Abstract

The diffraction by one or an arbitrary number of cascaded anisotropic planar gratings with slanted fringes is analyzed by using rigorous three-dimensional vector coupled-wave theory. Arbitrary angle of incidence and polarization are treated. The existence of uniaxial external regions and the treatment of both phase and amplitude anisotropic slanted gratings are included in the analysis. The anisotropy and the three-dimensionality of the problem cause coupling between orthogonally polarized waves. The Bragg conditions for various combinations of ordinary (O) and extraordinary (E) polarized waves are quantified. Sample calculations are presented for single anisotropic gratings (a lithium niobate hologram in air and an interdigitated-electrode-induced electro-optic grating in an optical waveguide), for two cascaded anisotropic gratings (a pair of interdigitated-electrode-induced gratings satisfying the OOO forward Bragg condition, the EEE forward Bragg condition, and the OOO backward Bragg condition), and for multiple cascaded gratings (a lithium niobate hologram with depth modulation). The same analysis applies in the limiting cases of isotropic materials, a grating vector lying in the plane of incidence, etc. Applications for this analysis include optical storage, switching, modulation, deflection, and data processing.

© 1987 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. W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction,”IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).
    [CrossRef]
  3. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
  4. P. Phariseau, “On the diffraction of light by progressive supersonic waves,” Proc. Indian Acad. Sci. Sect. A 44, 165–170 (1965).
  5. G. L. Fillmore, R. F. Tynan, “Sensitometric characteristics of hardened dichromated gelatin films,”J. Opt. Soc. Am. 61, 199–203 (1974).
    [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, T. K. Gaylord, “Analysis of multiwave diffraction of thick gratings,”J. Opt. Soc. Am. 67, 1165–1170 (1977).
    [CrossRef]
  8. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,”J. Opt. Soc. Am. 71, 811–818 (1981).
    [CrossRef]
  9. 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]
  10. 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]
  11. K. Rokushima, J. Yamakita, “Analysis of anisotropic dielectric gratings,”J. Opt. Soc. Am. 73, 901–908 (1983).
    [CrossRef]
  12. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [CrossRef]
  13. 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]
  14. 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]
  15. 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]
  16. C. B. Burckhardt, “Diffraction of a plane wave at a sinusoidally stratified dielectric grating,”J. Opt. Soc. Am. 56, 1502–1509 (1966).
    [CrossRef]
  17. L. Bergstein, D. Kermisch, “Image storage and reconstruction in volume holography,” Proc. Symp. Mod. Opt. 17, 655–680 (1967).
  18. R. S. Chu, T. Tamir, “Guided wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Tech. MTT-18, 486–504 (1970).
  19. R. S. Chu, T. Tamir, “Wave propagation and dispersion in space time periodic media,” Proc. IEE, 119, 797–806 (1972).
  20. F. G. Kaspar, “Diffraction by thick periodically stratified gratings with complex dielectric constant,”J. Opt. Soc. Am. 63, 37–45 (1973).
    [CrossRef]
  21. S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric wavelengths,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
    [CrossRef]
  22. R. S. Chu, J. A. Kong, “Modal theory of spatially periodic media,” IEEE Trans. Microwave Theory Tech. MTT-25, 18–24 (1977).
  23. D. Yevick, L. Thylén, “Analysis of gratings by the beam-propagation method,”J. Opt. Soc. Am. 72, 1084–1089 (1982).
    [CrossRef]
  24. L. Thylén, D. Yevick, “Beam propagation method in anisotropic media,” Appl. Opt. 21, 2751–2754 (1982).
    [CrossRef] [PubMed]
  25. D. Yevick, B. Hermansson, “Soliton analysis with the propagating beam method,” Opt. Commun. 47, 101–106 (1983).
    [CrossRef]
  26. L. Thylén, “The beam propagation method: an analysis of its applicability,” Opt. Quantum Electron. 15, 433–439 (1983).
    [CrossRef]
  27. R. V. Johnson, A. R. Tanguay, “Optical beam propagation method for birefringent phase grating diffraction,” Opt. Eng. 25, 235–249 (1986).
    [CrossRef]
  28. C. M. Verber, “Integrated-optical approaches to numerical optical computing,” Proc. IEEE 72, 942–953 (1984).
    [CrossRef]
  29. M. M. Mirsalehi, T. K. Gaylord, “Truth-table look-up parallel data processing using an optical content-addressable memory,” Appl. Opt. 25, 2277–2283 (1986).
    [CrossRef] [PubMed]
  30. 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]
  31. D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 85–88.
  32. 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]
  33. M. G. Moharam, T. K. Gaylord, “Chain-matrix analysis of arbitrary thickness dielectric reflection gratings,”J. Opt. Soc. Am. 72, 187–190 (1982).
    [CrossRef]
  34. R. S. Weis, T. K. Gaylord, “Rigorous analysis of birefringent networks,” J. Opt. Soc. Am. A 3, (13), P105 (1986).
  35. T. K. Gaylord, M. G. Moharam, “Planar dielectric grating diffraction theories,” Appl. Phys. B 28, 1–14 (1982).
    [CrossRef]
  36. Program eigcc from the International Mathematics and Statistics Library, Houston, Texas.
  37. R. W. Dixon, “Acoustic diffraction of light in anisotropic media,” IEEE J. Quantum Electron. QE-3, 85–93 (1967).
    [CrossRef]
  38. P. Yeh, “Electromagnetic propagation in birefringent layered media,”J. Opt. Soc. Am. 69, 742–756 (1979).
    [CrossRef]
  39. I. P. Kaminow, An Introduction to Electrooptic Devices (Academic, New York, 1974).
  40. R. S. Weis, T. K. Gaylord, “Lithium niobate: summary of physical properties and crystal structure,” Appl. Phys. A 37, 191–203 (1985).
    [CrossRef]
  41. M. G. Moharam, T. K. Gaylord, R. Magnusson, “Criteria for Bragg regime diffraction by phase gratings,” Opt. Commun. 32, 14–18 (1980).
    [CrossRef]
  42. 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]
  43. Program leq2c from the International Mathematics and Statistics Library, Houston, Texas.
  44. M. M. Mirsalehi, T. K. Gaylord, E. I. Verriest, “Integrated optical Givens rotation device,” Appl. Opt. 25, 1608–1614 (1986).
    [CrossRef] [PubMed]

