Abstract

The perfectly matched layer (PML) boundary condition is generally employed to prevent spurious reflections from numerical boundaries in wave propagation methods. However, PML requires additional computational resources. We have examined the performance of the PML by changing the distribution of sampling points and the PML’s absorption profile with a view to optimizing the PML’s efficiency. We used the collocation method in our study. We found that equally spaced field sampling points give better absorption of beams under both optimal and nonoptimal conditions for low PML widths. At high PML widths, unequally spaced basis points may be equally efficient. The efficiency of various PML absorption profiles, including new ones, has been studied, and we conclude that for better numerical efficiency it is important to choose an appropriate profile.

© 2004 Optical Society of America

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  1. M. D. Feit, J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990–3998 (1978).
    [CrossRef] [PubMed]
  2. C. L. Xu, W. P. Huang, “Finite-difference beam propagation methods for guided-wave optics,” in Methods for Modeling and Simulation of Optical Guided-Wave Devices, W. Huang, ed., Vol. 11 of Progress in Electromagnetic Research (EMW, Cambridge, Mass., 1995), pp. 1–49.
  3. A. Sharma, “Collocation method for wave propagation through optical waveguiding structures,” in Methods for Modeling and Simulation of Optical Guided-Wave Devices, W. Huang, ed., Vol. 11 of Progress in Electromagnetic Research (EMW, Cambridge, Mass., 1995), pp. 143–198.
  4. S. Banerjee, A. Sharma, “Propagation characteristics of optical waveguiding structures by direct solution of the Helmholtz equation for total fields,” J. Opt. Soc. Am. A 6, 1884–1894 (1989).
    [CrossRef]
  5. A. Sharma, S. Banerjee, “Method for propagation of total fields or beams through optical waveguides,” Opt. Lett. 14, 94–96 (1989).
    [CrossRef]
  6. A. Sharma, A. Taneja, “Unconditionally stable procedure to propagate beams through optical waveguides using the collocation method,” Opt. Lett. 16, 1162–1164 (1991).
    [CrossRef] [PubMed]
  7. A. Sharma, A. Taneja, “Variable-transformed collocation method for field propagation through waveguiding structures,” Opt. Lett. 17, 804–806 (1992).
    [CrossRef] [PubMed]
  8. E. L. Lindman, “Free space boundary conditions of the time dependent wave equation,” J. Comput. Phys. 18, 66–78 (1975).
    [CrossRef]
  9. B. Engquist, A. Majda, “Absorbing boundary conditions for the numerical simulation of waves,” Math. Comput. 31, 629–651 (1977).
    [CrossRef]
  10. G. Mur, “Absorbing boundary condition for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat. EMC-23, 377–382 (1981).
    [CrossRef]
  11. C. Vasallo, F. Collino, “Highly efficient absorbing boundary conditions for the beam propagation method,” J. Lightwave Technol. 14, 1570–1577 (1996).
    [CrossRef]
  12. G. R. Hadley, “Transparent boundary condition for beam propagation,” Opt. Lett. 16, 624–626 (1991).
    [CrossRef] [PubMed]
  13. G. R. Hadley, “Transparent boundary condition for the beam propagation method,” Opt. Lett. 28, 624–626 (1992).
  14. G. R. Hadley, “Transparent boundary condition for the beam propagation method,” IEEE J. Quantum Electron. 28, 363–370 (1992).
    [CrossRef]
  15. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]
  16. W. P. Huang, C. L. Xu, W. Lui, K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett. 8, 649–651 (1996).
    [CrossRef]
  17. W. P. Huang, C. L. Xu, W. Lui, K. Yokoyama, “The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon Technol. Lett. 8, 652–654 (1996).
    [CrossRef]
  18. D. Zhou, W. P. Huang, C. L. Xu, D. G. Fang, B. Chen, “The perfectly matched layer boundary condition for scalar finite-difference time-domain method,” IEEE Photon. Technol. Lett. 13, 454–456 (2001).
    [CrossRef]
  19. W. C. Chew, 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]
  20. B. Chen, D. G. Fang, B. H. Zhou, “Modified Berenger PML absorbing boundary condition for FD-TD meshes,” IEEE Microwave Guided Wave Lett. 5, 399–401 (1995).
    [CrossRef]
  21. J. C. Chen, K. Li, “Quartic perfectly matched layers for dielectric waveguides and gratings,” Microwave Opt. Technol. Lett. 10, 319–323 (1995).
    [CrossRef]

