Abstract

Some mode characteristics are obtained by the full vector supercell overlap method that has been developed to model triangular lattice elliptical hole photonic crystal fibers regardless of whether the light is guided by total internal reflection or a photonic bandgap mechanism. When the central defect hole is large enough, the modes are disordered. Birefringence (Δn) dependence on the central defect is discussed in detail by numerical analysis.

© 2003 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |

  1. J. Broeng, D. Mogilevstev, S. E. Barkou, and A. Bjarklev, �??Photonic crystal fibers: a new class of optical waveguides,�?? Opt. Fiber Technol. 5, 305�??330 (1999).
    [CrossRef]
  2. 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 fibers and step index fibers,�?? in Optical Fiber Communication Conference, Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), paper FG4, pp. 114�??116.
  3. 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]
  4. N. A. Mortensen, �??Effective area of photonic crystal fibers,�?? Opt. Express 10, 341�??348 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-341">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-341</a>.
    [CrossRef] [PubMed]
  5. N. A. Mortensen, J. R. Folken, P. M. W. Skovgaard, and J. Broeng, �??Numerical aperture of single-mode photonic crystal fibers,�?? IEEE Photon. Technol. Lett. 14, 1094�??1096 (2002).
    [CrossRef]
  6. J. M. Dudley and S. Coen, �??Numerical simulations and coherence properties of supercontinuum generation in photonic crystal and tapered optical fibers,�?? IEEE J. Sel. Top. Quantum Electron. 8, 651�??659 (2002).
    [CrossRef]
  7. S. E. Barkou, J. Broeng, and A. Bjarklev, �??Dispersion properties of photonic bandgap guiding fibers,�?? in Optical Fiber Communication Conference, Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), paper FG5, pp. 117�??119.
  8. T. A. Birk, D. Mogilevtsev, J. C. Knight, and P. St. J. Russell, �??Single material fibers for dispersion compensation,�?? in Optical Fiber Communication Conference (Optical Society of America, Washington, D.C., 1999), paper FG2, pp. 108�??110.
  9. A. Bjarklev, J. Broeng, K. Dridi, and S. E. Barkou, �??Dispersion properties of photonic crystal fibres,�?? in European Conference on Optical Communication (Madrid, Spain, 1998), pp. 135�??136.
  10. R. G. Bin, L. S. Qin, W. Zhi, and S. S. Jian, �??Study on dispersion properties of photonic crystal fiber by effective-index model,�?? Acta Opt. Sin. 24, (2004), in Chinese (to be published).
  11. J. Broeng, S. E. B. Libori, T. Sondergaard, and A. Bjarklev, �??Analysis of air-guiding photonic bandgap fibers,�?? Opt. Lett. 25, 96�??98 (2000).
    [CrossRef]
  12. J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, �??Photonic band gap guidance in optical fibers,�?? Science 282, 1476�??1478 (1998).
    [CrossRef] [PubMed]
  13. R.F. Cregan, B. J. Mangan, J. C.Knight, T. A.Birks, P. St. J. Russell, P. J. Roberts, and D. C.Allan, �??Single-mode photonic band gap guidance of light in air,�?? Science 285, 1537�??1539 (1999).
    [CrossRef] [PubMed]
  14. S. E. B. Libori, J. Broeng, E. Knudsen, A. Bjarklev, and H. Simonsen, �??High-birefringent photonic crystal fibers,�?? in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper TuM2.
  15. T. P. Hansen, J. Broeng, S. E. B. Libori, E. Knudsen, A. Bjarklev, J. R. Jensen, and H. Simonsen, �??Highly birefringent index-guiding photonic crystal fibers,�?? IEEE Photon. Technol. Lett. 13, 588�??590 (2001).
    [CrossRef]
  16. J. Broeng, D. Mogilevtsev, S. E. B. Libori, and A. Bjarklev, �??Polarization-preserving holey fibers,�?? in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper MA1.3.
  17. M. J. Steel and P. M. Osgood, Jr, �??Elliptical-hole photonic crystal fibers,�?? Opt. Lett. 26, 229�??231 (2001).
    [CrossRef]
  18. M. J. Steel and R. M. Osgood, �??Polarization and dispersive properties of elliptical-hole photonic crystal fibers,�?? J. Lightwave Technol. 19, 495�??503 (2001).
    [CrossRef]
  19. K. Tajima, J. Zhou, K. Nakajima, and K. Sato, �??Ultra low loss and long length photonic crystal fiber,�?? in Optical Fiber Communication Conference, Vol. 86 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2003), pp. PD1-1�??PD1-3.
