Abstract

A novel full vector model based on the supercell lattice method is presented to analyze photonic crystal fiber (PCF). From symmetry analysis waveguide modes, we classify PCF modes into nondegenerate or degenerate pairs according to the minimum waveguide sectors and their appropriate boundary conditions. We describe how the modes of the PCF can be labeled by step-index fiber analogs, with the exception of modes that have the same symmetry as the PCF. When the doublet of the degenerate pairs both have the same symmetry as the PCF, they will be split into two nondegenerate modes.

© 2003 Optical Society of America

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References

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  1. J. C. Night and P. St. Russell, “New way to guide light,” Science 296, 276–277 (2002).
    [Crossref]
  2. T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997).
    [Crossref] [PubMed]
  3. A. Ferrando and E. Silvestre, “Nearly zero ultraflattened dispersion in photonic crystal fibers,” Opt. Lett. 25, 790–792 (2000).
    [Crossref]
  4. A. L. Gaeta, “Nonlinear propagation and continuum generation in microstructured optical fibers,” Opt. Lett. 27, 924–926 (2002).
    [Crossref]
  5. T. A. Birks, D. Mogilevtsev, J. C. Knight, P. St. J. Russell, J. Broeng, P. J. Roberts, J. A. West, D. C. Allan, and J. C. Fajardo, “The analogy between photonic crystal fibres and step index fibres,” in Optical Fiber Communication Conference (Optical Society of America, Washington, D.C., 1998), FG4, pp. 114–116.
  6. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, New York, 1995).
  7. S. Guo and S. Albin, “Simple plane wave implementation for photonic crystal calculations,” Opt. Express 11, 167–175 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-2-167
    [Crossref] [PubMed]
  8. T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Modeling large air fraction holey optical fibers,” J. Lightwave Technol. 18, 50–56 (2000).
    [Crossref]
  9. D. Mogilevtsev, T. A. Birks, and P. St. Russell, “Localized function method for modeling defect mode in 2-d photonic crystal,” J. Lightwave Technol. 17, 2078–2081 (1999).
    [Crossref]
  10. M. Koshiba, “Full vector analysis of photonic crystal fibers using the finite elelment method,” IEICE Electron. E85-C,  4, 881–888 (2002).
  11. F. Fogli, L. Saccomandi, P. Bassi, G. Bellanca, and S. Trillo, “Full vectorial BPM modeling of index-guiding photonic crystal fibers and couplers,” Opt. Express 10, 54–59 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-1-54
    [Crossref] [PubMed]
  12. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B 19, 2322–2330 (2002).
    [Crossref]
  13. B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. Martijn de Sterke, and R. C. McPhedran, “Multipole method for microstructured optical fibers. II. Implementation and results,” J. Opt. Soc. Am. B 192331–2340 (2002).
    [Crossref]
  14. F. G. Omenetto, A. J. Taylor, M. D. Moores, J. Arriaga, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Simultaneous generation of spectrally distinct third harmonics in a photonic crystal fiber,” Opt. Lett. 26, 1158 (2001).
    [Crossref]
  15. A. Efimov, A. J. Taylor, F. G. Omenetto, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Nonlinear generation of very high-order UV modes in microstructured fibers,” Opt. Express 11, 910–918 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-8-910
    [Crossref] [PubMed]
  16. F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, “Complete analysis of the characteristics of propagation into photonic crystal fibers, by the finite element method,” Opt. Fiber Technol. 6, 181–191 (2000).
    [Crossref]
  17. A. Ferrando, E. Silvestre, J. J. Miret, P. Andrés, and M. V. Andrés, “Vector description of higher-order modes in photonic crystal fibers,” J. Opt. Soc. Am. B 17, 1333–1340 (2000).
    [Crossref]
  18. M. J. Steel, T. P. White, C. M. Sterke, R. C. McPhedran, and L. C. Botten, “Symmetry and degeneracy in microstructured optical fibers,” Opt. Lett. 26, 488–490 (2001).
    [Crossref]
  19. M. J. Steel, T. P. White, C. M. Sterke, R. C. McPhedran, and L. C. Botten, “Symmetry and birefringence in microstructured optical fibers,” Optical Fiber Communication Conference 2001 (Optical Society of America, Washington, D.C., 2001), WDD88-1-WDD88-3.
  20. A. W. Snyder, Optical Waveguide Theory (Chapman and Hall, New York, 1983).
  21. W. Zhi, R. Guobin, L. Shuqin, and J. Shuisheng, “Novel supercell lattice method for the photonic crystal fibers,” Opt. Express 11, 980–991 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-980.
    [Crossref] [PubMed]
  22. W. Zhi, R. Guobin, L. Shuqin, and J. Shuisheng are preparing a manuscript to be called “A novel supercell overlapping method for different photonic crystal fibers.”
  23. I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1994).
  24. I. Kimel and L. R. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2562–2567 (1993).
    [Crossref]
  25. P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides. I. Summary of results,” IEEE Trans. Microwave Theory Tech. MTT-23, 421–429 (1975).
    [Crossref]

