Abstract

With the Lippman–Schwinger equation, dyadic Green’s functions, and the vector coherent transfer function method, an electromagnetic theory of a waveguide multilayered optical memory is first developed for the static case, from which a theory describing a conventional multilayered optical memory with bits stored as a refractive index change is also derived. In addition, the formulas for readout signals and cross talk are given, and some problems of numerical calculations are discussed. The theories can be used effectively for optimum design of a multilayered optical memory with bits stored as a refractive index change.

© 2007 Optical Society of America

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2006 (5)

2005 (1)

2004 (2)

O. Haeberlé, "Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part II: confocal and multiphoton microscopy," Opt. Commun. 235, 1-10 (2004).
[CrossRef]

Z. Liang, J. S. Chen, T. Yang, Y. Xie, J. Chen, and S. Zhuang, "Principles and experiments of the waveguide multilayered optical memory," J. Optoelectron., Laser 15, 315-317 (2004), (in Chinese).

2003 (2)

O. Haeberlé, "Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part I: conventional microscopy," Opt. Commun. 216, 55-63 (2003).
[CrossRef]

L. Tsang, C. Ong, C. C. Huang, and V. Jandhyala, "Evaluation of the Green's function for the mixed potential integral equation (MPIE) method in the time domain for layered media," IEEE Trans. Antennas Propag. 51, 1559-1571 (2003).
[CrossRef]

2002 (1)

Z. Liang, T. Yang, H. Ming, and J. Xie, "A novel 3D multilayered waveguide memory," Proc. SPIE 4930, 134-137 (2002).
[CrossRef]

2001 (2)

T. Yu and W. Cai, "High-order window functions and fast algorithms for calculating dyadic electromagnetic Green's functions in multilayered media," Radio Sci. 36, 559-569 (2001).
[CrossRef]

C. Li, D. Wang, L. Luo, H. Yang, Z. Xia, and Q. Gong, "Feasibility of femtosecond laser writing multi-layered bit planes in fused silica for three-dimensional optical data storage," Chin. Phys. Lasers 18, 541-543 (2001).

1999 (1)

T. J. Cui and W. C. Chew, "Fast evaluation of Sommerfeld integrals for EM scattering and radiation by three-dimensional buried objects," IEEE Trans. Geosci. Remote Sens. 37, 887-900 (1999).
[CrossRef]

1998 (4)

P. Török, P. D. Higdon and T. Wilson, "Theory for confocal and conventional microscopes imaging small dielectric scatters," J. Mod. Opt. 45, 1681-1698 (1998).
[CrossRef]

M. Watanabe, H. Sun, S. Juodkazis, T. Takahashi, S. Matsuo, Y. Suzuki, J. Nishii, and H. Misawa, "Three-dimensional optical data storage in vitreous silica," Jpn. J. Appl. Phys., Part 1 37, L1527-L1530 (1998).
[CrossRef]

M. Ishikawa, Y. Kawata, C. Egami, O. Sugihara, and N. Okamoto, "Reflection-type confocal readout for multilayered optical memory," Opt. Lett. 23, 1781-1783 (1998).
[CrossRef]

T. Tsujioka and M. Irie, "Fluorescence readout of near-field photochromic memory," Appl. Opt. 37, 4419-4424 (1998).
[CrossRef]

1997 (2)

P. Török and P. Varga, "Electromagnetic diffraction of light focused through a stratified medium," Appl. Opt. 36, 2305-2312 (1997).
[CrossRef] [PubMed]

P. Török and T. Wilson, "Rigorous theory for axial resolution in confocal microscopes," Opt. Commun. 137, 127-135 (1997).
[CrossRef]

1996 (1)

1994 (2)

1993 (1)

1992 (2)

G. A. Rakuljic, V. Leyva, and A. Yariv, "Optical data storage by using orthogonal wavelength-multiplexed volume holograms," Opt. Lett. 17, 1471-1473 (1992).
[CrossRef] [PubMed]

S. Barkeshli and P. H. Pathak, "On the dyadic Green's function for a planar multilayered dielectric/magnetic media," IEEE Trans. Microwave Theory Tech. 40, 128-142 (1992).
[CrossRef]

1991 (2)

1989 (1)

D. A. Parthenopoulos and P. M. Rentzepis, "Three-dimensional optical storage memory," Science 245, 843-845 (1989).
[CrossRef] [PubMed]

1959 (1)

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system," Proc. R. Soc. London, Ser. A 253, 358-370 (1959).
[CrossRef]

Barkeshli, S.

