Abstract

The diffraction by two planar slanted fringe gratings superposed in the same volume of an anisotropic medium is treated using rigorous 3-D vector coupled wave analysis. Arbitrary angle of incidence and polarization are included. Both phase and/or amplitude slanted gratings in anisotropic media are treated in the analysis. The external boundary regions can be either isotropic (for bulk applications) or uniaxial anisotropic (for integrated applications). Both forward- and backward-diffracted orders are characterized by a number pair (i1,i2), where i1 and i2 are integers. The Floquet condition is discussed for the case of two superposed gratings. When the external regions are anisotropic, each diffracted order has an ordinary (O), and an extraordinary (E) component. The analysis is also generalized for an arbitrary number of superposed gratings. The numerical complexity is discussed. In the case of equal grating periodicities along the boundaries, the diffracted orders become degenerate in the external regions. In this case, an alternative analysis that utilizes a cascaded stack of unslanted gratings can be used. Limiting cases are also presented. The various Bragg conditions are identified and quantified. Sample calculations presented include the quantification of the crosstalk between two superposed gratings, the evaluation of the effects of coupled Bragg conditions in beam combining applications, design and analysis of a beam splitter and a beam combiner, demonstration of the use of a cascaded stack of unslanted gratings of constant modulation to represent two superposed gratings that have the same periodicity along the boundaries, and finally evaluation of the effect of the phase difference between two gratings. The same analysis applies in the limiting cases of isotropic materials, single slanted gratings, etc. Applications of this analysis include optical storage, optical digital truth table look-up processing, neural nets, optical interconnects, beam splitting, and beam combining.

© 1989 Optical Society of America

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1988

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/OSA J. Lightwave Technol. LT-6, 1083–1104 (1988).
[CrossRef]

H. Lee, “Cross-Talk Effects in Multiplexed Volume Holograms,” Opt. Lett. 13, 874–876 (1988).
[CrossRef] [PubMed]

M. R. Feldman, S. C. Esener, C. C. Guest, S. H. Lee, “Comparison Between Optical and Electrical Interconnects Based on Power and Speed Considerations,” Appl. Opt. 27, 1742–1751 (1988).
[CrossRef] [PubMed]

D. Psaltis, D. Brady, K. Wagner, “Adaptive Optical Networks Using Photorefractive Crystals,” Appl. Opt. 27, 1752–1759 (1988).
[CrossRef]

1987

1986

1985

H. Mada, “Architecture for Optical Computing Using Holographic Associative Memories,” Appl. Opt. 24, 2063–2066 (1985).
[CrossRef] [PubMed]

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]

1984

C. M. Verber, “Integrated-Optical Approaches to Numerical Optical Computing,” Proc. IEEE 72, 942–953 (1984).
[CrossRef]

1983

1982

1981

1980

M. G. Moharam, T. K. Gaylord, R. Magnusson, “Criteria for Bragg Regime Diffraction by Phase Gratings,” Opt. Commun. 32, 14–18 (1980).
[CrossRef]

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

1979

1978

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. 44, 324–326 (1978).

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]

1977

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 of Thick Gratings,” J. Opt. Soc. Am. 67, 1165–1170 (1977).
[CrossRef]

1976

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

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]

S. K. Case, “Coupled-Wave Theory for Multiply Exposed Thick Holographic Gratings,” J. Opt. Soc. Am. 65, 724–729 (1975).
[CrossRef]

R. C. Alferness, S. K. Case, “Coupling in Doubly Exposed, Thick Holographic Gratings,” J. Opt. Soc. Am. 65, 730–739 (1975).
[CrossRef]

1974

1973

1972

R. S. Chu, T. Tamir, “Wave Propagation and Dispersion in Space Time Periodic Media,” Proc. Inst. Electr. Eng. 119, 797–806 (1972).
[CrossRef]

1971

1970

M. Sakaguchi, N. Nishida, T. Nemoto, “A New Associative Memory System Utilizing Holography,” IEEE Trans. Comput. C-19, 1174–1181 (1970).
[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).

1969

H. Kogelnik, “Coupled Wave Theory for Thick Hologram Gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

1967

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. Modern 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

T. Tamir, H. C. Wang, “Scattering and 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

P. Phariseau, “On the Diffraction of Light by Progressive Supersonic Waves,” Proc. Indian Acad. Sci. Sect. A 44, 165–170 (1965).

1964

T. Tamir, H. C. Wang, A. A. Oliner, “Wave Propagation in Sinusoidally Stratified Dielectric Media,” IEEE Trans. Microwave Theory Tech. MTT-12, 323–335 (1964).
[CrossRef]

1950

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).

Alferness, R. C.

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. Modern Opt. 17, 655–680 (1967).

Bertoni, H. L.

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

Bloch, P. D.

P. D. Bloch, L. Solymar, “Analysis of a 4-Port Bragg Device,” Proc. Inst. Electr. Eng. 127, Pt. H, 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]

Brady, D.

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]

Esener, S. C.

Feldman, M. R.

M. R. Feldman, S. C. Esener, C. C. Guest, S. H. Lee, “Comparison Between Optical and Electrical Interconnects Based on Power and Speed Considerations,” Appl. Opt. 27, 1742–1751 (1988).
[CrossRef] [PubMed]

M. R. Feldman, C. C. Guest, “Optical Interconnect Complexity Limitations for Holograms Fabricated with Electron Beam Lithography,” in Technical Digest of Topical Meeting on Optical Computing (Optical Society of America, Washington, DC, 1987).

Fillmore, G. L.

Gaylord, T. K.

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/OSA 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]

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]

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, “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, “Anisotropic Guided-Wave Diffraction by Interdigitated-Electrode-Induced Phase Gratings,” Appl. Opt. 27, 5035–5050 (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]

Gu, X. G.

