Abstract

The previously developed full-vectorial optical waveguide eigenmode solvers using pseudospectral frequency-domain (PSFD) formulations for optical waveguides with arbitrary step-index profile is further implemented with the uniaxial perfectly matched layer (UPML) absorption boundary conditions for treating leaky waveguides and calculating their complex modal effective indices. The role of the UPML reflection coefficient in achieving high-accuracy mode solution results is particularly investigated. A six-air-hole microstructured fiber is analyzed as an example to compare with published high-accuracy multipole method results for both the real and imaginary parts of the effective indices. It is shown that by setting the UPML reflection coefficient values as small as on the order of 10−40 ∼ 10−70, relative errors in the calculated complex effective indices can be as small as on the order of 10−12.

© 2011 OSA

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    [Crossref]
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  12. C. H. Teng, B. Y. Lin, H. C. Chang, H. C. Hsu, C. N. Lin, and K. A. Feng, “A Legendre pseudospectral penalty scheme for solving time-domain Maxwell’s equations,” J. Sci. Comput. 36, 351–390 (2008).
    [Crossref]
  13. B. Y. Lin, H. C. Hsu, C. H. Teng, H. C. Chang, J. K. Wang, and Y. L. Wang, “Unraveling near-field origin of electromagnetic waves scattered from silver nanorod arrays using pseudo-spectral time-domain calculation,” Opt. Express17, 14211–14228 (2009). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-14211 .
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    [Crossref]
  23. P. J. Chiang and H. C. Chang, “Analysis of leaky optical waveguides using pseudospectral methods,” OSA 2006 Integrated Photonics Research and Applications (IPRA ’06) Technical Digest (Optical Society of America, 2005), paper ITuA3.
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    [Crossref]
  27. W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microwave Opt. Technol. Lett. 7, 599–604 (1994).
    [Crossref]
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    [Crossref]

2010 (1)

P. J. Chiang and Y. C. Chiang, “Pseudospectral frequency-domain formulae based on modified perfectly matched layers for calculating both guided and leaky modes,” IEEE Photon. Technol. Lett. 22, 908–910 (2010).
[Crossref]

2008 (2)

P. J. Chiang, C. L. Wu, C. H. Teng, C. S. Yang, and H. C. Chang, “Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations,” IEEE J. Quantum Electron. 44, 56–66 (2008).
[Crossref]

C. H. Teng, B. Y. Lin, H. C. Chang, H. C. Hsu, C. N. Lin, and K. A. Feng, “A Legendre pseudospectral penalty scheme for solving time-domain Maxwell’s equations,” J. Sci. Comput. 36, 351–390 (2008).
[Crossref]

2007 (1)

P. J. Chiang, C. P. Yu, and H. C. Chang, “Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method,” Phys. Rev. E 75, 026703 (2007).
[Crossref]

2005 (1)

C. C. Huang, C. C. Huang, and J. Y. Yang, “A full-vectorial pseudospectral modal analysis of dielectric optical waveguides with stepped refractive index profiles,” IEEE J. Sel. Top. Quantum Electron. 11, 457–465 (2005).
[Crossref]

2004 (1)

G. Zhao and Q. H. Liu, “The 3-D multidomain pseudospectral time-domain algorithm for inhomogeneous conductive media,” IEEE Trans. Antennas Propagat. 52, 742–749 (2004).
[Crossref]

2003 (1)

P. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003).
[Crossref] [PubMed]

2002 (7)

2001 (1)

2000 (2)

1999 (2)

B. Yang and J. S. Hesthaven, “A pseudospectral method for time-domain computation of electromagnetic scattering by bodies of revolution,” IEEE Trans. Antennas Propagat. 47, 132–141 (1999).
[Crossref]

J. S. Hesthaven, P. G. Dinesen, and J. P. Lynov, “Spectral collocation time-domain modeling of diffractive optical elements,” J. Comput. Phys. 155, 287–306 (1999).
[Crossref]