S. Barkeshli and P. H. Pathak, "On the dyadic Green's function for a planar multilayered dielectric/magnetic media," IEEE Trans. Microwave Theory Tech. 40, 128-142 (1992).
[CrossRef]

Bashaw, M. C.

J. E. Heanue, M. C. Bashaw, and L. Hesselink, "Volume holographic storage and retrieval of digital data," Science 265, 749-752 (1994).
[CrossRef] [PubMed]

Cai, W.

T. Yu and W. Cai, "High-order window functions and fast algorithms for calculating dyadic electromagnetic Green's functions in multilayered media," Radio Sci. 36, 559-569 (2001).
[CrossRef]

Callan, J. P.

Chen, J.

Chen, J. S.

Z. Liang, J. S. Chen, T. Yang, Y. Xie, J. Chen, and S. Zhuang, "Principles and experiments of the waveguide multilayered optical memory," J. Optoelectron., Laser 15, 315-317 (2004), (in Chinese).

Chew, W. C.

T. J. Cui and W. C. Chew, "Fast evaluation of Sommerfeld integrals for EM scattering and radiation by three-dimensional buried objects," IEEE Trans. Geosci. Remote Sens. 37, 887-900 (1999).
[CrossRef]

Crowe, D. G.

Cui, T. J.

T. J. Cui and W. C. Chew, "Fast evaluation of Sommerfeld integrals for EM scattering and radiation by three-dimensional buried objects," IEEE Trans. Geosci. Remote Sens. 37, 887-900 (1999).
[CrossRef]

Dereux, A.

Egami, C.

Finlay, R. J.

Girard, C.

Glezer, E. N.

Gong, Q.

C. Li, D. Wang, L. Luo, H. Yang, Z. Xia, and Q. Gong, "Feasibility of femtosecond laser writing multi-layered bit planes in fused silica for three-dimensional optical data storage," Chin. Phys. Lasers 18, 541-543 (2001).

Guo, H.

Haeberlé, O.

O. Haeberlé, "Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part II: confocal and multiphoton microscopy," Opt. Commun. 235, 1-10 (2004).
[CrossRef]

O. Haeberlé, "Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part I: conventional microscopy," Opt. Commun. 216, 55-63 (2003).
[CrossRef]

Heanue, J. E.

J. E. Heanue, M. C. Bashaw, and L. Hesselink, "Volume holographic storage and retrieval of digital data," Science 265, 749-752 (1994).
[CrossRef] [PubMed]

Her, T.-H.

Hesselink, L.

J. E. Heanue, M. C. Bashaw, and L. Hesselink, "Volume holographic storage and retrieval of digital data," Science 265, 749-752 (1994).
[CrossRef] [PubMed]

Higdon, P. D.

P. Török, P. D. Higdon and T. Wilson, "Theory for confocal and conventional microscopes imaging small dielectric scatters," J. Mod. Opt. 45, 1681-1698 (1998).
[CrossRef]

Huang, C. C.

L. Tsang, C. Ong, C. C. Huang, and V. Jandhyala, "Evaluation of the Green's function for the mixed potential integral equation (MPIE) method in the time domain for layered media," IEEE Trans. Antennas Propag. 51, 1559-1571 (2003).
[CrossRef]

Huang, L.

Irie, M.

Ishikawa, M.

Jandhyala, V.

L. Tsang, C. Ong, C. C. Huang, and V. Jandhyala, "Evaluation of the Green's function for the mixed potential integral equation (MPIE) method in the time domain for layered media," IEEE Trans. Antennas Propag. 51, 1559-1571 (2003).
[CrossRef]

Juodkazis, S.

M. Watanabe, H. Sun, S. Juodkazis, T. Takahashi, S. Matsuo, Y. Suzuki, J. Nishii, and H. Misawa, "Three-dimensional optical data storage in vitreous silica," Jpn. J. Appl. Phys., Part 1 37, L1527-L1530 (1998).
[CrossRef]

Kawata, Y.

Leyva, V.

Li, C.

