Abstract

A modified formulation of Maxwell’s equations is presented that includes a complex and nonlinear coordinate transform along one or two Cartesian coordinates. The added degrees of freedom in the modified Maxwell’s equations allow one to map an infinite space to a finite space and to specify graded perfectly matched absorbing boundaries that allow the outgoing wave condition to be satisfied. The approach is validated by numerical results obtained by using Fourier-modal methods and shows enhanced convergence rate and accuracy.

© 2005 Optical Society of America

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  1. J. P. Bérenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]
  2. Z. Sacks, D. Kingsland, R. Lee, J. Lee, “A perfectly matched anisotropic absorber for use as an absorbers boundary conditions,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995).
    [CrossRef]
  3. 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]
  4. J. M. Elson, P. Tran, “ R-matrix propagator with perfectly matched layers for the study of integral optical components,” J. Opt. Soc. Am. A 16, 2983–2989 (1999).
    [CrossRef]
  5. P. Bienstman, R. Baets, “Advanced boundary conditions for eigenmode expansion models,” Opt. Quantum Electron. 34, 523–540 (2002).
    [CrossRef]
  6. E. Silberstein, Ph. Lalanne, J. P. Hugonin, Q. Cao, “On the use of grating theory in integrated optics,” J. Opt. Soc. Am. A 18, 2865–2875 (2001).
    [CrossRef]
  7. Q. Cao, Ph. Lalanne, J. P. Hugonin, “Stable and efficient Bloch-mode computational method for one-dimensional grating waveguide,” J. Opt. Soc. Am. A 19, 335–338 (2002).
    [CrossRef]
  8. J. C. Chen, K. Li, “Quartic perfectly matched layers for dielectric waveguides and gratings,” Microwave Opt. Technol. Lett. 10, 319–323 (1995).
    [CrossRef]
  9. E. A. Marengo, C. M. Rappaport, E. L. Miller, “Optimum PML ABC conductivity profile in FDTD,” IEEE Trans. Magn. 35, 1506–1509 (1999).
    [CrossRef]
  10. E. Bahar, “Electromagnetic wave propagation in inhomogeneous multilayered structures of arbitrary varying thickness—generalized field transforms,” J. Math. Phys. 14, 1024–1029 (1972).
    [CrossRef]
  11. D. Maystre, “General study of grating anomalies from electromagnetic surface modes,” in Electromagnetic Surface modes, A. D. Boardman, ed. (Wiley, 1982), Chap. 17.
  12. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68, 1206–1210 (1978).
    [CrossRef]
  13. T. K. Gaylord, M. G. Moharam, “Analysis and application of optical diffraction by gratings,” Proc. IEEE 73, 894–936 (1985).
    [CrossRef]
  14. F. Montiel, M. Nevière, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity through the R-matrix propagation algorithm,” J. Opt. Soc. Am. A 11, 3241–3250 (1994).
    [CrossRef]
  15. Ph. Lalanne, E. Silberstein, “Fourier-modal method applied to waveguide computational problems,” Opt. Lett. 25, 1092–1094 (2000).
    [CrossRef]
  16. J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, M. Leppihalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer’s star product,” Opt. Commun. 198, 265–272 (2001).
    [CrossRef]
  17. S. J. Hewlett, F. Ladouceur, “Fourier decomposition method applied to mapped infinite domains: scalar analysis of dielectric waveguides down to modal cutoff,” J. Lightwave Technol. 13, 375–383 (1995).
    [CrossRef]
  18. P. Lalanne, J. P. Hugonin, J. M. Gérard, “Electromagnetic study of the Q of pillar microcavities in the small limit diameter,” Appl. Phys. Lett. 84, 4726–4728 (2004).
    [CrossRef]
  19. C. Sauvan, G. Lecamp, P. Lalanne, J. P. Hugonin, “Modal-reflectivity enhancement by geometry tuning in photonic crystal microcavities,” Opt. Express 13, 245–255 (2005).
    [CrossRef] [PubMed]
  20. C. Sauvan, P. Lalanne, J. C. Rodier, J. P. Hugonin, A. Talneau, “Accurate modeling of line-defect photonic crystal waveguides,” IEEE Photonics Technol. Lett. 15, 1243–1245 (2003).
    [CrossRef]
  21. J. P. Hugonin, P. Lalanne, I. Del Villar, I. R. Matias, “Fourier modal methods for modeling optical dielectric waveguides,” Opt. Quantum Electron. 37, 107–119 (2005).
    [CrossRef]