1997 (1)

B. Yang, D. Gottlieb, and J. S. Hesthaven, “Spectral simulations of electromagnetic wave scattering,” J. Comput. Phys. 134, 216–230 (1997).
[Crossref]

1995 (1)

Z. S. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propagat. 43, 1460–1463 (1995).
[Crossref]

1994 (2)

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microwave Opt. Technol. Lett. 7, 599–604 (1994).
[Crossref]

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comp. Phys. 114, 185–200 (1994).
[Crossref]

1973 (1)

W. J. Gordon and C. A. Hall, “Transfinite element methods: blending-function interpolation over arbitrary curved element domains,” Numer. Math. 21, 109–129 (1973).
[Crossref]

Benson, T. M.

Berenger, J. P.

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comp. Phys. 114, 185–200 (1994).
[Crossref]

Botten, L. C.

Chang, H. C.

C. H. Teng, B. Y. Lin, H. C. Chang, H. C. Hsu, C. N. Lin, and K. A. Feng, “A Legendre pseudospectral penalty scheme for solving time-domain Maxwell’s equations,” J. Sci. Comput. 36, 351–390 (2008).
[Crossref]

P. J. Chiang, C. L. Wu, C. H. Teng, C. S. Yang, and H. C. Chang, “Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations,” IEEE J. Quantum Electron. 44, 56–66 (2008).
[Crossref]

P. J. Chiang, C. P. Yu, and H. C. Chang, “Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method,” Phys. Rev. E 75, 026703 (2007).
[Crossref]

Y. C. Chiang, Y. P. Chiou, and H. C. Chang, “Improved full-vectorial finite-difference mode solver for optical waveguides with step-index profiles,” J. Lightwave Technol. 20, 1609–1618, (2002).
[Crossref]

P. J. Chiang and H. C. Chang, “Analysis of leaky optical waveguides using pseudospectral methods,” OSA 2006 Integrated Photonics Research and Applications (IPRA ’06) Technical Digest (Optical Society of America, 2005), paper ITuA3.

P. J. Chiang, C. S. Yang, C. L. Wu, C. H. Teng, and H. C. Chang, “Application of pseudospectral methods to optical waveguide mode solvers,” OSA 2005 Integrated Photonics Research and Applications (IPRA ’05) Technical Digest (Optical Society of America, 2005), paper IMG4.

Chew, W. C.

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microwave Opt. Technol. Lett. 7, 599–604 (1994).
[Crossref]

Chiang, P. J.

P. J. Chiang and Y. C. Chiang, “Pseudospectral frequency-domain formulae based on modified perfectly matched layers for calculating both guided and leaky modes,” IEEE Photon. Technol. Lett. 22, 908–910 (2010).
[Crossref]

P. J. Chiang, C. L. Wu, C. H. Teng, C. S. Yang, and H. C. Chang, “Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations,” IEEE J. Quantum Electron. 44, 56–66 (2008).
[Crossref]

P. J. Chiang, C. P. Yu, and H. C. Chang, “Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method,” Phys. Rev. E 75, 026703 (2007).
[Crossref]

P. J. Chiang, C. S. Yang, C. L. Wu, C. H. Teng, and H. C. Chang, “Application of pseudospectral methods to optical waveguide mode solvers,” OSA 2005 Integrated Photonics Research and Applications (IPRA ’05) Technical Digest (Optical Society of America, 2005), paper IMG4.

P. J. Chiang and H. C. Chang, “Analysis of leaky optical waveguides using pseudospectral methods,” OSA 2006 Integrated Photonics Research and Applications (IPRA ’06) Technical Digest (Optical Society of America, 2005), paper ITuA3.

Chiang, Y. C.

P. J. Chiang and Y. C. Chiang, “Pseudospectral frequency-domain formulae based on modified perfectly matched layers for calculating both guided and leaky modes,” IEEE Photon. Technol. Lett. 22, 908–910 (2010).
[Crossref]

Y. C. Chiang, Y. P. Chiou, and H. C. Chang, “Improved full-vectorial finite-difference mode solver for optical waveguides with step-index profiles,” J. Lightwave Technol. 20, 1609–1618, (2002).
[Crossref]

Chiou, Y. P.

de Sterke, C. M.

