Abstract

Although perfectly matched layers (PMLs) have been widely used to truncate numerical simulations of electromagnetism and other wave equations, we point out important cases in which a PML fails to be reflectionless even in the limit of infinite resolution. In particular, the underlying coordinate-stretching idea behind PML breaks down in photonic crystals and in other structures where the material is not an analytic function in the direction perpendicular to the boundary, leading to substantial reflections. The alternative is an adiabatic absorber, in which reflections are made negligible by gradually increasing the material absorption at the boundaries, similar to a common strategy to combat discretization reflections in PMLs. We demonstrate the fundamental connection between such reflections and the smoothness of the absorption profile via coupled-mode theory, and show how to obtain higher-order and even exponential vanishing of the reflection with absorber thickness (although further work remains in optimizing the constant factor).

© 2008 Optical Society of America

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    [CrossRef]
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  4. M. Koshiba, Y. Tsuji, and S. Sasaki, "High-performance absorbing boundary conditions for photonic crystal waveguide simulations," IEEE Microwave Wirel. Compon. Lett. 11, 152-154 (2001).
    [CrossRef]
  5. Y. Tsuji and M. Koshiba, "Finite element method using port truncation by perfectly matched layer boundary conditions for optical waveguide discontinuity problems," J. Lightwave Technol. 20, 463-468 (2002).
    [CrossRef]
  6. E. P. Kosmidou, T. I. Kosmani, and T. D. Tsiboukis, "A comparative FDTD study of various PML configurations for the termination of nonlinear photonic bandgap waveguide structures," IEEE Trans. Magn. 39, 1191-1194 (2003).
    [CrossRef]
  7. A. R. Weily, L. Horvath, K. P. Esselle, and B. C. Sanders, "Performance of PML absorbing boundary conditions in 3d photonic crystal waveguides," Microwave Opt. Technol. Lett. 40, 1-3 (2004).
    [CrossRef]
  8. N. Kono and M. Koshiba, "General finite-element modeling of 2-D magnetophotonic crystal waveguides," IEEE Photon. Tech. Lett. 17, 1432-1434 (2005).
    [CrossRef]
  9. W. C. Chew and J. M. Jin, "Perfectly matched layers in the discretized space: An analysis and optimization," Electromagnetics 16, 325-340 (1996).
    [CrossRef]
  10. S. G. Johnson, P. Bienstman, M. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. Joannopoulos, "Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals," Phys. Rev. E 66, 066608 (2002).
  11. E. A. Marengo, C. M. Rappaport, and E. L. Miller, "Optimum PML ABC conductivity profile in FDFD," IEEE Trans. Magn. 35, 1506-1509 (1999).
    [CrossRef]
  12. J. S. Juntunen, N. V. Kantartzis, and T. D. Tsiboukis, "Zero reflection coefficient in discretized PML," IEEE Microwave Wirel. Compon. Lett. 11, 155-157 (2001).
    [CrossRef]
  13. Y. S. Rickard and N. K. Nikolova, "Enhancing the PML absorbing boundary conditions for the wave equation," IEEE Trans. Antennas Propag. 53, 1242-1246 (2005).
    [CrossRef]
  14. Z. Chen and X. Liu, "An adaptive perfectly matched layer technique for time-harmonic scattering problems," SIAM J. Num. Anal. 43, 645-671 (2005).
    [CrossRef]
  15. Z. 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 Propag. 43, 1460-1463 (1995).
    [CrossRef]
  16. 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]
  17. C. M. Rappaport, "Perfectly matched absorbing boundary conditions based on anisotropic lossy mapping of space," IEEE Microwave and Guided Wave Lett. 5, 90-92 (1995).
    [CrossRef]
  18. F. L. Teixeira and W. C. Chew, "General close-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media," IEEE Microwave and Guided Wave Lett. 8, 223-225 (1998).
    [CrossRef]
  19. F. Collino and P. B. Monk, "Optimizing the perfectly matched layer," Comput. Methods Appl. Mech. Engrg. 164, 157-171 (1998).
    [CrossRef]
  20. A. J. Ward and J. B. Pendry, "Refraction and geometry in Maxwell???s equations," J. Mod. Opt. 43, 773-793 (1996).
    [CrossRef]
  21. W. Huang, C. Xu,W. Lui, and K. Yokoyama, "The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations," IEEE Photon. Tech. Lett. 8, 652-654 (1996).
    [CrossRef]
  22. D. Pissoort and F. Olyslager, "Termination of periodic waveguides by PMLs in time-harmonic integral equation-like techniques," IEEE Antennas and Wireless Propagation Lett. 2, 281-284 (2003).
    [CrossRef]
  23. A. Mekis, S. Fan, and J. D. Joannopoulos, "Absorbing boundary conditions for FDTD simulations of photonic crystal waveguides," IEEE Microwave and Guided Wave Lett. 9, 502-504 (1999).
    [CrossRef]
  24. E. Moreno, D. Erni, and C. Hafner, "Modeling of discontinuities in photonic crystal waveguides with the multiple multipole method," Phys. Rev. E 66, 036618 (2002).
  25. Z.-Y. Li and K.-M. Ho, "Light propagation in semi-infinite photonic crystals and related waveguide structures," Phys. Rev. B 68 (2003).
  26. D. Pissoort, B. Denecker, P. Bienstman, F. Olyslager, and D. De Zutter, "Comparative study of three methods for the simulation of two-dimensional photonic crystals," J. Opt. Soc. Am. A 21, 2186-2195 (2004).
    [CrossRef]
  27. M. Lassas and E. Somersalo, "On the existence and convergence of the solution of PML equations," Computing 60, 229-241 (1998).
    [CrossRef]
  28. L. Zschiedrich, R. Klose, A. Sch¨adle, and F. Schmidt, "A new finite element realization of the perfectly matched layer method for helmholtz scattering problems on polygonal domains in two dimensions," J. Comput. Appl. Math. 188, 12-32 (2006).
    [CrossRef]
  29. J. Fang and Z. Wu, "Generalized perfectly matched layer for the absorption of propagating and evanescent waves in lossless and lossy media," IEEE Trans. Microwave Theory Tech. 44, 2216-2222 (1998).
    [CrossRef]
  30. X. T. Dong, X. S. Rao, Y. B. Gan, B. Guo, and W. Y. Yin, "Perfectly matched layer-absorbing boundary condition for left-handed materials," IEEE Microwave Wirel. Compon. Lett. 14, 301-303 (2004).
    [CrossRef]
  31. A. Dranov, J. Kellendonk, and R. Seller, "Discrete time adiabatic theorems for quantum mechanical systems," J. Math. Phys. 39, 1340-1349 (1998).
    [CrossRef]
  32. A. Christ and H. L. Hartnagel, "Three-dimensional finite-difference method for the analysis of microwave-device embedding," IEEE Trans. Microwave Theory Tech. 35, 688-696 (1987).
    [CrossRef]
  33. M. Povinelli, S. Johnson, and J. Joannopoulos, "Slow-light, band-edge waveguides for tunable time delays," Opt. Express 13, 7145-7159 (2005).
    [CrossRef] [PubMed]
  34. D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd Edition (Academic Press, San Diego, 1991).
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    [CrossRef]
  36. J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd Edition (Springer, 1989).
  37. D. Elliott, "The evaluation and estimation of the coefficients in the Chebyshev series expansion of a function," Mathematics of Computation 18, 274-284 (1964).
    [CrossRef]
  38. J. P. Boyd, "The optimization of convergence for Chebyshev polynomial methods in an unbounded domain," J. Comput. Phys. 45, 43-79 (1982).
    [CrossRef]
  39. H. Cheng, Advanced Analytic Methods in Applied Mathematics, Science and Engineering (Luban Press, Boston, 2006).