2005 (2)

C. Sauvan, G. Lecamp, P. Lalanne, J. P. Hugonin, “Modal-reflectivity enhancement by geometry tuning in photonic crystal microcavities,” Opt. Express 13, 245–255 (2005).
[CrossRef] [PubMed]

J. P. Hugonin, P. Lalanne, I. Del Villar, I. R. Matias, “Fourier modal methods for modeling optical dielectric waveguides,” Opt. Quantum Electron. 37, 107–119 (2005).
[CrossRef]

2004 (1)

P. Lalanne, J. P. Hugonin, J. M. Gérard, “Electromagnetic study of the Q of pillar microcavities in the small limit diameter,” Appl. Phys. Lett. 84, 4726–4728 (2004).
[CrossRef]

2003 (1)

C. Sauvan, P. Lalanne, J. C. Rodier, J. P. Hugonin, A. Talneau, “Accurate modeling of line-defect photonic crystal waveguides,” IEEE Photonics Technol. Lett. 15, 1243–1245 (2003).
[CrossRef]

2002 (2)

P. Bienstman, R. Baets, “Advanced boundary conditions for eigenmode expansion models,” Opt. Quantum Electron. 34, 523–540 (2002).
[CrossRef]

Q. Cao, Ph. Lalanne, J. P. Hugonin, “Stable and efficient Bloch-mode computational method for one-dimensional grating waveguide,” J. Opt. Soc. Am. A 19, 335–338 (2002).
[CrossRef]

2001 (2)

E. Silberstein, Ph. Lalanne, J. P. Hugonin, Q. Cao, “On the use of grating theory in integrated optics,” J. Opt. Soc. Am. A 18, 2865–2875 (2001).
[CrossRef]

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, M. Leppihalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer’s star product,” Opt. Commun. 198, 265–272 (2001).
[CrossRef]

2000 (1)

1999 (2)

E. A. Marengo, C. M. Rappaport, E. L. Miller, “Optimum PML ABC conductivity profile in FDTD,” IEEE Trans. Magn. 35, 1506–1509 (1999).
[CrossRef]

J. M. Elson, P. Tran, “ R-matrix propagator with perfectly matched layers for the study of integral optical components,” J. Opt. Soc. Am. A 16, 2983–2989 (1999).
[CrossRef]

1995 (3)

Z. Sacks, D. Kingsland, R. Lee, J. Lee, “A perfectly matched anisotropic absorber for use as an absorbers boundary conditions,” IEEE Trans. Antennas Propag. 43, 1460–1463 (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]

S. J. Hewlett, F. Ladouceur, “Fourier decomposition method applied to mapped infinite domains: scalar analysis of dielectric waveguides down to modal cutoff,” J. Lightwave Technol. 13, 375–383 (1995).
[CrossRef]

1994 (3)

F. Montiel, M. Nevière, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity through the R-matrix propagation algorithm,” J. Opt. Soc. Am. A 11, 3241–3250 (1994).
[CrossRef]

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

1985 (1)

T. K. Gaylord, M. G. Moharam, “Analysis and application of optical diffraction by gratings,” Proc. IEEE 73, 894–936 (1985).
[CrossRef]

1978 (1)

1972 (1)

E. Bahar, “Electromagnetic wave propagation in inhomogeneous multilayered structures of arbitrary varying thickness—generalized field transforms,” J. Math. Phys. 14, 1024–1029 (1972).
[CrossRef]

Aalto, T.