2001 (1)

D. Zhou, W. P. Huang, C. L. Xu, D. G. Fang, B. Chen, “The perfectly matched layer boundary condition for scalar finite-difference time-domain method,” IEEE Photon. Technol. Lett. 13, 454–456 (2001).
[CrossRef]

1996 (3)

C. Vasallo, F. Collino, “Highly efficient absorbing boundary conditions for the beam propagation method,” J. Lightwave Technol. 14, 1570–1577 (1996).
[CrossRef]

W. P. Huang, C. L. Xu, W. Lui, K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett. 8, 649–651 (1996).
[CrossRef]

W. P. Huang, C. L. Xu, W. Lui, K. Yokoyama, “The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon Technol. Lett. 8, 652–654 (1996).
[CrossRef]

1995 (2)

B. Chen, D. G. Fang, B. H. Zhou, “Modified Berenger PML absorbing boundary condition for FD-TD meshes,” IEEE Microwave Guided Wave Lett. 5, 399–401 (1995).
[CrossRef]

J. C. Chen, K. Li, “Quartic perfectly matched layers for dielectric waveguides and gratings,” Microwave Opt. Technol. Lett. 10, 319–323 (1995).
[CrossRef]

1994 (2)

W. C. Chew, 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]

1992 (3)

A. Sharma, A. Taneja, “Variable-transformed collocation method for field propagation through waveguiding structures,” Opt. Lett. 17, 804–806 (1992).
[CrossRef] [PubMed]

G. R. Hadley, “Transparent boundary condition for the beam propagation method,” Opt. Lett. 28, 624–626 (1992).

G. R. Hadley, “Transparent boundary condition for the beam propagation method,” IEEE J. Quantum Electron. 28, 363–370 (1992).
[CrossRef]

1991 (2)

1989 (2)

1981 (1)

G. Mur, “Absorbing boundary condition for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat. EMC-23, 377–382 (1981).
[CrossRef]

1978 (1)

1977 (1)

B. Engquist, A. Majda, “Absorbing boundary conditions for the numerical simulation of waves,” Math. Comput. 31, 629–651 (1977).
[CrossRef]

1975 (1)

E. L. Lindman, “Free space boundary conditions of the time dependent wave equation,” J. Comput. Phys. 18, 66–78 (1975).
[CrossRef]

Banerjee, S.

Berenger, J. P.

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

Chen, B.

D. Zhou, W. P. Huang, C. L. Xu, D. G. Fang, B. Chen, “The perfectly matched layer boundary condition for scalar finite-difference time-domain method,” IEEE Photon. Technol. Lett. 13, 454–456 (2001).
[CrossRef]

B. Chen, D. G. Fang, B. H. Zhou, “Modified Berenger PML absorbing boundary condition for FD-TD meshes,” IEEE Microwave Guided Wave Lett. 5, 399–401 (1995).
[CrossRef]

Chen, J. C.

J. C. Chen, K. Li, “Quartic perfectly matched layers for dielectric waveguides and gratings,” Microwave Opt. Technol. Lett. 10, 319–323 (1995).
[CrossRef]

Chew, W. C.

W. C. Chew, 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]

Collino, F.

C. Vasallo, F. Collino, “Highly efficient absorbing boundary conditions for the beam propagation method,” J. Lightwave Technol. 14, 1570–1577 (1996).
[CrossRef]

Engquist, B.

B. Engquist, A. Majda, “Absorbing boundary conditions for the numerical simulation of waves,” Math. Comput. 31, 629–651 (1977).
[CrossRef]

Fang, D. G.