2003 (3)

2002 (6)

2001 (2)

2000 (4)

F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, “Complete analysis of the characteristics of propagation into photonic crystal fibers, by the finite element method,” Opt. Fiber Technol. 6, 181–191 (2000).
[Crossref]

A. Ferrando, E. Silvestre, J. J. Miret, P. Andrés, and M. V. Andrés, “Vector description of higher-order modes in photonic crystal fibers,” J. Opt. Soc. Am. B 17, 1333–1340 (2000).
[Crossref]

T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Modeling large air fraction holey optical fibers,” J. Lightwave Technol. 18, 50–56 (2000).
[Crossref]

A. Ferrando and E. Silvestre, “Nearly zero ultraflattened dispersion in photonic crystal fibers,” Opt. Lett. 25, 790–792 (2000).
[Crossref]

1999 (1)

1997 (1)

1993 (1)

I. Kimel and L. R. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2562–2567 (1993).
[Crossref]

1975 (1)

P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides. I. Summary of results,” IEEE Trans. Microwave Theory Tech. MTT-23, 421–429 (1975).
[Crossref]

Albin, S.

Allan, D. C.

T. A. Birks, D. Mogilevtsev, J. C. Knight, P. St. J. Russell, J. Broeng, P. J. Roberts, J. A. West, D. C. Allan, and J. C. Fajardo, “The analogy between photonic crystal fibres and step index fibres,” in Optical Fiber Communication Conference (Optical Society of America, Washington, D.C., 1998), FG4, pp. 114–116.

Andrés, M. V.

A. Ferrando, E. Silvestre, J. J. Miret, P. Andrés, and M. V. Andrés, “Vector description of higher-order modes in photonic crystal fibers,” J. Opt. Soc. Am. B 17, 1333–1340 (2000).
[Crossref]

Andrés, P.

A. Ferrando, E. Silvestre, J. J. Miret, P. Andrés, and M. V. Andrés, “Vector description of higher-order modes in photonic crystal fibers,” J. Opt. Soc. Am. B 17, 1333–1340 (2000).
[Crossref]

Arriaga, J.

Bassi, P.

Bellanca, G.

Bennett, P. J.

Birks, T. A.

D. Mogilevtsev, T. A. Birks, and P. St. Russell, “Localized function method for modeling defect mode in 2-d photonic crystal,” J. Lightwave Technol. 17, 2078–2081 (1999).
[Crossref]

T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997).
[Crossref] [PubMed]

T. A. Birks, D. Mogilevtsev, J. C. Knight, P. St. J. Russell, J. Broeng, P. J. Roberts, J. A. West, D. C. Allan, and J. C. Fajardo, “The analogy between photonic crystal fibres and step index fibres,” in Optical Fiber Communication Conference (Optical Society of America, Washington, D.C., 1998), FG4, pp. 114–116.

Botten, L. C.