Dinesen, P. G.

J. S. Hesthaven, P. G. Dinesen, and J. P. Lynov, “Spectral collocation time-domain modeling of diffractive optical elements,” J. Comput. Phys. 155, 287–306 (1999).
[Crossref]

Feng, K. A.

C. H. Teng, B. Y. Lin, H. C. Chang, H. C. Hsu, C. N. Lin, and K. A. Feng, “A Legendre pseudospectral penalty scheme for solving time-domain Maxwell’s equations,” J. Sci. Comput. 36, 351–390 (2008).
[Crossref]

Gordon, W. J.

W. J. Gordon and C. A. Hall, “Transfinite element methods: blending-function interpolation over arbitrary curved element domains,” Numer. Math. 21, 109–129 (1973).
[Crossref]

Gottlieb, D.

B. Yang, D. Gottlieb, and J. S. Hesthaven, “Spectral simulations of electromagnetic wave scattering,” J. Comput. Phys. 134, 216–230 (1997).
[Crossref]

Hadley, G. R.

Hall, C. A.

W. J. Gordon and C. A. Hall, “Transfinite element methods: blending-function interpolation over arbitrary curved element domains,” Numer. Math. 21, 109–129 (1973).
[Crossref]

Hesthaven, J. S.

B. Yang and J. S. Hesthaven, “A pseudospectral method for time-domain computation of electromagnetic scattering by bodies of revolution,” IEEE Trans. Antennas Propagat. 47, 132–141 (1999).
[Crossref]

J. S. Hesthaven, P. G. Dinesen, and J. P. Lynov, “Spectral collocation time-domain modeling of diffractive optical elements,” J. Comput. Phys. 155, 287–306 (1999).
[Crossref]

B. Yang, D. Gottlieb, and J. S. Hesthaven, “Spectral simulations of electromagnetic wave scattering,” J. Comput. Phys. 134, 216–230 (1997).
[Crossref]

Hsu, H. C.

C. H. Teng, B. Y. Lin, H. C. Chang, H. C. Hsu, C. N. Lin, and K. A. Feng, “A Legendre pseudospectral penalty scheme for solving time-domain Maxwell’s equations,” J. Sci. Comput. 36, 351–390 (2008).
[Crossref]

Huang, C. C.

C. C. Huang, C. C. Huang, and J. Y. Yang, “A full-vectorial pseudospectral modal analysis of dielectric optical waveguides with stepped refractive index profiles,” IEEE J. Sel. Top. Quantum Electron. 11, 457–465 (2005).
[Crossref]

C. C. Huang, C. C. Huang, and J. Y. Yang, “A full-vectorial pseudospectral modal analysis of dielectric optical waveguides with stepped refractive index profiles,” IEEE J. Sel. Top. Quantum Electron. 11, 457–465 (2005).
[Crossref]

Kingsland, D. M.

Z. S. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propagat. 43, 1460–1463 (1995).
[Crossref]

Koshiba, M.

Kuhlmey, B. T.

Lee, J. F.

Z. S. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propagat. 43, 1460–1463 (1995).
[Crossref]

Lee, R.

Z. S. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propagat. 43, 1460–1463 (1995).
[Crossref]

Lin, B. Y.

C. H. Teng, B. Y. Lin, H. C. Chang, H. C. Hsu, C. N. Lin, and K. A. Feng, “A Legendre pseudospectral penalty scheme for solving time-domain Maxwell’s equations,” J. Sci. Comput. 36, 351–390 (2008).
[Crossref]

Lin, C. N.