D. Zhou, W. P. Huang, C. L. Xu, D. G. Fang, B. Chen, “The perfectly matched layer boundary condition for scalar finite-difference time-domain method,” IEEE Photon. Technol. Lett. 13, 454–456 (2001).
[CrossRef]

B. Chen, D. G. Fang, B. H. Zhou, “Modified Berenger PML absorbing boundary condition for FD-TD meshes,” IEEE Microwave Guided Wave Lett. 5, 399–401 (1995).
[CrossRef]

Feit, M. D.

Fleck, J. A.

Hadley, G. R.

G. R. Hadley, “Transparent boundary condition for the beam propagation method,” Opt. Lett. 28, 624–626 (1992).

G. R. Hadley, “Transparent boundary condition for the beam propagation method,” IEEE J. Quantum Electron. 28, 363–370 (1992).
[CrossRef]

G. R. Hadley, “Transparent boundary condition for beam propagation,” Opt. Lett. 16, 624–626 (1991).
[CrossRef] [PubMed]

Huang, W. P.

D. Zhou, W. P. Huang, C. L. Xu, D. G. Fang, B. Chen, “The perfectly matched layer boundary condition for scalar finite-difference time-domain method,” IEEE Photon. Technol. Lett. 13, 454–456 (2001).
[CrossRef]

W. P. Huang, C. L. Xu, W. Lui, K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett. 8, 649–651 (1996).
[CrossRef]

W. P. Huang, C. L. Xu, W. Lui, K. Yokoyama, “The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon Technol. Lett. 8, 652–654 (1996).
[CrossRef]

C. L. Xu, W. P. Huang, “Finite-difference beam propagation methods for guided-wave optics,” in Methods for Modeling and Simulation of Optical Guided-Wave Devices, W. Huang, ed., Vol. 11 of Progress in Electromagnetic Research (EMW, Cambridge, Mass., 1995), pp. 1–49.

Li, K.

J. C. Chen, K. Li, “Quartic perfectly matched layers for dielectric waveguides and gratings,” Microwave Opt. Technol. Lett. 10, 319–323 (1995).
[CrossRef]

Lindman, E. L.

E. L. Lindman, “Free space boundary conditions of the time dependent wave equation,” J. Comput. Phys. 18, 66–78 (1975).
[CrossRef]

Lui, W.

W. P. Huang, C. L. Xu, W. Lui, K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett. 8, 649–651 (1996).
[CrossRef]

W. P. Huang, C. L. Xu, W. Lui, K. Yokoyama, “The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon Technol. Lett. 8, 652–654 (1996).
[CrossRef]

Majda, A.

B. Engquist, A. Majda, “Absorbing boundary conditions for the numerical simulation of waves,” Math. Comput. 31, 629–651 (1977).
[CrossRef]

Mur, G.

G. Mur, “Absorbing boundary condition for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat. EMC-23, 377–382 (1981).
[CrossRef]

Sharma, A.

A. Sharma, A. Taneja, “Variable-transformed collocation method for field propagation through waveguiding structures,” Opt. Lett. 17, 804–806 (1992).
[CrossRef] [PubMed]

A. Sharma, A. Taneja, “Unconditionally stable procedure to propagate beams through optical waveguides using the collocation method,” Opt. Lett. 16, 1162–1164 (1991).
[CrossRef] [PubMed]

S. Banerjee, A. Sharma, “Propagation characteristics of optical waveguiding structures by direct solution of the Helmholtz equation for total fields,” J. Opt. Soc. Am. A 6, 1884–1894 (1989).
[CrossRef]

A. Sharma, S. Banerjee, “Method for propagation of total fields or beams through optical waveguides,” Opt. Lett. 14, 94–96 (1989).
[CrossRef]

A. Sharma, “Collocation method for wave propagation through optical waveguiding structures,” in Methods for Modeling and Simulation of Optical Guided-Wave Devices, W. Huang, ed., Vol. 11 of Progress in Electromagnetic Research (EMW, Cambridge, Mass., 1995), pp. 143–198.