2006

L. Zschiedrich, R. Klose, A. Sch¨adle, and F. Schmidt, "A new finite element realization of the perfectly matched layer method for helmholtz scattering problems on polygonal domains in two dimensions," J. Comput. Appl. Math. 188, 12-32 (2006).
[CrossRef]

2005

M. Povinelli, S. Johnson, and J. Joannopoulos, "Slow-light, band-edge waveguides for tunable time delays," Opt. Express 13, 7145-7159 (2005).
[CrossRef] [PubMed]

N. Kono and M. Koshiba, "General finite-element modeling of 2-D magnetophotonic crystal waveguides," IEEE Photon. Tech. Lett. 17, 1432-1434 (2005).
[CrossRef]

Y. S. Rickard and N. K. Nikolova, "Enhancing the PML absorbing boundary conditions for the wave equation," IEEE Trans. Antennas Propag. 53, 1242-1246 (2005).
[CrossRef]

Z. Chen and X. Liu, "An adaptive perfectly matched layer technique for time-harmonic scattering problems," SIAM J. Num. Anal. 43, 645-671 (2005).
[CrossRef]

2004

A. R. Weily, L. Horvath, K. P. Esselle, and B. C. Sanders, "Performance of PML absorbing boundary conditions in 3d photonic crystal waveguides," Microwave Opt. Technol. Lett. 40, 1-3 (2004).
[CrossRef]

D. Pissoort, B. Denecker, P. Bienstman, F. Olyslager, and D. De Zutter, "Comparative study of three methods for the simulation of two-dimensional photonic crystals," J. Opt. Soc. Am. A 21, 2186-2195 (2004).
[CrossRef]

X. T. Dong, X. S. Rao, Y. B. Gan, B. Guo, and W. Y. Yin, "Perfectly matched layer-absorbing boundary condition for left-handed materials," IEEE Microwave Wirel. Compon. Lett. 14, 301-303 (2004).
[CrossRef]

2003

D. Pissoort and F. Olyslager, "Termination of periodic waveguides by PMLs in time-harmonic integral equation-like techniques," IEEE Antennas and Wireless Propagation Lett. 2, 281-284 (2003).
[CrossRef]

Z.-Y. Li and K.-M. Ho, "Light propagation in semi-infinite photonic crystals and related waveguide structures," Phys. Rev. B 68 (2003).

E. P. Kosmidou, T. I. Kosmani, and T. D. Tsiboukis, "A comparative FDTD study of various PML configurations for the termination of nonlinear photonic bandgap waveguide structures," IEEE Trans. Magn. 39, 1191-1194 (2003).
[CrossRef]

2002

S. G. Johnson, P. Bienstman, M. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. Joannopoulos, "Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals," Phys. Rev. E 66, 066608 (2002).

Y. Tsuji and M. Koshiba, "Finite element method using port truncation by perfectly matched layer boundary conditions for optical waveguide discontinuity problems," J. Lightwave Technol. 20, 463-468 (2002).
[CrossRef]

E. Moreno, D. Erni, and C. Hafner, "Modeling of discontinuities in photonic crystal waveguides with the multiple multipole method," Phys. Rev. E 66, 036618 (2002).