C. Li, D. Wang, L. Luo, H. Yang, Z. Xia, and Q. Gong, "Feasibility of femtosecond laser writing multi-layered bit planes in fused silica for three-dimensional optical data storage," Chin. Phys. Lasers 18, 541-543 (2001).

Liang, Z.

H. Guo, S. Zhuang, J. Chen, and Z. Liang, "Imaging theory of an aplanatic system with a stratified medium based on the method for a vector coherent transfer function," Opt. Lett. 31, 2978-2980 (2006).
[CrossRef] [PubMed]

Z. Liang, J. S. Chen, T. Yang, Y. Xie, J. Chen, and S. Zhuang, "Principles and experiments of the waveguide multilayered optical memory," J. Optoelectron., Laser 15, 315-317 (2004), (in Chinese).

Z. Liang, T. Yang, H. Ming, and J. Xie, "A novel 3D multilayered waveguide memory," Proc. SPIE 4930, 134-137 (2002).
[CrossRef]

Liu, Q. H.

E. Simsek, Q. H. Liu, and B. Wei, "Singularity subtraction for evaluation of Green's functions for multilayer media," IEEE Trans. Microwave Theory Tech. 54, 216-224 (2006).
[CrossRef]

Luo, L.

C. Li, D. Wang, L. Luo, H. Yang, Z. Xia, and Q. Gong, "Feasibility of femtosecond laser writing multi-layered bit planes in fused silica for three-dimensional optical data storage," Chin. Phys. Lasers 18, 541-543 (2001).

Martin, O. J. F.

Matsuo, S.

M. Watanabe, H. Sun, S. Juodkazis, T. Takahashi, S. Matsuo, Y. Suzuki, J. Nishii, and H. Misawa, "Three-dimensional optical data storage in vitreous silica," Jpn. J. Appl. Phys., Part 1 37, L1527-L1530 (1998).
[CrossRef]

Mazur, E.

Milosavljevic, M.

Ming, H.

Z. Liang, T. Yang, H. Ming, and J. Xie, "A novel 3D multilayered waveguide memory," Proc. SPIE 4930, 134-137 (2002).
[CrossRef]

Misawa, H.

M. Watanabe, H. Sun, S. Juodkazis, T. Takahashi, S. Matsuo, Y. Suzuki, J. Nishii, and H. Misawa, "Three-dimensional optical data storage in vitreous silica," Jpn. J. Appl. Phys., Part 1 37, L1527-L1530 (1998).
[CrossRef]

Mok, F. H.

Nakano, M.

Nishii, J.

M. Watanabe, H. Sun, S. Juodkazis, T. Takahashi, S. Matsuo, Y. Suzuki, J. Nishii, and H. Misawa, "Three-dimensional optical data storage in vitreous silica," Jpn. J. Appl. Phys., Part 1 37, L1527-L1530 (1998).
[CrossRef]

Okamoto, N.

Ong, C.

L. Tsang, C. Ong, C. C. Huang, and V. Jandhyala, "Evaluation of the Green's function for the mixed potential integral equation (MPIE) method in the time domain for layered media," IEEE Trans. Antennas Propag. 51, 1559-1571 (2003).
[CrossRef]

Parthenopoulos, D. A.

D. A. Parthenopoulos and P. M. Rentzepis, "Three-dimensional optical storage memory," Science 245, 843-845 (1989).
[CrossRef] [PubMed]

Pathak, P. H.

S. Barkeshli and P. H. Pathak, "On the dyadic Green's function for a planar multilayered dielectric/magnetic media," IEEE Trans. Microwave Theory Tech. 40, 128-142 (1992).
[CrossRef]

Rakuljic, G. A.

Rentzepis, P. M.

D. A. Parthenopoulos and P. M. Rentzepis, "Three-dimensional optical storage memory," Science 245, 843-845 (1989).
[CrossRef] [PubMed]

Richards, B.

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system," Proc. R. Soc. London, Ser. A 253, 358-370 (1959).
[CrossRef]

Salazar-Palma, M.

M. Yuan, T. K. Sarkar, and M. Salazar-Palma, "A direct discrete complex image method from the closed-form Green's functions in multilayered media," IEEE Trans. Microwave Theory Tech. 54, 1025-1031 (2006).
[CrossRef]

Sarkar, T. K.