1987 (1)

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]

1986 (4)

M. M. Mirsalehi, T. K. Gaylord, E. I. Verriest, “Integrated optical Givens rotation device,” Appl. Opt. 25, 1608–1614 (1986).
[CrossRef] [PubMed]

M. M. Mirsalehi, T. K. Gaylord, “Truth-table look-up parallel data processing using an optical content-addressable memory,” Appl. Opt. 25, 2277–2283 (1986).
[CrossRef] [PubMed]

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

R. S. Weis, T. K. Gaylord, “Rigorous analysis of birefringent networks,” J. Opt. Soc. Am. A 3, (13), P105 (1986).

1985 (3)

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]

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

1984 (1)

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

1983 (5)

1982 (5)

1981 (1)

1980 (1)

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

1979 (1)

1977 (3)

1975 (1)

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

1974 (1)

1973 (1)

1972 (1)

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

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

Bertoni, H. L.

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

Burckhardt, C. B.

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. IEE, 119, 797–806 (1972).

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

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]

Fillmore, G. L.

Gaylord, T. K.

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]

M. M. Mirsalehi, T. K. Gaylord, E. I. Verriest, “Integrated optical Givens rotation device,” Appl. Opt. 25, 1608–1614 (1986).
[CrossRef] [PubMed]

R. S. Weis, T. K. Gaylord, “Rigorous analysis of birefringent networks,” J. Opt. Soc. Am. A 3, (13), P105 (1986).

M. M. Mirsalehi, T. K. Gaylord, “Truth-table look-up parallel data processing using an optical content-addressable memory,” Appl. Opt. 25, 2277–2283 (1986).
[CrossRef] [PubMed]

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

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

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]

T. K. Gaylord, M. G. Moharam, “Planar dielectric grating diffraction theories,” Appl. Phys. B 28, 1–14 (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 of thick gratings,”J. Opt. Soc. Am. 67, 1165–1170 (1977).
[CrossRef]

Glytsis, E. N.

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]

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.

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, 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]

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 of 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.

Mirsalehi, M. M.

Moharam, M. G.

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, “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]

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

Rokushima, K.

Tamir, T.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric wavelengths,” 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. IEE, 119, 797–806 (1972).

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

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

T. Tamir, “Scattering of electromagnetic waves by a sinusoidally stratified half space: II. Diffraction aspects 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]

Thylén, L.

Tynan, R. F.

Verber, C. M.

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

Verriest, E. I.

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, “Rigorous analysis of birefringent networks,” J. Opt. Soc. Am. A 3, (13), P105 (1986).

