the propagating beam method,” Opt. Commun. 47, 101–106 (1983).
[CrossRef]

Johnson, R. V.

R. V. Johnson, A. R. Tanguay, “Optical beam propagation method for birefringent phase grating diffraction,” Opt. Eng. 25, 235–249 (1986).
[CrossRef]

Kaminow, I. P.

I. P. Kaminow, An Introduction to Electrooptic Devices (Academic, New York, 1974).

Kaspar, F. G.

Kazankova, V. V.

V. V. Kazankova, V. I. Protasevich, Y. A. Pryakhin, “Superposition of holograms taking into account the limits of the dynamic range of the photographic layer,” Opt. Spectrosc. (USSR) 44, 324–326 (1978).

Kenan, R. P.

R. P. Kenan, “Theory of diffraction of guided optical waves by thick holograms,” J. Appl. Phys. 46, 4545–4551 (1975).
[CrossRef]

Kermisch, D.

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

Klein, W. R.

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

Knoesen, A.

A. Knoesen, T. K. Gaylord, M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar waveguides,” IEEE J. Lightwave Technol. LT-6, 1083–1104 (1988).
[CrossRef]

A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic propagation at interfaces and in waveguides in uniaxial crystals: surface impedance/admittance approach,” Appl. Phys. B 38, 171–178 (1985).
[CrossRef]

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

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

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

Kowarschik, R.

R. Kowarschik, “Diffraction efficiency of sequentially stored gratings in transmission volume holograms,” Opt. Acta 25, 67–81 (1978).
[CrossRef]

R. Kowarschik, “Diffraction efficiency of sequentially stored gratings in reflection volume holograms,” Opt. Quantum Electron. 10, 171–178 (1978).
[CrossRef]

Lee, H.

Lin-Chung, P. J.

Magnusson, R.

M. G. Moharam, T. K. Gaylord, R. Magnusson, “Criteria for Bragg regime diffraction by phase gratings,” Opt. Commun. 32, 14–18 (1980).
[CrossRef]

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

Maystre, D.

D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 85–88.

Moharam, M. G.

A. Knoesen, T. K. Gaylord, M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar waveguides,” IEEE J. Lightwave Technol. LT-6, 1083–1104 (1988).
[CrossRef]

E. N. Glytsis, T. K. Gaylord, M. G. Moharam, “Electric field, permittivity, and strain distributions induced by interdigitated electrodes on electro-optic waveguides,” IEEE J. Lightwave Technol. LT-5, 668–683 (1987).
[CrossRef]

A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic propagation at interfaces and in waveguides in uniaxial crystals: surface impedance/admittance approach,” Appl. Phys. B 38, 171–178 (1985).
[CrossRef]

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

M. G. Moharam, T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 73, 1105–1112 (1983).
[CrossRef]

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

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

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

M. G. Moharam, T. K. Gaylord, R. Magnusson, “Criteria for Bragg regime diffraction by phase gratings,” Opt. Commun. 32, 14–18 (1980).
[CrossRef]

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.

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

Phariseau, P.

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

Protasevich, V. I.

V. V. Kazankova, V. I. Protasevich, Y. A. Pryakhin, “Superposition of holograms taking into account the limits of the dynamic range of the photographic layer,” Opt. Spectrosc. (USSR) 44, 324–326 (1978).

Pryakhin, Y. A.

V. V. Kazankova, V. I. Protasevich, Y. A. Pryakhin, “Superposition of holograms taking into account the limits of the dynamic range of the photographic layer,” Opt. Spectrosc. (USSR) 44, 324–326 (1978).

Rabson, T. A.

Rokushima, K.

Sheng, P.

W. J. Burke, P. Sheng, “Crosstalk noise from multiple thick-phase holograms,” J. Appl. Phys. 48, 681–685 (1977).
[CrossRef]

Slinger, C. W.

Solymar, L.

C. W. Slinger, L. Solymar, “Grating interactions in holograms recorded with two object waves,” Appl. Opt. 25, 3283–3287 (1986).
[CrossRef] [PubMed]

P. D. Bloch, L. Solymar, “Analysis of a 4-port Bragg device,” Proc. Inst. Electr. Eng. Part H 127, 133–137 (1980).

B. Benlarbi, L. Solymar, “The effect of the relative intensity of the reference beam on the reconstructing properties of volume phase holograms,” Opt. Acta 26, 271–278 (1979).
[CrossRef]

Tamir, T.

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

R. S. Chu, T. Tamir, “Wave propagation and dispersion in space time periodic media,” Proc. Inst. Electr. Eng. 119, 797–806 (1972).
[CrossRef]

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 at 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]

Tanguay, A. R.

R. V. Johnson, A. R. Tanguay, “Optical beam propagation method for birefringent phase grating diffraction,” Opt. Eng. 25, 235–249 (1986).
[CrossRef]

Teitler, S.

Thylén, L.

Tittel, F. K.

Tomishima, K.

Tsujinishi, R.

Tsukada, N.

Tynan, R. F.

Verber, C. M.

C. M. Verber, “Integrated-optical approaches to numerical optical computing,” Proc. IEEE 72, 942–953 (1984).
[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]

Weis, R. S.