2001

M. Koshiba, Y. Tsuji, and S. Sasaki, "High-performance absorbing boundary conditions for photonic crystal waveguide simulations," IEEE Microwave Wirel. Compon. Lett. 11, 152-154 (2001).
[CrossRef]

J. S. Juntunen, N. V. Kantartzis, and T. D. Tsiboukis, "Zero reflection coefficient in discretized PML," IEEE Microwave Wirel. Compon. Lett. 11, 155-157 (2001).
[CrossRef]

1999

E. A. Marengo, C. M. Rappaport, and E. L. Miller, "Optimum PML ABC conductivity profile in FDFD," IEEE Trans. Magn. 35, 1506-1509 (1999).
[CrossRef]

A. Mekis, S. Fan, and J. D. Joannopoulos, "Absorbing boundary conditions for FDTD simulations of photonic crystal waveguides," IEEE Microwave and Guided Wave Lett. 9, 502-504 (1999).
[CrossRef]

1998

M. Lassas and E. Somersalo, "On the existence and convergence of the solution of PML equations," Computing 60, 229-241 (1998).
[CrossRef]

A. Dranov, J. Kellendonk, and R. Seller, "Discrete time adiabatic theorems for quantum mechanical systems," J. Math. Phys. 39, 1340-1349 (1998).
[CrossRef]

J. Fang and Z. Wu, "Generalized perfectly matched layer for the absorption of propagating and evanescent waves in lossless and lossy media," IEEE Trans. Microwave Theory Tech. 44, 2216-2222 (1998).
[CrossRef]

F. L. Teixeira and W. C. Chew, "General close-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media," IEEE Microwave and Guided Wave Lett. 8, 223-225 (1998).
[CrossRef]

F. Collino and P. B. Monk, "Optimizing the perfectly matched layer," Comput. Methods Appl. Mech. Engrg. 164, 157-171 (1998).
[CrossRef]

1996

A. J. Ward and J. B. Pendry, "Refraction and geometry in Maxwell???s equations," J. Mod. Opt. 43, 773-793 (1996).
[CrossRef]

W. Huang, C. Xu,W. Lui, and K. Yokoyama, "The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations," IEEE Photon. Tech. Lett. 8, 652-654 (1996).
[CrossRef]

W. C. Chew and J. M. Jin, "Perfectly matched layers in the discretized space: An analysis and optimization," Electromagnetics 16, 325-340 (1996).
[CrossRef]

1995

Z. 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 Propag. 43, 1460-1463 (1995).
[CrossRef]

C. M. Rappaport, "Perfectly matched absorbing boundary conditions based on anisotropic lossy mapping of space," IEEE Microwave and Guided Wave Lett. 5, 90-92 (1995).
[CrossRef]

1994

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. Comput. Phys. 114, 185-200 (1994).
[CrossRef]

1987

A. Christ and H. L. Hartnagel, "Three-dimensional finite-difference method for the analysis of microwave-device embedding," IEEE Trans. Microwave Theory Tech. 35, 688-696 (1987).
[CrossRef]

1982

J. P. Boyd, "The optimization of convergence for Chebyshev polynomial methods in an unbounded domain," J. Comput. Phys. 45, 43-79 (1982).
[CrossRef]

1973

K. O. Mead and L. M. Delves, "On the convergence rate of generalized fourier expansions," IMA J. Appl. Math. 12, 247-259 (1973).
[CrossRef]

1964

D. Elliott, "The evaluation and estimation of the coefficients in the Chebyshev series expansion of a function," Mathematics of Computation 18, 274-284 (1964).
[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]

Bienstman, P.

D. Pissoort, B. Denecker, P. Bienstman, F. Olyslager, and D. De Zutter, "Comparative study of three methods for the simulation of two-dimensional photonic crystals," J. Opt. Soc. Am. A 21, 2186-2195 (2004).
[CrossRef]

S. G. Johnson, P. Bienstman, M. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. Joannopoulos, "Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals," Phys. Rev. E 66, 066608 (2002).

Boyd, J. P.

J. P. Boyd, "The optimization of convergence for Chebyshev polynomial methods in an unbounded domain," J. Comput. Phys. 45, 43-79 (1982).
[CrossRef]

Chen, Z.

Z. Chen and X. Liu, "An adaptive perfectly matched layer technique for time-harmonic scattering problems," SIAM J. Num. Anal. 43, 645-671 (2005).
[CrossRef]

Chew, W. C.

F. L. Teixeira and W. C. Chew, "General close-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media," IEEE Microwave and Guided Wave Lett. 8, 223-225 (1998).
[CrossRef]

W. C. Chew and J. M. Jin, "Perfectly matched layers in the discretized space: An analysis and optimization," Electromagnetics 16, 325-340 (1996).
[CrossRef]

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]

Christ, A.

A. Christ and H. L. Hartnagel, "Three-dimensional finite-difference method for the analysis of microwave-device embedding," IEEE Trans. Microwave Theory Tech. 35, 688-696 (1987).
[CrossRef]

Collino, F.

F. Collino and P. B. Monk, "Optimizing the perfectly matched layer," Comput. Methods Appl. Mech. Engrg. 164, 157-171 (1998).
[CrossRef]

De Zutter, D.

Delves, L. M.

K. O. Mead and L. M. Delves, "On the convergence rate of generalized fourier expansions," IMA J. Appl. Math. 12, 247-259 (1973).
[CrossRef]

Denecker, B.

Dong, X. T.

X. T. Dong, X. S. Rao, Y. B. Gan, B. Guo, and W. Y. Yin, "Perfectly matched layer-absorbing boundary condition for left-handed materials," IEEE Microwave Wirel. Compon. Lett. 14, 301-303 (2004).
[CrossRef]

Dranov, A.

A. Dranov, J. Kellendonk, and R. Seller, "Discrete time adiabatic theorems for quantum mechanical systems," J. Math. Phys. 39, 1340-1349 (1998).
[CrossRef]

Elliott, D.

