Abstract

In this paper we present a full-vectorial finite-difference analysis of microstructured optical fibers. A new mode solver is described which uses Yee’s 2-D mesh and an index averaging technique. The modal characteristics are calculated for both conventional optical fibers and microstructured optical fibers. Comparison with previous finite difference mode solvers and other numerical methods is made and excellent agreement is achieved.

©2002 Optical Society of America

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    [Crossref]
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    [Crossref]
  24. W. P. Huang, C. L. Xu, S. T. Chu, and S. K. Chaudhuri, “The finite-difference vector beam propagation method. Analysis and Assessment,” J. Lightwave Technol. 10, 295–305 (1992).
    [Crossref]
  25. K. Bierwirth, N. Schulz, and F. Arndt, “Finite-difference analysis of rectangular dielectric waveguide structures,” IEEE Trans. Microwave Theory Tech. 34, 1104–1113 (1986).
    [Crossref]
  26. H. Dong, A. Chronopoulos, J. Zou, and A. Gopinath, “Vectorial integrated finite-difference analysis of dielectric waveguides,” J. Lightwave Technol. 11, 1559–1563 (1993).
    [Crossref]
  27. P. Lüsse, P. Stuwe, J. Schüle, and H. G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12, 487–493 (1994).
    [Crossref]
  28. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propagat. 14, 302–307 (1966).
    [Crossref]
  29. K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics, (CRC, Boca Raton, 1993).
  30. S. Dey and R. Mittra, “A conformal finite-difference time-domain technique for modeling cylindrical dielectric resonators,” IEEE Trans. Microwave Theory Tech. 47, 1717–1739 (1999).
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    [Crossref] [PubMed]
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    [Crossref]
  33. M. Midrio, M. P. Singh, and C. G. Someda, “The space filling mode of holey fibers: an analytical vectorial solution,” J. Lightwave Technol. 18, 1031–1037 (2000).
    [Crossref]
  34. Z. Zhu and T. G. Brown, “Analysis of the space filling modes of photonic crystal fibers,” Opt. Express 8, 547–554 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-10-547
    [Crossref] [PubMed]
  35. M. J. Steel, T. P. White, C. M. de Sterke, R. C. McPhedran, and L. C. Botten, “Symmetry and degeneracy in microstructured optical fibers,” Opt. Lett. 26, 488–490 (2001).
    [Crossref]
  36. M. Koshiba and K. Saitoh, “Numerical verification of degeneracy in hexagonal photonic crystal fibers,” IEEE Photon. Technol. Lett. 13, 1313–1315 (2001).
    [Crossref]
  37. M. J. Steel and R. M. Osgood, “Elliptical-hole photonic crystal fibers,” Opt. Lett. 26, 229–231 (2001).
    [Crossref]

2002 (2)

2001 (11)

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173
[Crossref] [PubMed]

M. Koshiba and K. Saitoh, “Numerical verification of degeneracy in hexagonal photonic crystal fibers,” IEEE Photon. Technol. Lett. 13, 1313–1315 (2001).
[Crossref]

G. E. Town and J. T. Lizer, “Tapered holey fibers for spot size and numerical-aperture conversion,” Opt. Lett. 26, 1042–1044 (2001).
[Crossref]

Z. Zhu and T. G. Brown, “Analysis of the space filling modes of photonic crystal fibers,” Opt. Express 8, 547–554 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-10-547
[Crossref] [PubMed]

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]

T. P. White, R. C. McPhedran, C. M. de Sterke, L. C. Botten, and M. J. Steel, “Confinement losses in microstructured optical fibers,” Opt. Lett. 26, 1660–1662 (2001).
[Crossref]

B. J. Eggleton, C. Kerbage, P. Westbrook, R. S. Windeler, and A. Hale, “Microstructured optical fiber devices,” Opt. Express 9, 698–713 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-698
[Crossref] [PubMed]

T. Hasegawa, E. Sasaoka, M. Onishi, M. Nishimura, Y. Tsuji, and M. Koshiba, “Hole-assisted lightguide fiber for large anomalous dispersion and low optical loss,” Opt. Express 9, 681–686 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-681
[Crossref] [PubMed]

M. Qiu, “Analysis of guided modes in photonic crystal fibers using the finite-difference time-domain method,” Microwave Opt. Technol. Lett. 30, 327–330 (2001).
[Crossref]

M. J. Steel and R. M. Osgood, “Elliptical-hole photonic crystal fibers,” Opt. Lett. 26, 229–231 (2001).
[Crossref]

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

2000 (7)