R. S. Weis, T. K. Gaylord, “Electromagnetic transmission and reflection characteristics of anisotropic multilayered structures,” J. Opt. Soc. Am. A 3, 1720–1740 (1987).
[CrossRef]

R. S. Weis, T. K. Gaylord, “Lithium niobate: summary of physical properties and crystal structure,” Appl. Phys. A 37, 191–203 (1985).
[CrossRef]

Woodbury, D. A.

Yamakita, J.

Yeh, P.

Yevick, D.

Appl. Opt. (6)

Appl. Phys. A (1)

R. S. Weis, T. K. Gaylord, “Lithium niobate: summary of physical properties and crystal structure,” Appl. Phys. A 37, 191–203 (1985).
[CrossRef]

Appl. Phys. B (1)

A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic propagation at interfaces and in waveguides in uniaxial crystals: surface impedance/admittance approach,” Appl. Phys. B 38, 171–178 (1985).
[CrossRef]

Appl. Phys. Lett. (1)

R. A. Bartolini, A. Bloom, J. S. Escher, “Multiple storage of holograms in an organic medium,” Appl. Phys. Lett. 28, 506–507 (1976).
[CrossRef]

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, 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 J. Lightwave Technol. (2)

A. Knoesen, T. K. Gaylord, M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar waveguides,” IEEE J. Lightwave Technol. LT-6, 1083–1104 (1988).
[CrossRef]

E. N. Glytsis, T. K. Gaylord, M. G. Moharam, “Electric field, permittivity, and strain distributions induced by interdigitated electrodes on electro-optic waveguides,” IEEE J. Lightwave Technol. LT-5, 668–683 (1987).
[CrossRef]

IEEE J. Quantum Electron. (1)

R. W. Dixon, “Acoustic diffraction of light in anisotropic media,” IEEE J. Quantum Electron. QE-3, 85–93 (1967).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (4)

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

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

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

J. Appl. Phys. (2)

R. P. Kenan, “Theory of diffraction of guided optical waves by thick holograms,” J. Appl. Phys. 46, 4545–4551 (1975).
[CrossRef]

W. J. Burke, P. Sheng, “Crosstalk noise from multiple thick-phase holograms,” J. Appl. Phys. 48, 681–685 (1977).
[CrossRef]

J. Opt. Soc. Am. (16)

N. Tsukada, R. Tsujinishi, K. Tomishima, “Effects of the relative phase relationships of gratings on diffraction from thick holograms,” J. Opt. Soc. Am. 69, 705–711 (1979).
[CrossRef]

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

D. W. Berreman, “Optics in stratified and anisotropic media,” J. Opt. Soc. Am. 62, 502–510 (1972).
[CrossRef]

P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69, 742–756 (1979).
[CrossRef]

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

G. L. Fillmore, R. F. Tynan, “Sensitometric characteristics of hardened dichromated gelatin films,” J. Opt. Soc. Am. 61, 199–203 (1974).
[CrossRef]

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

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

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

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

M. G. Moharam, T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 73, 1105–1112 (1983).
[CrossRef]

K. Rokushima, J. Yamakita, “Analysis of anisotropic dielectric gratings,” J. Opt. Soc. Am. 73, 901–908 (1983).
[CrossRef]

D. Yevick, L. Thylén, “Analysis of gratings by the beam propagation method,” J. Opt. Soc. Am. 72, 1084–1089 (1982).
[CrossRef]

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

R. Alferness, S. K. Case, “Coupling in doubly exposed, thick holographic gratings,” J. Opt. Soc. Am. 65, 730–739 (1975).
[CrossRef]

S. K. Case, “Coupled-wave theory for multiply exposed thick holographic gratings,” J. Opt. Soc. Am. 65, 724–729 (1975).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Acta (2)

B. Benlarbi, L. Solymar, “The effect of the relative intensity of the reference beam on the reconstructing properties of volume phase holograms,” Opt. Acta 26, 271–278 (1979).
[CrossRef]

R. Kowarschik, “Diffraction efficiency of sequentially stored gratings in transmission volume holograms,” Opt. Acta 25, 67–81 (1978).
[CrossRef]

Opt. Commun. (2)

D. Yevick, B. Hermansson, “Soliton analysis with the propagating beam method,” Opt. Commun. 47, 101–106 (1983).
[CrossRef]

M. G. Moharam, T. K. Gaylord, R. Magnusson, “Criteria for Bragg regime diffraction by phase gratings,” Opt. Commun. 32, 14–18 (1980).
[CrossRef]

Opt. Eng. (1)

R. V. Johnson, A. R. Tanguay, “Optical beam propagation method for birefringent phase grating diffraction,” Opt. Eng. 25, 235–249 (1986).
[CrossRef]

Opt. Lett. (1)

Opt. Quantum Electron. (2)

L. Thylén, “The beam propagation method: an analysis of its applicability,” Opt. Quantum Electron. 15, 433–439 (1983).
[CrossRef]

R. Kowarschik, “Diffraction efficiency of sequentially stored gratings in reflection volume holograms,” Opt. Quantum Electron. 10, 171–178 (1978).
[CrossRef]

Opt. Spectrosc. (USSR) (1)

V. V. Kazankova, V. I. Protasevich, Y. A. Pryakhin, “Superposition of holograms taking into account the limits of the dynamic range of the photographic layer,” Opt. Spectrosc. (USSR) 44, 324–326 (1978).