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

Yamakita, J.

Yeh, P.

Yevick, D.

Appl. Opt. (3)

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 (2)

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, “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).

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. (1)

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 wavelengths,” 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).

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

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. Opt. Soc. Am. (12)

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]

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[CrossRef]

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[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

A three-dimensional perspective view of the diffraction geometry of multiple, cascaded, anisotropic gratings.

Fig. 2
Fig. 2

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

Fig. 3
Fig. 3

Geometry of forward 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 diffraction, (b) EE diffraction, (c) OE diffraction for negative birefringent material, (d) OE diffraction for positive birefringent material, (e) EO diffraction for negative birefringent material, and (f) EO diffraction for positive birefringent material. In all cases the superscripts (1) and (2) correspond to the two possible solutions.

Fig. 5
Fig. 5

(a) The xy-plane (top-view) diffraction geometry of the LiNbO3 hologram. (b) The diffraction efficiencies of the 0-order, +1-order, and +2-order forward-diffracted waves, as functions of the grating thickness. (c) The diffraction efficiencies of the 0-order, +-order, and +2-order backward-diffracted waves as functions of the grating thickness.

Fig. 6
Fig. 6

(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. (b)–(d) Diffraction efficiencies of the zero-order and first-order Bragg forward-diffracted waves, as functions of the grating thickness for (b) OO diffraction, (c) OE diffraction for Λ1 = 1.202 λm, and (d) OE diffraction for Λ1 = 3.072 μm.

Fig. 7
Fig. 7

Interdigitated-electrode-induced grating [geometry of Fig. 6(a)] diffraction efficiencies of the zero-order and first-order Bragg forward-diffracted waves, as a function of the grating thickness for (a) EE diffraction, (b) EO diffraction for Λ1 = 0.754 μm, and (c) EO diffraction for Λ1 = 4.927 μm.

Fig. 8
Fig. 8

(a) The xy-plane (top-view) geometry of two cascaded anisotropic gratings for forward Bragg diffraction. 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. (b) Diffraction efficiencies of the 0-order, +1-order, and +2-order O diffracted waves, for OOO forward Bragg diffraction, as a function of the total grating thickness.

Fig. 9
Fig. 9

Diffraction efficiencies of the 0-order, +1-order, and +2-order E diffracted waves and of the 0-order O-wave in the two-grating example [geometry of Fig. 8(a)] for EEE diffraction, as a function of the total grating thickness.

Fig. 10
Fig. 10

(a) The xy-plane (top-view) geometry of two cascaded anisotropic gratings for backward Bragg diffraction. (b), (c) Diffraction efficienciesof the 0-order and +1-order O forward-diffracted waves and of the +1-order and +2-order O backward-diffracted waves, in the two-grating backward-diffraction example, for OOO backward Bragg diffraction for (b) s1 = 1100 μm and (c) s1 = 550 μm.

Fig. 11
Fig. 11

The 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

The 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, and (c) +-order forward-diffracted wave.

Tables (3)

Tables Icon

Table 1 Comparison for the 000-Type Diffraction of the Rigorous Analysis of Double Grating Diffraction with the Results from Analyzing the Gratings Individually: Diffraction Efficiency of the +1-Order Forward-Diffracted Ordinary Wave

Tables Icon

Table 2 Comparison for the EEE-Type Diffraction of the Rigorous Analysis of Double Grating Diffraction with the Results from Analyzing the Gratings Individually: Diffraction Efficiency of the +1-Order Forward-Diffracted Extraordinary Wave

Tables Icon

Table 3 Efficiencies of Transmitted, +1-Order Forward, and 0-Order Backward Diffraction of the Varying Modulation Grating as a Function of the Number of Approximating Gratings (N) and the Number of Spatial Harmonics Retained (M)

Equations (134)