C. H. Teng, B. Y. Lin, H. C. Chang, H. C. Hsu, C. N. Lin, and K. A. Feng, “A Legendre pseudospectral penalty scheme for solving time-domain Maxwell’s equations,” J. Sci. Comput. 36, 351–390 (2008).
[Crossref]

Liu, Q. H.

G. Zhao and Q. H. Liu, “The 3-D multidomain pseudospectral time-domain algorithm for inhomogeneous conductive media,” IEEE Trans. Antennas Propagat. 52, 742–749 (2004).
[Crossref]

Q. H. Liu, “A pseudospectral frequency-domain (PSFD) method for computational electromagnetics,” IEEE Antennas Wireless Propagat. Lett. 1, 131–134 (2002).
[Crossref]

Lynov, J. P.

J. S. Hesthaven, P. G. Dinesen, and J. P. Lynov, “Spectral collocation time-domain modeling of diffractive optical elements,” J. Comput. Phys. 155, 287–306 (1999).
[Crossref]

Maystre, D.

Mcphedran, R. C.

Renversez, G.

Russell, P.

P. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003).
[Crossref] [PubMed]

Sacks, Z. S.

Z. S. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propagat. 43, 1460–1463 (1995).
[Crossref]

Saitoh, K.

Sewell, P.

Teng, C. H.

C. H. Teng, B. Y. Lin, H. C. Chang, H. C. Hsu, C. N. Lin, and K. A. Feng, “A Legendre pseudospectral penalty scheme for solving time-domain Maxwell’s equations,” J. Sci. Comput. 36, 351–390 (2008).
[Crossref]

P. J. Chiang, C. L. Wu, C. H. Teng, C. S. Yang, and H. C. Chang, “Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations,” IEEE J. Quantum Electron. 44, 56–66 (2008).
[Crossref]

P. J. Chiang, C. S. Yang, C. L. Wu, C. H. Teng, and H. C. Chang, “Application of pseudospectral methods to optical waveguide mode solvers,” OSA 2005 Integrated Photonics Research and Applications (IPRA ’05) Technical Digest (Optical Society of America, 2005), paper IMG4.

Thomas, N.

Tsuji, Y.

Weedon, W. H.

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microwave Opt. Technol. Lett. 7, 599–604 (1994).
[Crossref]

White, T. P.

Wu, C. L.

P. J. Chiang, C. L. Wu, C. H. Teng, C. S. Yang, and H. C. Chang, “Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations,” IEEE J. Quantum Electron. 44, 56–66 (2008).
[Crossref]

P. J. Chiang, C. S. Yang, C. L. Wu, C. H. Teng, and H. C. Chang, “Application of pseudospectral methods to optical waveguide mode solvers,” OSA 2005 Integrated Photonics Research and Applications (IPRA ’05) Technical Digest (Optical Society of America, 2005), paper IMG4.

Yang, B.

B. Yang and J. S. Hesthaven, “A pseudospectral method for time-domain computation of electromagnetic scattering by bodies of revolution,” IEEE Trans. Antennas Propagat. 47, 132–141 (1999).
[Crossref]

B. Yang, D. Gottlieb, and J. S. Hesthaven, “Spectral simulations of electromagnetic wave scattering,” J. Comput. Phys. 134, 216–230 (1997).
[Crossref]

Yang, C. S.

P. J. Chiang, C. L. Wu, C. H. Teng, C. S. Yang, and H. C. Chang, “Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations,” IEEE J. Quantum Electron. 44, 56–66 (2008).
[Crossref]

P. J. Chiang, C. S. Yang, C. L. Wu, C. H. Teng, and H. C. Chang, “Application of pseudospectral methods to optical waveguide mode solvers,” OSA 2005 Integrated Photonics Research and Applications (IPRA ’05) Technical Digest (Optical Society of America, 2005), paper IMG4.

Yang, J. Y.

C. C. Huang, C. C. Huang, and J. Y. Yang, “A full-vectorial pseudospectral modal analysis of dielectric optical waveguides with stepped refractive index profiles,” IEEE J. Sel. Top. Quantum Electron. 11, 457–465 (2005).
[Crossref]

Yu, C. P.