Taneja, A.

Vasallo, C.

C. Vasallo, F. Collino, “Highly efficient absorbing boundary conditions for the beam propagation method,” J. Lightwave Technol. 14, 1570–1577 (1996).
[CrossRef]

Weedon, W. H.

W. C. Chew, 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]

Xu, C. L.

D. Zhou, W. P. Huang, C. L. Xu, D. G. Fang, B. Chen, “The perfectly matched layer boundary condition for scalar finite-difference time-domain method,” IEEE Photon. Technol. Lett. 13, 454–456 (2001).
[CrossRef]

W. P. Huang, C. L. Xu, W. Lui, K. Yokoyama, “The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon Technol. Lett. 8, 652–654 (1996).
[CrossRef]

W. P. Huang, C. L. Xu, W. Lui, K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett. 8, 649–651 (1996).
[CrossRef]

C. L. Xu, W. P. Huang, “Finite-difference beam propagation methods for guided-wave optics,” in Methods for Modeling and Simulation of Optical Guided-Wave Devices, W. Huang, ed., Vol. 11 of Progress in Electromagnetic Research (EMW, Cambridge, Mass., 1995), pp. 1–49.

Yokoyama, K.

W. P. Huang, C. L. Xu, W. Lui, K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett. 8, 649–651 (1996).
[CrossRef]

W. P. Huang, C. L. Xu, W. Lui, K. Yokoyama, “The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon Technol. Lett. 8, 652–654 (1996).
[CrossRef]

Zhou, B. H.

B. Chen, D. G. Fang, B. H. Zhou, “Modified Berenger PML absorbing boundary condition for FD-TD meshes,” IEEE Microwave Guided Wave Lett. 5, 399–401 (1995).
[CrossRef]

Zhou, D.

D. Zhou, W. P. Huang, C. L. Xu, D. G. Fang, B. Chen, “The perfectly matched layer boundary condition for scalar finite-difference time-domain method,” IEEE Photon. Technol. Lett. 13, 454–456 (2001).
[CrossRef]

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

G. R. Hadley, “Transparent boundary condition for the beam propagation method,” IEEE J. Quantum Electron. 28, 363–370 (1992).
[CrossRef]

IEEE Microwave Guided Wave Lett. (1)

B. Chen, D. G. Fang, B. H. Zhou, “Modified Berenger PML absorbing boundary condition for FD-TD meshes,” IEEE Microwave Guided Wave Lett. 5, 399–401 (1995).
[CrossRef]

IEEE Photon Technol. Lett. (1)

W. P. Huang, C. L. Xu, W. Lui, K. Yokoyama, “The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon Technol. Lett. 8, 652–654 (1996).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

D. Zhou, W. P. Huang, C. L. Xu, D. G. Fang, B. Chen, “The perfectly matched layer boundary condition for scalar finite-difference time-domain method,” IEEE Photon. Technol. Lett. 13, 454–456 (2001).
[CrossRef]

W. P. Huang, C. L. Xu, W. Lui, K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett. 8, 649–651 (1996).
[CrossRef]

IEEE Trans. Electromagn. Compat. (1)

G. Mur, “Absorbing boundary condition for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat. EMC-23, 377–382 (1981).
[CrossRef]

J. Comput. Phys. (2)

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

E. L. Lindman, “Free space boundary conditions of the time dependent wave equation,” J. Comput. Phys. 18, 66–78 (1975).
[CrossRef]

J. Lightwave Technol. (1)

C. Vasallo, F. Collino, “Highly efficient absorbing boundary conditions for the beam propagation method,” J. Lightwave Technol. 14, 1570–1577 (1996).
[CrossRef]

J. Opt. Soc. Am. A (1)

Math. Comput. (1)

B. Engquist, A. Majda, “Absorbing boundary conditions for the numerical simulation of waves,” Math. Comput. 31, 629–651 (1977).
[CrossRef]