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ɛ ¯ l ( x , y ) = h ɛ ¯ l h exp ( j h K l · r ) ,             l = 1 , 2 , , N ,
K l = K x l x ^ + K y l y ^ = K l sin ϕ l x ^ - K l cos ϕ l y ^ ,
K l sin ϕ l = K m sin ϕ m = K x ,
E inc = u ^ E exp ( - j k 1 · r ) ,
k 1 = k 1 x x ^ + k 1 y y ^ + k 1 z z ^ = k 1 [ ( - cos α sin δ ) x ^ - ( cos α cos δ ) y ^ - ( sin α ) z ^ ] ,
k 1 = k 0 n O 1 ,
k 1 = 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 ,
u ^ D = u D x x ^ + u D y y ^ + u D z z ^ = ( cos ψ cos δ + sin ψ sin α sin δ ) x ^ + ( - cos ψ sin δ + sin ψ sin α cos δ ) y ^ + ( - sin ψ cos α ) z ^ .
u E = ( ɛ ¯ 1 ) - 1 u ^ D / 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 i exp ( - j k 1 i · r ) + R i exp ( - j k 1 i · r ) ] ,
E 3 = i { T i exp [ - j k 3 i · ( r + s y ^ ) ] + T i exp [ - j k 3 i · ( r + s y ^ ) ] } ,
k q i = ( k 1 x - i K x ) x ^ + k y q i y ^ + k 1 z z ^ = k x i x ^ + k y q i y ^ + k 1 z z ^ ,
k q i = ( k 1 x - i K x ) x ^ + k y q i y ^ + k 1 z z ^ = k x i x ^ + k y q i y ^ + k 1 z z ^ ,
( k q i · k q i ) = k 0 2 n O q 2 ,             q = 1 , 3 ,
n O q 2 ( k q i · k q i ) + ( n E q 2 - n O q 2 ) ( k q i · c ^ q ) 2 = k 0 2 n O q 2 n E q 2 ,             q = 1 , 3 ,
c ^ q = c q x x ^ + c q y y ^ + c q z z ^ ,             q = 1 , 3.
tan δ q i = r q k x i / k y q i ,             tan δ q i = r q k x i / k y q i ,
tan α q i = - k 1 z / ( k x i 2 + k y q i 2 ) , tan α q i = - k 1 z / ( k x i 2 + k y q i 2 ) .
H q = ( j / ω μ 0 ) × E q ,             q = 1 , 3 ,
E 2 l = i { S x i l ( y ) x ^ + S y i l ( y ) y ^ + S z i l ( y ) z ^ } × exp [ - j ( k x i x - i K y l y + k 1 z z ) ] ,
H 2 l = ( 0 μ 0 ) 1 / 2 i { U x i l ( y ) x ^ + U y i l ( y ) y ^ + U z i l ( y ) z ^ } × exp [ - j ( k x i x - i K y l y + k 1 z z ) ] ,
× E 2 l = - j ω μ 0 H 2 l ,
× H 2 l = j ω 0 ɛ ¯ l ( x , y ) E 2 l .
d S z i l d y + j i K y l S z i l + j k 1 z S y i l = - j k 0 U x i l ,
j k x i S z i l - j k 1 z S x i l = - j k 0 U y i l ,
- d S x i l d y - j i K y l S x i l - j k x i S y i l = - j k 0 U z i l ,
d U z i l d y + j i K y l U z i l + j k 1 z U y i l = j k 0 ρ [ ɛ x x l i - ρ S x ρ l + ɛ x y l i - ρ S y ρ l + ɛ x z l i - ρ S z ρ l ] ,
j k x i U z i l - j k 1 z U x i l = j k 0 ρ [ ɛ y x l i - ρ S x ρ l + ɛ y y l i - ρ S y ρ l + ɛ y z l i - ρ S z ρ l ] ,
- d U x i l d y - j i K y l U x i l - j k x i U y i l = j k 0 ρ [ ɛ z x l i - ρ S x ρ l + ɛ z y l i - ρ S y ρ l + ɛ z z l i - ρ S z ρ l ] ,