D. Psaltis, J. Yu, X. G. Gu, H. Lee, “Optical Neural Nets Implemented with Volume Holograms,” in Technical Digest of Topical Meeting on Optical Computing (Optical Society of America, Washington, DC, 1987).

Guest, C. C.

M. R. Feldman, S. C. Esener, C. C. Guest, S. H. Lee, “Comparison Between Optical and Electrical Interconnects Based on Power and Speed Considerations,” Appl. Opt. 27, 1742–1751 (1988).
[CrossRef] [PubMed]

M. R. Feldman, C. C. Guest, “Optical Interconnect Complexity Limitations for Holograms Fabricated with Electron Beam Lithography,” in Technical Digest of Topical Meeting on Optical Computing (Optical Society of America, Washington, DC, 1987).

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]

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. 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. Modern 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]

Knight, G. R.

Knoesen, A.

A. Knoesen, T. K. Gaylord, M. G. Moharam, “Hybrid Guided Modes in Uniaxial Dielectric Planar Waveguides,” IEEE/OSA 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]

J. A. Kong, “Second-Order Coupled-Mode Equations for Spatially Periodic Media,” J. Opt. Soc. Am. 67, 825–829 (1977).
[CrossRef]

R. S. Chu, J. A. Kong, “Modal Theory of Spatially Periodic Media,” IEEE Trans. Microwave Theory Tech. MTT-25, 18–24 (1977).

Kowarschik, R.

R. Kowarschik, “Diffraction Efficiency of Sequentially Stored Gratings in Reflection Volume Holograms,” Opt. Quantum Electron. 10, 171–178 (1978).
[CrossRef]

R. Kowarschik, “Diffraction Efficiency of Sequentially Stored Gratings in Transmission Volume Holograms,” Opt. Acta 25, 67–81 (1978).
[CrossRef]

Lee, H.

H. Lee, “Cross-Talk Effects in Multiplexed Volume Holograms,” Opt. Lett. 13, 874–876 (1988).
[CrossRef] [PubMed]

D. Psaltis, J. Yu, X. G. Gu, H. Lee, “Optical Neural Nets Implemented with Volume Holograms,” in Technical Digest of Topical Meeting on Optical Computing (Optical Society of America, Washington, DC, 1987).

Lee, S. H.

Leger, J. R.

Mada, H.

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.

A. Knoesen, T. K. Gaylord, M. G. Moharam, “Hybrid Guided Modes in Uniaxial Dielectric Planar Waveguides,” IEEE/OSA J. Lightwave Technol. LT-6, 1083–1104 (1988).
[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, “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, “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]

Nemoto, T.

M. Sakaguchi, N. Nishida, T. Nemoto, “A New Associative Memory System Utilizing Holography,” IEEE Trans. Comput. C-19, 1174–1181 (1970).
[CrossRef]

Nishida, N.

M. Sakaguchi, N. Nishida, T. Nemoto, “A New Associative Memory System Utilizing Holography,” IEEE Trans. Comput. C-19, 1174–1181 (1970).
[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. 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. 44, 324–326 (1978).

Psaltis, D.

D. Psaltis, D. Brady, K. Wagner, “Adaptive Optical Networks Using Photorefractive Crystals,” Appl. Opt. 27, 1752–1759 (1988).
[CrossRef]

D. Psaltis, J. Yu, X. G. Gu, H. Lee, “Optical Neural Nets Implemented with Volume Holograms,” in Technical Digest of Topical Meeting on Optical Computing (Optical Society of America, Washington, DC, 1987).

Rabson, T. A.

Rokushima, K.

Sakaguchi, M.

M. Sakaguchi, N. Nishida, T. Nemoto, “A New Associative Memory System Utilizing Holography,” IEEE Trans. Comput. C-19, 1174–1181 (1970).
[CrossRef]

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. 127, Pt. H, 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]

Swanson, G. J.

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 and 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]

Thylen, L.

Tittel, F. K.

Tomishima, K.

Tsujinishi, R.

Tsukada, N.

Tynan, R. F.

Veldkamp, W. B.

Verber, C. M.

C. M. Verber, “Integrated-Optical Approaches to Numerical Optical Computing,” Proc. IEEE 72, 942–953 (1984).
[CrossRef]

Wagner, K.

Wang, H. C.

T. Tamir, H. C. Wang, “Scattering and 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]

Woodbury, D. A.

Yamakita, J.

Yeh, P.

Yevick, D.

Yu, J.

D. Psaltis, J. Yu, X. G. Gu, H. Lee, “Optical Neural Nets Implemented with Volume Holograms,” in Technical Digest of Topical Meeting on Optical Computing (Optical Society of America, Washington, DC, 1987).

Appl. Opt.

E. N. Glytsis, T. K. Gaylord, “Anisotropic Guided-Wave Diffraction by Interdigitated-Electrode-Induced Phase Gratings,” Appl. Opt. 27, 5035–5050 (1988).
[CrossRef]

G. R. Knight, “Page-Oriented Associative Holographic Memory,” Appl. Opt. 13, 904–912 (1974).
[CrossRef] [PubMed]

D. A. Woodbury, F. K. Tittel, T. A. Rabson, “Hologram Indexing in LiNbO3 with a Tunable Pulsed Laser Source,” Appl. Opt. 18, 2555–2558 (1979).
[CrossRef] [PubMed]

L. Thylen, D. Yevick, “Beam Propagation Method in Anisotropic Media,” Appl. Opt. 21, 2751–2754 (1982).
[CrossRef] [PubMed]

H. Mada, “Architecture for Optical Computing Using Holographic Associative Memories,” Appl. Opt. 24, 2063–2066 (1985).
[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]

C. W. Slinger, L. Solymar, “Grating Interactions in Holograms Recorded with Two Object Waves,” Appl. Opt. 25, 3283–3287 (1986).
[CrossRef] [PubMed]

J. R. Leger, G. J. Swanson, W. B. Veldkamp, “Coherent Laser Addition Using Binary Phase Gratings,” Appl. Opt. 26, 4391–4399 (1987).
[CrossRef] [PubMed]

M. R. Feldman, S. C. Esener, C. C. Guest, S. H. Lee, “Comparison Between Optical and Electrical Interconnects Based on Power and Speed Considerations,” Appl. Opt. 27, 1742–1751 (1988).
[CrossRef] [PubMed]

D. Psaltis, D. Brady, K. Wagner, “Adaptive Optical Networks Using Photorefractive Crystals,” Appl. Opt. 27, 1752–1759 (1988).
[CrossRef]

Appl. Phys. B

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.