M. Yuan, T. K. Sarkar, and M. Salazar-Palma, "A direct discrete complex image method from the closed-form Green's functions in multilayered media," IEEE Trans. Microwave Theory Tech. 54, 1025-1031 (2006).
[CrossRef]

Simsek, E.

E. Simsek, Q. H. Liu, and B. Wei, "Singularity subtraction for evaluation of Green's functions for multilayer media," IEEE Trans. Microwave Theory Tech. 54, 216-224 (2006).
[CrossRef]

Strickler, J. H.

Sugihara, O.

Sun, H.

M. Watanabe, H. Sun, S. Juodkazis, T. Takahashi, S. Matsuo, Y. Suzuki, J. Nishii, and H. Misawa, "Three-dimensional optical data storage in vitreous silica," Jpn. J. Appl. Phys., Part 1 37, L1527-L1530 (1998).
[CrossRef]

Suzuki, Y.

M. Watanabe, H. Sun, S. Juodkazis, T. Takahashi, S. Matsuo, Y. Suzuki, J. Nishii, and H. Misawa, "Three-dimensional optical data storage in vitreous silica," Jpn. J. Appl. Phys., Part 1 37, L1527-L1530 (1998).
[CrossRef]

Takahashi, T.

M. Watanabe, H. Sun, S. Juodkazis, T. Takahashi, S. Matsuo, Y. Suzuki, J. Nishii, and H. Misawa, "Three-dimensional optical data storage in vitreous silica," Jpn. J. Appl. Phys., Part 1 37, L1527-L1530 (1998).
[CrossRef]

Török, P.

P. Török, P. D. Higdon and T. Wilson, "Theory for confocal and conventional microscopes imaging small dielectric scatters," J. Mod. Opt. 45, 1681-1698 (1998).
[CrossRef]

P. Török and P. Varga, "Electromagnetic diffraction of light focused through a stratified medium," Appl. Opt. 36, 2305-2312 (1997).
[CrossRef] [PubMed]

P. Török and T. Wilson, "Rigorous theory for axial resolution in confocal microscopes," Opt. Commun. 137, 127-135 (1997).
[CrossRef]

Tsang, L.

L. Tsang, C. Ong, C. C. Huang, and V. Jandhyala, "Evaluation of the Green's function for the mixed potential integral equation (MPIE) method in the time domain for layered media," IEEE Trans. Antennas Propag. 51, 1559-1571 (2003).
[CrossRef]

Tsujioka, T.

Varga, P.

Wang, D.

C. Li, D. Wang, L. Luo, H. Yang, Z. Xia, and Q. Gong, "Feasibility of femtosecond laser writing multi-layered bit planes in fused silica for three-dimensional optical data storage," Chin. Phys. Lasers 18, 541-543 (2001).

Watanabe, M.

M. Watanabe, H. Sun, S. Juodkazis, T. Takahashi, S. Matsuo, Y. Suzuki, J. Nishii, and H. Misawa, "Three-dimensional optical data storage in vitreous silica," Jpn. J. Appl. Phys., Part 1 37, L1527-L1530 (1998).
[CrossRef]

Webb, W. W.

Wei, B.

E. Simsek, Q. H. Liu, and B. Wei, "Singularity subtraction for evaluation of Green's functions for multilayer media," IEEE Trans. Microwave Theory Tech. 54, 216-224 (2006).
[CrossRef]

Wilson, T.

P. Török, P. D. Higdon and T. Wilson, "Theory for confocal and conventional microscopes imaging small dielectric scatters," J. Mod. Opt. 45, 1681-1698 (1998).
[CrossRef]

P. Török and T. Wilson, "Rigorous theory for axial resolution in confocal microscopes," Opt. Commun. 137, 127-135 (1997).
[CrossRef]

Wolf, E.

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system," Proc. R. Soc. London, Ser. A 253, 358-370 (1959).
[CrossRef]

Xia, Z.

C. Li, D. Wang, L. Luo, H. Yang, Z. Xia, and Q. Gong, "Feasibility of femtosecond laser writing multi-layered bit planes in fused silica for three-dimensional optical data storage," Chin. Phys. Lasers 18, 541-543 (2001).

Xie, J.

Z. Liang, T. Yang, H. Ming, and J. Xie, "A novel 3D multilayered waveguide memory," Proc. SPIE 4930, 134-137 (2002).
[CrossRef]

Xie, Y.