d V ˜ l d y = j A ˜ l V ˜ l ,
V ˜ l ( y ) = W ˜ l exp [ Λ ˜ l ( y + Ψ l ) ] C ˜ l ,
u x δ i 0 + R x i + R x i = S x i 1 ( 0 ) ,
u z δ i 0 + R z i + R z i = S z i 1 ( 0 ) ,
( k 1 y u z - k 1 z u y ) δ i 0 + k y 1 i R z i + k y 1 i R z i - k 1 z ( R y i + R y i ) = k 0 U x i 1 ( 0 ) ,
( k 1 x u y - k 1 y u x ) δ i 0 + k x i ( R y i + R y i ) - k y 1 i R x i - k y 1 i R x i = k 0 U z i 1 ( 0 ) ,
T x i + T x i = S x i N ( - Ψ N + 1 ) exp ( - j i K y N Ψ N + 1 ) = S ^ x i N ( - Ψ N + 1 ) ,
T z i + T z i = S z i N ( - Ψ N + 1 ) exp ( - j i K y N Ψ N + 1 ) = S ^ z i N ( - Ψ N + 1 ) ,
k y 3 i T z i + k y 3 i T z i - k 1 z ( T y i + T y i ) = k 0 U x i N ( - Ψ N + 1 ) × exp ( - j i K y N Ψ N + 1 ) = k 0 U ^ x i N ( - Ψ N + 1 ) ,
k x i ( T y i + T y i ) - k y 3 i T x i - k y 3 i T x i = k 0 U z i N ( - Ψ N + 1 ) × exp ( - j i K y N Ψ N + 1 ) = k 0 U ^ z i N ( - Ψ N + 1 ) .
k 1 i · ( ɛ ¯ 1 R i ) = 0 ,
k 1 i · ( ɛ ¯ 1 R i ) = 0 ,
k 3 i · ( ɛ ¯ 3 T i ) = 0 ,
k 3 i · ( ɛ ¯ 3 T i ) = 0 ,
c ^ 1 · R i = 0 ,
c ^ 1 · ( k 1 i × R i ) = 0 ,
c ^ 3 · T i = 0 ,
c ^ 3 · ( k 3 i × T i ) = 0.
D ˜ l - 1 , l W ˜ l - 1 exp ( - Λ ˜ l - 1 s l - 1 ) C ˜ l - 1 - W ˜ l C ˜ l = 0 ˜ ,
D ˜ l - 1 , l = diag [ d ˜ l - 1 , l , d ˜ l - 1 , l , d ˜ l - 1 , l , d ˜ l - 1 , l ] ,
= 3 M + 3 M + 3 M + 3 M + 4 M N = 12 M + 4 M N ;
= 4 M + 4 M + 4 M + 4 M + ( N - 1 ) 4 M , = 12 M + 4 M N .
[ P ˜ 11 0 ˜ P ˜ 12 0 ˜ 0 ˜ P ˜ 13 0 ˜ P ˜ 14 ] [ C ˜ 1 C ˜ N ] = p ˜ ,
P ˜ C ˜ 1 = p ˜ ,
C ˜ N = B ˜ C ˜ 1 ,
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 ˜ 12 W ˜ 1 exp ( - Λ ˜ 1 s 1 ) ] .
DE q i = τ q Re { E q i 2 k y q i * - ( E q i · k q i ) E q i y * } / Re { k 1 y - ( u ^ · k 1 ) u y } ,
DE q i = τ q Re { E q i 2 k y q i * - ( E q i · k q i ) E q i y * } / Re { k 1 y - ( u ^ · k 1 ) u y } ,
i ( DE 1 i + DE 1 i + DE 3 i + DE 3 i ) = 1.
σ 1 = σ l - 1 - i l K l ,
σ l - 1 = k 1 - i 1 K 1 - i 2 K 2 - - i l - 1 K l - 1 ,
- m i = i 1 + i 2 + + i N + m ,
( σ l - 1 · σ l - 1 ) = k 0 2 n O 2
n O 2 ( σ l - 1 · σ l - 1 ) + ( n E 2 - n O 2 ) ( σ l - 1 · c ^ ) 2 = k 0 2 n O 2 n E
( σ l · σ l ) = k 0 2 n O 2
n O 2 ( σ l · σ l ) + ( n E 2 - n O 2 ) ( σ l · c ^ ) 2 = k 0 2 n O 2 n E 2 .
( σ l - 1 - i l K l ) · ( σ l - 1 - i l K l ) = k 0 2 n O 2 .
n O 2 ( σ l - 1 - i l K l ) · ( σ l - 1 - i l K l ) + ( n E 2 - n O 2 ) [ ( σ l - 1 - i l K l ) · c ^ ] 2 = k 0 2 n O 2 n E 2 .