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.

H. Kogelnik, “Coupled Wave Theory for Thick Hologram Gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Can. J. Phys.

T. Tamir, H. C. Wang, “Scattering and 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. Quantum Electron.

R. W. Dixon, “Acoustic Diffraction of Light in Anisotropic Media,” IEEE J. Quantum Electron. QE-3, 85–93 (1967).
[CrossRef]

IEEE Trans. Comput.

M. Sakaguchi, N. Nishida, T. Nemoto, “A New Associative Memory System Utilizing Holography,” IEEE Trans. Comput. C-19, 1174–1181 (1970).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

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]

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).

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.

W. R. Klein, B. D. Cook, “Unified Approach to Ultrasonic Light Diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).
[CrossRef]

IEEE/OSA J. Lightwave Technol.

A. Knoesen, T. K. Gaylord, M. G. Moharam, “Hybrid Guided Modes in Uniaxial Dielectric Planar Waveguides,” IEEE/OSA J. Lightwave Technol. LT-6, 1083–1104 (1988).
[CrossRef]

J. Appl. Phys.

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.

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 (1971).
[CrossRef]

F. G. Kaspar, “Diffraction by Thick Periodically Stratified Gratings with Complex Dielectric Constant,” J. Opt. Soc. Am. 63, 37–45 (1973).
[CrossRef]

S. K. Case, “Coupled-Wave Theory for Multiply Exposed Thick Holographic Gratings,” J. Opt. Soc. Am. 65, 724–729 (1975).
[CrossRef]

R. C. Alferness, S. K. Case, “Coupling in Doubly Exposed, Thick Holographic Gratings,” J. Opt. Soc. Am. 65, 730–739 (1975).
[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 of 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]

D. Yevick, L. Thylen, “Analysis of Gratings by the Beam-Propagation Method,” J. Opt. Soc. Am. 72, 1084–1089 (1982).
[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]

K. Rokushima, J. Yamakita, “Analysis of Anisotropic Dielectric Gratings,” J. Opt. Soc. Am. 73, 901–908 (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]

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]

P. Yeh, “Electromagnetic Propagation in Birefringent Layered Media,” J. Opt. Soc. Am. 69, 742–756 (1979).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Acta

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.

M. G. Moharam, T. K. Gaylord, R. Magnusson, “Criteria for Bragg Regime Diffraction by Phase Gratings,” Opt. Commun. 32, 14–18 (1980).
[CrossRef]

D. Yevick, B. Hermansson, “Soliton Analysis with the Propagating Beam Method,” Opt. Commun. 47, 101–106 (1983).
[CrossRef]

Opt. Eng.

R. V. Johnson, A. R. Tanguay, “Optical Beam Propagation Method for Birefringent Phase Grating Diffraction,” Opt. Eng. 25, 235–249 (1986).
[CrossRef]

Opt. Lett.

Opt. Quantum Electron.

R. Kowarschik, “Diffraction Efficiency of Sequentially Stored Gratings in Reflection Volume Holograms,” Opt. Quantum Electron. 10, 171–178 (1978).
[CrossRef]

L. Thylen, “The Beam Propagation Method: an Analysis of Its Applicability,” Opt. Quantum Electron. 15, 433–439 (1983).
[CrossRef]

Opt. Spectrosc.

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. 44, 324–326 (1978).

Proc. IEEE

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.

R. R. Aggrawal, “Diffraction of Light by Ultrasonic Waves,” Proc. Indian Acad. Sci. 31, 417–426 (1950).

Proc. Indian Acad. Sci. Sect. A

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.

R. S. Chu, T. Tamir, “Wave Propagation and Dispersion in Space Time Periodic Media,” Proc. Inst. Electr. Eng. 119, 797–806 (1972).
[CrossRef]

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

Proc. Symp. Modern Opt.

L. Bergstein, D. Kermisch, “Image Storage and Reconstruction in Volume Holography,” Proc. Symp. Modern Opt. 17, 655–680 (1967).

Radio Sci.

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

Program EIGCC from the International Mathematics and Statistics Library, Houston, TX.

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

M. R. Feldman, C. C. Guest, “Optical Interconnect Complexity Limitations for Holograms Fabricated with Electron Beam Lithography,” in Technical Digest of Topical Meeting on Optical Computing (Optical Society of America, Washington, DC, 1987).

D. Psaltis, J. Yu, X. G. Gu, H. Lee, “Optical Neural Nets Implemented with Volume Holograms,” in Technical Digest of Topical Meeting on Optical Computing (Optical Society of America, Washington, DC, 1987).

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Figures (14)

Fig. 1
Fig. 1

Three-dimensional perspective view of the diffraction geometry of the two superposed anisotropic gratings.

Fig. 2
Fig. 2

The x-y plane (top view) of the two superposed gratings geometry. The grating vectors K 1 and K 2, and the ordinary (O) and extraordinary (E) forward- and backward-diffracted wavevector x-y projections are shown with their corresponding diffraction angles δ 1 i 1 i 2 , δ 1 i 1 i 2 , δ 3 i 1 i 2, and δ 3 i 1 i 2.

Fig. 3
Fig. 3

(a) The x-y plane projection of the Floquet condition in the case of the two superposed anisotropic gratings. (b) The Floquet condition in the case of a single anisotropic grating. The dashed lines represent the y axis of the coordinate system.