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, M. Leppihalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer’s star product,” Opt. Commun. 198, 265–272 (2001).
[CrossRef]

Baets, R.

P. Bienstman, R. Baets, “Advanced boundary conditions for eigenmode expansion models,” Opt. Quantum Electron. 34, 523–540 (2002).
[CrossRef]

Bahar, E.

E. Bahar, “Electromagnetic wave propagation in inhomogeneous multilayered structures of arbitrary varying thickness—generalized field transforms,” J. Math. Phys. 14, 1024–1029 (1972).
[CrossRef]

Bérenger, J. P.

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

Bienstman, P.

P. Bienstman, R. Baets, “Advanced boundary conditions for eigenmode expansion models,” Opt. Quantum Electron. 34, 523–540 (2002).
[CrossRef]

Cao, Q.

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]

Del Villar, I.

J. P. Hugonin, P. Lalanne, I. Del Villar, I. R. Matias, “Fourier modal methods for modeling optical dielectric waveguides,” Opt. Quantum Electron. 37, 107–119 (2005).
[CrossRef]

Elson, J. M.

Gaylord, T. K.

T. K. Gaylord, M. G. Moharam, “Analysis and application of optical diffraction by gratings,” Proc. IEEE 73, 894–936 (1985).
[CrossRef]

Gérard, J. M.

P. Lalanne, J. P. Hugonin, J. M. Gérard, “Electromagnetic study of the Q of pillar microcavities in the small limit diameter,” Appl. Phys. Lett. 84, 4726–4728 (2004).
[CrossRef]

Heimala, P.

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, M. Leppihalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer’s star product,” Opt. Commun. 198, 265–272 (2001).
[CrossRef]

Hewlett, S. J.

S. J. Hewlett, F. Ladouceur, “Fourier decomposition method applied to mapped infinite domains: scalar analysis of dielectric waveguides down to modal cutoff,” J. Lightwave Technol. 13, 375–383 (1995).
[CrossRef]

Hugonin, J. P.

C. Sauvan, G. Lecamp, P. Lalanne, J. P. Hugonin, “Modal-reflectivity enhancement by geometry tuning in photonic crystal microcavities,” Opt. Express 13, 245–255 (2005).
[CrossRef] [PubMed]

J. P. Hugonin, P. Lalanne, I. Del Villar, I. R. Matias, “Fourier modal methods for modeling optical dielectric waveguides,” Opt. Quantum Electron. 37, 107–119 (2005).
[CrossRef]

P. Lalanne, J. P. Hugonin, J. M. Gérard, “Electromagnetic study of the Q of pillar microcavities in the small limit diameter,” Appl. Phys. Lett. 84, 4726–4728 (2004).
[CrossRef]

C. Sauvan, P. Lalanne, J. C. Rodier, J. P. Hugonin, A. Talneau, “Accurate modeling of line-defect photonic crystal waveguides,” IEEE Photonics Technol. Lett. 15, 1243–1245 (2003).
[CrossRef]

Q. Cao, Ph. Lalanne, J. P. Hugonin, “Stable and efficient Bloch-mode computational method for one-dimensional grating waveguide,” J. Opt. Soc. Am. A 19, 335–338 (2002).
[CrossRef]

E. Silberstein, Ph. Lalanne, J. P. Hugonin, Q. Cao, “On the use of grating theory in integrated optics,” J. Opt. Soc. Am. A 18, 2865–2875 (2001).
[CrossRef]

Kingsland, D.

Z. Sacks, D. Kingsland, R. Lee, J. Lee, “A perfectly matched anisotropic absorber for use as an absorbers boundary conditions,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995).
[CrossRef]

Knop, K.

Kuittinen, M.

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, M. Leppihalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer’s star product,” Opt. Commun. 198, 265–272 (2001).
[CrossRef]

Ladouceur, F.