Brechet, F.

F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, “Complete analysis of the characteristics of propagation into photonic crystal fibers, by the finite element method,” Opt. Fiber Technol. 6, 181–191 (2000).
[Crossref]

Broderick, N. G. R.

Broeng, J.

T. A. Birks, D. Mogilevtsev, J. C. Knight, P. St. J. Russell, J. Broeng, P. J. Roberts, J. A. West, D. C. Allan, and J. C. Fajardo, “The analogy between photonic crystal fibres and step index fibres,” in Optical Fiber Communication Conference (Optical Society of America, Washington, D.C., 1998), FG4, pp. 114–116.

Efimov, A.

Elias, L. R.

I. Kimel and L. R. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2562–2567 (1993).
[Crossref]

Fajardo, J. C.

T. A. Birks, D. Mogilevtsev, J. C. Knight, P. St. J. Russell, J. Broeng, P. J. Roberts, J. A. West, D. C. Allan, and J. C. Fajardo, “The analogy between photonic crystal fibres and step index fibres,” in Optical Fiber Communication Conference (Optical Society of America, Washington, D.C., 1998), FG4, pp. 114–116.

Ferrando, A.

A. Ferrando and E. Silvestre, “Nearly zero ultraflattened dispersion in photonic crystal fibers,” Opt. Lett. 25, 790–792 (2000).
[Crossref]

A. Ferrando, E. Silvestre, J. J. Miret, P. Andrés, and M. V. Andrés, “Vector description of higher-order modes in photonic crystal fibers,” J. Opt. Soc. Am. B 17, 1333–1340 (2000).
[Crossref]

Fogli, F.

Gaeta, A. L.

Gradshtein, I. S.

I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1994).

Guo, S.

Guobin, R.

W. Zhi, R. Guobin, L. Shuqin, and J. Shuisheng, “Novel supercell lattice method for the photonic crystal fibers,” Opt. Express 11, 980–991 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-980.
[Crossref] [PubMed]

W. Zhi, R. Guobin, L. Shuqin, and J. Shuisheng are preparing a manuscript to be called “A novel supercell overlapping method for different photonic crystal fibers.”

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, New York, 1995).

Kimel, I.

I. Kimel and L. R. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2562–2567 (1993).
[Crossref]

Knight, J. C.

Koshiba, M.

M. Koshiba, “Full vector analysis of photonic crystal fibers using the finite elelment method,” IEICE Electron. E85-C,  4, 881–888 (2002).

Kuhlmey, B. T.

Marcou, J.

F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, “Complete analysis of the characteristics of propagation into photonic crystal fibers, by the finite element method,” Opt. Fiber Technol. 6, 181–191 (2000).
[Crossref]

Maystre, D.

McIsaac, P. R.

P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides. I. Summary of results,” IEEE Trans. Microwave Theory Tech. MTT-23, 421–429 (1975).
[Crossref]

McPhedran, R. C.

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, New York, 1995).

Miret, J. J.

A. Ferrando, E. Silvestre, J. J. Miret, P. Andrés, and M. V. Andrés, “Vector description of higher-order modes in photonic crystal fibers,” J. Opt. Soc. Am. B 17, 1333–1340 (2000).
[Crossref]

Mogilevtsev, D.

D. Mogilevtsev, T. A. Birks, and P. St. Russell, “Localized function method for modeling defect mode in 2-d photonic crystal,” J. Lightwave Technol. 17, 2078–2081 (1999).
[Crossref]

T. A. Birks, D. Mogilevtsev, J. C. Knight, P. St. J. Russell, J. Broeng, P. J. Roberts, J. A. West, D. C. Allan, and J. C. Fajardo, “The analogy between photonic crystal fibres and step index fibres,” in Optical Fiber Communication Conference (Optical Society of America, Washington, D.C., 1998), FG4, pp. 114–116.

Monro, T. M.

Moores, M. D.

Night, J. C.