Proc. IEEE (2)

C. M. Verber, “Integrated-optical approaches to numerical optical computing,” Proc. IEEE 72, 942–953 (1984).
[CrossRef]

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

Proc. Indian Acad. Sci. A (1)

R. R. Aggrawal, “Diffraction of light by ultrasonic waves,” Proc. Indian Acad. Sci. A 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. Inst. Electr. Eng. (1)

R. S. Chu, T. Tamir, “Wave propagation and dispersion in space time periodic media,” Proc. Inst. Electr. Eng. 119, 797–806 (1972).
[CrossRef]

Proc. Inst. Electr. Eng. Part H (1)

P. D. Bloch, L. Solymar, “Analysis of a 4-port Bragg device,” Proc. Inst. Electr. Eng. Part H 127, 133–137 (1980).

Proc. Symp. Mod. Opt. (1)

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

Radio Sci. (1)

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

Other (2)

D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 85–88.

I. P. Kaminow, An Introduction to Electrooptic Devices (Academic, New York, 1974).

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Figures (13)

Fig. 1
Fig. 1

Three-dimensional perspective view of the single anisotropic grating-diffraction geometry.

Fig. 2
Fig. 2

The xy plane (top view) of the single-anisotropic-grating-diffraction problem. The grating vector K, and the O and E, forward- and backward-diffracted wave-vector xy projections are shown.

Fig. 3
Fig. 3

Geometry of forward-diffracted O and E waves, showing the double conical nature of the diffraction with external anisotropic regions. All forward-diffracted wave vectors (i = −1, 0, +1) have the same z component.

Fig. 4
Fig. 4

Various types of Bragg conditions in the uniaxial anisotropic case, (a) OO-type diffraction, (b) EE-type diffraction, (c) OE-type diffraction for negative birefringent material, (d) OE-type diffraction for positive birefringent material, (e) EO-type diffraction for negative birefringent material, (f) EO-type diffraction for positive birefringent material. In all cases the superscripts (1) and (2) correspond to the two possible solutions.

Fig. 5
Fig. 5

The xy plane (top view) of the multiple-cascaded-anisotropic-gratings diffraction problem. The grating vectors Kl, and the ordinary and extraordinary, forward- and backward-diffracted wave-vector xy projections are shown.

Fig. 6
Fig. 6

The xy plane (top view) of the two superposed-grating geometries. The grating vectors K1 and K2, and the ordinary and extraordinary forward- and backward diffracted wave-vector xy projections are shown with their corresponding diffraction angles δ O 1 i 1 i 2 , δ E 1 i 1 i 2 , δ O 3 i 1 i 2, and δ E 1 i 1 i 2.

Fig. 7
Fig. 7

The xy plane projection of the Floquet condition in the case of the two superposed anisotropic gratings. The dashed line represents the y axis of the coordinate system.

Fig. 8
Fig. 8

(a) xy-plane (top view) diffraction geometry of the lithium niobate hologram. (b) Diffraction efficiencies of the 0-order, +1-order, and +2-order forward-diffracted waves, as a function of the grating thickness. (c) Diffraction efficiencies of the 0-order, +1-order, and +2-order backward-diffracted waves as a function of the grating thickness.

Fig. 9
Fig. 9

(a) The xy-plane (top-view) diffraction geometry of the interdigitated-electrode-induced grating. The positions of the O and E wave vectors are as shown for an E incident wave. In the case of an O incident wave these positions are interchanged. Diffraction efficiencies of the forward, 0-order and +1-order Bragg diffracted waves, as a function of the grating thickness: (b) OO-type diffraction, (c) OE-type diffraction for Λ = 1.202 μm, (d) OE-type diffraction for Λ = 3.072 μm.

Fig. 10
Fig. 10

Interdigitated-electrode-induced grating [geometry of Fig. 9(a)] diffraction efficiencies of the forward, 0-order and +1-order Bragg diffracted waves, as a function of the grating thickness: (a) EE-type diffraction, (b) EO-type diffraction for Λ = 0.754 μm, (c) EO-type diffraction for Λ = 4.927 μm.

Fig. 11
Fig. 11

Exponential variation of the amplitude of the space-charge electric field with grating thickness together with a stair-step approximation for N = 5.

Fig. 12
Fig. 12

Diffraction efficiencies as a function of the number of approximating gratings N, with the number of orders M retained in the analysis as a parameter: (a) 0-order backward-diffracted wave, (b) 0-order forward-diffracted wave, (c) +1-order forward-diffracted wave.

Fig. 13
Fig. 13

(a) The xy-plane (top view) diffraction geometry of the two superposed anisotropic gratings in lithium niobate. The positions of the ordinary components of the (0, 0)-, (1, 0)-, (−1, 1)-order foward-diffracted waves are shown. (b)–(e) Diffraction efficiencies of forward-diffracted ordinary waves as functions of the slant angle of the second grating: (b) (0, 0)-order, (c) (1, 0)-order, (d) (−1, 1)-order, (e) (1, 1)-order. The isolated points at ϕ2 = 90 deg (gratings coincidence) correspond to the collapse of an infinite number of double index orders (i1, i2) (where i1 + i2 = i) to a single index order i of the single (in this case) grating.