P. J. Chiang, C. P. Yu, and H. C. Chang, “Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method,” Phys. Rev. E 75, 026703 (2007).
[Crossref]

Zhao, G.

G. Zhao and Q. H. Liu, “The 3-D multidomain pseudospectral time-domain algorithm for inhomogeneous conductive media,” IEEE Trans. Antennas Propagat. 52, 742–749 (2004).
[Crossref]

IEEE Antennas Wireless Propagat. Lett. (1)

Q. H. Liu, “A pseudospectral frequency-domain (PSFD) method for computational electromagnetics,” IEEE Antennas Wireless Propagat. Lett. 1, 131–134 (2002).
[Crossref]

IEEE J. Quantum Electron. (1)

P. J. Chiang, C. L. Wu, C. H. Teng, C. S. Yang, and H. C. Chang, “Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations,” IEEE J. Quantum Electron. 44, 56–66 (2008).
[Crossref]

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

C. C. Huang, C. C. Huang, and J. Y. Yang, “A full-vectorial pseudospectral modal analysis of dielectric optical waveguides with stepped refractive index profiles,” IEEE J. Sel. Top. Quantum Electron. 11, 457–465 (2005).
[Crossref]

IEEE Photon. Technol. Lett. (1)

P. J. Chiang and Y. C. Chiang, “Pseudospectral frequency-domain formulae based on modified perfectly matched layers for calculating both guided and leaky modes,” IEEE Photon. Technol. Lett. 22, 908–910 (2010).
[Crossref]

IEEE Trans. Antennas Propagat. (3)

Z. S. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propagat. 43, 1460–1463 (1995).
[Crossref]

G. Zhao and Q. H. Liu, “The 3-D multidomain pseudospectral time-domain algorithm for inhomogeneous conductive media,” IEEE Trans. Antennas Propagat. 52, 742–749 (2004).
[Crossref]

B. Yang and J. S. Hesthaven, “A pseudospectral method for time-domain computation of electromagnetic scattering by bodies of revolution,” IEEE Trans. Antennas Propagat. 47, 132–141 (1999).
[Crossref]

J. Comp. Phys. (1)

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comp. Phys. 114, 185–200 (1994).
[Crossref]

J. Comput. Phys. (2)

J. S. Hesthaven, P. G. Dinesen, and J. P. Lynov, “Spectral collocation time-domain modeling of diffractive optical elements,” J. Comput. Phys. 155, 287–306 (1999).
[Crossref]

B. Yang, D. Gottlieb, and J. S. Hesthaven, “Spectral simulations of electromagnetic wave scattering,” J. Comput. Phys. 134, 216–230 (1997).
[Crossref]

J. Lightwave Technol. (7)

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

J. Sci. Comput. (1)

C. H. Teng, B. Y. Lin, H. C. Chang, H. C. Hsu, C. N. Lin, and K. A. Feng, “A Legendre pseudospectral penalty scheme for solving time-domain Maxwell’s equations,” J. Sci. Comput. 36, 351–390 (2008).
[Crossref]

Microwave Opt. Technol. Lett. (1)

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microwave Opt. Technol. Lett. 7, 599–604 (1994).
[Crossref]

Numer. Math. (1)

W. J. Gordon and C. A. Hall, “Transfinite element methods: blending-function interpolation over arbitrary curved element domains,” Numer. Math. 21, 109–129 (1973).
[Crossref]

Phys. Rev. E (1)

P. J. Chiang, C. P. Yu, and H. C. Chang, “Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method,” Phys. Rev. E 75, 026703 (2007).
[Crossref]

Science (1)

P. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003).
[Crossref] [PubMed]

Other (4)

B. Y. Lin, H. C. Hsu, C. H. Teng, H. C. Chang, J. K. Wang, and Y. L. Wang, “Unraveling near-field origin of electromagnetic waves scattered from silver nanorod arrays using pseudo-spectral time-domain calculation,” Opt. Express17, 14211–14228 (2009). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-14211 .
[Crossref] [PubMed]

P. J. Chiang, C. S. Yang, C. L. Wu, C. H. Teng, and H. C. Chang, “Application of pseudospectral methods to optical waveguide mode solvers,” OSA 2005 Integrated Photonics Research and Applications (IPRA ’05) Technical Digest (Optical Society of America, 2005), paper IMG4.

