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 Optical Society of America

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  1. G. R. Hadley, “High-accuracy finite-difference equations for dielectric waveguide analysis I: Uniform regions and dielectric interfaces,” J. Lightwave Technol. 20, 1210–1218 (2002).
    [CrossRef]
  2. G. R. Hadley, “High-accuracy finite-difference equations for dielectric waveguide analysis II: Dielectric corners,” J. Lightwave Technol. 20, 1219–1231 (2002).
    [CrossRef]
  3. N. Thomas, P. Sewell, and T. M. Benson, “A new full-vectorial higher order finite-difference scheme for the modal analysis of rectangular dielectric waveguides,” J. Lightwave Technol. 25, 2563–2570 (2002).
    [CrossRef]
  4. 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]
  5. M. Koshiba, and Y. Tsuji, “Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightwave Technol. 18, 737–743 (2000).
    [CrossRef]
  6. 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.
  7. 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]
  8. B. Yang, D. Gottlieb, and J. S. Hesthaven, “Spectral simulations of electromagnetic wave scattering,” J. Comput. Phys. 134, 216–230 (1997).
    [CrossRef]
  9. B. Yang, and J. S. Hesthaven, “A pseudospectral method for time-domain computation of electromagnetic scattering by bodies of revolution,” IEEE Trans. Antenn. Propag. 47, 132–141 (1999).
    [CrossRef]
  10. 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]
  11. G. Zhao, and Q. H. Liu, “The 3-D multidomain pseudospectral time-domain algorithm for inhomogeneous conductive media,” IEEE Trans. Antenn. Propag. 52, 742–749 (2004).
    [CrossRef]
  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. Express 17, 14211–14228 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-14211.
    [CrossRef] [PubMed]
  14. Q. H. Liu, “A pseudospectral frequency-domain (PSFD) method for computational electromagnetics,” IEEE Antennas Wirel. Propag. Lett. 1, 131–134 (2002).
    [CrossRef]
  15. P. J. Chiang, C. P. Yu, and H. C. Chang, “Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75, 026703 (2007).
    [CrossRef]
  16. 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]
  17. 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]
  18. 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]
  19. 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]
  20. P. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003).
    [CrossRef] [PubMed]
  21. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]
  22. 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. Antenn. Propag. 43, 1460–1463 (1995).
    [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.
  24. 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. Express 12, 6165–6177 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-25-6165.
    [CrossRef] [PubMed]
  25. Y. Tsuji, and M. Koshiba, “Guided-mode and leaky-mode analysis by imaginary distance beam propagation method based on finite element scheme,” J. Lightwave Technol. 18, 618–623 (2000).
    [CrossRef]
  26. 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]
  27. W. C. Chew, and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994).
    [CrossRef]
  28. K. Saitoh, and M. Koshiba, “Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides,” J. Lightwave Technol. 19, 405–413 (2001).
    [CrossRef]

2010

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]

2009

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. Express 17, 14211–14228 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-14211.
[CrossRef] [PubMed]

2008

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]

2007

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

2005

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

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

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. Express 12, 6165–6177 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-25-6165.
[CrossRef] [PubMed]

2003

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

2002

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

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]

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]

G. R. Hadley, “High-accuracy finite-difference equations for dielectric waveguide analysis I: Uniform regions and dielectric interfaces,” J. Lightwave Technol. 20, 1210–1218 (2002).
[CrossRef]

G. R. Hadley, “High-accuracy finite-difference equations for dielectric waveguide analysis II: Dielectric corners,” J. Lightwave Technol. 20, 1219–1231 (2002).
[CrossRef]

N. Thomas, P. Sewell, and T. M. Benson, “A new full-vectorial higher order finite-difference scheme for the modal analysis of rectangular dielectric waveguides,” J. Lightwave Technol. 25, 2563–2570 (2002).
[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]

2001

K. Saitoh, and M. Koshiba, “Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides,” J. Lightwave Technol. 19, 405–413 (2001).
[CrossRef]

2000

Y. Tsuji, and M. Koshiba, “Guided-mode and leaky-mode analysis by imaginary distance beam propagation method based on finite element scheme,” J. Lightwave Technol. 18, 618–623 (2000).
[CrossRef]

M. Koshiba, and Y. Tsuji, “Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightwave Technol. 18, 737–743 (2000).
[CrossRef]

1999

B. Yang, and J. S. Hesthaven, “A pseudospectral method for time-domain computation of electromagnetic scattering by bodies of revolution,” IEEE Trans. Antenn. Propag. 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

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

1995

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. Antenn. Propag. 43, 1460–1463 (1995).
[CrossRef]

1994

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

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

1973

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.

N. Thomas, P. Sewell, and T. M. Benson, “A new full-vectorial higher order finite-difference scheme for the modal analysis of rectangular dielectric waveguides,” J. Lightwave Technol. 25, 2563–2570 (2002).
[CrossRef]

Berenger, J. P.

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

Botten, L. C.

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]

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]

Chang, H. C.

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. Express 17, 14211–14228 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-14211.
[CrossRef] [PubMed]

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 Stat. Nonlin. Soft Matter Phys. 75, 026703 (2007).
[CrossRef]

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. Express 12, 6165–6177 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-25-6165.
[CrossRef] [PubMed]

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]

Chew, W. C.

W. C. Chew, and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microw. 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 Stat. Nonlin. Soft Matter Phys. 75, 026703 (2007).
[CrossRef]

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.

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]

de Sterke, C. M.

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]

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]

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.