1999 (4)

1998 (1)

1997 (2)

1996 (1)

1994 (2)

K. S. Chiang, “Review of numerical and approximate methods for the modal analysis of general optical dielectric waveguides,” Opt. Quantum Electron. 26, s113–s134 (1994).
[Crossref]

P. Lüsse, P. Stuwe, J. Schüle, and H. G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12, 487–493 (1994).
[Crossref]

1993 (2)

H. Dong, A. Chronopoulos, J. Zou, and A. Gopinath, “Vectorial integrated finite-difference analysis of dielectric waveguides,” J. Lightwave Technol. 11, 1559–1563 (1993).
[Crossref]

W. P. Huang and C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. 29, 2639–2649 (1993).
[Crossref]

1992 (1)

W. P. Huang, C. L. Xu, S. T. Chu, and S. K. Chaudhuri, “The finite-difference vector beam propagation method. Analysis and Assessment,” J. Lightwave Technol. 10, 295–305 (1992).
[Crossref]

1988 (1)

M. S. Stern, “Semivectorial polarized finite difference method for optical waveguides with arbitrary index profiles,” IEE Proc. J. Optoelectron. 135, 56–63 (1988).
[Crossref]

1986 (1)

K. Bierwirth, N. Schulz, and F. Arndt, “Finite-difference analysis of rectangular dielectric waveguide structures,” IEEE Trans. Microwave Theory Tech. 34, 1104–1113 (1986).
[Crossref]

1966 (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propagat. 14, 302–307 (1966).
[Crossref]

Andres, M. V.

Andres, P.

Arndt, F.

K. Bierwirth, N. Schulz, and F. Arndt, “Finite-difference analysis of rectangular dielectric waveguide structures,” IEEE Trans. Microwave Theory Tech. 34, 1104–1113 (1986).
[Crossref]

Arriaga, J.

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809 (2000).
[Crossref]

Atkin, D. M.

Bennett, P. J.

Bierwirth, K.

K. Bierwirth, N. Schulz, and F. Arndt, “Finite-difference analysis of rectangular dielectric waveguide structures,” IEEE Trans. Microwave Theory Tech. 34, 1104–1113 (1986).
[Crossref]

Birks, T. A.

Bjarklev, A.

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]

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. 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]

Brown, T. G.

Chaudhuri, S. K.

W. P. Huang, C. L. Xu, S. T. Chu, and S. K. Chaudhuri, “The finite-difference vector beam propagation method. Analysis and Assessment,” J. Lightwave Technol. 10, 295–305 (1992).
[Crossref]

Chiang, K. S.

K. S. Chiang, “Review of numerical and approximate methods for the modal analysis of general optical dielectric waveguides,” Opt. Quantum Electron. 26, s113–s134 (1994).
[Crossref]

Chronopoulos, A.

H. Dong, A. Chronopoulos, J. Zou, and A. Gopinath, “Vectorial integrated finite-difference analysis of dielectric waveguides,” J. Lightwave Technol. 11, 1559–1563 (1993).
[Crossref]

Chu, S. T.

W. P. Huang, C. L. Xu, S. T. Chu, and S. K. Chaudhuri, “The finite-difference vector beam propagation method. Analysis and Assessment,” J. Lightwave Technol. 10, 295–305 (1992).
[Crossref]

de Sandro, J. P.

de Sterke, C. M.

Dey, S.

S. Dey and R. Mittra, “A conformal finite-difference time-domain technique for modeling cylindrical dielectric resonators,” IEEE Trans. Microwave Theory Tech. 47, 1717–1739 (1999).

Dong, H.

H. Dong, A. Chronopoulos, J. Zou, and A. Gopinath, “Vectorial integrated finite-difference analysis of dielectric waveguides,” J. Lightwave Technol. 11, 1559–1563 (1993).
[Crossref]

Eggleton, B. J.

Ferrando, A.

Gopinath, A.

H. Dong, A. Chronopoulos, J. Zou, and A. Gopinath, “Vectorial integrated finite-difference analysis of dielectric waveguides,” J. Lightwave Technol. 11, 1559–1563 (1993).
[Crossref]

Hale, A.

Hansen, T. P.

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]

Hasegawa, T.

Huang, W. P.

W. P. Huang and C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. 29, 2639–2649 (1993).
[Crossref]

W. P. Huang, C. L. Xu, S. T. Chu, and S. K. Chaudhuri, “The finite-difference vector beam propagation method. Analysis and Assessment,” J. Lightwave Technol. 10, 295–305 (1992).
[Crossref]

Jensen, J. R.