  20. R. G. Bin, W. Zhi, L. S. Qin, and S. S. Jian, �??Study on dispersion properties of photonic crystal fiber by effective-index model,�?? accepted by OECC�??2003, Shanghai. (to be published by the Chinese Optical Society).
  21. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding theFlow of Light ( Princeton University, Princeton, N.J., 1995).
  22. S. Guo and S. Albin, �??Simple plane wave implementation for photonic crystal calculations,�?? Opt. Express 11, 167�??175 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-2-167">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-2-167</a>.
    [CrossRef] [PubMed]
  23. S. G. Johnson and J. D. Joannopoulos, �??Block-iterative frequency-domain methods for Maxwell's equations in a plane wave basis,�?? Opt. Express 8, 173�??190 (2001), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173.</a>
    [CrossRef] [PubMed]
  24. 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]
  25. T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, �??Holey optical fibers: an efficient modal model,�?? J. Lightwave Technol. 17, 1093�??1102 (1999).
    [CrossRef]
  26. T. M. Monro, D. J. Richardson, and N. G. R. Broderick, �??Efficient modeling of holey fibers,�?? in Optical Fiber Communication Conference (Optical Society of America, Washington, D.C., 1999), paper FG3, pp. 111�??113.
  27. W. Zhi, R.G. Bin, L. S. Qin, and S. S. Jian, �??Supercell lattice method for photonic crystal fibers,�?? Opt. Express 11, 980�??991 (2003),<a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-980">http://www.opticsexpress.org/abstract.cfm? URI=OPEX-11-9-980</a>.
    [CrossRef] [PubMed]
  28. D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, �??Localized function method for modeling defect mode in 2-d photonic crystal,�?? J. Lightwave Technol. 17, 2078�??2081 (1999).
    [CrossRef]
  29. M. Koshiba, �??Full vector analysis of photonic crystal fibers using the finite element method,�?? IEICE Electron. E85-C, 881�??888 (2002).
  30. Z. Zhu and T. G. Brown, �??Full-vectorial finite difference analysis of microstructured optical fibers,�?? Opt. Express 10, 853�??864 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-853">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-853</a>.
    [CrossRef] [PubMed]
  31. F. Fogli, L. Saccomandi, P. Bassi, G. Bellanca, and S. Trillo, �??Full vectorial BPM modeling of indexguiding photonic crystal fibers and couplers,�?? Opt. Express 10, 54�??59 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-1-54">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-1-54</a>.
    [CrossRef] [PubMed]
  32. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, �??Multipole method for microstructured optical fibers. I. Formulation,�?? J. Opt. Soc. Am. B 19, 2322�??2330 (2002).
    [CrossRef]
  33. B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. M. de Sterke, and R. C. McPhedran, �??Multipole method for microstructured optical fibers. II. Implementation and results,�?? J. Opt. Soc. Am. B 19, 2331�??2340 (2002).
    [CrossRef]
  34. A.W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  35. R. Guobin, W. Zhi, L. Shuqin, and J. Shuisheng, �??Mode classification and degeneracy in photonic crystal fiber,�?? Opt. Express 11, 1310�??1321 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-11-1310">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-11-1310</a>.
    [CrossRef] [PubMed]
  36. W. Zhi, R. G. Bin, L. S. Qin, and S. S. Jian, �??Novel supercell overlapping method for different photonic crystal fibers,�?? J. Lightwave Technol. submitted.
  37. I. S. Gradshtein, and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1994).
  38. I. Kimel and L. R. Elias, �??Relations between Hermite and Laguerre Gaussian modes,�?? IEEE J. Quantum Electron. 29, 2562�??2567 (1993).
    [CrossRef]
  39. 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]
  40. P. R. McIsaac, �??Symmetry-induced modal characteristics of uniform waveguides- II: theory,�?? IEEE Trans. Microwave Theory Tech. MTT-23, 429�??433 (1975).
    [CrossRef]
  41. 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]
  42. R. G. Bin, W. Zhi, L. S. Qin, and S. S. Jian, �??Birefringence properties of elliptical-hole photonic crystal fiber,�?? Chin. J. Lasers 31 (2004), in Chinese (to be published).