D. Elliott, "The evaluation and estimation of the coefficients in the Chebyshev series expansion of a function," Mathematics of Computation 18, 274-284 (1964).
[CrossRef]

Erni, D.

E. Moreno, D. Erni, and C. Hafner, "Modeling of discontinuities in photonic crystal waveguides with the multiple multipole method," Phys. Rev. E 66, 036618 (2002).

Esselle, K. P.

A. R. Weily, L. Horvath, K. P. Esselle, and B. C. Sanders, "Performance of PML absorbing boundary conditions in 3d photonic crystal waveguides," Microwave Opt. Technol. Lett. 40, 1-3 (2004).
[CrossRef]

Fan, S.

A. Mekis, S. Fan, and J. D. Joannopoulos, "Absorbing boundary conditions for FDTD simulations of photonic crystal waveguides," IEEE Microwave and Guided Wave Lett. 9, 502-504 (1999).
[CrossRef]

Fang, J.

J. Fang and Z. Wu, "Generalized perfectly matched layer for the absorption of propagating and evanescent waves in lossless and lossy media," IEEE Trans. Microwave Theory Tech. 44, 2216-2222 (1998).
[CrossRef]

Gan, Y. B.

X. T. Dong, X. S. Rao, Y. B. Gan, B. Guo, and W. Y. Yin, "Perfectly matched layer-absorbing boundary condition for left-handed materials," IEEE Microwave Wirel. Compon. Lett. 14, 301-303 (2004).
[CrossRef]

Guo, B.

X. T. Dong, X. S. Rao, Y. B. Gan, B. Guo, and W. Y. Yin, "Perfectly matched layer-absorbing boundary condition for left-handed materials," IEEE Microwave Wirel. Compon. Lett. 14, 301-303 (2004).
[CrossRef]

Hafner, C.

E. Moreno, D. Erni, and C. Hafner, "Modeling of discontinuities in photonic crystal waveguides with the multiple multipole method," Phys. Rev. E 66, 036618 (2002).

Hartnagel, H. L.

A. Christ and H. L. Hartnagel, "Three-dimensional finite-difference method for the analysis of microwave-device embedding," IEEE Trans. Microwave Theory Tech. 35, 688-696 (1987).
[CrossRef]

Ho, K.-M.

Z.-Y. Li and K.-M. Ho, "Light propagation in semi-infinite photonic crystals and related waveguide structures," Phys. Rev. B 68 (2003).

Horvath, L.

A. R. Weily, L. Horvath, K. P. Esselle, and B. C. Sanders, "Performance of PML absorbing boundary conditions in 3d photonic crystal waveguides," Microwave Opt. Technol. Lett. 40, 1-3 (2004).
[CrossRef]

Huang, W.

W. Huang, C. Xu,W. Lui, and K. Yokoyama, "The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations," IEEE Photon. Tech. Lett. 8, 652-654 (1996).
[CrossRef]

Ibanescu, M.

S. G. Johnson, P. Bienstman, M. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. Joannopoulos, "Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals," Phys. Rev. E 66, 066608 (2002).

Jin, J. M.

W. C. Chew and J. M. Jin, "Perfectly matched layers in the discretized space: An analysis and optimization," Electromagnetics 16, 325-340 (1996).
[CrossRef]

Joannopoulos, J.

M. Povinelli, S. Johnson, and J. Joannopoulos, "Slow-light, band-edge waveguides for tunable time delays," Opt. Express 13, 7145-7159 (2005).
[CrossRef] [PubMed]

S. G. Johnson, P. Bienstman, M. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. Joannopoulos, "Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals," Phys. Rev. E 66, 066608 (2002).

Joannopoulos, J. D.

A. Mekis, S. Fan, and J. D. Joannopoulos, "Absorbing boundary conditions for FDTD simulations of photonic crystal waveguides," IEEE Microwave and Guided Wave Lett. 9, 502-504 (1999).
[CrossRef]

Johnson, S.

Johnson, S. G.

S. G. Johnson, P. Bienstman, M. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. Joannopoulos, "Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals," Phys. Rev. E 66, 066608 (2002).

Juntunen, J. S.

J. S. Juntunen, N. V. Kantartzis, and T. D. Tsiboukis, "Zero reflection coefficient in discretized PML," IEEE Microwave Wirel. Compon. Lett. 11, 155-157 (2001).
[CrossRef]

Kantartzis, N. V.

J. S. Juntunen, N. V. Kantartzis, and T. D. Tsiboukis, "Zero reflection coefficient in discretized PML," IEEE Microwave Wirel. Compon. Lett. 11, 155-157 (2001).
[CrossRef]

Kellendonk, J.

A. Dranov, J. Kellendonk, and R. Seller, "Discrete time adiabatic theorems for quantum mechanical systems," J. Math. Phys. 39, 1340-1349 (1998).
[CrossRef]

Kingsland, D. M.

Z. 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 Propag. 43, 1460-1463 (1995).
[CrossRef]

Klose, R.

L. Zschiedrich, R. Klose, A. Sch¨adle, and F. Schmidt, "A new finite element realization of the perfectly matched layer method for helmholtz scattering problems on polygonal domains in two dimensions," J. Comput. Appl. Math. 188, 12-32 (2006).
[CrossRef]

Kono, N.

N. Kono and M. Koshiba, "General finite-element modeling of 2-D magnetophotonic crystal waveguides," IEEE Photon. Tech. Lett. 17, 1432-1434 (2005).
[CrossRef]

Koshiba, M.