ϕ 1 + ϕ 2 = 180 ° ,
K 1 = K 2 .
K 1 = ( 2 / i 1 ) ( k 1 x sin ϕ 1 - k 1 y cos ϕ 1 ) ,
K 2 = ( 2 / i 2 ) ] [ ( k 1 x - i 1 K 1 sin ϕ 1 ) sin ϕ 2 - ( k 1 y + i 1 K 1 × cos ϕ 1 ) cos ϕ 2 ] .
( 1 + I 12 ) tan δ tan 3 ϕ 1 - ( 1 + 3 I 12 ) tan 2 ϕ 1 + ( 1 - 3 I 12 ) × tan δ tan ϕ 1 - ( 1 - I 12 ) = 0 ,
K 1 = 2 B E E ( 0 ) / i 1 A E E ( 0 ) ,
K 2 = 2 B E E ( 1 ) / i 2 A E E ( 1 ) ,
a 3 tan 3 ϕ 1 + a 2 tan 2 ϕ 1 + a 1 tan ϕ 1 + a 0 = 0 ,
ɛ x x ɛ E - ɛ 11 sin ( K 1 · r ) , ɛ y y ɛ z z ɛ O - ɛ 22 sin ( K 1 · r ) , ɛ x y = ɛ y x ɛ 12 sin ( K 1 · r ) , ɛ x z = ɛ z x 0 , ɛ y z = ɛ z y ɛ 23 sin ( K 1 · r ) ,
ɛ x x l ɛ y y l ɛ O + ( ɛ 11 l V l / Λ e l ) sin ( K l · r ) , ɛ z z l ɛ E + ( ɛ 33 l V l / Λ e l ) sin ( K l · r ) , ɛ x y l = ɛ y x l ( ɛ 12 l V l / Λ e l ) cos ( K l · r ) , ɛ x z l = ɛ z x l ( ɛ 13 l V l / Λ e ) cos ( K l · r ) , ɛ y z l = ɛ z y l 0.
ɛ x x l ɛ E - ɛ 11 F l sin ( K l · r ) , ɛ y y l ɛ z z l ɛ O - ɛ 22 F l sin ( K l · r ) , ɛ x y l = ɛ y x l ɛ 12 F l sin ( K l · r ) , ɛ x z l = ɛ z x l 0 , ɛ y z l = ɛ z y l ɛ 23 F l sin ( K 1 · r ) ,
δ = ϕ 1 ± cos - 1 ( 2 π i 1 / 2 k 1 Λ 1 cos α ) ,
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 ɛ y y q - ɛ y y q ɛ x x q - ɛ x y q 2 n O q 2 n E q 2 k x i 2 - ɛ y y q ɛ z z q - ɛ y z q 2 n O q 2 n E q 2 k 1 z 2 + 2 ɛ x y q ɛ y z q - ɛ y y q ɛ x z q n O q 2 n E q 2 k 1 z k x i ,
A ˜ l = [ 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 ] ,
a ˜ 11 = K ˜ x ɛ ˜ y y l - 1 ɛ ˜ y x l - K ˜ y l ,             a ˜ 12 = K ˜ x ɛ ˜ y y l - 1 ɛ ˜ y z l , α ˜ 13 = 1 k 0 K ˜ x ɛ ˜ y y l - 1 K ˜ z ,             a ˜ 14 = I ˜ k 0 - 1 k 0 K ˜ x ɛ ˜ y y l - 1 K ˜ x , a ˜ 21 = K ˜ z ɛ ˜ y y l - 1 ɛ ˜ y x l ,             a ˜ 22 = K ˜ z ɛ ˜ y y l - 1 ɛ ˜ y z l - K ˜ y l , a ˜ 23 = 1 k 0 K ˜ z ɛ ˜ y y l - 1 K ˜ z - I ˜ k 0 ,             a ˜ 24 = - 1 k 0 K ˜ z ɛ ˜ y y l - 1 K ˜ x , a ˜ 31 = k 0 ɛ ˜ y z l ɛ ˜ y y l - 1 ɛ ˜ y x l - k 0 ɛ ˜ x z l - 1 k 0 K ˜ x K ˜ z , a ˜ 32 = k 0 ɛ ˜ y z l ɛ ˜ y y l - 1 ɛ ˜ z y l - k 0 ɛ ˜ z z l + 1 k 0 K ˜ x K ˜ x , a ˜ 33 = ɛ ˜ z y l ɛ ˜ y y l - 1 K ˜ z - K ˜ y l ,             a ˜ 34 = - ɛ ˜ y z l ɛ ˜ y y l - 1 K ˜ x , a ˜ 41 = k 0 ɛ ˜ x x l - k 0 ɛ ˜ x y l ɛ ˜ y y l - 1 ɛ ˜ y x l - 1 k 0 K ˜ z K ˜ z , a ˜ 42 = k 0 ɛ ˜ x z l - k 0 ɛ ˜ x y l ɛ ˜ y y l - 1 ɛ ˜ y z l + 1 k 0 K ˜ z K ˜ x , a ˜ 43 = - ɛ ˜ x y l ɛ ˜ y y l - 1 K ˜ z ,             a ˜ 44 = ɛ ˜ x y l ɛ ˜ y y l - 1 K ˜ x - K ˜ y l ,