Fig. 4
Fig. 4

Geometry of forward O and E waves, showing the double conical nature of the diffraction with uniaxial external regions (regions 1 and 3). All forward- and backward-diffracted waves have the same z component. Only forward O and E diffracted waves are shown, and specifically the (0,0)-, (−1, −1)-, and (1,1)-order waves.

Fig. 5
Fig. 5

Equivalence of the two superposed anisotropic gratings (a) (in general N superposed anisotropic gratings) to a single anisotropic grating of varying modulation (b), and the equivalence of the single grating to a stack of cascaded single anisotropic gratings of constant modulation (c), in the case of equal periodicity along the boundaries, i.e., Λ1x = Λ2x = Λx.

Fig. 6
Fig. 6

Generalized wavevector diagram for the Bragg condition of the (i1,i2)-order wave. The surfaces S0, S11, S12, and S2 are parts of the ordinary or extraordinary wavevector surfaces.

Fig. 7
Fig. 7

(a) The x-y 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 forward-diffracted waves are shown, (b)–(e) Diffraction efficiencies of the (0,0)-, (1,0)-, (−1,1)-, and (1,1)-order forward-diffracted ordinary waves as functions of the slant angle of the second grating.

Fig. 8
Fig. 8

(a) The x-y plane (top view) of the diffraction geometry of the coherent beam combining problem, (b) The wavevector diagram that corresponds to the coherent beam combining problem.

Fig. 9
Fig. 9

(a) Diffraction efficiencies of the first-order forward-diffracted ordinary waves as a function of the angle of incidence using the rigorous coupled wave analysis of single anisotropic gratings. The D E 1 ( 1 ) corresponds to the diffraction efficiency of the first-order forward-diffracted ordinary wave due to the first grating, and D E 1 ( 2 ) is the diffraction efficiency of the first-order forward-diffracted ordinary wave due to the second grating. (b),(c) The diffraction efficiencies of the (1,0)- and (0.l)-order forward-diffracted ordinary waves as a function of the angle of incidence using the rigorous coupled wave analysis of superposed anisotropic gratings.

Fig. 10
Fig. 10

(a) Diffraction efficiencies of the (0,0)-, (1,0)-, and (1,−1)-order forward-diffracted ordinary waves as functions of the grating region thickness for an angle of incidence δ = δB1 = 1°. (b) The diffraction efficiencies of the (0,0)-, (0,1)-, and (−1,1)-order forward-diffracted ordinary waves as functions of the grating region thickness for an angle of incidence δ = δB2 = 4°.

Fig. 11
Fig. 11

(a) The x-y plane (top view) of the diffraction geometry of the coherent beam splitter problem. Only the positions of the (1,0)-, (0,0)-, and (0,−1)-order forward-diffracted extraordinary waves are shown, (b) The diffraction efficiencies of the (0,0)-, (1,0)-, and (0,−1)-order forward-diffracted extraordinary waves as functions of the grating region thickness.

Fig. 12
Fig. 12

The x-y plane (top view) of the diffraction geometry of the coherent beam combining problem. The three plane waves incident at angles δ1, δ2, and δ3 produce the groups of plane wave components shown at the right-hand side of the grating. Again, only some of the positions of forward-diffracted waves are shown denoted by (i1,i2)j where j = 1,2,3.

Fig. 13
Fig. 13

(a) Normalized profile of the ε x x relative permittivity element [gxx(y)] as function of y. In addition, the modulation of the i-th cascaded approximating grating is shown. (b) and (c) Diffraction efficiency of the (1,0)-order forward-diffracted extraordinary wave (straight line) calculated using the rigorous coupled wave analysis of superposed anisotropic gratings, and the diffraction efficiency of the equivalent first-order forward-diffracted extraordinary wave of the stack of cascaded single anisotropic gratings as a function of the number of approximating gratings (b) for grating thickness s = 20 μm and (c) for grating thickness s = 40 μm.

Fig. 14
Fig. 14

The x-y plane (top view) diffraction geometry of the coherent beam combining of two plane waves (of wavevectors k 1 and k 2 by two superposed anisotropic gratings which have a spatial phase difference in their fringes which is denoted as Δϕ (normalized in the range from 0 to 2π).

Equations (116)