N. Kono and M. Koshiba, "General finite-element modeling of 2-D magnetophotonic crystal waveguides," IEEE Photon. Tech. Lett. 17, 1432-1434 (2005).
[CrossRef]

Y. Tsuji and M. Koshiba, "Finite element method using port truncation by perfectly matched layer boundary conditions for optical waveguide discontinuity problems," J. Lightwave Technol. 20, 463-468 (2002).
[CrossRef]

M. Koshiba, Y. Tsuji, and S. Sasaki, "High-performance absorbing boundary conditions for photonic crystal waveguide simulations," IEEE Microwave Wirel. Compon. Lett. 11, 152-154 (2001).
[CrossRef]

Kosmani, T. I.

E. P. Kosmidou, T. I. Kosmani, and T. D. Tsiboukis, "A comparative FDTD study of various PML configurations for the termination of nonlinear photonic bandgap waveguide structures," IEEE Trans. Magn. 39, 1191-1194 (2003).
[CrossRef]

Kosmidou, E. P.

E. P. Kosmidou, T. I. Kosmani, and T. D. Tsiboukis, "A comparative FDTD study of various PML configurations for the termination of nonlinear photonic bandgap waveguide structures," IEEE Trans. Magn. 39, 1191-1194 (2003).
[CrossRef]

Lassas, M.

M. Lassas and E. Somersalo, "On the existence and convergence of the solution of PML equations," Computing 60, 229-241 (1998).
[CrossRef]

Lee, J. F.

Z. 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 Propag. 43, 1460-1463 (1995).
[CrossRef]

Lee, R.

Z. 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 Propag. 43, 1460-1463 (1995).
[CrossRef]

Li, Z.-Y.

Z.-Y. Li and K.-M. Ho, "Light propagation in semi-infinite photonic crystals and related waveguide structures," Phys. Rev. B 68 (2003).

Lidorikis, E.

S. G. Johnson, P. Bienstman, M. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. Joannopoulos, "Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals," Phys. Rev. E 66, 066608 (2002).

Liu, X.

Z. Chen and X. Liu, "An adaptive perfectly matched layer technique for time-harmonic scattering problems," SIAM J. Num. Anal. 43, 645-671 (2005).
[CrossRef]

Lui, W.

W. Huang, C. Xu,W. Lui, and K. Yokoyama, "The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations," IEEE Photon. Tech. Lett. 8, 652-654 (1996).
[CrossRef]

Marengo, E. A.

E. A. Marengo, C. M. Rappaport, and E. L. Miller, "Optimum PML ABC conductivity profile in FDFD," IEEE Trans. Magn. 35, 1506-1509 (1999).
[CrossRef]

Mead, K. O.

K. O. Mead and L. M. Delves, "On the convergence rate of generalized fourier expansions," IMA J. Appl. Math. 12, 247-259 (1973).
[CrossRef]

Mekis, A.

A. Mekis, S. Fan, and J. D. Joannopoulos, "Absorbing boundary conditions for FDTD simulations of photonic crystal waveguides," IEEE Microwave and Guided Wave Lett. 9, 502-504 (1999).
[CrossRef]

Miller, E. L.

E. A. Marengo, C. M. Rappaport, and E. L. Miller, "Optimum PML ABC conductivity profile in FDFD," IEEE Trans. Magn. 35, 1506-1509 (1999).
[CrossRef]

Monk, P. B.

F. Collino and P. B. Monk, "Optimizing the perfectly matched layer," Comput. Methods Appl. Mech. Engrg. 164, 157-171 (1998).
[CrossRef]

Moreno, E.

E. Moreno, D. Erni, and C. Hafner, "Modeling of discontinuities in photonic crystal waveguides with the multiple multipole method," Phys. Rev. E 66, 036618 (2002).

Nikolova, N. K.

Y. S. Rickard and N. K. Nikolova, "Enhancing the PML absorbing boundary conditions for the wave equation," IEEE Trans. Antennas Propag. 53, 1242-1246 (2005).
[CrossRef]

Olyslager, F.

D. Pissoort, B. Denecker, P. Bienstman, F. Olyslager, and D. De Zutter, "Comparative study of three methods for the simulation of two-dimensional photonic crystals," J. Opt. Soc. Am. A 21, 2186-2195 (2004).
[CrossRef]

D. Pissoort and F. Olyslager, "Termination of periodic waveguides by PMLs in time-harmonic integral equation-like techniques," IEEE Antennas and Wireless Propagation Lett. 2, 281-284 (2003).
[CrossRef]

Pendry, J. B.

A. J. Ward and J. B. Pendry, "Refraction and geometry in Maxwell???s equations," J. Mod. Opt. 43, 773-793 (1996).
[CrossRef]

Pissoort, D.

D. Pissoort, B. Denecker, P. Bienstman, F. Olyslager, and D. De Zutter, "Comparative study of three methods for the simulation of two-dimensional photonic crystals," J. Opt. Soc. Am. A 21, 2186-2195 (2004).
[CrossRef]

D. Pissoort and F. Olyslager, "Termination of periodic waveguides by PMLs in time-harmonic integral equation-like techniques," IEEE Antennas and Wireless Propagation Lett. 2, 281-284 (2003).
[CrossRef]

Povinelli, M.

Rao, X. S.

X. T. Dong, X. S. Rao, Y. B. Gan, B. Guo, and W. Y. Yin, "Perfectly matched layer-absorbing boundary condition for left-handed materials," IEEE Microwave Wirel. Compon. Lett. 14, 301-303 (2004).
[CrossRef]

Rappaport, C. M.