( P ˜ 11 ) k n = { E 1 i W 11 , k n + Z 1 i W 21 , k n - k 0 W 31 , k n if k y 1 i k y 1 i α i W 11 , k n + β i W 21 , k n - k 0 W 31 , k n if k y 1 i = k y 1 i ,
( P ˜ 12 ) k n = { E 2 i W 11 , k n + Z 2 i W 21 , k n - k 0 W 41 , k n if k y 1 i k y 1 i γ i W 11 , k n + α i W 21 , k n + k 0 W 41 , k n if k y 1 i = k y 1 i ,
( P ˜ 13 ) k n = { ( E 3 i W 1 N , k n + Z 3 i W 2 N , k n - k 0 W 3 N , k n ) ω n if k y 3 i k y 3 i ( θ i W 1 N , k n + μ i W 2 N , k n - k 0 W 3 N , k n ) ω n if k y 3 i = k y 3 i ,
( P ˜ 14 ) k n = { ( E 4 i W 1 N , k n + Z 4 i W 2 N , k n - k 0 W 4 N , k n ) ω n if k y 3 i k y 3 i ( ν i W 1 N , k n + θ i W 2 N , k n + k 0 W 4 N , k n ) ω n if k y 3 i = k 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 = { Θ 1 if k y 10 k y 10 ζ 1 if k y 10 = k y 10
P 3 m + 2 = { Θ 2 if k y 30 k y 30 η 1 if k y 30 = k y 30 .
E 1 i = ( A 1 i / Ξ 1 i ) P 1 i + k 1 z ( l x i 1 / l y i 1 ) ,
E 2 i = ( A 1 i / Ξ 1 i ) Q 1 i - k x i ( l x i 1 / l y i 1 ) - k y 1 i .
E 3 i = ( A 3 i / Ξ 3 i ) P 3 i + k 1 z ( l x i 3 / l y i 3 ) ,
E 4 i = ( A 3 i / Ξ 3 i ) Q 3 i - k x i ( l x i 3 / l y i 3 ) - k y 3 i ,
Z 1 i = ( B 1 i / Ξ 1 i ) P 1 i + k 1 z ( l z i 1 / l y i 1 ) + k y 1 i ,
Z 2 i = ( B 1 i / Ξ 1 i ) Q 1 i - k x i ( l z i 1 / l y i 1 ) ,
Z 3 i = ( B 3 i / Ξ 3 i ) P 3 i + k 1 z ( l z i 3 / l y i 3 ) + k y 3 i ,
Z 4 i = ( B 3 i / Ξ 3 i ) Q 3 i - k x i ( l z i 3 / l y i 3 ) ,
P q i = k 1 z ( l x i q l y i q - l x i q l y i q ) Δ q i + [ k 1 z ( l z i q l y i q - l z i q l y i q ) - k y q i + k y q i ] Γ q i ,
Q q i = k x i ( l z i q l y i q - l z i q l y i q ) Γ q i + [ k x i ( l x i q l y i q - l x i q l y i q ) - k y q i + k y q i ] Δ q i ,
Ξ q i = A q i Δ q i - B q i Γ q i             ( q = 1 , 3 )
A q i = r q y i l x i q l y i q - r q x i ,             B q i = r q y i l z i q l y i q - r q z i ,
Γ q i = c q x - c q y l x i q l y i q ,             Δ q i = c q z - c q z l z i q l y i q ,
l w i q = k x i ɛ x w q + k y q i ɛ y w q + k 1 z ɛ z w q ,
l w i q = k x i ɛ x w q + k y q i ɛ y w q + k 1 z ɛ z w q ,
Θ 1 = k 1 z l x i 1 l y i 1 u x + k 1 z u y + ( k y 1 i - k 1 y + k 1 z l z i 1 l y i l ) u z - Π Ξ 1 i P 1 i ,
Θ 2 = ( k 1 y - k x i l x i 1 l y i 1 - k y 1 i ) u x - k 1 x u y - k x i l z i 1 l y i l u z - Π i Ξ 1 i Q 1 i ,
Π i = ( r 1 x i - r 1 y i l x i 1 l y i l ) u x + ( r 1 z i - r 1 y i l z i l l y i 1 ) u z .