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

ε ˜ ( x , y ) = ε ˜ 0 + h 1 ε ˜ 1 h 1 exp ( j h 1 K 1 · r ) + h 2 ε ˜ 2 h 2 exp ( j h 2 K 2 · r ) ,
K l = K l x x ˆ + K l y y ˆ = K l sin ϕ l x ˆ K l cos ϕ l y ˆ ,
E inc = u ˆ E 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 | = 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 ˆ D = u D x x ˆ + u D y y ˆ + u 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 .
u E = ( ε ˜ 1 ) 1 u ˆ 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 2 i 2 [ R i 1 i 2 exp ( j k 1 i 1 i 2 · r ) + R i 1 i 2 exp ( j k 1 i 1 i 2 · r ) ] ,
E 3 = i 2 i 2 { T i 1 i 2 exp [ j k 3 i 1 i 2 · ( r + s y ˆ ) ] + T i 1 i 2 exp [ j k 3 i 1 i 2 · ( r + s y ˆ ) ] } .
k q i 1 i 2 = ( k 1 x i 1 K 1 x i 2 K 2 x ) x ˆ + k y q i 1 i z y ˆ + k 1 z z ˆ = k x i 1 i 2 x ˆ + k y q i 1 i 2 y ˆ + k 1 z z ˆ ,
k q i 1 i 2 = ( k 1 x i 1 K 1 x i 2 K 2 x ) x ˆ + k y q i 1 i 2 y ˆ + k 1 z z ˆ = k x i 1 i 2 x ˆ + k y q i 1 i 2 y ˆ + k 1 z z ˆ ,
( k q i 1 i 2 · k q i 1 i 1 ) = k 0 2 n O q 2 q = 1 , 3 ,
n O q 2 ( k q i 1 i 2 · k q i 1 i 2 ) + ( n E q 2 n O q 2 ) ( k q i 1 i 2 · c ˆ q ) 2 = k 0 2 n O q 2 n E q 2 q = 1 , 3 ,
tan δ q i 1 i 2 = r q k x i 1 i 2 / k y q i 1 i 2 , tan δ q i 1 i 2 = r q k x i 1 i 2 / k y q i 1 i 2 ,
tan α q i 1 i 2 = k 1 z / ( k x i 1 i 2 2 + k y q i 1 i 2 2 ) , tan α q i 1 i 2 = k 1 z / ( k x i 1 i 2 2 + k y q i 1 i 2 2 ) .
H q = ( j / ω μ 0 ) × E q q = 1 , 3 ,
E 2 = i 1 i 2 { S x i 1 i 2 ( y ) x ˆ + S y i 1 i 2 ( y ) y ˆ + S x i 1 i 2 ( y ) } f ( x , y , z ) ,
H 2 = ( 0 μ 0 ) 1 / 2 i 1 i 2 { U x i 1 i 2 ( y ) x ˆ + U y i 1 i 2 ( y ) y ˆ + U z i 1 i 2 ( y ) z ˆ } f ( x , y , z ) ,
× E 2 = j ω μ 0 H 2 ,
× H 2 = j ω 0 ε ˜ ( x , y ) E 2 .
d S z i 1 i 2 d y + j ( i 1 K 1 y + i 2 K 2 y ) S z i 1 i 2 + j k 1 z S y i 1 i 2 = j k 0 U x i 1 i 2 ,
j k x i 1 i 2 S z i 1 i 2 j k 1 z S x i 1 i 2 = j k 0 U y i 1 i 2 ,
d S x i 1 i 2 d y j ( i 1 K 1 y + i 2 K 2 y ) S x i 1 i 2 j k x i 1 i 2 S y i 1 i 2 = j k 0 U z i 1 i 2 ,
d U z i 1 i 2 d y + j ( i 1 K 1 y + i 2 K 2 y ) U z i 1 i 2 + j k 1 z U y i 1 i 2 = j k 0 ρ 1 [ ε 1 x x ( i 1 ρ 1 ) S x ρ 1 i 2 + ε 1 x y ( i 1 ρ 1 ) S y ρ 1 i 2 + ε 1 x z ( i 1 ρ 1 ) S z ρ 1 i 2 ] + j k 0 ρ 2 [ ε 2 x x ( i 2 ρ 2 ) S x i 1 ρ 2 + ε 2 x y ( i 2 ρ 2 ) S y i 1 ρ 2 + ε 2 x z ( i 2 ρ 2 ) S z i 1 ρ 2 ] ,
j k x i 1 i 2 U z i 1 i 2 j k 1 z U x i 1 i 2 = j k 0 ρ 1 [ ε 1 y x ( i 1 ρ 1 ) S x ρ 1 i 2 + ε 1 y y ( i 1 ρ 1 ) S y ρ 1 i 2 + ε 1 y z ( i 1 ρ 1 ) S z ρ 1 i 2 ] + j k 0 ρ 2 [ ε 2 y x ( i 2 ρ 2 ) S x i 1 ρ 2 ) + ε 2 y y ( i 2 ρ 2 ) S y i 1 ρ 2 ) + ε 2 y z ( i 2 ρ 2 ) S z i 1 ρ 2 ] ,
d U x i 1 i 2 d y j ( i 1 K 1 y + i 2 K 2 y ) U x i 1 i 2 j k x i 1 i 2 U y i 1 i 2 = j k 0 ρ 1 [ ε 1 z x ( i 1 ρ 1 ) S x ρ 1 i 2 + ε 1 z y ( i 1 ρ 1 ) S y ρ 1 i 2 + ε 1 z z ( i 1 ρ 1 ) S z ρ 1 i 2 ] + j k 0 ρ 2 [ ε 2 z x ( i 2 ρ 2 ) S x i 1 ρ 2 + ε 2 z y ( i 2 ρ 2 ) S y i 1 ρ 2 + ε 2 z z ( i 2 ρ 2 ) S z i 1 ρ 2 ] ,
S ˜ w = [ S w 1 , S w 2 , , S w M 2 ] T [ S w ( m , m ) , S w ( m , m , + 1 ) , , S w ( m , m ) ] T ,