J. C. Night and P. St. Russell, “New way to guide light,” Science 296, 276–277 (2002).
[Crossref]

Omenetto, F. G.

Pagnoux, D.

F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, “Complete analysis of the characteristics of propagation into photonic crystal fibers, by the finite element method,” Opt. Fiber Technol. 6, 181–191 (2000).
[Crossref]

Renversez, G.

Richardson, D. J.

Roberts, P. J.

T. A. Birks, D. Mogilevtsev, J. C. Knight, P. St. J. Russell, J. Broeng, P. J. Roberts, J. A. West, D. C. Allan, and J. C. Fajardo, “The analogy between photonic crystal fibres and step index fibres,” in Optical Fiber Communication Conference (Optical Society of America, Washington, D.C., 1998), FG4, pp. 114–116.

Roy, P.

F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, “Complete analysis of the characteristics of propagation into photonic crystal fibers, by the finite element method,” Opt. Fiber Technol. 6, 181–191 (2000).
[Crossref]

Russell, P. St.

Russell, P. St. J.

Ryzhik, I. M.

I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1994).

Saccomandi, L.

Shuisheng, J.

W. Zhi, R. Guobin, L. Shuqin, and J. Shuisheng, “Novel supercell lattice method for the photonic crystal fibers,” Opt. Express 11, 980–991 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-980.
[Crossref] [PubMed]

W. Zhi, R. Guobin, L. Shuqin, and J. Shuisheng are preparing a manuscript to be called “A novel supercell overlapping method for different photonic crystal fibers.”

Shuqin, L.

W. Zhi, R. Guobin, L. Shuqin, and J. Shuisheng, “Novel supercell lattice method for the photonic crystal fibers,” Opt. Express 11, 980–991 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-980.
[Crossref] [PubMed]

W. Zhi, R. Guobin, L. Shuqin, and J. Shuisheng are preparing a manuscript to be called “A novel supercell overlapping method for different photonic crystal fibers.”

Silvestre, E.

A. Ferrando and E. Silvestre, “Nearly zero ultraflattened dispersion in photonic crystal fibers,” Opt. Lett. 25, 790–792 (2000).
[Crossref]

A. Ferrando, E. Silvestre, J. J. Miret, P. Andrés, and M. V. Andrés, “Vector description of higher-order modes in photonic crystal fibers,” J. Opt. Soc. Am. B 17, 1333–1340 (2000).
[Crossref]

Snyder, A. W.

A. W. Snyder, Optical Waveguide Theory (Chapman and Hall, New York, 1983).

Steel, M. J.

M. J. Steel, T. P. White, C. M. Sterke, R. C. McPhedran, and L. C. Botten, “Symmetry and degeneracy in microstructured optical fibers,” Opt. Lett. 26, 488–490 (2001).
[Crossref]

M. J. Steel, T. P. White, C. M. Sterke, R. C. McPhedran, and L. C. Botten, “Symmetry and birefringence in microstructured optical fibers,” Optical Fiber Communication Conference 2001 (Optical Society of America, Washington, D.C., 2001), WDD88-1-WDD88-3.

Sterke, C. M.

M. J. Steel, T. P. White, C. M. Sterke, R. C. McPhedran, and L. C. Botten, “Symmetry and degeneracy in microstructured optical fibers,” Opt. Lett. 26, 488–490 (2001).
[Crossref]

M. J. Steel, T. P. White, C. M. Sterke, R. C. McPhedran, and L. C. Botten, “Symmetry and birefringence in microstructured optical fibers,” Optical Fiber Communication Conference 2001 (Optical Society of America, Washington, D.C., 2001), WDD88-1-WDD88-3.

Sterke, C. Martijn de

Taylor, A. J.

Trillo, S.

Wadsworth, W. J.

West, J. A.

T. A. Birks, D. Mogilevtsev, J. C. Knight, P. St. J. Russell, J. Broeng, P. J. Roberts, J. A. West, D. C. Allan, and J. C. Fajardo, “The analogy between photonic crystal fibres and step index fibres,” in Optical Fiber Communication Conference (Optical Society of America, Washington, D.C., 1998), FG4, pp. 114–116.