Z. Liang, J. S. Chen, T. Yang, Y. Xie, J. Chen, and S. Zhuang, "Principles and experiments of the waveguide multilayered optical memory," J. Optoelectron., Laser 15, 315-317 (2004), (in Chinese).

Yang, H.

C. Li, D. Wang, L. Luo, H. Yang, Z. Xia, and Q. Gong, "Feasibility of femtosecond laser writing multi-layered bit planes in fused silica for three-dimensional optical data storage," Chin. Phys. Lasers 18, 541-543 (2001).

Yang, T.

Z. Liang, J. S. Chen, T. Yang, Y. Xie, J. Chen, and S. Zhuang, "Principles and experiments of the waveguide multilayered optical memory," J. Optoelectron., Laser 15, 315-317 (2004), (in Chinese).

Z. Liang, T. Yang, H. Ming, and J. Xie, "A novel 3D multilayered waveguide memory," Proc. SPIE 4930, 134-137 (2002).
[CrossRef]

Yariv, A.

Yu, T.

T. Yu and W. Cai, "High-order window functions and fast algorithms for calculating dyadic electromagnetic Green's functions in multilayered media," Radio Sci. 36, 559-569 (2001).
[CrossRef]

Yuan, M.

M. Yuan, T. K. Sarkar, and M. Salazar-Palma, "A direct discrete complex image method from the closed-form Green's functions in multilayered media," IEEE Trans. Microwave Theory Tech. 54, 1025-1031 (2006).
[CrossRef]

Zhuang, S.

Appl. Opt. (4)

Chin. Phys. Lasers (1)

C. Li, D. Wang, L. Luo, H. Yang, Z. Xia, and Q. Gong, "Feasibility of femtosecond laser writing multi-layered bit planes in fused silica for three-dimensional optical data storage," Chin. Phys. Lasers 18, 541-543 (2001).

IEEE Trans. Antennas Propag. (1)

L. Tsang, C. Ong, C. C. Huang, and V. Jandhyala, "Evaluation of the Green's function for the mixed potential integral equation (MPIE) method in the time domain for layered media," IEEE Trans. Antennas Propag. 51, 1559-1571 (2003).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

T. J. Cui and W. C. Chew, "Fast evaluation of Sommerfeld integrals for EM scattering and radiation by three-dimensional buried objects," IEEE Trans. Geosci. Remote Sens. 37, 887-900 (1999).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (3)

M. Yuan, T. K. Sarkar, and M. Salazar-Palma, "A direct discrete complex image method from the closed-form Green's functions in multilayered media," IEEE Trans. Microwave Theory Tech. 54, 1025-1031 (2006).
[CrossRef]

E. Simsek, Q. H. Liu, and B. Wei, "Singularity subtraction for evaluation of Green's functions for multilayer media," IEEE Trans. Microwave Theory Tech. 54, 216-224 (2006).
[CrossRef]

S. Barkeshli and P. H. Pathak, "On the dyadic Green's function for a planar multilayered dielectric/magnetic media," IEEE Trans. Microwave Theory Tech. 40, 128-142 (1992).
[CrossRef]

J. Mod. Opt. (1)

P. Török, P. D. Higdon and T. Wilson, "Theory for confocal and conventional microscopes imaging small dielectric scatters," J. Mod. Opt. 45, 1681-1698 (1998).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Optoelectron., Laser (1)

Z. Liang, J. S. Chen, T. Yang, Y. Xie, J. Chen, and S. Zhuang, "Principles and experiments of the waveguide multilayered optical memory," J. Optoelectron., Laser 15, 315-317 (2004), (in Chinese).

Jpn. J. Appl. Phys., Part 1 (1)

M. Watanabe, H. Sun, S. Juodkazis, T. Takahashi, S. Matsuo, Y. Suzuki, J. Nishii, and H. Misawa, "Three-dimensional optical data storage in vitreous silica," Jpn. J. Appl. Phys., Part 1 37, L1527-L1530 (1998).
[CrossRef]

Opt. Commun. (3)

P. Török and T. Wilson, "Rigorous theory for axial resolution in confocal microscopes," Opt. Commun. 137, 127-135 (1997).
[CrossRef]

O. Haeberlé, "Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part I: conventional microscopy," Opt. Commun. 216, 55-63 (2003).
[CrossRef]

O. Haeberlé, "Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part II: confocal and multiphoton microscopy," Opt. Commun. 235, 1-10 (2004).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

(a) Schematic diagram of a waveguide multilayered optical memory. (b) Structures of a waveguide multilayered disc without the transparent cement layer and with a different cladding and substrate.