1998 OSA Technical Digest Series (2)

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 fibers and step index fibers,�?? in Optical Fiber Communication Conference, Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), paper FG4, pp. 114�??116.

S. E. Barkou, J. Broeng, and A. Bjarklev, �??Dispersion properties of photonic bandgap guiding fibers,�?? in Optical Fiber Communication Conference, Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), paper FG5, pp. 117�??119.

Acta Opt. Sin. (1)

R. G. Bin, L. S. Qin, W. Zhi, and S. S. Jian, �??Study on dispersion properties of photonic crystal fiber by effective-index model,�?? Acta Opt. Sin. 24, (2004), in Chinese (to be published).

Chin. J. Lasers (1)

R. G. Bin, W. Zhi, L. S. Qin, and S. S. Jian, �??Birefringence properties of elliptical-hole photonic crystal fiber,�?? Chin. J. Lasers 31 (2004), in Chinese (to be published).

European Conf. on Optical Communication (1)

A. Bjarklev, J. Broeng, K. Dridi, and S. E. Barkou, �??Dispersion properties of photonic crystal fibres,�?? in European Conference on Optical Communication (Madrid, Spain, 1998), pp. 135�??136.

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 J. Sel. Top. Quantum Electron. (1)

J. M. Dudley and S. Coen, �??Numerical simulations and coherence properties of supercontinuum generation in photonic crystal and tapered optical fibers,�?? IEEE J. Sel. Top. Quantum Electron. 8, 651�??659 (2002).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

N. A. Mortensen, J. R. Folken, P. M. W. Skovgaard, and J. Broeng, �??Numerical aperture of single-mode photonic crystal fibers,�?? IEEE Photon. Technol. Lett. 14, 1094�??1096 (2002).
[CrossRef]

T. P. Hansen, J. Broeng, S. E. B. Libori, E. Knudsen, A. Bjarklev, J. R. Jensen, and H. Simonsen, �??Highly birefringent index-guiding photonic crystal fibers,�?? IEEE Photon. Technol. Lett. 13, 588�??590 (2001).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

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]

P. R. McIsaac, �??Symmetry-induced modal characteristics of uniform waveguides- II: theory,�?? IEEE Trans. Microwave Theory Tech. MTT-23, 429�??433 (1975).
[CrossRef]

IEICE Electron. (1)

M. Koshiba, �??Full vector analysis of photonic crystal fibers using the finite element method,�?? IEICE Electron. E85-C, 881�??888 (2002).

J. Lightwave Technol. (5)

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

OECC (1)

R. G. Bin, W. Zhi, L. S. Qin, and S. S. Jian, �??Study on dispersion properties of photonic crystal fiber by effective-index model,�?? accepted by OECC�??2003, Shanghai. (to be published by the Chinese Optical Society).

Opt. Express (6)

S. Guo and S. Albin, �??Simple plane wave implementation for photonic crystal calculations,�?? Opt. Express 11, 167�??175 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-2-167">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-2-167</a>.
[CrossRef] [PubMed]

S. G. Johnson and J. D. Joannopoulos, �??Block-iterative frequency-domain methods for Maxwell's equations in a plane wave basis,�?? Opt. Express 8, 173�??190 (2001), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173.</a>
[CrossRef] [PubMed]

Z. Zhu and T. G. Brown, �??Full-vectorial finite difference analysis of microstructured optical fibers,�?? Opt. Express 10, 853�??864 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-853">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-853</a>.
[CrossRef] [PubMed]

F. Fogli, L. Saccomandi, P. Bassi, G. Bellanca, and S. Trillo, �??Full vectorial BPM modeling of indexguiding photonic crystal fibers and couplers,�?? Opt. Express 10, 54�??59 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-1-54">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-1-54</a>.
[CrossRef] [PubMed]

W. Zhi, R.G. Bin, L. S. Qin, and S. S. Jian, �??Supercell lattice method for photonic crystal fibers,�?? Opt. Express 11, 980�??991 (2003),<a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-980">http://www.opticsexpress.org/abstract.cfm? URI=OPEX-11-9-980</a>.
[CrossRef] [PubMed]