P. J. Chiang and H. C. Chang, “Analysis of leaky optical waveguides using pseudospectral methods,” OSA 2006 Integrated Photonics Research and Applications (IPRA ’06) Technical Digest (Optical Society of America, 2005), paper ITuA3.

C. P. Yu and H. C. Chang, “Yee-mesh-based finite difference eigenmode solver with PML absorbing boundary conditions for optical waveguides and photonic crystal fibers,” Opt. Express12, 6165–6177 (2004). http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-25-6165 .
[Crossref] [PubMed]

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Figures (8)

Fig. 1
Fig. 1

Interface between two homogeneous regions, Regions a and b, with relative permittivity and permeability tensors, ([ɛa], [μa]) and ([ɛb], [μb]), in the waveguide cross-section. is a unit vector normal to the interface.

Fig. 2
Fig. 2

Cross-section of an arbitrary leaky waveguide problem with the computational domain surrounded by UPML regions.

Fig. 3
Fig. 3

(a) The cross-section of the triangular holey fiber with one ring of six air holes. The computational domain with UPML regions I, II, and III is shown to contain a quarter of the cross-section. (b) Mesh division profile of the computational domain.

Fig. 4
Fig. 4

(a) Relative errors in the real part of the effective index for the sixth mode in the holey fiber of Fig. 3 versus the number of unknowns for different UPML reflection coefficient values. (b) Relative errors in the imaginary part of the effective index for the sixth mode in the holey fiber of Fig. 3 versus the number of unknowns for different UPML reflection coefficient values.

Fig. 5
Fig. 5

(a) Total relative errors in the effective index for the sixth mode in the holey fiber of Fig. 3 versus the number of unknowns for different UPML reflection coefficient values. (b) Total relative errors in the effective index for the sixth mode in the holey fiber of Fig. 3 versus the UPML reflection coefficient using 28224 unknowns.

Fig. 6
Fig. 6

(a) Real part and (b) imaginary part of the effective index of the sixth mode of the holey fiber of Fig. 3 versus the width (Wx = Wy) of the computational domain for R =107 and 1070.

Fig. 7
Fig. 7

Total relative errors in the effective index of the sixth mode in the holey fiber of Fig. 3 versus the number of unknowns for two different computational domain sizes, Wx = Wy = 13.75 μm and Wx = Wy = 15.75 μm.

Fig. 8
Fig. 8

Normalized |Ex|, |Ez|, |Hx|, and |Hy| field profiles of the first six modes of the holey fiber of Fig. 3.

Tables (6)

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Table 1 Definition of sx and sy values in the UPML and non-UPML regions.

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Table 2 Real and imaginary parts of the effective index of the sixth mode of the holey fiber of Fig. 3 obtained using R = 10−70 and different numbers of unknowns.

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Table 3 Real and imaginary parts of the effective index of the sixth mode of the holey fiber of Fig. 3 obtained using 28224 unknowns and different values for the UPML reflection coefficient.

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Table 4 Real and imaginary parts of the effective index of the fundamental mode of the holey fiber of Fig. 3 obtained using 28224 unknowns and three different values for the UPML reflection coefficient.

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Table 5 Real and imaginary parts of the effective index of the fundamental mode of the holey fiber of Fig. 3 obtained with R = 10−70 and different numbers of unknowns.

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Table 6 Real and imaginary parts of the effective indices and losses of the first six modes of the holey fiber of Fig. 3 obtained using 28224 unknowns and R = 10−70.