Equations (138)

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ε ¯ ( x , y ) = h ε ¯ h exp ( j h K · r ) ,
K = K x x ˆ + K y y ˆ = K sin ϕ x ˆ K cos ϕ y ˆ ,
E inc = E ˆ exp ( j k 1 · r ) ,
k 1 = k 1 x x ˆ + k 1 y y ˆ + k 1 z z ˆ = | k 1 | ( cos α sin δ x ˆ cos α cos δ y ˆ sin α z ˆ ) ,
| k 1 | O = k 0 n O 1
| k 1 | E = k 0 n O 1 n E 1 [ n O 1 2 + ( n E 1 2 n O 1 2 ) ( c 1 x cos α sin δ + c 1 y cos α cos δ + c 1 z sin α ) 2 ] 1 / 2 ,
D ˆ = D x x ˆ + D y y ˆ + D z z ˆ = ( cos ψ cos δ sin ψ sin α sin δ ) x ˆ + ( cos ψ sin δ + sin ψ sin α cos δ ) y ˆ + ( sin ψ cos α ) z ˆ .
E = ( ε ¯ 1 ) 1 D ˆ / 0 .
tan ψ O = ( c 1 x cos δ + c 1 y sin δ ) [ sin α ( c 1 x sin δ c 1 y cos δ ) + c 1 z cos α ]
tan ψ E = 1 / tan ψ O .
E 1 = E inc + i [ R O i exp ( j k O 1 i · r ) + R E i exp ( j k E 1 i · r ) ] ,
E 3 = i { T O i exp [ j k O 3 i · ( r + s y ˆ ) ] + T E i exp [ j k E 3 i · ( r + s y ˆ ) ] } .
k O q i = ( k 1 x i K x ) x ˆ + k O y q i y ˆ + k 1 z z ˆ = k x i x ˆ + k O y q i y ˆ + k 1 z z ˆ ,
k E q i = ( k 1 x i K x ) x ˆ + k E y q i y ˆ + k 1 z z ˆ = k x i x ˆ + k E y q i y ˆ + k 1 z z ˆ ,
( k O q i · k O q i ) = k 0 2 n O q 2 , q = 1,3 ,
n O q 2 ( k E q i · k E q i ) + ( n E q 2 n O q 2 ) ( k E q i · c ˆ q ) 2 = k 0 2 n O q 2 n E q 2 , q = 1 , 3 ,
c ˆ q = c q x x ˆ + c q y y ˆ + c q z z ˆ , q = 1,3.
tan δ O q i = r q k x i / k O y q i , tan δ E q i = r q k x i / k E y q i ,
tan α O q i = k 1 z / ( k x i 2 + k O y q i 2 ) 1 / 2 , tan α E q i = k 1 z / ( k x i 2 + k E y q i 2 ) 1 / 2 .
H q = ( j / ω μ 0 ) × E q , q = 1 , 3 ,
E 2 = i [ S x i ( y ) x ˆ + S y i ( y ) y ˆ + S z i ( y ) z ˆ ] × exp [ j ( k x i x i K y y + k 1 z z ) ] ,
H 2 = ( 0 μ 0 ) 1 / 2 i [ U x i ( y ) x ˆ + U y i ( y ) y ˆ + U z i ( y ) z ˆ ] × exp [ j ( k x i x i K y y + k 1 z z ) ] ,
× E 2 = j ω μ 0 H 2 ,
× H 2 = j ω 0 ε ¯ ( x , y ) E 2 .
d S z i d y + j i K y S z i + j k 1 z S y i = j k 0 U x i ,
j k x i S z i j k 1 z S x i = j k 0 U y i ,
d S x i d y j i K y S x i j k x i S y i = j k 0 U z i ,
d U z i d y + j i K y U z i + j k 1 z U y i = j k 0 ρ ( ε x x 1 ρ S x ρ + ε x y i ρ S y ρ + ε x z i ρ S z ρ ) ,
j k x i U z i j k 1 z U x i = j k 0 ρ ( ε y x i ρ S x ρ + ε y y i ρ S y ρ + ε y z i ρ S z ρ ) ,
d U x i d y j i K y U x i j k x i U y i = j k 0 ρ ( ε z x i ρ S x ρ + ε z y i ρ S y ρ + ε z z i ρ S z ρ ) ,
d V ˜ d y = j A ˜ V ˜ ,
V ˜ ( y ) = W ˜ exp [ Λ ˜ y ] C ˜ ,
u x δ i 0 + R O x i + R E x i = S x i ( 0 ) ,
u z δ i 0 + R O z i + R E z i = S z i ( 0 ) ,
( k 1 y u z k 1 z u y ) δ i 0 + k O y 1 i R O z i + k E y 1 i R E z i k 1 z ( R O y i + R E y i ) = k 0 U x i ( 0 ) ,