N. A. Mortensen, �??Effective area of photonic crystal fibers,�?? Opt. Express 10, 341�??348 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-341">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-341</a>.
[CrossRef] [PubMed]

Opt. Fiber Technol. (1)

J. Broeng, D. Mogilevstev, S. E. Barkou, and A. Bjarklev, �??Photonic crystal fibers: a new class of optical waveguides,�?? Opt. Fiber Technol. 5, 305�??330 (1999).
[CrossRef]

Opt. Lett. (4)

Optical Fiber Communication Conf. 1999 (1)

T. A. Birk, D. Mogilevtsev, J. C. Knight, and P. St. J. Russell, �??Single material fibers for dispersion compensation,�?? in Optical Fiber Communication Conference (Optical Society of America, Washington, D.C., 1999), paper FG2, pp. 108�??110.

Optical Fiber Communication Conference (2)

T. M. Monro, D. J. Richardson, and N. G. R. Broderick, �??Efficient modeling of holey fibers,�?? in Optical Fiber Communication Conference (Optical Society of America, Washington, D.C., 1999), paper FG3, pp. 111�??113.

K. Tajima, J. Zhou, K. Nakajima, and K. Sato, �??Ultra low loss and long length photonic crystal fiber,�?? in Optical Fiber Communication Conference, Vol. 86 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2003), pp. PD1-1�??PD1-3.

OSA Trends in Optics and Photonics (2)

J. Broeng, D. Mogilevtsev, S. E. B. Libori, and A. Bjarklev, �??Polarization-preserving holey fibers,�?? in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper MA1.3.

S. E. B. Libori, J. Broeng, E. Knudsen, A. Bjarklev, and H. Simonsen, �??High-birefringent photonic crystal fibers,�?? in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper TuM2.

Science (2)

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, �??Photonic band gap guidance in optical fibers,�?? Science 282, 1476�??1478 (1998).
[CrossRef] [PubMed]

R.F. Cregan, B. J. Mangan, J. C.Knight, T. A.Birks, P. St. J. Russell, P. J. Roberts, and D. C.Allan, �??Single-mode photonic band gap guidance of light in air,�?? Science 285, 1537�??1539 (1999).
[CrossRef] [PubMed]

Other (4)

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding theFlow of Light ( Princeton University, Princeton, N.J., 1995).

A.W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

R. Guobin, W. Zhi, L. Shuqin, and J. Shuisheng, �??Mode classification and degeneracy in photonic crystal fiber,�?? Opt. Express 11, 1310�??1321 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-11-1310">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-11-1310</a>.
[CrossRef] [PubMed]

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (13)

Fig. 1.
Fig. 1.

Schematic of the way in which the transverse dielectric structure is constructed. PC1 is a perfect triangular lattice constructed by the blue elliptical holes, the dielectric constant is εair in the hole and εsi outside the hole. PC2 is another perfect triangular lattice constructed by the coaxial elliptical rings with which the inner major axis is dc and the outer major axis is d, the dielectric constant is 0 in dc or outside d and εsi -εair between dc and d.

Fig. 2.
Fig. 2.

Simulation results of the dielectric constant profiles of the PCF and both virtually perfect PCs with parameters of D=2.3 µm, d=0.8D, dc =0.4D, η=2, and N=4.

Fig. 3.
Fig. 3.

Minimum sectors for waveguides with C 2ν symmetry. The waveguide modes are classified into four classes (p=1,2,3,4). The solid lines indicate short-circuit boundary conditions, the dashed lines indicate open-circuit boundary conditions.

Fig. 4.
Fig. 4.

Modal index of the first 12 modes of the triangular lattice EHPCF with the parameters D=2. 3 µm, d=0.8D, dc =D, η=2, λ=D/0.8.

Fig. 5.
Fig. 5.

Electric field vector of the first six modes in the PCF with parameters as in Fig. 4.

Fig. 6.
Fig. 6.

Modal index of the first 12 modes of the triangular lattice EHPCF with parameters D=2.3 µm, d=0.8D, dc =0.2D, η=2, λ=D/0.8.

Fig. 7.
Fig. 7.