U ˜ w = [ U w 1 , U w 2 , , U w M 2 ] T [ U w ( m , m ) , U w ( m , m , + 1 ) , , U w ( m , m ) ] T ,
d V ˜ d y = j à V ˜ ,
V ˜ ( y ) = W ˜ exp [ λ ˜ y ] C ˜ ,
u x δ i 1 i 2 00 + R x i 1 i 2 + R x i 1 i 2 = S x i 1 i 2 ( 0 ) ,
U z δ i 1 i 2 00 + R z i 1 i 2 + R z i 1 i 2 = S z i 1 i 2 ( 0 ) ,
( k 1 y u z k 1 z u y ) δ i 1 i 2 00 + k y 1 i 1 i 2 R z i 1 i 2 + k y 1 i 1 i 2 R z i 1 i 2 k 1 z ( R y i 1 i 2 + R y i 1 i 2 ) = k 0 U x i 1 i 2 ( 0 ) ,
( k 1 x u y k 1 y u x ) δ i 1 i 2 00 + k x i 1 i 2 ( R y i 1 i 2 + R y i 1 i 2 ) k y 1 i 1 i 2 R x i 1 i 2 k y 1 i 1 i 2 R x i 1 i 2 = k 0 U z i 1 i 2 ( 0 ) ,
T x i 1 i 2 + T x i 1 i 2 = S x i 1 i 2 ( s ) exp [ j ( i 1 K 1 y + i 2 K 2 y ) s ] = S ˆ x i 1 i 2 ( s ) ,
T z i 1 i 2 + T z i 1 i 2 = S z i 1 i 2 ( s ) exp [ j ( i 1 K 1 y + i 2 K 2 y ) s ] = S ˆ z i 1 i 2 ( s ) ,
k y 3 i 1 i 2 T z i 1 i 2 + k y 3 i 1 i 2 T z i 1 i 2 k 1 ( T y i 1 i 2 + T y i 1 i 2 ) = k 0 U x i 1 i 2 ( s ) exp [ j ( i 1 K 1 y + i 2 K 2 y ) s ] = k 0 U ˆ x i 1 i 2 ( s ) ,
k x i 1 i 2 ( T y i 1 i 2 + T y i 1 i 2 ) k y 3 i 1 i 2 T x i 1 i 2 k y 3 i 1 i 2 T x i 1 i 2 = k 0 U z i 1 i 2 ( s ) exp [ j ( i 1 K 1 y + i 2 K 2 y ) s ] = k 0 U ˆ z 1 i 2 ( s ) ,
k 1 i 1 i 2 · ( ε ˜ 1 R i 1 i 2 ) = 0 ,
k 1 i 1 i 2 · ( ε ˜ 1 R i 1 i 2 ) = 0 ,
k 3 i 1 i 2 · ( ε ˜ 3 T i 1 i 2 ) = 0 ,
k 3 i 1 i 2 · ( ε ˜ 3 T i 1 i 2 = 0 ,
c ˆ 1 · R i 1 i 2 = 0 ,
c ˆ 1 · ( k 1 i 1 i 2 × R i 1 i 2 ) = 0 ,
c ˆ 3 · T i 1 i 2 = 0 ,
c ˆ 3 · ( k 3 i 1 i 2 × T i 1 i 2 ) = 0 ,
P ˜ C ˜ = p ˜ ,
D E q i 1 i 2 = τ q Re { | E q i 1 i 2 | 2 k * y q i 1 i 2 ( E q i 1 i 2 · k q i 1 i 2 ) E * q i 1 i 2 y } / Re { k 1 y ( u ˆ · k 1 ) u y } ,
D E q i 1 i 2 = τ q Re { | E q i 1 i 2 | 2 k * y q i 1 i 2 ( E q i 1 i 2 · k q i 1 i 2 ) E * q i 1 i 2 y } / Re { k 1 y ( u ˆ · k 1 ) u y } ,
i 1 i 2 ( D E 1 i 2 + D E 1 i 1 i 2 + D E 3 i 2 + D E 3 i 1 i 2 ) = 1 .
ε ˜ ( x , y ) = ε ˜ 0 + h f ˜ h ( y ) exp ( j h K x x ) = ε ˜ 0 + Φ ˜ ( x ; y ) ,
F i = i 1 F i 1 , i i 1 ,
ε x x = ε y y = ε 0 + ε 11 sin ( K 1 · r ) + ε 11 sin ( K 2 · r ) , ε z z = ε E + ε 33 sin ( K 1 · r ) + ε 33 sin ( K 2 · r ) , ε x y = ε y x = ε 12 cos ( K 1 · r ) + ε 12 cos ( K 2 · r ) , ε x z = ε z x = ε 13 cos ( K 1 · r ) + ε 13 cos ( K 2 · r ) , ε y z = ε x y = 0 ,
ε x x = ε E ε 11 sin ( K 1 · r ) ε 11 sin ( K 2 · r ) , ε y y = ε z z = ε O ε 22 sin ( K 1 · r ) + ε 22 sin ( K 2 · r ) , ε x y = ε y x = ε 12 sin ( K 1 · r ) ε 12 sin ( K 2 · r ) , ε x z = ε z x = 0 , ε y z = ε z y = ε 23 sin ( K 1 · r ) ε 23 sin ( K 2 · r ) ,
C 1 = T 1 ( 1 ) , A 1 = T 0 ( 1 ) , B 1 = T 1 ( 1 ) C 2 = T 2 ( 2 ) , A 2 = T 1 ( 2 ) , B 2 = T 0 ( 2 ) , C 3 = T 0 ( 3 ) , A 3 = T 1 ( 3 ) , B 3 = T 2 ( 3 ) ,
Δ O q i 1 i 2 = k 0 2 n O q 2 k x i 1 i 2 2 k 1 z 2
Δ E q i 1 i 2 = k 0 2 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 1 i 2 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 1 i 2 ,
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 if ( I / M ) N , I int ( I / M ) M m 1 if ( I / M ) N ,
I = I ( i 1 , i 2 , , i N ) = M N 1 ( m + i 1 ) + M N 2 ( m + i 2 ) + + M ( m + i N 1 ) + ( m + 1 ) + i N ,