( k 1 x u y k 1 y u x ) δ i 0 + k x i ( R O y i + R E y i ) k O y 1 i R O x i k E y 1 i R E x i = k 0 U z i ( 0 ) ,
T O x i + T E x i = S x i ( s ) exp ( j i K y s ) = S ˆ x i ( s ) ,
T O z i + T E z i = S z i ( s ) exp ( j i K y s ) = S ˆ z i ( s ) ,
k O y 3 i T O z i + k E y 3 i T E z i k 1 z ( T O y i + T E y i ) = k 0 U x i ( s ) exp ( j i K y s ) = k 0 U ˆ x i ( s ) ,
k x i ( T O y i + T E y i ) k O y 3 i T O x i k E y 3 i T E x i = k 0 U z i ( s ) exp ( j i K y s ) = k 0 U ˆ z i ( s ) .
k O 1 i · ( ε ¯ 1 R O i ) = 0 ,
k E 1 i · ( ε ¯ 1 R E i ) = 0 ,
k O 3 i · ( ε ¯ 3 T O i ) = 0 ,
k E 3 i · ( ε ¯ 3 T E i ) = 0 ,
c ˆ 1 · R O i = 0 ,
c ˆ 1 · ( k E 1 i × R E i ) = 0 ,
c ˆ 3 · T O i = 0 ,
c ˆ 3 · ( k E 3 i × T E i ) = 0.
P ˜ C ˜ = p ˜ ,
D E O q i = τ q Re [ | E O q i | 2 k O y q i * ( E O q i · k O q i ) E O q i y * ] / Re [ k 1 y + ( u ˆ · k 1 ) u y ] ,
D E E q i = τ q Re [ | E E q i | 2 k E y q i * ( E E q i · k E q i ) E E q i y * ] / Re [ k 1 y ( u ˆ · k 1 ) u y ] ,
i ( D E O 1 i + D E E 1 i + D E O 3 i + D E E 3 i ) = 1.
A ˜ = [ K ˜ y 0 ˜ 0 ˜ a ˜ 14 0 ˜ K ˜ y a ˜ 23 0 ˜ 0 ˜ a ˜ 32 K ˜ y 0 ˜ a ˜ 41 0 ˜ 0 ˜ K ˜ y ] ,
d 2 S ˜ x d y 2 = a ˜ 14 a ˜ 41 S ˜ x K ˜ y d S ˜ x d y = B ˜ 1 S ˜ x K ˜ y d S ˜ x d y , d 2 S ˜ z d y 2 = a ˜ 23 a ˜ 32 S ˜ z K ˜ y d S ˜ z d y = B ˜ 2 S ˜ z K ˜ y d S ˜ z d y , d 2 U ˜ x d y 2 = a ˜ 32 a ˜ 23 U ˜ x K ˜ y d U ˜ x d y = B ˜ 3 U ˜ x K ˜ y d U ˜ x d y , d 2 U ˜ z d y 2 = a ˜ 41 a ˜ 14 U ˜ z K ˜ y d U ˜ z d y = B ˜ 4 U ˜ z K ˜ y d U ˜ z d y .
X ˜ u ( y ) = A ˜ 11 ( y ) X ˜ u ( 0 ) + A ˜ 12 ( y ) a ˜ k l Y ˜ w ( 0 ) , u w , X ˜ Y ˜ ,
A ˜ t i = [ 0 ˜ I ˜ B ˜ i K ˜ y ] , exp ( A ˜ t i y ) = [ A ˜ 11 ( y ) A ˜ 12 ( y ) A ˜ 21 ( y ) A ˜ 22 ( y ) ] ,
A ˜ = [ K ˜ y 0 ˜ a ˜ 13 a ˜ 14 0 ˜ K ˜ y a ˜ 23 a ˜ 24 a ˜ 31 a ˜ 32 K ˜ y 0 ˜ a ˜ 41 a ˜ 42 0 ˜ K ˜ y ] ,
d 2 S ˜ x d y 2 = ( a ˜ 13 a ˜ 31 + a ˜ 14 a ˜ 41 ) S ˜ x + ( a ˜ 13 a ˜ 32 + a ˜ 14 a ˜ 42 ) S ˜ z K ˜ y d S ˜ x d y , d 2 S ˜ z d y 2 = ( a ˜ 23 a ˜ 31 + a ˜ 24 a ˜ 41 ) S ˜ x + ( a ˜ 23 a ˜ 32 + a ˜ 24 a ˜ 42 ) S ˜ z K ˜ y d S ˜ z d y , d 2 U ˜ x d y 2 = ( a ˜ 31 a ˜ 13 + a ˜ 32 a ˜ 23 ) U ˜ x + ( a ˜ 31 a ˜ 14 + a ˜ 32 a ˜ 24 ) U ˜ z K ˜ y d U ˜ x d y , d 2 U ˜ z d y 2 = ( a ˜ 41 a ˜ 13 + a ˜ 42 a ˜ 23 ) U ˜ x + ( a ˜ 41 a ˜ 14 + a ˜ 42 a ˜ 24 ) U ˜ z K ˜ y d U ˜ z d y .
X ˜ u ( y ) = A ˜ 11 ( y ) X ˜ u ( 0 ) + A ˜ 12 ( y ) X ˜ w ( 0 ) + A ˜ 13 ( y ) [ a ˜ k 1 l 1 Y ˜ u ( 0 ) + a ˜ k 2 l 2 Y ˜ w ( 0 ) ] + A ˜ 14 ( y ) [ a ˜ k 3 l 3 Y ˜ u ( 0 ) + a ˜ k 4 l 4 Y ˜ w ( 0 ) ] ,