Fig. 2
Fig. 2

Schematic diagram of reading data from a waveguide multilayered optical memory.

Fig. 3
Fig. 3

Schematic diagram of a conventional multilayered optical memory, BS, beam splitter.

Equations (67)

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ε ( r ) = ε w ( r ) + Δ ε p ( r ) .
× × E ( r ) + k 0 2 ε ( r ) E ( r ) = 0 ,
× × E r ( r ) + k 0 2 ε w ( r ) E r ( r ) = 0
E ( r ) = E r ( r ) + D d r G ( r ; r ) V ( r ) E ( r ) ,
E p ( r ) = E r ( r ) + k = 1 N p W k G ( r ; r k ) V ( r k ) E p ( r k ) ,
E s ( x , y , z u ) = k = 1 N p W k G ( x , y , z u ; r k ) V ( r k ) E p ( r k ) .
G m , n ( r ; r ) = z z k n 2 δ ( r r ) + U ( z z ) j 4 π 2 d k x d k y × exp ( j κ m z m ) exp ( j κ n z n ) 1 2 κ n { Δ Υ m , n > n n × [ exp ( j k m > r ) + R m > exp ( j 2 κ m z m ) exp ( j k m < r ) ] × [ exp ( j k n > r ) + R n < exp ( j 2 κ n z n 1 ) exp ( j k n < r ) ] + Δ Υ m , n > k m κ n k n κ m [ n m > exp ( j k m > r ) + R m > exp ( j 2 κ m z m ) n m < exp ( j k m < r ) ] × [ n n > exp ( j k n > r ) + R n < exp ( j 2 κ n z n 1 ) n n < exp ( j k n < r ) ] } + U ( z z ) j 4 π 2 d k x d k y exp ( j κ m z m ) exp ( j κ n z n 1 ) 1 2 κ n { Δ Υ m , n < n n [ exp ( j k m < r ) + R m < exp ( j 2 κ m z m ) exp ( j k m > r ) ] [ exp ( j k n < r ) + R n > exp ( j 2 κ n z n ) exp ( j k n > r ) ] + Δ Υ m , n < k m κ n k n κ m [ n m < exp ( j k m < r ) + R m < exp ( j 2 κ m z m ) n m > exp ( j k m > r ) ] × [ n n < exp ( j k n < r ) + R n > exp ( j 2 κ n z n ) n n > exp ( j k n > r ) ] } ,
U ( ξ ) = { 1 ξ > 0 0 ξ < 0 }
k m > , < = x k x + y k y ± z κ m ,
n = x k y y k x k t ,
n m > , < = ( x k x + y k y ) κ m ± z k t 2 k m k t ,
k m = ω ( μ m ε m ) 1 2 , k t 2 = k x 2 + k y 2 .
Υ m , n > = i = n m 1 T i > ( z i ) exp ( j κ i + 1 l i + 1 ) ,
T i > ( z i ) = 1 + Γ i > 1 + Γ i > R i + 1 > ( z i + 1 ) exp ( j 2 κ i + 1 l i + 1 ) ,
R i > ( z i ) = Γ i > + R i + 1 > ( z i + 1 ) exp ( j 2 κ i + 1 l i + 1 ) 1 + Γ i > R i + 1 > ( z i + 1 ) exp ( j 2 κ i + 1 l i + 1 ) ,
Γ i > = η i + 1 η i η i + 1 + η i .
Υ m , n < = i = n m + 1 T i < ( z i ) exp ( j κ i 1 l i 1 ) ,
T i < ( z i ) = 1 + Γ i < 1 + Γ i < R i 1 < ( z i 2 ) exp ( j 2 κ i 1 l i 1 ) ,
R i < ( z i 1 ) = Γ i < + R i 1 < ( z i 2 ) exp ( j 2 κ i 1 l i 1 ) 1 + Γ i < R i 1 < ( z i 2 ) exp ( j 2 κ i 1 l i 1 ) ,
Γ i < = η i 1 η i η i 1 + η i ,
Δ = [ 1 R n > ( x n ) R n < ( x n 1 ) exp ( j 2 κ n l n ) ] 1 ,