Equations (38)

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ɛ 0 [ ɛ ] E = 0 ,
μ 0 [ μ ] H = 0 ,
× E = j ω μ 0 [ μ ] H ,
× H = j ω ɛ 0 [ ɛ ] E ,
[ ɛ ] = [ ɛ x ( x , y ) 0 0 0 ɛ y ( x , y ) 0 0 0 ɛ z ( x , y ) ] and [ μ ] = [ μ x ( x , y ) 0 0 0 μ y ( x , y ) 0 0 0 μ z ( x , y ) ] .
× ( [ ɛ ] 1 × H ) k 0 2 [ μ ] H = 0 ,
H ( x , y , z ) = [ x ^ H x ( x , y ) + y ^ H y ( x , y ) + z ^ H z ( x , y ) ] e j β z ,
H z z = 1 μ z [ ( μ x H x ) x + ( μ y H y ) y ] .
[ P x x P x y P y x P y y ] [ H x H y ] = β 2 [ H x H y ] ,
P x x H x = k 0 2 ɛ y μ x H x + ɛ y { y [ 1 ɛ z ( H x y ) ] } + x { 1 μ z [ ( μ x H x ) x ] } ,
P x y H y = ɛ y { y [ 1 ɛ z ( H y x ) ] } + x { 1 μ z [ ( μ y H y ) y ] } ,
P y y H y = k 0 2 ɛ x μ y H y + ɛ x { x [ 1 ɛ z ( H y x ) ] } + y { 1 μ z [ ( μ y H y ) y ] } ,
P y x H x = ɛ x { x [ 1 ɛ z ( H x y ) ] } + y { 1 μ z [ ( μ x H x ) x ] } .
[ H x a H y a ] = [ n y n x μ x a n x μ y a n y ] 1 [ n y n x μ x b n x μ y b n y ] [ H x b H y b ] ,
n x μ x a H x a x + n y H x a y = n x [ μ y a H y a y + ( μ z a μ z b ) ( H x b x + H y b y ) ] + n y [ H y a x ɛ z a ɛ z b ( H y b x H x b y ) ] ,
n x H y a x + n y μ y a H y a y = n y [ μ x a H x a y + ( μ z a μ z b ) ( H x b x + H y b y ) ] + n x [ H x a y + ɛ z a ɛ z b ( H y b x H x b y ) ] ,
[ μ ] = [ μ x 0 0 0 μ y 0 0 0 μ z ] , [ ɛ ] = [ ɛ x 0 0 0 ɛ y 0 0 0 ɛ z ] ,
[ μ ] 1 [ μ ] UPML = [ ɛ ] 1 [ ɛ ] UPML = [ s y s x 0 0 0 s x s y 0 0 0 s y s x ] .
[ μ ] = [ s y s x μ x 0 0 0 s x s y μ y 0 0 0 s y s x μ z ] ,
[ ɛ ] = [ s y s x ɛ x 0 0 0 s x s y ɛ y 0 0 0 s y s x ɛ z ] ,
s x = 1 j α E x ,
s y = 1 j α E y ,
α x = α E x ( ρ x ) = α max ( x ) ( ρ x d x ) m ,
α y = α E y ( ρ y ) = α max ( y ) ( ρ y d y ) m ,
α max ( x ) = ( m + 1 ) λ 4 π n d x ln 1 R ,
α max ( y ) = ( m + 1 ) λ 4 π n d y ln 1 R ,
s x = 1 j 3 λ 4 π n d x ( ρ x d x ) 2 ln 1 R ,
s y = 1 j 3 λ 4 π n d y ( ρ y d y ) 2 ln 1 R .
P x x H ¯ x = k 0 2 ɛ y μ y H ¯ x + ɛ y ɛ z s y [ s y H ¯ x y + 1 s y 2 H ¯ x y 2 ] + μ x μ z { [ ( s x ) 2 + s x s x ] H ¯ x + ( 3 s x s x ) H ¯ x x + ( 1 s x ) 2 2 H ¯ x x 2 } ,