A ˜ t i = [ 0 ˜ 0 ˜ I ˜ 0 ˜ 0 ˜ 0 ˜ 0 ˜ I ˜ B ˜ i 1 B ˜ i 2 K ˜ y 0 ˜ B ˜ i 3 B ˜ i 4 0 ˜ K ˜ y ] , exp ( A ˜ t i y ) = [ A ˜ 11 A ˜ 12 A ˜ 13 A ˜ 14 A ˜ 21 A ˜ 22 A ˜ 23 A ˜ 24 A ˜ 31 A ˜ 32 A ˜ 33 A ˜ 34 A ˜ 41 A ˜ 42 A ˜ 43 A ˜ 44 ] .
E q = i E i q = i ( S x i q x ˆ + S y i q y ˆ + S z i q z ˆ ) exp ( j k i q · r ) ,
H q = i H i q = ( 0 μ 0 ) 1 / 2 i ( U x i q x ˆ + U y i q y ˆ + U z i q z ˆ ) × exp ( j k i q · r ) ,
V ˜ i q = ( S x i q S z i q U x i q U z i q ) = W ˜ i q exp ( Λ ˜ i q y ) C ˜ i q ,
V ˜ q = ( S ˜ x q S ˜ z q U ˜ x q U ˜ z q ) = W ˜ q exp [ Λ ˜ q ( y + s q ) ] C ˜ q ,
W ˜ 1 C ˜ 1 = W ˜ C ˜ ,
W ˜ exp ( Λ ˜ s ) C ˜ = W ˜ 3 C ˜ 3 .
σ 2 , i = σ 0 i K ,
( σ 0 · σ 0 ) = k 0 2 n O 2 ,
n O 2 ( σ 0 · σ 0 ) + ( n E 2 n O 2 ) ( σ 0 · c ˆ ) 2 = k 0 2 n O 2 n E 2 ,
( σ 2 , i · σ 2 , i ) = k 0 2 n O 2 ,
n O 2 ( σ 2 , i · σ 2 , i ) + ( n E 2 n O 2 ) ( σ 2 , i · c ˆ ) 2 = k 0 2 n O 2 n E 2 .
( σ 0 i K ) · ( σ 0 i K ) = k 0 2 n O 2 .
n O 2 ( σ 0 i K ) · ( σ 0 i K ) + ( n E 2 + n O 2 ) [ ( σ 0 i K ) · c ˆ ] 2 = k 0 2 n O 2 n E 2 .
K k x = K k sin ϕ k = K l x = K l sin ϕ l = K x ,
V ˜ l ( y ) = W ˜ l exp [ Λ ˜ l ( y + Ψ l ) ] C ˜ l , l = 1 , 2 , , N ,
D ˜ l 1 , l W ˜ l 1 exp [ Λ ˜ l 1 s l 1 ] C ˜ l 1 = W ˜ l C ˜ l ,
P ˜ C ˜ 1 = p ˜ ,
B ˜ = W ˜ N 1 D ˜ N 1 , N W ˜ N 1 exp ( Λ ˜ N 1 s N 1 ) ] × [ W ˜ N 1 1 D ˜ N 2 , N 1 W ˜ N 2 exp ( Λ ˜ N 2 s N 2 ) ] × [ W ˜ 2 1 D ˜ 1,2 W ˜ 1 exp ( Λ ˜ 1 s 1 ) ]
σ l = σ l 1 i l K l ,
ε ¯ ( x , y ) = ε ¯ 0 + h 1 ε ¯ 1 h 1 exp ( j h 1 K 1 · r ) + h 2 ε ¯ 1 h 1 exp ( j h 2 K 2 · r ) ,
F i = i 1 F i 1 , i i 1 ,
ε x x = ε 2 x x ε E r 11 sin ( K · r ) , ε y y = ε z z = ε 2 y y ε 2 z z ε O r 22 sin ( K · r ) , ε x y = ε y x = ε 2 x y = ε 2 y x r 12 sin ( K · r ) , ε x z = ε z x = ε 2 x z = ε 2 z x 0 , ε y z = ε z y = ε 2 y z = ε 2 z y r 23 sin ( K · r ) ,
ε x x = ε y y = ε 2 x x = ε 2 y y ε O + ( r 11 V / Λ e ) sin ( K · r ) , ε z z = ε 2 z z ε E + ( r 33 V / Λ e ) sin ( K · r ) , ε x y = ε y x = ε 2 x y = ε 2 y x ( r 12 V / Λ e ) cos ( K · r ) , ε x z = ε z x = ε 2 x z = ε 2 z x ( r 13 V / Λ e ) cos ( K · r ) , ε y z = ε z y = ε 2 y z = ε 2 z y 0 ,
ε x x l = ε 2 x x l ε E r 11 F l sin ( K · r ) , ε y y l = ε z z l = ε 2 y y l = ε 2 z z l ε O r 22 F l sin ( K · r ) , ε x y l = ε y x l = ε 2 x y l = ε 2 y x l r 12 F l sin ( K · r ) , ε x z l = ε z x l = ε 2 x z = ε 2 z x l 0 , ε y z l = ε z y l = ε 2 y z l = ε 2 z y l r 23 F l sin ( K · r ) .
ε x x = ε y y = ε 2 x x = ε 2 y y = ε O + ε 11 sin ( K 1 · r ) + ε 11 sin ( K 2 · r ) , ε z z = ε 2 z z = ε E + ε 33 sin ( K 1 · r ) + ε 33 sin ( K 2 · r ) , ε x y = ε y x = ε 2 x y = ε 2 y x = ε 12 cos ( K 1 · r ) + ε 12 cos ( K 2 · r ) , ε x z = ε z x = ε 2 x z = ε 2 z x = ε 13 cos ( K 1 · r ) + ε 13 cos ( K 2 · r ) , ε y z = ε z y = ε 2 y z = ε 2 z y = 0 ,