White, T. P.

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, New York, 1995).

Zhi, W.

W. Zhi, R. Guobin, L. Shuqin, and J. Shuisheng, “Novel supercell lattice method for the photonic crystal fibers,” Opt. Express 11, 980–991 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-980.
[Crossref] [PubMed]

W. Zhi, R. Guobin, L. Shuqin, and J. Shuisheng are preparing a manuscript to be called “A novel supercell overlapping method for different photonic crystal fibers.”

IEEE J. Quantum Electron. (1)

I. Kimel and L. R. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2562–2567 (1993).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides. I. Summary of results,” IEEE Trans. Microwave Theory Tech. MTT-23, 421–429 (1975).
[Crossref]

IEICE Electron. E85-C (1)

M. Koshiba, “Full vector analysis of photonic crystal fibers using the finite elelment method,” IEICE Electron. E85-C,  4, 881–888 (2002).

J. Lightwave Technol. (2)

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

Opt. Express (4)

Opt. Fiber Technol. (1)

F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, “Complete analysis of the characteristics of propagation into photonic crystal fibers, by the finite element method,” Opt. Fiber Technol. 6, 181–191 (2000).
[Crossref]

Opt. Lett. (5)

Science (1)

J. C. Night and P. St. Russell, “New way to guide light,” Science 296, 276–277 (2002).
[Crossref]

Other (6)

T. A. Birks, D. Mogilevtsev, J. C. Knight, P. St. J. Russell, J. Broeng, P. J. Roberts, J. A. West, D. C. Allan, and J. C. Fajardo, “The analogy between photonic crystal fibres and step index fibres,” in Optical Fiber Communication Conference (Optical Society of America, Washington, D.C., 1998), FG4, pp. 114–116.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, New York, 1995).

M. J. Steel, T. P. White, C. M. Sterke, R. C. McPhedran, and L. C. Botten, “Symmetry and birefringence in microstructured optical fibers,” Optical Fiber Communication Conference 2001 (Optical Society of America, Washington, D.C., 2001), WDD88-1-WDD88-3.

A. W. Snyder, Optical Waveguide Theory (Chapman and Hall, New York, 1983).

W. Zhi, R. Guobin, L. Shuqin, and J. Shuisheng are preparing a manuscript to be called “A novel supercell overlapping method for different photonic crystal fibers.”

I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1994).

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Figures (12)

Fig. 1.
Fig. 1.

Reconstruction of the dielectric structure of a triangular lattice TIR-PCF with the parameters Λ=2.3µm, d/Λ=0.6, P 1=50, P 2=500. (a) Three-dimensional (3-D) refractive-index reconstruction. (b) Cross section along the y=0 axis of index reconstruction.

Fig. 2.
Fig. 2.

Minimum sectors for waveguides with C symmetry. The modes of waveguides are classified into eight classes (p=1,….8). Solid lines indicate short-circuit boundary conditions, dashed lines indicate open-circuit boundary conditions.

Fig. 3.
Fig. 3.

Mode index of the first 24 modes in TIR-PCF with structural parameters Λ=2.3µm, d/Λ=0.8 at λ=633nm.

Fig. 4.
Fig. 4.

Total modal intensity distribution of HE11 mode (a) and 2-D electric vector distributions of polarization doublet modes (b), (c) with parameters Λ=2.3µm, d/Λ=0.8, at wavelength λ=633nm.

Fig. 5.
Fig. 5.

Electric vector distributions for modes 3, 4, 5, 6 corresponding to (a) TE01, (b), (c) HE21, and (d) TM01 mode of step-index fiber.

Fig. 6.
Fig. 6.

Intensity distribution of the combination of modes 4 and 5 (HE21).

Fig. 7.
Fig. 7.

Electric vector distributions of mode 7–14.