E. A. Marengo, C. M. Rappaport, and E. L. Miller, "Optimum PML ABC conductivity profile in FDFD," IEEE Trans. Magn. 35, 1506-1509 (1999).
[CrossRef]

C. M. Rappaport, "Perfectly matched absorbing boundary conditions based on anisotropic lossy mapping of space," IEEE Microwave and Guided Wave Lett. 5, 90-92 (1995).
[CrossRef]

Rickard, Y. S.

Y. S. Rickard and N. K. Nikolova, "Enhancing the PML absorbing boundary conditions for the wave equation," IEEE Trans. Antennas Propag. 53, 1242-1246 (2005).
[CrossRef]

Sacks, Z.

Z. 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 Propag. 43, 1460-1463 (1995).
[CrossRef]

Sanders, B. C.

A. R. Weily, L. Horvath, K. P. Esselle, and B. C. Sanders, "Performance of PML absorbing boundary conditions in 3d photonic crystal waveguides," Microwave Opt. Technol. Lett. 40, 1-3 (2004).
[CrossRef]

Sasaki, S.

M. Koshiba, Y. Tsuji, and S. Sasaki, "High-performance absorbing boundary conditions for photonic crystal waveguide simulations," IEEE Microwave Wirel. Compon. Lett. 11, 152-154 (2001).
[CrossRef]

Sch¨adle, A.

L. Zschiedrich, R. Klose, A. Sch¨adle, and F. Schmidt, "A new finite element realization of the perfectly matched layer method for helmholtz scattering problems on polygonal domains in two dimensions," J. Comput. Appl. Math. 188, 12-32 (2006).
[CrossRef]

Schmidt, F.

L. Zschiedrich, R. Klose, A. Sch¨adle, and F. Schmidt, "A new finite element realization of the perfectly matched layer method for helmholtz scattering problems on polygonal domains in two dimensions," J. Comput. Appl. Math. 188, 12-32 (2006).
[CrossRef]

Seller, R.

A. Dranov, J. Kellendonk, and R. Seller, "Discrete time adiabatic theorems for quantum mechanical systems," J. Math. Phys. 39, 1340-1349 (1998).
[CrossRef]

Skorobogatiy, M.

S. G. Johnson, P. Bienstman, M. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. Joannopoulos, "Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals," Phys. Rev. E 66, 066608 (2002).

Somersalo, E.

M. Lassas and E. Somersalo, "On the existence and convergence of the solution of PML equations," Computing 60, 229-241 (1998).
[CrossRef]

Teixeira, F. L.

F. L. Teixeira and W. C. Chew, "General close-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media," IEEE Microwave and Guided Wave Lett. 8, 223-225 (1998).
[CrossRef]

Tsiboukis, T. D.

E. P. Kosmidou, T. I. Kosmani, and T. D. Tsiboukis, "A comparative FDTD study of various PML configurations for the termination of nonlinear photonic bandgap waveguide structures," IEEE Trans. Magn. 39, 1191-1194 (2003).
[CrossRef]

J. S. Juntunen, N. V. Kantartzis, and T. D. Tsiboukis, "Zero reflection coefficient in discretized PML," IEEE Microwave Wirel. Compon. Lett. 11, 155-157 (2001).
[CrossRef]

Tsuji, Y.

Y. Tsuji and M. Koshiba, "Finite element method using port truncation by perfectly matched layer boundary conditions for optical waveguide discontinuity problems," J. Lightwave Technol. 20, 463-468 (2002).
[CrossRef]

M. Koshiba, Y. Tsuji, and S. Sasaki, "High-performance absorbing boundary conditions for photonic crystal waveguide simulations," IEEE Microwave Wirel. Compon. Lett. 11, 152-154 (2001).
[CrossRef]

Ward, A. J.

A. J. Ward and J. B. Pendry, "Refraction and geometry in Maxwell???s equations," J. Mod. Opt. 43, 773-793 (1996).
[CrossRef]

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]

Weily, A. R.

A. R. Weily, L. Horvath, K. P. Esselle, and B. C. Sanders, "Performance of PML absorbing boundary conditions in 3d photonic crystal waveguides," Microwave Opt. Technol. Lett. 40, 1-3 (2004).
[CrossRef]

Wu, Z.

J. Fang and Z. Wu, "Generalized perfectly matched layer for the absorption of propagating and evanescent waves in lossless and lossy media," IEEE Trans. Microwave Theory Tech. 44, 2216-2222 (1998).
[CrossRef]

Xu, C.

W. Huang, C. Xu,W. Lui, and K. Yokoyama, "The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations," IEEE Photon. Tech. Lett. 8, 652-654 (1996).
[CrossRef]

Yin, W. Y.

X. T. Dong, X. S. Rao, Y. B. Gan, B. Guo, and W. Y. Yin, "Perfectly matched layer-absorbing boundary condition for left-handed materials," IEEE Microwave Wirel. Compon. Lett. 14, 301-303 (2004).
[CrossRef]

Yokoyama, K.

W. Huang, C. Xu,W. Lui, and K. Yokoyama, "The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations," IEEE Photon. Tech. Lett. 8, 652-654 (1996).
[CrossRef]

Zschiedrich, L.

L. Zschiedrich, R. Klose, A. Sch¨adle, and F. Schmidt, "A new finite element realization of the perfectly matched layer method for helmholtz scattering problems on polygonal domains in two dimensions," J. Comput. Appl. Math. 188, 12-32 (2006).
[CrossRef]

Comput. Methods Appl. Mech. Engrg.

F. Collino and P. B. Monk, "Optimizing the perfectly matched layer," Comput. Methods Appl. Mech. Engrg. 164, 157-171 (1998).
[CrossRef]

Computing

M. Lassas and E. Somersalo, "On the existence and convergence of the solution of PML equations," Computing 60, 229-241 (1998).
[CrossRef]

Electromagnetics

W. C. Chew and J. M. Jin, "Perfectly matched layers in the discretized space: An analysis and optimization," Electromagnetics 16, 325-340 (1996).
[CrossRef]

IEEE Antennas and Wireless Propagation Lett.