G. R. Hadley, “High-accuracy finite-difference equations for dielectric waveguide analysis II: Dielectric corners,” J. Lightwave Technol. 20, 1219–1231 (2002).
[CrossRef]

G. R. Hadley, “High-accuracy finite-difference equations for dielectric waveguide analysis I: Uniform regions and dielectric interfaces,” J. Lightwave Technol. 20, 1210–1218 (2002).
[CrossRef]

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. Antenn. Propag. 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.

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. Express 17, 14211–14228 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-14211.
[CrossRef] [PubMed]

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. Antenn. Propag. 43, 1460–1463 (1995).
[CrossRef]

Koshiba, M.

K. Saitoh, and M. Koshiba, “Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides,” J. Lightwave Technol. 19, 405–413 (2001).
[CrossRef]

M. Koshiba, and Y. Tsuji, “Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightwave Technol. 18, 737–743 (2000).
[CrossRef]

Y. Tsuji, and M. Koshiba, “Guided-mode and leaky-mode analysis by imaginary distance beam propagation method based on finite element scheme,” J. Lightwave Technol. 18, 618–623 (2000).
[CrossRef]

Kuhlmey, B. T.

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]

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]

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. Antenn. Propag. 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. Antenn. Propag. 43, 1460–1463 (1995).
[CrossRef]

Lin, B. Y.

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. Express 17, 14211–14228 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-14211.
[CrossRef] [PubMed]

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. Antenn. Propag. 52, 742–749 (2004).
[CrossRef]

Q. H. Liu, “A pseudospectral frequency-domain (PSFD) method for computational electromagnetics,” IEEE Antennas Wirel. Propag. 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.

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]

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]

Mcphedran, R. C.

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]

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]

Renversez, G.

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]

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]

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. Antenn. Propag. 43, 1460–1463 (1995).
[CrossRef]

Saitoh, K.

K. Saitoh, and M. Koshiba, “Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides,” J. Lightwave Technol. 19, 405–413 (2001).
[CrossRef]

Sewell, P.

N. Thomas, P. Sewell, and T. M. Benson, “A new full-vectorial higher order finite-difference scheme for the modal analysis of rectangular dielectric waveguides,” J. Lightwave Technol. 25, 2563–2570 (2002).
[CrossRef]

Teng, C. H.

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. Express 17, 14211–14228 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-14211.
[CrossRef] [PubMed]

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]

Thomas, N.

N. Thomas, P. Sewell, and T. M. Benson, “A new full-vectorial higher order finite-difference scheme for the modal analysis of rectangular dielectric waveguides,” J. Lightwave Technol. 25, 2563–2570 (2002).
[CrossRef]

Tsuji, Y.

M. Koshiba, and Y. Tsuji, “Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightwave Technol. 18, 737–743 (2000).
[CrossRef]

Y. Tsuji, and M. Koshiba, “Guided-mode and leaky-mode analysis by imaginary distance beam propagation method based on finite element scheme,” J. Lightwave Technol. 18, 618–623 (2000).
[CrossRef]

Wang, J. K.

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. Express 17, 14211–14228 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-14211.
[CrossRef] [PubMed]

Wang, Y. L.

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. Express 17, 14211–14228 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-14211.
[CrossRef] [PubMed]

Weedon, W. H.

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

White, T. P.

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]

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]

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]

Yang, B.

B. Yang, and J. S. Hesthaven, “A pseudospectral method for time-domain computation of electromagnetic scattering by bodies of revolution,” IEEE Trans. Antenn. Propag. 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]

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 Stat. Nonlin. Soft Matter Phys. 75, 026703 (2007).
[CrossRef]

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. Express 12, 6165–6177 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-25-6165.
[CrossRef] [PubMed]

Zhao, G.

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

IEEE Antennas Wirel. Propag. Lett.

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

IEEE J. Quantum Electron.

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.

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.

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. Antenn. Propag.

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. Antenn. Propag. 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. Antenn. Propag. 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. Antenn. Propag. 47, 132–141 (1999).
[CrossRef]

J. Comput. Phys.

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

J. Lightwave Technol.

Y. Tsuji, and M. Koshiba, “Guided-mode and leaky-mode analysis by imaginary distance beam propagation method based on finite element scheme,” J. Lightwave Technol. 18, 618–623 (2000).
[CrossRef]

K. Saitoh, and M. Koshiba, “Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides,” J. Lightwave Technol. 19, 405–413 (2001).
[CrossRef]

G. R. Hadley, “High-accuracy finite-difference equations for dielectric waveguide analysis I: Uniform regions and dielectric interfaces,” J. Lightwave Technol. 20, 1210–1218 (2002).
[CrossRef]

G. R. Hadley, “High-accuracy finite-difference equations for dielectric waveguide analysis II: Dielectric corners,” J. Lightwave Technol. 20, 1219–1231 (2002).
[CrossRef]

N. Thomas, P. Sewell, and T. M. Benson, “A new full-vectorial higher order finite-difference scheme for the modal analysis of rectangular dielectric waveguides,” J. Lightwave Technol. 25, 2563–2570 (2002).
[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]

M. Koshiba, and Y. Tsuji, “Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightwave Technol. 18, 737–743 (2000).
[CrossRef]

J. Opt. Soc. Am. B

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]

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]

J. Sci. Comput.

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]

Microw. Opt. Technol. Lett.

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

Numer. Math.

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]

Opt. Express

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. Express 12, 6165–6177 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-25-6165.
[CrossRef] [PubMed]

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. Express 17, 14211–14228 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-14211.
[CrossRef] [PubMed]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

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

Science

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

Other

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.

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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)

Tables Icon

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

Tables Icon

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.

Tables Icon

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.

Tables Icon

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.

Tables Icon

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.

Tables Icon

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)

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

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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