Δ O q i = k 0 2 n O q 2 k x i 2 k 1 z 2 ,
Δ E q i = k 0 2 ε y y q ε y y q ε x x q ε x y q 2 n O q 2 n E q 2 k x i 2 ε y y q ε z z q ε y z q 2 n O q 2 n E q 2 k 1 z 2 + 2 ε x y q ε y z q ε y y q ε x z q n O q 2 n E q 2 k 1 z k x i ,
A ˜ = [ a ˜ 11 a ˜ 12 a ˜ 13 a ˜ 14 a ˜ 21 a ˜ 22 a ˜ 23 a ˜ 24 a ˜ 31 a ˜ 32 a ˜ 33 a ˜ 34 a ˜ 41 a ˜ 42 a ˜ 43 a ˜ 44 ] ,
( P ˜ 11 ) k n = { E 1 i W 1 , k n + Z 1 i W 2 , k n k 0 W 3 , k n if k O y 1 i k E y 1 i α i W 1 , k n + β i W 2 , k n k 0 W 3 , k n if k O y 1 i = k E y 1 i ,
( P ˜ 12 ) k n = { E 2 i W 1 , k n + Z 2 i W 2 , k n k 0 W 4 , k n if k O y 1 i k E y 1 i γ i W 1 , k n + α i W 2 , k n + k 0 W 4 , k n if k O y 1 i = k E y 1 i ,
( P ˜ 13 ) k n = { ( E 3 i W 1 , k n + Z 3 i W 2 , k n k 0 W 3 , k n ) ω n if k O y 3 i k E y 3 i ( θ i W 1 , k n + μ i W 2 , k n k 0 W 3 , k n ) ω n if k O y 3 i = k E y 3 i ,
( P ˜ 14 ) k n = { ( E 4 i W 1 , k n + Z 4 i W 2 , k n k 0 W 4 , k n ) ω n if k O y 3 i k E y 3 i ( ν i W 1 , k n + θ i W 2 , k n + k 0 W 4 , k n ) ω n if k O y 3 i = k E y 3 i ,
p ˜ T = [ 0 0 p m + 1 0 0 | 0 0 p 3 m + 2 0 0 | 0 0 | 0 0 ] ,
p m + 1 = { Ө 1 if k O y 10 k E y 10 ζ 1 if k O y 10 = k E y 10
p 3 m + 2 = { Ө 2 if k O y 30 k E y 30 η 1 if k O y 30 = k E y 30 .
E 1 i = ( A 1 i / Ξ 1 i ) P 1 i + k 1 z ( l E x i 1 / l E y i 1 ) ,
E 2 i = ( A 1 i / Ξ 1 i ) Q 1 i k x i ( l E x i 1 / l E y i 1 ) k E y 1 i ,
E 3 i = ( A 3 i / Ξ 3 i ) P 3 i + k 1 z ( l E x i 3 / l E y i 3 ) ,
E 4 i = ( A 3 i / Ξ 3 i ) Q 3 i k x i ( l E x i 3 / l E y i 3 ) k E y 3 i ,
Z 1 i = ( B 1 i / Ξ 1 i ) P 1 i + k 1 z ( l E z i 1 / l E y i 1 ) + k E y 1 i ,
Z 2 i = ( B 1 i / Ξ 1 i ) Q 1 i k x i ( l E z i 1 / l E y i 1 ) ,
Z 3 i = ( B 3 i / Ξ 3 i ) P 3 i + k 1 z ( l E z i 3 / l E y i 3 ) + k E y 3 i ,
Z 4 i = ( B 3 i / Ξ 3 i ) Q 3 i k x i ( l E z i 3 / l E y i 3 ) ,
P q i = k 1 z ( l O x i q l O y i q l E x i q l E y i q ) Δ q i + [ k 1 z ( l E z i q l E y i q l O z i q l O y i q ) k O y q i + k E y q i ] Γ q i ,
Q q i = k x i ( l O z i q l O y i q l E z i q l E y i q ) Γ q i + [ k x i ( l E x i q l E y i q l O x i q l O y i q ) k O y q i + k E y q i ] Δ q i ,
Ξ q i = A q i Δ q i B q i Γ q i with q = 1,3
A q i = r q y i l E x i q l E y i q r q x i , B q i = r q y i l E z i q l E y i q r q z i ,
Γ q i = c q x c q y l O x i q l O y i q , Δ q i = c q z c q y l O z i q l O y i q ,
l w i q = k x i ε x w q + k O y q i ε y w q + k 1 z ε z w q ,