D. Pissoort and F. Olyslager, "Termination of periodic waveguides by PMLs in time-harmonic integral equation-like techniques," IEEE Antennas and Wireless Propagation Lett. 2, 281-284 (2003).
[CrossRef]

IEEE Microwave and Guided Wave Lett.

A. Mekis, S. Fan, and J. D. Joannopoulos, "Absorbing boundary conditions for FDTD simulations of photonic crystal waveguides," IEEE Microwave and Guided Wave Lett. 9, 502-504 (1999).
[CrossRef]

C. M. Rappaport, "Perfectly matched absorbing boundary conditions based on anisotropic lossy mapping of space," IEEE Microwave and Guided Wave Lett. 5, 90-92 (1995).
[CrossRef]

F. L. Teixeira and W. C. Chew, "General close-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media," IEEE Microwave and Guided Wave Lett. 8, 223-225 (1998).
[CrossRef]

IEEE Microwave Wirel. Compon. Lett.

J. S. Juntunen, N. V. Kantartzis, and T. D. Tsiboukis, "Zero reflection coefficient in discretized PML," IEEE Microwave Wirel. Compon. Lett. 11, 155-157 (2001).
[CrossRef]

M. Koshiba, Y. Tsuji, and S. Sasaki, "High-performance absorbing boundary conditions for photonic crystal waveguide simulations," IEEE Microwave Wirel. Compon. Lett. 11, 152-154 (2001).
[CrossRef]

X. T. Dong, X. S. Rao, Y. B. Gan, B. Guo, and W. Y. Yin, "Perfectly matched layer-absorbing boundary condition for left-handed materials," IEEE Microwave Wirel. Compon. Lett. 14, 301-303 (2004).
[CrossRef]

IEEE Photon. Tech. Lett.

W. Huang, C. Xu,W. Lui, and K. Yokoyama, "The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations," IEEE Photon. Tech. Lett. 8, 652-654 (1996).
[CrossRef]

N. Kono and M. Koshiba, "General finite-element modeling of 2-D magnetophotonic crystal waveguides," IEEE Photon. Tech. Lett. 17, 1432-1434 (2005).
[CrossRef]

IEEE Trans. Antennas Propag.

Y. S. Rickard and N. K. Nikolova, "Enhancing the PML absorbing boundary conditions for the wave equation," IEEE Trans. Antennas Propag. 53, 1242-1246 (2005).
[CrossRef]

Z. 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 Propag. 43, 1460-1463 (1995).
[CrossRef]

IEEE Trans. Magn.

E. A. Marengo, C. M. Rappaport, and E. L. Miller, "Optimum PML ABC conductivity profile in FDFD," IEEE Trans. Magn. 35, 1506-1509 (1999).
[CrossRef]

E. P. Kosmidou, T. I. Kosmani, and T. D. Tsiboukis, "A comparative FDTD study of various PML configurations for the termination of nonlinear photonic bandgap waveguide structures," IEEE Trans. Magn. 39, 1191-1194 (2003).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

J. Fang and Z. Wu, "Generalized perfectly matched layer for the absorption of propagating and evanescent waves in lossless and lossy media," IEEE Trans. Microwave Theory Tech. 44, 2216-2222 (1998).
[CrossRef]

A. Christ and H. L. Hartnagel, "Three-dimensional finite-difference method for the analysis of microwave-device embedding," IEEE Trans. Microwave Theory Tech. 35, 688-696 (1987).
[CrossRef]

IMA J. Appl. Math.

K. O. Mead and L. M. Delves, "On the convergence rate of generalized fourier expansions," IMA J. Appl. Math. 12, 247-259 (1973).
[CrossRef]

J. Comput. Appl. Math.

L. Zschiedrich, R. Klose, A. Sch¨adle, and F. Schmidt, "A new finite element realization of the perfectly matched layer method for helmholtz scattering problems on polygonal domains in two dimensions," J. Comput. Appl. Math. 188, 12-32 (2006).
[CrossRef]

J. Comput. Phys.

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

J. P. Boyd, "The optimization of convergence for Chebyshev polynomial methods in an unbounded domain," J. Comput. Phys. 45, 43-79 (1982).
[CrossRef]

J. Lightwave Technol.

J. Math. Phys.

A. Dranov, J. Kellendonk, and R. Seller, "Discrete time adiabatic theorems for quantum mechanical systems," J. Math. Phys. 39, 1340-1349 (1998).
[CrossRef]

J. Mod. Opt.

A. J. Ward and J. B. Pendry, "Refraction and geometry in Maxwell???s equations," J. Mod. Opt. 43, 773-793 (1996).
[CrossRef]

J. Opt. Soc. Am. A

Mathematics of Computation

D. Elliott, "The evaluation and estimation of the coefficients in the Chebyshev series expansion of a function," Mathematics of Computation 18, 274-284 (1964).
[CrossRef]

Microwave Opt. Technol. Lett.

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]

A. R. Weily, L. Horvath, K. P. Esselle, and B. C. Sanders, "Performance of PML absorbing boundary conditions in 3d photonic crystal waveguides," Microwave Opt. Technol. Lett. 40, 1-3 (2004).
[CrossRef]

Opt. Express

Phys. Rev. B

Z.-Y. Li and K.-M. Ho, "Light propagation in semi-infinite photonic crystals and related waveguide structures," Phys. Rev. B 68 (2003).