α i = k 1 z k x i / k y 1 i ,             β i = ( k 1 z 2 / k y 1 i ) + k y 1 i , γ i = ( k x i 2 / k y 1 i ) + k y 1 i , ζ i = ( k 1 z k x i / k y 1 i ) u x + k 1 z u y + [ ( k 1 z 2 / k y 1 i ) + k y 1 i + k 1 y ] u z , η i = [ ( k x i 2 / k y 1 i ) + k y 1 i - k 1 y ] u x + k 1 x u y + ( k x i k 1 z / k y 1 i ) u z , θ i = k 1 z k x i k y 3 i ,             μ i = ( k 1 z 2 / k y 3 i ) + k y 3 i , ν i = ( k x i 2 / k y 3 i ) + k y 3 i .
R x i = ( Δ 1 i / Ξ 1 i ) [ Π i δ i 0 + A 1 i S x i 1 ( 0 ) + B 1 i S z i l ( 0 ) ] ,
R z i = ( - Γ 1 i / Ξ 1 i ) R x i ,
T x i = ( Δ 3 i / Ξ 3 i ) [ A 3 i S ^ x i N ( - Ψ N + 1 ) + B 3 i S ^ z i N ( - Ψ N + 1 ) ] ,
T z i = ( - Γ 3 i / Ξ 3 i ) T x i ,
R y i = - ( l x i 1 / l y i 1 ) R x i - ( l z i l / l y i l ) R z i ,
R y i = - ( l x i 1 / l y i 1 ) R x i - ( l z i 1 / l y i 1 ) R z i ,
T y i = - ( l x i 3 / l y i 3 ) T x i - ( l z i 3 / l y i 3 ) T z i ,
T y i = - ( l x i 3 / l y i 3 ) T x i - ( l z i 3 / l y i 3 ) T z i .
σ ( l - 1 ) x = k 1 x - i 1 K x - i 2 K x - - i l - 1 K x = k 1 x - i K x , σ ( l - 1 ) y = k 1 y - i 1 K y 1 - i 2 K y 2 - - i l - 1 K y ( l - 1 ) ,
σ ( l - 1 ) z = k 1 z .
Λ l = 2 π i l 2 ( σ ( l - 1 ) x sin ϕ l - σ ( l - 1 ) y cos ϕ l ) .
Λ l = 2 π i l A E E ( l - 1 ) 2 B E E ( l - 1 ) ,
A E E ( l - 1 ) = ɛ x x sin 2 ϕ l + ɛ y y cos 2 ϕ l - ɛ x y sin 2 ϕ l ,
B E E ( l - 1 ) = ( ɛ x x sin ϕ l - ɛ x y cos ϕ l ) σ ( l - 1 ) x - ( ɛ y y cos ϕ l - ɛ x y sin ϕ l ) σ ( l - 1 ) y + ( ɛ x z sin ϕ l - ɛ y z cos ϕ l ) σ ( l - 1 ) z .
Λ 1 = 2 π i l A O E ( l - 1 ) B O E ( l - 1 ) ± [ D O E ( l - 1 ) ] 1 / 2 ,
A O E ( l - 1 ) = A E E ( l - 1 ) ,
B O E ( l - 1 ) = B E E ( l - 1 ) ,
C O E ( l - 1 ) = ɛ x x σ ( l - 1 ) x 2 + ɛ y y σ ( l - 1 ) y 2 + ɛ z z σ ( l - 1 ) z 2 + 2 ɛ x y σ ( l - 1 ) x σ ( l - 1 ) y + 2 ɛ y z σ ( l - 1 ) y σ ( l - 1 ) z + 2 ɛ z x σ ( l - 1 ) z σ ( l - 1 ) x - k 0 2 n O 2 n E 2 = σ l - 1 · ( ɛ ¯ σ l - 1 ) - k 0 2 n O 2 n E 2 ,
D O E ( l - 1 ) = B O E 2 ( l - 1 ) - A O E ( l - 1 ) C O E ( l - 1 ) .
Λ 1 = 2 π i l B E O ( l - 1 ) ± [ D E O ( l - 1 ) ] 1 / 2 ,
B E O ( l - 1 ) = σ ( l - 1 ) x sin ϕ l - σ ( l - 1 ) y cos ϕ l ,
C E O ( l - 1 ) = σ ( l - 1 ) x 2 + σ ( l - 1 ) y 2 + σ ( l - 1 ) z 2 - k 0 2 n O 2 = σ l - 1 · σ l - 1 - k 0 2 n O 2 ,
D E O ( l - 1 ) = B E O 2 - C E O ( l - 1 ) .

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