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]

Joannopoulos, J. D.

Johnson, S. G.

Kerbage, C.

Knight, J. C.

Knudsen, E.

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]

Koshiba, M.

Kunz, K. S.

K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics, (CRC, Boca Raton, 1993).

Libori, S. E. B.

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]

Lizer, J. T.

Luebbers, R. J.

K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics, (CRC, Boca Raton, 1993).

Lüsse, P.

P. Lüsse, P. Stuwe, J. Schüle, and H. G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12, 487–493 (1994).
[Crossref]

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]

McPhedran, R. C.

Midrio, M.

Miret, J. J.

Mittra, R.

S. Dey and R. Mittra, “A conformal finite-difference time-domain technique for modeling cylindrical dielectric resonators,” IEEE Trans. Microwave Theory Tech. 47, 1717–1739 (1999).

Mogilevtsev, D.

Monro, T. M.

Monro, T.M.

Mortensen, N. A.

Nishimura, M.

Onishi, M.

Ortigosa-Blanch, A.

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809 (2000).
[Crossref]

Osgood, R. M.

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]

Qiu, M.

M. Qiu, “Analysis of guided modes in photonic crystal fibers using the finite-difference time-domain method,” Microwave Opt. Technol. Lett. 30, 327–330 (2001).
[Crossref]

Ranka, J. K.

Richardson, D. J.

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. J.

Saitoh, K.

M. Koshiba and K. Saitoh, “Numerical verification of degeneracy in hexagonal photonic crystal fibers,” IEEE Photon. Technol. Lett. 13, 1313–1315 (2001).
[Crossref]

Sasaoka, E.

Schüle, J.

P. Lüsse, P. Stuwe, J. Schüle, and H. G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12, 487–493 (1994).
[Crossref]

Schulz, N.

K. Bierwirth, N. Schulz, and F. Arndt, “Finite-difference analysis of rectangular dielectric waveguide structures,” IEEE Trans. Microwave Theory Tech. 34, 1104–1113 (1986).
[Crossref]

Silvestre, E.

Simonsen, H.

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]

Singh, M. P.

Someda, C. G.

Steel, M. J.

Stentz, A. J.

Stern, M. S.

M. S. Stern, “Semivectorial polarized finite difference method for optical waveguides with arbitrary index profiles,” IEE Proc. J. Optoelectron. 135, 56–63 (1988).
[Crossref]

Stuwe, P.

P. Lüsse, P. Stuwe, J. Schüle, and H. G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12, 487–493 (1994).
[Crossref]

Town, G. E.

Tsuji, Y.

Unger, H. G.

P. Lüsse, P. Stuwe, J. Schüle, and H. G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12, 487–493 (1994).
[Crossref]

Vassallo, C.

C. Vassallo, “1993-1995 optical mode solvers,” Opt. Quantum Electron. 29, 95–114 (1997).
[Crossref]

Wadsworth, W. J.

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809 (2000).
[Crossref]

Westbrook, P.

White, T. P.

Windeler, R. S.

Xu, C. L.

W. P. Huang and C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. 29, 2639–2649 (1993).
[Crossref]

W. P. Huang, C. L. Xu, S. T. Chu, and S. K. Chaudhuri, “The finite-difference vector beam propagation method. Analysis and Assessment,” J. Lightwave Technol. 10, 295–305 (1992).
[Crossref]

Yee, K. S.

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propagat. 14, 302–307 (1966).
[Crossref]

Zhu, Z.

Zou, J.

H. Dong, A. Chronopoulos, J. Zou, and A. Gopinath, “Vectorial integrated finite-difference analysis of dielectric waveguides,” J. Lightwave Technol. 11, 1559–1563 (1993).
[Crossref]

IEE Proc. J. Optoelectron. (1)

M. S. Stern, “Semivectorial polarized finite difference method for optical waveguides with arbitrary index profiles,” IEE Proc. J. Optoelectron. 135, 56–63 (1988).
[Crossref]

IEEE J. Quantum Electron. (1)

W. P. Huang and C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. 29, 2639–2649 (1993).
[Crossref]

IEEE Photon. Technol. Lett. (3)

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809 (2000).
[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]

M. Koshiba and K. Saitoh, “Numerical verification of degeneracy in hexagonal photonic crystal fibers,” IEEE Photon. Technol. Lett. 13, 1313–1315 (2001).
[Crossref]

IEEE Trans. Antennas Propagat. (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propagat. 14, 302–307 (1966).
[Crossref]

IEEE Trans. Microwave Theory Tech. (2)

S. Dey and R. Mittra, “A conformal finite-difference time-domain technique for modeling cylindrical dielectric resonators,” IEEE Trans. Microwave Theory Tech. 47, 1717–1739 (1999).