Phys. Rev. E

S. G. Johnson, P. Bienstman, M. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. Joannopoulos, "Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals," Phys. Rev. E 66, 066608 (2002).

E. Moreno, D. Erni, and C. Hafner, "Modeling of discontinuities in photonic crystal waveguides with the multiple multipole method," Phys. Rev. E 66, 036618 (2002).

SIAM J. Num. Anal.

Z. Chen and X. Liu, "An adaptive perfectly matched layer technique for time-harmonic scattering problems," SIAM J. Num. Anal. 43, 645-671 (2005).
[CrossRef]

Other

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech, 2000).

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd Edition (Princeton Univ. Press, 2008).

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd Edition (Academic Press, San Diego, 1991).

H. Cheng, Advanced Analytic Methods in Applied Mathematics, Science and Engineering (Luban Press, Boston, 2006).

J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd Edition (Springer, 1989).

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Figures (8)

Fig. 1.
Fig. 1.

(a) PML is still reflectionless for inhomogeneous media such as waveguides that are homogeneous in the direction perpenendicular to the PML. (b, c) PML is no longer reflectionless when the dielectric function is discontinuous (non-analytic) in the direction perpendicular to the PML, as in a photonic crystal (b) or a waveguide entering the PML at an angle (c).

Fig. 2.
Fig. 2.

Reflection coefficient as a function of discretization resolution for both a uniform medium and a periodic medium with PML and non-PML absorbing boundaries (insets). For the periodic medium, PML fails to be reflectionless even in the limit of high resolution, and does no better than a non-PML absorber. Inset: reflection as a function of absorber thickness L for fixed resolution ~ 50pixels/λ : as the absorber becomes thicker and the absorption is turned on more gradually, reflection goes to zero via the adiabatic theorem; PML for the uniform medium only improves the constant factor.

Fig. 3.
Fig. 3.

Reflectivity vs. PML thickness L for 1d vacuum (inset) at a resolution of 50pixels/λ for various shape functions s(u) ranging from linear [s(u) = u] to quintic [s(u) = u 5]. For reference, the corresponding asymptotic power laws are shown as dashed lines.

Fig. 4.
Fig. 4.

Reflectivity vs. pPML thickness L for the 1d periodic medium (inset) with period a, as in Fig. 2, at a resolution of 50pixels/a with a wavevector kx = 0.9π/a (vacuum wavelength λ = 0.9597a, just below the first gap) for various shape functions s(u) ranging from linear [s(u) = u] to quintic [s(u) = u 5]. For reference, the corresponding asymptotic power laws are shown as dashed lines.

Fig. 5.
Fig. 5.

Field convergence factor [Eq. (11)] (~ reflection/L 2) vs. PML thickness L for 2d vacuum (inset) at a resolution of 20pixels/λ for various shape functions s(u) ranging from linear [s(u) = u] to quintic [s(u) = u 5]. For reference, the corresponding asymptotic power laws are shown as dashed lines. Inset: ℜ[Ez ] field pattern for the (point) source at the origin (blue/white/red = positive/zero/negative).

Fig. 6.
Fig. 6.

Field convergence factor [Eq. (11)] (~ reflection/L 2) vs. pPML thickness L for the discontinuous 2d periodic medium (left inset: square lattice of square air holes in ε = 12) with period a, at a resolution of 10pixels/a with a vacuum wavelength λ = 0.6667a (not in a band gap) for various shape functions s(u) ranging from linear [s(u) = u] to quintic [s(u) = u 5]. For reference, the corresponding asymptotic power laws are shown as dashed lines. Right inset: ℜ[Ez ] field pattern for the (point) source at the origin (blue/white/red = positive/zero/negative).

Fig. 7.
Fig. 7.

Reflectivity vs. PML thickness L for 1d vacuum (blue circles) at a resolution of 50pixels/λ, and for pPML thickness L in the 1d periodic medium of Fig. 4 (red squares) with period a at a resolution of 50pixels/a with a wavevector kx = 0.9π/a (vacuum wavelength λ = 0.9597a. In both cases, a C (infinitely differentiable) shape function s(u) = e 1-1/u for u > 0 is used, leading to asymptotic convergence as e -α√L for some constants α.

Fig. 8.
Fig. 8.

Reflectivity vs. PML thickness L for 1d vacuum (inset) at a resolution of 50pixels/λ for s(u) =u 2, with the round-trip reflection either set to R 0 = 10-16 (upper blue line) or set to match the estimated transition reflection from Fig. 3 (lower red line). By matching the round-trip reflection R 0 to the estimated transition reflection, one can obtain a substantial reduction in the constant factor of the total reflection, although the asymptotic power law is only changed by a lnL factor

Equations (13)

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

× H = H y x H x y = iωεE z
E z y = iωH x
E z x = iωH y
x 1 1 + i σ ( x ) ω x ,
H y x ( 1 + ω ) H x y = iωεE z + σεE z
E z y = iωH x
E z x = iωH y + σH y
R round trip ~ e 4 k x ω 0 L σ ( x ) dx ,
σ ( x ) = σ 0 s ( x L )
σ 0 = ln R 0 4 Ln 0 1 s ( u ) du .
E z ( L + 1 ) ( x , y ) E z ( L ) ( x , y ) 2 E z ( L ) ( x , y ) 2
c r ( L ) = 0 1 s ( u ) M [ s ( u ) ] Δ β [ s ( u ) ] e iL 0 u Δ β [ s ( u ) ] du du
c r ( L ) = s ( d ) ( 0 + ) M ( 0 + ) Δ β ( 0 + ) [ iL Δ β ( 0 + ) ] d + O ( L ( d + 1 ) ) ,

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