Fig. 8
Fig. 8

Intensity distributions of nondegenerate mode HE311, degenerate mode EH11, HE12 and EH21.

Fig. 9.
Fig. 9.

Modal intensity distribution of EH311 mode (a) and 2-D electric vector distributions of EH311 and EH312 modes (b), (c) with parameters Λ=2.3µm, d/Λ=0.9, at wavelength λ=0.633µm.

Fig. 10.
Fig. 10.

Minimum sectors for waveguides with C symmetry. The waveguides modes are divided into eight classes (p=1,….6). Solid lines indicate short-circuit boundary conditions; dashed lines indicate open-circuit boundary conditions.

Fig. 11.
Fig. 11.

Electric vector distributions for modes 3–6 of the square lattice PCF.

Fig. 12.
Fig. 12.

intensity distributions of nondegenerate modes HE211 and TE01 of a square lattice PCF.

Tables (2)

Tables Icon

Table 1. Mode-index (n eff ), mode class (p), degeneracy, computation error (Δn) and label for first 14 modes

Tables Icon

Table 2. Mode index (n eff ), mode class (p), degeneracy, computation error (Δn), and label for first eight modes

Equations (16)

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

E j ( x , y , z ) = [ e j t ( x , y ) + e j z ( x , y ) ] e j ( β j z ω t ) ,
( t 2 β j 2 + k 2 n 2 ) e x = x ( e x In n 2 x + e y In n 2 y ) ,
( t 2 β j 2 + k 2 n 2 ) e y = y ( e x In n 2 x + e y In n 2 y ) ,
e x ( x , y ) = a , b = 0 F 1 ε a b x ψ a ( x ) ψ b ( y ) ,
e y ( x , y ) = a , b = 0 F 1 ε a b y ψ a ( x ) ψ b ( y ) ,
ψ i ( s ) = 2 i 2 π 1 4 ( i ) ! ω exp ( s 2 2 ω 2 ) H i ( s ω ) ,
n 2 ( x , y ) = a , b = 0 P 1 1 P 1 a b cos 2 π a x l 1 x cos 2 π b y l 1 y + a , b = 0 P 2 1 P 2 a b cos 2 π a x l 2 x cos 2 π b y l 2 y ,
In n 2 ( x , y ) = a , b = 0 P 1 1 P 1 a b In cos 2 π a x l 1 x cos 2 π b y l 1 y + a , b = 0 P 2 1 P 2 ab ln cos 2 π a x l 2 x cos 2 π b y l 2 y ,
L [ e x e y ] = [ I abcd ( 1 ) + k 2 I abcd ( 2 ) + I abcd ( 3 ) x I abcd ( 4 ) x I abcd ( 4 ) y I abcd ( 1 ) + k 2 I abcd ( 2 ) + I abcd ( 3 ) y ] [ e x e y ] = β j 2 [ e x e y ] ,
I abcd ( 1 ) = + ψ a ( x ) ψ b ( y ) t 2 [ ψ c ( x ) ψ d ( y ) ] dx dy ,
I abcd ( 2 ) = + n 2 ψ a ( x ) ψ b ( y ) ψ c ( x ) ψ d ( y ) dx dy ,
I abcd ( 3 ) x = + ψ a ( x ) ψ b ( y ) x [ ψ c ( x ) ψ d ( y ) In n 2 x ] dx dy ,
I abcd ( 3 ) y = + ψ a ( x ) ψ b ( y ) y [ ψ c ( x ) ψ d ( y ) In n 2 y ] dx dy ,
I abcd ( 4 ) x = + ψ a ( x ) ψ b ( y ) x [ ψ c ( x ) ψ d ( y ) In n 2 y ] dx dy ,
I abcd ( 4 ) y = + ψ a ( x ) ψ b ( y ) y [ ψ c ( x ) ψ d ( y ) In n 2 x ] dx dy .
e t = i k 2 n 2 β 2 { β t e z ( μ 0 ε 0 ) 1 2 k z ̂ × t h z } .

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