S. J. Hewlett, F. Ladouceur, “Fourier decomposition method applied to mapped infinite domains: scalar analysis of dielectric waveguides down to modal cutoff,” J. Lightwave Technol. 13, 375–383 (1995).
[CrossRef]

Lalanne, P.

C. Sauvan, G. Lecamp, P. Lalanne, J. P. Hugonin, “Modal-reflectivity enhancement by geometry tuning in photonic crystal microcavities,” Opt. Express 13, 245–255 (2005).
[CrossRef] [PubMed]

J. P. Hugonin, P. Lalanne, I. Del Villar, I. R. Matias, “Fourier modal methods for modeling optical dielectric waveguides,” Opt. Quantum Electron. 37, 107–119 (2005).
[CrossRef]

P. Lalanne, J. P. Hugonin, J. M. Gérard, “Electromagnetic study of the Q of pillar microcavities in the small limit diameter,” Appl. Phys. Lett. 84, 4726–4728 (2004).
[CrossRef]

C. Sauvan, P. Lalanne, J. C. Rodier, J. P. Hugonin, A. Talneau, “Accurate modeling of line-defect photonic crystal waveguides,” IEEE Photonics Technol. Lett. 15, 1243–1245 (2003).
[CrossRef]

Lalanne, Ph.

Lecamp, G.

Lee, J.

Z. Sacks, D. Kingsland, R. Lee, J. Lee, “A perfectly matched anisotropic absorber for use as an absorbers boundary conditions,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995).
[CrossRef]

Lee, R.

Z. Sacks, D. Kingsland, R. Lee, J. Lee, “A perfectly matched anisotropic absorber for use as an absorbers boundary conditions,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995).
[CrossRef]

Leppihalme, M.

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, M. Leppihalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer’s star product,” Opt. Commun. 198, 265–272 (2001).
[CrossRef]

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]

Marengo, E. A.

E. A. Marengo, C. M. Rappaport, E. L. Miller, “Optimum PML ABC conductivity profile in FDTD,” IEEE Trans. Magn. 35, 1506–1509 (1999).
[CrossRef]

Matias, I. R.

J. P. Hugonin, P. Lalanne, I. Del Villar, I. R. Matias, “Fourier modal methods for modeling optical dielectric waveguides,” Opt. Quantum Electron. 37, 107–119 (2005).
[CrossRef]

Maystre, D.

D. Maystre, “General study of grating anomalies from electromagnetic surface modes,” in Electromagnetic Surface modes, A. D. Boardman, ed. (Wiley, 1982), Chap. 17.

Miller, E. L.

E. A. Marengo, C. M. Rappaport, E. L. Miller, “Optimum PML ABC conductivity profile in FDTD,” IEEE Trans. Magn. 35, 1506–1509 (1999).
[CrossRef]

Moharam, M. G.

T. K. Gaylord, M. G. Moharam, “Analysis and application of optical diffraction by gratings,” Proc. IEEE 73, 894–936 (1985).
[CrossRef]

Montiel, F.

Nevière, M.

Rappaport, C. M.

E. A. Marengo, C. M. Rappaport, E. L. Miller, “Optimum PML ABC conductivity profile in FDTD,” IEEE Trans. Magn. 35, 1506–1509 (1999).
[CrossRef]

Rodier, J. C.

C. Sauvan, P. Lalanne, J. C. Rodier, J. P. Hugonin, A. Talneau, “Accurate modeling of line-defect photonic crystal waveguides,” IEEE Photonics Technol. Lett. 15, 1243–1245 (2003).
[CrossRef]

Sacks, Z.

Z. Sacks, D. Kingsland, R. Lee, J. Lee, “A perfectly matched anisotropic absorber for use as an absorbers boundary conditions,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995).
[CrossRef]

Sauvan, C.