( i 1 , i 2 , , i N ) = [ I 1 ( I ) , I 2 ( I ) , , I N ( I ) ] ,
i k = I k ( I ) = { int ( L k / M N k ) m 1 if ( L k / M N k ) N , int ( L k / M N K ) m if ( L k / M N k ) N ,
L k = { L k 1 [ int ( L k 1 / M N k + 1 ) 1 ] M N k + 1 if ( L k 1 / M N k + 1 ) N L k 1 int ( L k 1 / M N k + 1 ) M N k + 1 if ( L k 1 / M N k + 1 ) N ,
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 ε ˜ y y 1 ε ˜ y x K ˜ y , a ˜ 12 = K ˜ x ε ˜ y y 1 ε ˜ y z , a ˜ 13 = ( 1 / k 0 ) K ˜ x ε ˜ y y 1 K ˜ z , a ˜ 14 = I ˜ k 0 ( 1 / k 0 ) K ˜ x ε ˜ y y 1 K ˜ x , a ˜ 21 = K ˜ x ε ˜ y y 1 ε ˜ y x , a ˜ 22 = K ˜ z ε ˜ y y 1 ε ˜ y x , K ˜ y , a ˜ 23 = ( 1 / k 0 ) K ˜ z ε ˜ y y 1 K ˜ z , I ˜ k 0 , a ˜ 24 = ( 1 / k 0 ) K ˜ z ε ˜ y y 1 K ˜ x , a ˜ 31 = k o ε ˜ y z ε ˜ y y 1 ε ˜ y x k 0 ε ˜ x z ( 1 / k 0 ) K ˜ x K ˜ z , a ˜ 32 = k o ε ˜ y z ε ˜ y y 1 ε ˜ z y , k 0 ε ˜ z z + ( 1 / k 0 ) K ˜ x K ˜ x , a ˜ 33 = ε ˜ z y ε ˜ y y 1 K ˜ z , K ˜ y , a ˜ 34 = ε ˜ y z ε ˜ y y 1 K ˜ x , a ˜ 41 = k o ε ˜ x x k 0 ε ˜ x y ε ˜ y y 1 ε ˜ y x ( 1 / k 0 ) K ˜ z K ˜ z , a ˜ 42 = k o ε ˜ x z k 0 ε ˜ x y ε ˜ y y 1 ε ˜ y z + ( 1 / k 0 ) K ˜ z K ˜ x , a ¯ 43 = ε ˜ x y ε ˜ y y 1 K ˜ z , a ˜ 44 = ε ˜ x y ε ˜ y y 1 K ˜ x K ˜ y ,
( 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 ,
( P ¯ 11 ) k n = { E 1 i 1 i 2 W 1 , k n + Z 1 i 1 i 2 W 2 , k n k 0 W 3 , k n if k y 1 i 1 i 2 k y 1 i 1 i 2 , α i 1 i 2 W 1 , k n + β i 1 i 2 W 2 , k n k 0 W 3 , k n if k y 1 i 1 i 2 k y 1 i 1 i 2 ,
( P ¯ 12 ) k n = { E 2 i 1 i 2 W 1 , k n + Z 2 i 1 i 2 W 2 , k n k 0 W 4 , k n if k y 1 i 1 i 2 k y 1 i 1 i 2 , γ i 1 i 2 W 1 , k n + α i 1 i 2 W 2 , k n + k 0 W 4 , k n if k y 1 i 1 i 2 k y 1 i 1 i 2 ,
( P ¯ 13 ) k n = { ( E 3 i 1 i 2 W 1 , k n + Z 3 i 1 i 2 W 2 , k n k 0 W 3 , k n ) ω n if k y 3 i 1 i 2 k y 3 i 1 i 2 , ( θ i 1 i 2 W 1 , k n + μ i 1 i 2 W 2 , k n k 0 W 3 , k n ) ω n if k y 3 i 1 i 2 = k y 3 i 1 i 2 ,
( P ¯ 14 ) k n = { ( E 4 i 1 i 2 W 1 , k n + Z 4 i 1 i 2 W 2 , k n k 0 W 4 , k n ) ω n if k y 3 i 1 i 2 k y 3 i 1 i 2 , ( ν i 1 i 2 W 1 , k n + θ i 1 i 2 W 2 , k n k 0 W 4 , k n ) ω n if k y 3 i 1 i 2 = k y 3 i 1 i 2 ,
p ¯ T = [ 0 0 p I ( 0 , 0 ) 0 0 | 0 0 p I ( 0 , 0 ) + M 2 0 0 | 0 0 | 0 0 ] ,
p I ( 0 , 0 ) = { Ө 1 if k y 100 k y 100 , ζ 1 if k y 100 = k y 100 ,
p I ( 0 , 0 ) + M 2 = { Ө 1 if k y 300 k y 300 , η 1 if k y 300 = k y 300 .
E 1 i 1 i 2 = ( A 1 i 1 i 2 / Ξ 1 i 1 i 2 ) P 1 i 1 i 2 + k 1 z ( l x i 1 i 2 1 / l y i 1 i 2 1 ) ,
E 2 i 1 i 2 = ( A 1 i 1 i 2 / Ξ 1 i 1 i 2 ) Q 1 i 1 i 2 k x i 1 i 2 ( l x i 1 i 2 1 / l y i 1 i 2 1 ) k y 1 i 1 i 2 ,
E 3 i 1 i 2 = ( A 3 i 1 i 2 / Ξ 3 i 1 i 2 ) P 3 i 1 i 2 + k 1 z ( l x i 1 i 2 3 / l y i 1 i 2 3 ) ,
E 4 i 1 i 2 = ( A 3 i 1 i 2 / Ξ 3 i 1 i 2 ) Q 3 i 1 i 2 k x i 1 i 2 ( l x i 1 i 2 3 / l y i 1 i 2 3 ) k y 3 i 1 i 2 ,
Z 1 i 1 i 2 = ( B 1 i 1 i 2 / Ξ 1 i 1 i 2 ) P 1 i 1 i 2 + k 1 z ( l z i 1 i 2 1 / l y i 1 i 2 1 ) + k y 1 i 1 i 2 ,
Z 2 i 1 i 2 = ( B 1 i 1 i 2 / Ξ 1 i 1 i 2 ) Q 1 i 1 i 2 k x i 1 i 2 1 ( l z i 1 i 2 1 / l y i 1 i 2 1 ) ,