R N > ( x N ) = η N + 1 η N η N + 1 + η N ,
R 1 < ( x 0 ) = η 0 η 1 η 0 + η 1 ,
η i = κ i ω ε i , η i = ω μ i κ i ,
l i = z i z i 1 ,
E ( r ) = λ 2 E ̃ ( s ) exp ( j k s r ) d s x d s y ,
E ̃ ( s ) = [ x E x + y E y z s z 1 ( s x E x + s y E y ) ] exp [ j k ( s x x + s y y ) ] d x d y ,
E ̃ s ( s o x , s o y , z u ) = k = 1 N p W k G ̃ ( s o x , s o y , z u ; r k ) V ( r k ) E p ( r k ) ,
κ m = k m s m z = [ k m 2 k u + 1 2 ( s o x 2 + s o y 2 ) ] 1 2 ,
G ̃ u + 1 , n ( s o x , s o y , z u ; r k ) = j 2 k n s n z exp ( j k n s n z z n ) exp { j k u + 1 [ s o x ( x k x o ) + s o y y k ] } { Δ Υ u + 1 , n > n n [ exp ( j k n s n z z k ) + R n < exp ( j 2 k n s n z z n 1 ) exp ( j k n s n z z k ) ] + Δ Υ u + 1 , n > s n z s o z 1 n u + 1 > [ n n > exp ( j k n s n z z k ) + n n < R n < exp ( j 2 k n s n z z n 1 ) exp ( j k n s n z z k ) ] } .
E ̃ o ( s o x , s o y , z o ) = E ̃ s ( s o x , s o y , z u ) exp [ j k ( u + 1 ) s o z ( z u z o ) ] .
E ̃ o x ( s o x , s o y , z o ) = j k = 1 N p A exp { j k u + 1 [ s o x ( x k x o ) + s o y y k ] } × { Δ Υ u + 1 , n > k n s o y ( s o y E p x s o x E p y ) A + Δ Υ u + 1 , n > s n z s o x [ ( s o x E p x + s o y E p y ) k n s n z B + k u + 1 ( s o x 2 + s o y 2 ) E p z B ] } ,
E ̃ o y ( s o x , s o y , z o ) = j k = 1 N p A exp { j k u + 1 [ s o x ( x k x o ) + s o y y k ] } { Δ Υ u + 1 , n > k n s o x ( s o y E p x s o x E p y ) A + Δ Υ u + 1 , n > s n z s o y [ ( s o x E p x + s o y E p y ) k n s n z B + k u + 1 ( s o x 2 + s o y 2 ) E p z B ] } ,
A = 1 2 k n 2 s n z ( s o x 2 + s o y 2 ) W k V ( r k ) exp ( j k n s n z z n ) exp [ j k ( u + 1 ) s o z ( z u z o ) ] ,
A = exp ( j k n s n z z k ) + R n < exp ( j 2 k n s n z z n 1 ) exp ( j k n s n z z k ) ,
B ± = exp ( j k n s n z z k ) ± R n < exp ( j 2 k n s n z z n 1 ) exp ( j k n s n z z k ) ,
s o x = ( M n i n o ) sin θ i cos φ ,
s o y = ( M n i n o ) sin θ i sin φ ,
s o z = cos θ o ,
s m z = cos θ m = [ 1 ( k u + 1 M n i sin θ i k m n o ) 2 ] 1 2 .
E ̃ i ( s i x , s i y ) = H ̃ c 1 ( s i x , s i y ) E ̃ o x ( s o x , s o y ) + H ̃ c 2 ( s i x , s i y ) E ̃ o y ( s o x , s o y ) ,
H ̃ c 1 x ( s i x , s i y ) = j Γ ( cos θ o sin 2 φ + cos θ i cos 2 φ ) ,
H ̃ c 1 y ( s i x , s i y ) = j Γ ( cos θ i cos θ o ) sin φ cos φ ,
H ̃ c 1 z ( s i x , s i y ) = j Γ sin θ i cos φ ,
H ̃ c 2 x ( s i x , s i y ) = H ̃ c 1 y ( s i x , s i y ) ,