C. Sauvan, G. Lecamp, P. Lalanne, J. P. Hugonin, “Modal-reflectivity enhancement by geometry tuning in photonic crystal microcavities,” Opt. Express 13, 245–255 (2005).
[CrossRef] [PubMed]

C. Sauvan, P. Lalanne, J. C. Rodier, J. P. Hugonin, A. Talneau, “Accurate modeling of line-defect photonic crystal waveguides,” IEEE Photonics Technol. Lett. 15, 1243–1245 (2003).
[CrossRef]

Silberstein, E.

Talneau, A.

C. Sauvan, P. Lalanne, J. C. Rodier, J. P. Hugonin, A. Talneau, “Accurate modeling of line-defect photonic crystal waveguides,” IEEE Photonics Technol. Lett. 15, 1243–1245 (2003).
[CrossRef]

Tervo, J.

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, M. Leppihalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer’s star product,” Opt. Commun. 198, 265–272 (2001).
[CrossRef]

Tran, P.

Turunen, J.

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, M. Leppihalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer’s star product,” Opt. Commun. 198, 265–272 (2001).
[CrossRef]

Vahimaa, P.

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, M. Leppihalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer’s star product,” Opt. Commun. 198, 265–272 (2001).
[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]

Appl. Phys. Lett. (1)

P. Lalanne, J. P. Hugonin, J. M. Gérard, “Electromagnetic study of the Q of pillar microcavities in the small limit diameter,” Appl. Phys. Lett. 84, 4726–4728 (2004).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

C. Sauvan, P. Lalanne, J. C. Rodier, J. P. Hugonin, A. Talneau, “Accurate modeling of line-defect photonic crystal waveguides,” IEEE Photonics Technol. Lett. 15, 1243–1245 (2003).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

Z. Sacks, D. Kingsland, R. Lee, J. Lee, “A perfectly matched anisotropic absorber for use as an absorbers boundary conditions,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995).
[CrossRef]

IEEE Trans. Magn. (1)

E. A. Marengo, C. M. Rappaport, E. L. Miller, “Optimum PML ABC conductivity profile in FDTD,” IEEE Trans. Magn. 35, 1506–1509 (1999).
[CrossRef]

J. Comput. Phys. (1)

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

J. Lightwave Technol. (1)

S. J. Hewlett, F. Ladouceur, “Fourier decomposition method applied to mapped infinite domains: scalar analysis of dielectric waveguides down to modal cutoff,” J. Lightwave Technol. 13, 375–383 (1995).
[CrossRef]

J. Math. Phys. (1)

E. Bahar, “Electromagnetic wave propagation in inhomogeneous multilayered structures of arbitrary varying thickness—generalized field transforms,” J. Math. Phys. 14, 1024–1029 (1972).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

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. Commun. (1)

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, M. Leppihalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer’s star product,” Opt. Commun. 198, 265–272 (2001).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Opt. Quantum Electron. (2)

P. Bienstman, R. Baets, “Advanced boundary conditions for eigenmode expansion models,” Opt. Quantum Electron. 34, 523–540 (2002).
[CrossRef]

J. P. Hugonin, P. Lalanne, I. Del Villar, I. R. Matias, “Fourier modal methods for modeling optical dielectric waveguides,” Opt. Quantum Electron. 37, 107–119 (2005).
[CrossRef]

Proc. IEEE (1)

T. K. Gaylord, M. G. Moharam, “Analysis and application of optical diffraction by gratings,” Proc. IEEE 73, 894–936 (1985).
[CrossRef]

Other (1)

D. Maystre, “General study of grating anomalies from electromagnetic surface modes,” in Electromagnetic Surface modes, A. D. Boardman, ed. (Wiley, 1982), Chap. 17.

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Figures (4)

Fig. 1
Fig. 1

Nonlinear transform for mapping an infinite space onto a finite one: (a) example of a diffraction problem “isolated in space” ( x ] , [ ) , (b) equivalent diffraction problem in a bounded real space, ( x ] d 2 , d 2 [ ) , (c) same as (b) but with an artificial periodization of period d.