l w i q = k x i ε x w q + k E y q i ε y w q + k 1 z ε z w q ,
Ө 1 = k 1 z l E x i 1 l E y i 1 u x + k 1 z u y + ( k E y 1 i k 1 y + k 1 z l E z i 1 l E y i 1 ) u z Π i Ξ 1 i P 1 i ,
Ө 2 = ( k 1 y k x i l E x i 1 l E y i 1 k E y 1 i ) u x k 1 x u y k x i l E z i 1 l E y i 1 u z Π i Ξ 1 i Q 1 i ,
Π i = ( r 1 x i r 1 y i l E x i 1 l E y i 1 ) u x + ( r 1 z i r 1 y i l E z i 1 l E y i 1 ) u z .
α i = k 1 z k x i / k O y 1 i , β i = ( k 1 z 2 / k O y 1 i ) + k O y 1 i , γ i = ( k x i 2 / k O y 1 i ) + k O y 1 i , ζ i = ( k 1 z k x i / k O y 1 i ) u x + k 1 z u y + [ ( k 1 z 2 / k O y 1 i ) + k O y 1 i k 1 y ] u z , η i = [ ( k x i 2 / k O y 1 i ) + k O y 1 i k 1 y ] u x + k 1 x u y + ( k x i k 1 z / k O y 1 i ) u z , θ i = k 1 z k x i / k O y 3 i , μ i = ( k 1 z 2 / k O y 3 i ) + k O y 3 i , ν i = ( k x i 2 / k O y 3 i ) + k O y 3 i .
S ˜ y = ε ˜ y y 1 ε ˜ x y S ˜ x ε ˜ y y 1 ε ˜ y z S ˜ z + k 0 1 ε ˜ y y 1 K ˜ x U ˜ z k 0 1 ε ˜ y y 1 K ˜ z U ˜ x ,
U ˜ y = k 0 1 ( K ˜ z S ˜ x K ˜ x S ˜ z ) .
R O x i = ( Δ 1 i / Ξ 1 i ) [ Π i δ i 0 + A 1 i S x i ( 0 ) + B 1 i S z i ( 0 ) ] ,
R O z i = ( Γ 1 i / Ξ 1 i ) R O x i ,
T O x i = ( Δ 3 i / Ξ 3 i ) [ A 3 i S ˆ x i ( s ) + B 3 i S ˆ z i ( s ) ] ,
T O z i = ( Γ 3 i / Ξ 3 i ) T O x i ,
R O y i = ( l O x i 1 / l O y i 1 ) R O x i ( l O z i 1 / l O y i 1 ) R O z i ,
R E y i = ( l E x i 1 / l E y i 1 ) R E x i ( l E z i 1 / l E y i 1 ) R E z i ,
T O y i = ( l O x i 3 / l O y i 3 ) T O x i ( l O z i 3 / l O y i 3 ) T O z i ,
T E y i = ( l E x i 3 / l E y i 3 ) T E x i ( l E z i 3 / l E y i 3 ) T E z i .
W ˜ q = [ w ˜ 11 q w ˜ 12 q w ˜ 13 q w ˜ 14 q w ˜ 21 q w ˜ 22 q w ˜ 23 q w ˜ 24 q w ˜ 31 q w ˜ 32 q w ˜ 33 q w ˜ 34 q w ˜ 41 q w ˜ 42 q w ˜ 43 q w ˜ 44 q ] ,
Λ i = 2 π i 2 ( σ x sin ϕ σ y cos ϕ ) .
Λ i = 2 π i A E E 2 B E E ,
Λ i = 2 π i A O E B O E ± D O E 1 / 2 ,
Λ i = 2 π i B E O ± D E O 1 / 2 ,
( ε ˜ u υ ) k n = { ( D ˜ u υ ) i j if l 1 = l 2 [ ε ˜ 1 u υ ( l 1 l 2 ) ] i j if l 1 l 2
i = { k Int ( k / M ) if ( k / M ) N k [ Int ( k / M ) 1 ] M if ( k / M ) N ,
j = { n Int ( n / M ) if ( n / M ) N n [ Int ( n / M ) 1 ] M if ( n / M ) N ,
l 1 = { Int ( k / M ) + 1 if ( k / M ) N Int ( k / M ) if ( k / M ) N ,
l 2 = { Int ( n / M ) + 1 if ( n / M ) N Int ( n / M ) if ( n / M ) N ,
I = I ( i 1 , i 2 ) = M ( m + i 1 ) + ( m + 1 ) + i 2 ,
( i 1 , i 2 ) = [ I 1 ( I ) , I 2 ( I ) ] ,
i 1 = I 1 ( I ) = { Int ( I / M ) m 1 if ( I / M ) N Int ( I / M ) m if ( I / M ) N
i 2 = I 2 ( I ) = { I [ Int ( I / M ) 1 ] M m 1 i f ( I / M ) N I Int ( I / M ) M m 1 i f ( I / M ) N ,

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