K. Bierwirth, N. Schulz, and F. Arndt, “Finite-difference analysis of rectangular dielectric waveguide structures,” IEEE Trans. Microwave Theory Tech. 34, 1104–1113 (1986).
[Crossref]

J. Lightwave Technol. (6)

H. Dong, A. Chronopoulos, J. Zou, and A. Gopinath, “Vectorial integrated finite-difference analysis of dielectric waveguides,” J. Lightwave Technol. 11, 1559–1563 (1993).
[Crossref]

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Figures (6)

Fig. 1.
Fig. 1. (a) Yee’s 2-D mesh; (b) Mesh cells across a curved interface.

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Fig. 2.
Fig. 2. Relative error in the calculated fundamental mode index neff of a step-index circular optical fiber. The fiber has a core diameter 6 μm, a core refractive index of 1.45 at wavelength 1.5 μm, and air cladding with unity refractive index. The calculation window is chosen to be the first quadrant of the fiber cross section with a computation window size of 6μm by 6 μm. The left boundary is magnetic wall, the bottom is electric wall; all others are zero-value boundaries.

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Fig. 3.
Fig. 3. (a) Schematic of an air-hole assisted optical fiber; (b) Relative errors of calculated fundamental mode index from different mode solvers when the number of grids along the x-axis is varied (fiber parameters are the same as in Table 2); (c) Magnetic field plots of the fundamental mode (y-polarization); (d) Calculated GVD curves (material dispersion not considered).

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Fig. 4.
Fig. 4. (a) Calculation window of the holey fiber; (b) Comparison between different FD mode solvers (Λ=2.3 μm, r a=0.5 μm, silica refractive index of 1.45 is assumed at λ=1.5 μm, calculation window of 3Λ by 3Λ is the first quadrant bounded by C1~C4. C1: magnetic wall, C2~C4: electric wall); (c) Field plots of the y-polarized fundamental mode; (d) Calculated effective modal index β/k 0, nFSM and GVD (Silica refractive index of 1.45 is assumed at all wavelengths).

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Fig. 5.
Fig. 5. Convergence of modal birefringence of the fundamental modes in (a) air-hole assisted optical fiber and in (b) holey optical fiber. The parameters for the two fibers are listed in Table 2 and Fig. 4(b), respectively.

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Fig. 6.
Fig. 6. (a) Calculated effective index and GVD of the x- and y-polarized fundamental modes for a holey fiber with elliptical air holes. The air holes have their major axis along the y-direction. The semimajor axis is 0.8 μm; the semiminor axis is 0.5 μm. Other calculation parameters are same as in Fig. 4. The effective indices of x-polarized and y-polarized FSM are also shown. (b) Contour plots of major magnetic field components for x-polarized (left) and y-polarized (right) fundamental modes at wavelength of 1.5 μm.

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Tables (3)

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Table 1. Calculated fundamental mode indices of a step-index fiber from different FD mode solvers. The fiber parameters are the same as in Fig. 2. The analytical solution for the fundamental mode index is 1.438604.

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Table 2. Calculated fundamental mode indices for air-hole-assisted optical fiber shown in Fig. 3(a). Core index 1.45, silica cladding index 1.42, r 0=2μm, r a=2μm, Λ=5μm, wavelength=1.5 μm, first quadrant window size 8μm by 8μm, C1: magnetic wall, C2~C4: electric wall. The multipole analysis [32] gives neff =1.4353607.

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Table 3. Calculated fundamental mode indices for the holey fiber shown in Fig. 4(a) from different methods. The fiber parameters are same as in Fig. 4(b). The number of grids along both x-axis and y-axis is 120.

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Equations (32)