H ̃ c 2 y ( s i x , s i y ) = j Γ ( cos θ o cos 2 φ + cos θ i sin 2 φ ) ,
H ̃ c 2 z ( s i x , s i y ) = j Γ sin θ i sin φ .
E ( r ) = λ i 2 0 Φ i sin θ i cos θ i d θ i 0 2 π E ̃ i ( s i ) exp [ j k i ( s i x x + s i y y + s i z z ) ] d φ ,
E ̃ o x o ( s o x , s o y , z o ) = j A { Δ Υ u + 1 , n > k n s o y ( s o y E p x s o x E p y ) A + Δ Υ u + 1 , n > s n z s o x [ ( s o x E p x + s o y E p y ) k n s n z B + k u + 1 ( s o x 2 + s o y 2 ) E p z B ] } ,
E ̃ o y o ( s o x , s o y , z o ) = j A { Δ Υ u + 1 , n > k n s o x ( s o y E p x s o x E p y ) A + Δ Υ u + 1 , n > s n z s o y [ ( s o x E p x + s o y E p y ) k n s n z B + k u + 1 ( s o x 2 + s o y 2 ) E p z B ] } ,
E ̃ o k ( s o x , s o y , z o ) = E ̃ o o ( s o x , s o y , z o ) exp { j k u + 1 [ s o x ( x k x o ) + s o y y k ] } ,
0 2 π cos n φ exp [ j k i ρ sin θ i cos ( φ ϕ ) ] d φ = 2 π j n J n ( k i ρ sin θ i ) cos n ϕ ,
0 2 π sin n φ exp [ j k i ρ sin θ i cos ( φ ϕ ) ] d φ = 2 π j n J n ( k i ρ sin θ i ) sin n ϕ ,
E i x o = π M 2 k n 2 λ i 2 W k V ( r k ) [ k n E p x A 0 + j 2 M k i E p z cos ϕ A 1 + k n ( E p x cos 2 ϕ + E p y sin 2 ϕ ) A 2 ] ,
E i y o = π M 2 k n 2 λ i 2 W k V ( r k ) [ k n E p y A 0 + j 2 M k i E p z sin ϕ A 1 + k n ( E p x sin 2 ϕ E p y cos 2 ϕ ) A 2 ] ,
E i z o = π M k n 2 λ i 2 W k V ( r k ) [ M k i E p z B 0 + j k n ( cos ϕ E p x + sin ϕ E p y ) B 1 ] ,
A 0 = 0 Φ i cos 1 2 θ o cos 1 2 θ i sin θ i cos 1 θ n × ( cos θ o Δ Υ u + 1 , n > A + cos θ i cos 2 θ n Δ Υ u + 1 , n > B + ) × J 0 ( ) exp ( ) exp ( j Ψ ) d θ i ,
A 1 = 0 Φ i cos 3 2 θ o cos 3 2 θ i sin 2 θ i × Δ Υ u + 1 , n > B J 1 ( ) exp ( ) exp ( j Ψ ) d θ i ,
A 2 = 0 Φ i cos 1 2 θ o cos 1 2 θ i sin θ i cos 1 θ n × ( cos θ o Δ Υ u + 1 , n > A cos θ i cos 2 θ n Δ Υ u + 1 , n > B + ) × J 2 ( ) exp ( ) exp ( j Ψ ) d θ i ,
B 0 = 0 Φ i cos 1 2 θ o cos 1 2 θ i sin 3 θ i × Δ Υ u + 1 , n > B J 0 ( ) exp ( ) exp ( j Ψ ) d θ i ,
B 1 = 0 Φ i cos 1 2 θ o cos 1 2 θ i sin 2 θ i cos θ n × Δ Υ u + 1 , n > B + J 1 ( ) exp ( ) exp ( j Ψ ) d θ i ,
E i k ( x , y , z p = 0 ) = E i o [ x M ( x k x o ) , y M y k , z p = 0 ] ,
E i ( x , y , z p = 0 ) = k = 1 N p E i k ( x , y , z p = 0 ) .
P d = 1 2 S E i ( x , y , z p = 0 ) 2 d x d y ,
E A ( x , y , z p = 0 ) = N A E i k ( x , y , z p = 0 ) ,
P A = 1 2 S E A ( x , y , z p = 0 ) 2 d x d y .
Cross talk = 20 log [ ( P d P A ) P A ] .

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