Z 3 i 1 i 2 = ( B 3 i 1 i 2 / Ξ 3 i 1 i 2 ) P 3 i 1 i 2 + k 1 z ( l z i 1 i 2 3 / l y i 1 i 2 3 ) + k y 3 i 1 i 2 ,
Z 4 i 1 i 2 = ( B 3 i 1 i 2 / Ξ 3 i 1 i 2 ) Q 3 i 1 i 2 k x i 1 i 2 ( l z i 1 i 2 3 / l y i 1 i 2 3 ) ,
P q i 1 i 2 = k 1 z ( l x i 1 i 2 q l y i 1 i 2 q l x i 1 i 2 q l y i 1 i 2 q ) Δ q i 1 i 2 + [ k 1 z ( l z i 1 i 2 q l y i 1 i 2 q l z i 1 i 2 q l y i 1 i 2 q ) k y q i 1 i 2 + k y q i 1 i 2 ] Γ q i 1 i 2 ,
Q q i 1 i 2 = k x i 1 i 2 ( l z i 1 i 2 q l y i 1 i 2 q l z i 1 i 2 q l y i 1 i 2 q ) Γ q i 1 i 2 + [ k x i 1 i 2 ( l x i 1 i 2 q l y i 1 i 2 q l x i 1 i 2 q l y i 1 i 2 q ) k y q i 1 i 2 + k y q i 1 i 2 ] Δ q i 1 i 2
Ξ q i 1 i 2 = A q i 1 i 2 Δ q i 1 i 2 B q i 1 i 2 Γ q i 1 i 2 with q = 1 , 3 ,
A q i 1 i 2 = r q y i 1 i 2 l x i 1 i 2 q l y i 1 i 2 q r q x i 1 i 2 , B q i = r q y i 1 i 2 l z i 1 i 2 q l y i 1 i 2 q r q z i 1 i 2 ,
Γ q i 1 i 2 = c q x c q y l x i 1 i 2 l y i 1 i 2 , Δ q i 1 i 2 = c q x c q y l z i 1 i 2 l y i 1 i 2 ,
l w i 1 i 2 q = k x i 1 i 2 ε x w q + k y q i 1 i 2 ε y w q + k 1 z ε z w q ,
l wiq = k x i 1 i 2 ε x w q + k yq i 1 i 2 ε y w q + k 1 z ε x w q ,
Ө 1 = k 1 z l x i 1 i 2 1 l y i 1 i 2 1 u x + k 1 z u y + [ k y 1 i 1 i 2 k 1 y + k 1 z l z i 1 i 2 1 l y i 1 i 2 1 ] u z Π i 1 i 2 Ξ 1 i 1 i 2 P 1 i 1 i 2 ,
Ө 2 = [ k 1 y k x i 1 i 2 l x i 1 i 2 1 l y i 1 i 2 1 k y 1 i 1 i 2 ] u x k 1 x u y k x i 1 i 2 l z i 1 i 2 1 l y i 1 i 2 1 u z Π i 1 i 2 Ξ 1 i 1 i 2 Q 1 i 1 i 2 ,
Π i 1 i 2 = ( r 1 x i 1 i 2 r 1 y i 1 i 2 l x i 1 i 2 1 l y i 1 i 2 1 ) u x + ( r 1 x i 1 i 2 r 1 y i 1 i 2 l z i 1 i 2 1 l y i 1 i 2 1 ) u z .
α i 1 i 2 = k 1 z k x i 1 i 2 / k y 1 i 1 i 2 ,
β i 1 i 2 = ( k 1 z 2 / k y 1 i 1 i 2 ) + k y 1 i 1 i 2 ,
γ i 1 i 2 = ( k x i 1 i 2 2 / k y 1 i 1 i 2 ) + k y 1 i 1 i 2 ,
ζ i 1 i 2 = ( k 1 z k x i 1 i 2 / k y 1 i 1 i 2 ) u x + k 1 z u y + [ ( k 1 z 2 / k y 1 i 1 i 2 ) + k y 1 i 1 i 2 k 1 y ] u z ,
η i 1 i 2 = [ ( k x i 1 i 2 2 / k y 1 i 1 i 2 ) + k y 1 i 1 i 2 k 1 y ] u z , + k 1 x u y + ( k x i k 1 z / k y 1 i 1 i 2 ) u x ,
θ i 1 i 2 = k 1 z k x i 1 i 2 k y 3 i 1 i 2 ,
μ i 1 i 2 = ( k 1 z 2 / k y 3 i 1 i 2 ) + k y 3 i 1 i 2 ,
ν i 1 i 2 = ( k x i 1 i 2 2 / k y 3 i 1 i 2 ) + k y 3 i 1 i 2 .
R x i 1 i 2 = ( Δ 1 i 1 i 2 / Ξ 1 i 1 i 2 ) [ I I i 1 i 2 δ i 1 i 2 00 + A 1 i 1 i 2 S x i 1 i 2 ( 0 ) + B 1 i 1 i 2 S z i 1 i 2 ( 0 ) ] ,
R z i 1 i 2 = ( Γ 1 i 1 i 2 / Ξ 1 i 1 i 2 ) R x i 1 i 2 ,
T x i 1 i 2 = ( Δ 3 i 1 i 2 / Ξ 3 i 1 i 2 ) [ A 3 i 1 i 2 S x i 1 i 2 ( s ) + B 3 i 1 i 2 S z i 1 i 2 ( s ) ] ,
T 2 i 1 i 2 = ( Γ 3 i 1 i 2 / Ξ 3 i 1 i 2 ) T x i 1 i 2 ,
R y i 1 i 2 = ( l x i 1 i 2 1 / l y i 1 i 2 1 ) R x i 1 i 2 ( l z i 1 i 2 1 / l y i 1 i 2 1 ) R z i 1 i 2 ,
R y i 1 i 2 = ( l x i 1 i 2 1 / l y i 1 i 2 1 ) R x i 1 i 2 ( l z i 1 i 2 1 / l y i 1 i 2 1 ) R z i 1 i 2 ,
T y i 1 i 2 = ( l x i 1 i 2 3 / l y i 1 i 2 3 ) T x i 1 i 2 ( l z i 1 i 2 3 / l y i 1 i 2 3 ) T z i 1 i 2 ,
T y i 1 i 2 = ( l x i 1 i 2 3 / l y i 1 i 2 3 ) T x i 1 i 2 ( l z i 1 i 2 3 / l y i 1 i 2 3 ) T z i 1 i 2 ,

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