Electric field vector of the first three modes in the PCF with parameters as in Fig. 6.

Fig. 8.
Fig. 8.

Mode order of HE11y and HE11x of an EHPCF. The parameters are shown at top right.

Fig. 9.
Fig. 9.

(a) Mode orders of HE11y and HE11x of an EHPCF with the parameters shown at right. (b) Relationship between Δn and λ of fibers a, b, and c.

Fig. 10.
Fig. 10.

Schematic of fibers a, b, and c, which is helpful to understand the mode disorder between the fundamental doublets.

Fig. 11.
Fig. 11.

Relationship between Δn and central defect size dc /D at different wavelengths. The structure parameters of the PCF are D=2.3 µm, d=0.8D, and η=2.

Fig. 12.
Fig. 12.

Relationship between Δn and normalized frequency D/λ in the PCFs with different central defect sizes. The parameters are D=2.3 µm, d=0.8D, and η=2.

Fig. 13.
Fig. 13.

Relationship between Δn and normalized frequency D/λ in PCFs with different elliptical ratios. The parameters are D=2.3 µm and d=0.8D. The major axis of the elliptical hole is along the (a) y axis or (b) x axis. The major axis length is dc =0.4D.

Tables (1)

Tables Icon

Table 1. Parameters of Virtual PCs for Triangular Lattice PCFs

Equations (21)

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

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 ) = + ε 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 ) ln ε x ] dx dy ,
I abcd ( 3 ) y = + ψ a ( x ) ψ b ( y ) y [ ψ c ( x ) ψ d ( y ) ln ε y ] dx dy ,
I abcd ( 4 ) x = + ψ a ( x ) ψ b ( y ) x [ ψ c ( x ) ψ d ( y ) ln ε y ] dx dy ,
I abcd ( 4 ) y = + ψ a ( x ) ψ b ( y ) y [ ψ c ( x ) ψ d ( y ) ln ε x ] dx dy ,
ε ( x , y ) = ε | PC 1 + ε | PC 2 , ln ε = ( ln ε ) | PC 1 + ( ln ε ) | PC 2 ,
ε | PC 1 = a , b = 0 P 1 1 P 1 ab cos 2 π a x l 1 x cos 2 π b y l 1 y ,
ε PC 2 = a , b = 0 P 2 1 P 2 ab cos 2 π a x l 2 x cos 2 π b y l 2 y ,
( ln ε ) PC 1 = a , b = 0 P 1 1 P 1 ab ln cos 2 π a x l 1 x cos 2 π b y l 1 y ,
( ln ε ) PC 2 = 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 ,
f c = π d c 2 ( 2 η l 2 x l 2 y ) , f 2 = π d 2 ( 2 η l 2 x l 2 y ) ,
ε ( x , y ) PC 2 = m , n = ( P 2 1 ) P 2 1 F 2 ( K mn ) cos ( k 1 x ) cos ( k 2 y ) ,
F 2 ( K mn ) = 2 ( n si 2 n air 2 ) [ f . J 1 ( K mn d 2 ) K mn d 2 f c . J 1 ( K mn d c 2 ) K mn d c 2 ] , K mn 0 ,
F 2 ( 0 ) = ( f f c ) ( n si 2 n air 2 ) ,
K mn = ( m + n ) k x i ( m n ) k y , k x = 2 π l 2 x η , k y = 2 π l 2 y .
I abcd ( 4 ) x = f , g = 0 P 1 1 P 1 fg ln I fac ( 42 ) x I gbd ( 41 ) y f , g = 0 P 2 1 P 2 fg ln IN fac ( 42 ) x IN gbd ( 41 ) y ,
I abcd ( 4 ) y = f , g = 0 P 1 1 P 1 fg ln I fac ( 41 ) x I gbd ( 42 ) y f , g = 0 P 2 1 P 2 fg ln IN fac ( 41 ) x IN gbd ( 42 ) y ,
I i 1 i 2 i 3 ( 41 ) s = + cos ( 2 π i 1 s l 1 s ) s ψ i 2 ( s ) ψ i 3 ( s ) d s ,
I i 1 i 2 i 3 ( 42 ) s = + ψ i 3 ( s ) cos ( 2 π i 1 s l 1 s ) s ψ i 2 ( s ) d s .

Metrics