Fig. 2
Fig. 2

Nonlinear coordinate transform: (a) function f PML ( x ) , (b) function f ( x ) , (c) function X = F ( x ) for γ = 1 ( 1 i ) and f PML = 1 . Solid curves denote the real part, and dashed curves the imaginary part.

Fig. 3
Fig. 3

Nonlinear and piecewise-constant coordinate transforms on a simple geometry problem. All plots are shown in the mapped space ( x , z ) , where x = x if f = 1 . Numerical parameters are d = 1.1 μ m , e = 0.3 μ m , and M = 100 . (a) Refractive index distribution of the double-slit geometry. (b) H y 2 calculated for γ = 1 ( 1 i ) and f PML = 1 . (c)–(f) Transverse profile of Re ( H y ) [the horizontal axes correspond to the thick vertical line in (b), and the small vertical arrows correspond to x = d 2 ]: (c) real coordinate transform ( γ = 0 ) without PML ( f PML = 1 ) , (d) complex coordinate transform [ γ = 1 ( 1 i ) ] without PML ( f PML = 1 ) as in (b), (e) real coordinate transform ( γ = 0 ) with PML [ ( f PML ) 1 = 1 + i ] , (f) no nonlinear coordinate transform ( f = 1 ) with ( f PML ) 1 = 1 + i (solid curve) and ( f PML ) 1 = 5 ( 1 + i ) (dashed curve).

Fig. 4
Fig. 4

Convergence and accuracy performance for the modal reflectivity R associated into the different cases shown in Figs. 3d, 3e, 3f. The extrapolated values for R are R 0 = 0.3952113445 for TE polarization and 0.3554787 for TM polarization. The labels (d), (e), (f1), and (f5) refer to Figs. 3d, 3e, 3f [with ( f PML ) 1 = 1 + i ], and 3(f) [with ( f PML ) 1 = 5 ( 1 + i ) ], respectively.

Equations (18)

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z ( 1 ε x x H y z ) + x ( 1 ε z z H y x ) + k 0 2 μ H y = 0 ,
H y out = p a p h p ( z ) exp ( i k p x ) ,
H y in = p b p h p ( z ) exp ( i k p x ) ,
F ( x ) is real for x < e 2 ,
F ( x ) + i for x d 2 ,
F ( x ) i for x d 2 .
d x d X = f ( x ) f PML ( x ) .
z [ ( 1 f PML ε x x ) H y z ] + ( f x ) [ ( f PML ε z z ) ( f x ) H y ] + k 0 2 ( μ f PML ) H y = 0 .
H y = m = M M U m ( z ) exp ( i m K x ) ,
( f PML ε x x ) 1 H y z = m = M M V m ( z ) exp ( i m K x ) ,
d d z ( [ U ] [ V ] ) = ( 0 A 1 F x K x E 1 F x K x 0 ) ( [ U ] [ V ] ) ,
f ( x ) = 1 for x < e 2 ,
f ( x ) = [ 1 γ sin 2 ( π x e 2 q ) ] cos 2 ( π x e 2 q ) for e 2 < x < d 2 ,
f PML ( x ) = 1 for x < e 2 ,
f PML ( x ) = f PML for e 2 < x < d 2 ,
f n = δ n q 2 d ( 1 ) n [ ( 1 + γ 4 ) sinc ( n q d ) + 1 2 sinc ( n q d 1 ) + 1 2 sinc ( n q d + 1 ) γ 8 sinc ( n q d 2 ) γ 8 sinc ( n q d + 2 ) ] ,
F ( x ) = x for x < e 2 ,
F ( x ) = x x ( e 2 + q π ( 1 γ ) { tan ( π x e 2 q ) γ 1 γ a tan [ 1 γ tan ( π x e 2 q ) ] } ) for e 2 < x < d 2 .

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