Microwave Opt. Technol. Lett. (2)

W. C. Chew, 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. C. Chen, K. Li, “Quartic perfectly matched layers for dielectric waveguides and gratings,” Microwave Opt. Technol. Lett. 10, 319–323 (1995).
[CrossRef]

Opt. Lett. (5)

Other (2)

C. L. Xu, W. P. Huang, “Finite-difference beam propagation methods for guided-wave optics,” in Methods for Modeling and Simulation of Optical Guided-Wave Devices, W. Huang, ed., Vol. 11 of Progress in Electromagnetic Research (EMW, Cambridge, Mass., 1995), pp. 1–49.

A. Sharma, “Collocation method for wave propagation through optical waveguiding structures,” in Methods for Modeling and Simulation of Optical Guided-Wave Devices, W. Huang, ed., Vol. 11 of Progress in Electromagnetic Research (EMW, Cambridge, Mass., 1995), pp. 143–198.

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Figures (10)

Fig. 1
Fig. 1

Geometry of implementation of the PML technique.

Fig. 2
Fig. 2

Geometry of the definition of E R .

Fig. 3
Fig. 3

Energy remaining in the real window, E R (in %), as a function of the PML width (% of the total numerical window) for the square profile with the unequally spaced and equally spaced distributions of sample points.

Fig. 4
Fig. 4

Energy remaining in the real window, E R (in %), as a function of beam tilt angle for the square profile with unequally spaced and equally spaced distributions of sample points.

Fig. 5
Fig. 5

Energy remaining in the real window, E z (in %), as a function of propagation distance z. Results are shown for square profiles with unequally spaced and equally spaced distributions of sample points. The PML width is 3.5 μm (8% of the total window).

Fig. 6
Fig. 6

Energy remaining in the real window, E z (in %), as a function of propagation distance z. Results are shown for square profiles with unequally spaced and equally spaced distributions of sample points. The PML width is 6.7 μm (15% of the total window).

Fig. 7
Fig. 7

Square absorption profile in the unequally and the equally spaced point distributions for PML width 8%.

Fig. 8
Fig. 8

Square absorption profile in the unequally and the equally spaced point distributions for PML width 19%.

Fig. 9
Fig. 9

Energy remaining in the real window, E R (in %), as a function of PML width (%) for several absorption profiles with equally spaced point distributions.

Fig. 10
Fig. 10

Energy remaining in the real window, E R (in %), as a function of PML width (%) for several absorption profiles with unequally spaced point distributions.

Equations (22)

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

x=hσ,
hσ=σ  σ<xp=xp+ ξ1-ipξdξ xp<σ<xb,
pξ=p0ξq, q=2, 3, 4,.
pξ=p0 sinqπξ/2δ, q=2, 3, 4,,
2Ψx2+2Ψz2+k02n2x, zΨx, z=0,
Ψx, z=n=1N cnzϕnx,
d2Ψdz2+S0+RzΨz=0,
Ψz=Ψx1, zΨx2, zΨxN, z,Rz=k02n2x1, z0·00n2x2, z·····00·0n2xN, z,
dχdz=12ikS0+Rz-k2Iχz,
ϕnx=cosvnx, n=1, 3, 5, N-1,ϕnx=sinvnx n=2, 4, 6, N,
S0=AHA-1,
ϕnx=Nn-1Hn-1αxexp-½α2x2,
HNαxj=0, j=1, 2,, N.
S0=D1-AD2A-1,
D1=α4×diag-x12-x22  -xN2,D2=α2×diag1352N-1,A=Aij:Aij=ϕjxi.
x=hσ,ψx, z=hσUσ, z,
2Uz2+fσ2Uσ2+gσ+k02n2σ, zUσ, z=0,
fσ=hσ-2,
gσ=12h4hh-32 h2,
d2Udz2+Ŝ0+RzUz=0.
Ŝ0=FD1-FAD2A-1+G,
dχˆdz=12ikŜ0+Rz-k2Iχˆz.

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