P x y H y = ɛ y ɛ z s y ( s y H y x + 1 s y 2 H y y x ) + μ y μ z s y ( s y H y x + 1 s y 2 H y x y ) ,
P y y H y = k 0 2 ɛ x μ y H y + ɛ x ɛ z s x [ s x H y x + 1 s x 2 H y x 2 ] + μ y μ z { [ ( s y ) 2 + s y s y ] H y + ( 3 s y s y ) H y y + ( 1 s y ) 2 2 H y y 2 } ,
P y x H x = ɛ x ɛ z s x ( s x H x y + 1 s x 2 H x x y ) + μ x μ z s x ( s x H x y + 1 s x 2 H x y x ) ,
A ¯ ¯ UPML = [ D ¯ ¯ x 00 UPML + D ¯ ¯ y 00 UPML + k 0 2 ɛ y μ x I ¯ ¯ D ¯ ¯ x 01 UPML + D ¯ ¯ y 01 UPML D ¯ ¯ x 10 UPML + D ¯ ¯ y 10 UPML D ¯ ¯ x 11 UPML + D ¯ ¯ y 11 UPML + k 0 2 ɛ x μ y I ¯ ¯ ] 2 k × 2 k ,
D = x 00 UPML = μ x μ z ( S = x S = x + S = x S = x 1 + 3 S = x S = x 1 D = x 00 + S = x 1 S = x 1 D = x 00 2 ) , D = y 00 UPML = ɛ y ɛ z S = y 1 ( S = y D = y 00 + S = y 1 D = y 00 2 ) , D = x 01 UPML = ɛ y ɛ z S = y 1 ( S = y D = x 00 + S = y 1 D = y 00 D = x 00 ) , D = y 01 UPML = μ y μ z S = y 1 ( S = y D = x 00 + S = y 1 D = x 00 D = y 00 ) , D = x 10 UPML = ɛ x ɛ z S = x 1 ( S = x D = y 00 + S = x 1 D = x 00 D = y 00 ) , D = y 10 UPML = μ x μ z S = x 1 ( S = x D = y 00 + S = x 1 D = y 00 D = x 00 ) , D = x 11 UPML = ɛ x ɛ z S = x 1 ( S = x D = x 00 + S = x 1 D = x 00 2 ) , D = y 11 UPML = μ x μ z ( S = x S = x + S = x S = x 1 + 3 S = x S = x 1 D = x 00 + S = x 1 S = x 1 D = x 00 2 ) ,
S = x = ( 2 s x 11 1 x 2 0 0 0 0 , 0 0 2 s x k k 1 x 2 ) k × k S = y = ( 2 s y 11 1 y 2 0 0 0 0 , 0 0 2 s y k k 1 y 2 ) k × k
[ H ¯ x a H ¯ y a ] = [ n y n x μ x a s y a s x a n x μ y a s x a s y a n y ] 1 [ n y n x μ x b s y b s x b n x μ y b s x b s y b n y ] [ H ¯ x b H ¯ y b ] ,
n x μ x a s y a ( H ¯ x a s x a ) x + n y H ¯ x a y = n x [ μ y a s x a ( H ¯ y a s y a ) y + ( s x a s y a μ z a s x b s y b μ z b ) ( H ¯ x b x + H ¯ y b y ) ] + n y [ H ¯ y a x ( s x a s y a ɛ z a s x b s y b ɛ z b ) ( H ¯ y b x H ¯ x b y ) ] ,
n x H ¯ y a x + n y μ y a s x a ( H ¯ y a s y a ) y = n x [ μ x a s y a ( H ¯ x a s x a ) x + ( s x a s y a μ z a s x b s y b μ z b ) ( H ¯ x b x + H ¯ y b y ) ] + n x [ H ¯ x a y + ( s x a s y a ɛ z a s x b s y b ɛ z b ) ( H ¯ y b x H ¯ x b y ) ] .

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