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( ∇ t 2 + k 0 2 ε r ) E t + ∇ t ( ε r − 1 ∇ t ε r · E t ) = β 2 E t
( ∇ t 2 + k 0 2 ε r ) H t + ε r − 1 ∇ t ε r × ( ∇ t × H t ) = β 2 H t
i k 0 H x = ∂ E z ∂ y − iβ E y ,
i k 0 H y = iβ E x − ∂ E z ∂ x ,
i k 0 H z = ∂ E y ∂ x − ∂ E x ∂ y ,
− i k 0 ε r E x = ∂ H z ∂ y − iβ H y ,
− i k 0 ε r E y = iβ H x − ∂ H z ∂ x ,
− i k 0 ε r E z = ∂ H y ∂ x − ∂ H x ∂ y .
i k 0 H x ( j , l ) = [ E z ( j , l + 1 ) − E z ( j , l ) ] Δ y − iβ E y ( j , l ) ,
i k 0 H y ( j , l ) = iβ E x ( j , l ) − [ E z ( j + 1 , l ) − E z ( j , l ) ] Δ x ,
i k 0 H z ( j , l ) = [ E y ( j + 1 , l ) − E y ( j , l ) ] Δ x − [ E x ( j , l + 1 ) − E x ( j , l ) ] Δ y ,
− i k 0 ε rx ( j , l ) E x ( j , l ) = [ H z ( j , l ) − H z ( j , l − 1 ) ] Δ y − iβ H y ( j , l ) ,
− i k 0 ε ry ( j , l ) E y ( j , l ) = iβ H x ( j , l ) − [ H z ( j , l ) − H z ( j − 1 , l ) ] Δ x ,
− i k 0 ε rz ( j , l ) E z ( j , l ) = [ H y ( j , l ) − H y ( j − 1 , l ) ] Δ x − [ H x ( j , l ) − H x ( j , l − 1 ) ] Δ y ,
ε rx ( j , l ) = [ ε r ( j , l ) + ε r ( j , l − 1 ) ] 2 ,
ε ry ( j , l ) = [ ε r ( j , l ) + ε r ( j − 1 , l ) ] 2 ,
ε rz ( j , l ) = [ ε r ( j , l ) + ε r ( j − 1 , l − 1 ) + ε r ( j , l − 1 ) + ε r ( j − 1 , l ) ] 4 .
i k 0 [ H x H y H z ] = [ 0 − iβ I U y iβ I 0 − U x − U y U x 0 ] [ E x E y E z ] ,
− i k 0 [ ε rx 0 0 0 ε ry 0 0 0 ε rz ] [ E x E y E z ] = [ 0 − iβ I V y iβ I 0 − V x − V y V x 0 ] [ H x H y H z ] ,
U x = 1 Δ x [ − 1 1 − 1 1 ⋱ ⋱ ⋱ ⋱ − 1 1 − 1 ] , U y = 1 Δ y [ − 1 1 − 1 ⋱ ⋱ 1 ⋱ − 1 1 ] ,
V x = 1 Δ x [ 1 − 1 1 − 1 ⋱ ⋱ ⋱ − 1 1 − 1 1 ] , V y = 1 Δ y [ 1 1 ⋱ − 1 ⋱ ⋱ 1 − 1 1 ] .
P [ E x E y ] = [ P xx P xy P yx P yy ] [ E x E y ] = β 2 [ E x E y ] ,
P xx = − k 0 − 2 U x ε rz − 1 V y V x U y + ( k 0 2 I + U x ε rz − 1 V x ) ( ε rx + k 0 − 2 V y U y ) ,
P yy = − k 0 − 2 U y ε rz − 1 V x V y U x + ( k 0 2 I + U y ε rz − 1 V y ) ( ε ry + k 0 − 2 V x U x ) ,
P xy = U x ε rz − 1 V y ( ε ry + k 0 − 2 V x U x ) − k 0 − 2 ( k 0 2 I + U x ε rz − 1 V x ) V y U x ,
P yx = U y ε rz − 1 V x ( ε rx + k 0 − 2 V y U y ) − k 0 − 2 ( k 0 2 I + U y ε rz − 1 V y ) V x U y .
Q [ H x H y ] = [ Q xx Q xy Q yx Q yy ] [ H x H y ] = β 2 [ H x H y ] ,
Q xx = − k 0 − 2 V x U y U x ε rz − 1 V y + ( ε ry + k 0 − 2 V x U x ) ( k 0 2 I + U y ε rz − 1 V y ) ,
Q yy = − k 0 − 2 V y U x U y ε rz − 1 V x + ( ε rx + k 0 − 2 V y U y ) ( k 0 2 I + U x ε rz − 1 V x ) ,
Q xy = − ( ε ry + k 0 − 2 V x U x ) U y ε rz − 1 V x + k 0 − 2 V x U y ( k 0 2 I + U x ε rz − 1 V x ) ,
Q yx = − ( ε rx + k 0 − 2 V y U y ) U x ε rz − 1 V y + k 0 − 2 V y U x ( k 0 2 I + U y ε rz − 1 V y ) .
Q xx = P yy T , Q yy = P xx T , Q xy = − P xy T , Q yx = − P yx T .

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