Abstract

In this paper we present an extension of the modal method by Gegenbauer expansion (MMGE) [J. Opt. Soc. Am. A 28, 2006 (2011)], [Progress Electromagn. Res. 133, 17 (2013)] to the study of nonperiodic problems. The nonperiodicity is introduced through the perfectly matched layers (PMLs) concept, which can be introduced in an equivalent way either by a change of coordinates or by the use of a uniaxial anisotropic medium. These PMLs can generate strong irregularities of the electromagnetic fields that can significantly alter the convergence and stability of the numerical scheme. This is the case, e.g., for the famous Fourier modal method, especially when using complex stretching coordinates. In this work, it will be shown that the MMGE equipped with PMLs is a robust approach because of its natural immunity against spurious modes.

© 2013 Optical Society of America

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  1. K. Edee, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings,” J. Opt. Soc. Am. A 28, 2006–2013 (2011).
    [CrossRef]
  2. K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: weithing function, convergence and stability,” Progress Electromagn. Res. 133, 17–35 (2013).
  3. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68, 1206–1210 (1978).
    [CrossRef]
  4. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997).
    [CrossRef]
  5. R. H. Morf, “Exponentially convergent and numerically efficient solution of Maxwell’s equations for lamellar gratings,” J. Opt. Soc. Am. A 12, 1043–1056 (1995).
    [CrossRef]
  6. K. Edee, P. Schiavone, and G. Granet, “Analysis of defect in E.U.V lithography mask using a modal method by nodal B-spline expansion,” Jpn. J. Appl. Phys. 44, 6458–6462 (2005).
    [CrossRef]
  7. A. M. Armeanu, K. Edee, G. Granet, and P. Schiavone, “Modal method based on spline expansion for the electromagnetic analysis of the lamellar grating,” Progress Electromagn. Res. 106, 243–261 (2010).
    [CrossRef]
  8. G. Granet, L. B. Andriamanampisoa, K. Raniriharinosy, A. M. Armeanu, and K. Edee, “Modal analysis of lamellar gratings using the moment method with subsectional basis and adaptive spatial resolution,” J. Opt. Soc. Am. A 27, 1303–1310(2010).
    [CrossRef]
  9. J. C. Weeber, A. Dereux, C. Girard, G. Colas des Francs, J. R. Krenn, and J. P. Goudonnet, “Optical addressing at the subwavelength scale,” Phys. Rev. E. 62, 7381–7388 (2000).
    [CrossRef]
  10. B. Guizal, D. Barchiesi, and D. Felbacq, “Electromagnetic beam diffraction by a finite lamellar structure: an aperiodic coupled-wave method,” J. Opt. Soc. Am. A 20, 2274–2280 (2003).
  11. J. Bérenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200(1994).
    [CrossRef]
  12. Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE. Trans. Antennas Propag. 43, 1460–1463 (1995).
    [CrossRef]
  13. 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]
  14. W. C. Chew, J. M. Jin, and E. Michielssen, “Complex coordinate stretching as a generalized absorbing boundary condition,” Microw. Opt. Technol. Lett. 15, 363–369 (1997).
    [CrossRef]
  15. F. L. Teixeira and W. C. Chew, “Differential forms, metrics, and the reflectionless absorption of electromagnetic waves,” J. Electromagn. Waves Appl. 13, 665–686 (1999).
    [CrossRef]
  16. G. Granet, K. Edee, and D. Felbacq, “Scattering of a plane wave by rough surfaces: a new curvilinear coordinate system based approach,” Progress Electromagn. Res. 41, 235–250 (2003).
    [CrossRef]
  17. C. Baudier, R. Dusséeaux, K. Edee, and G. Granet, “Scattering of a plane wave by one-dimensional dielectric random surfaces. Study with the curvilinear method,” Waves Random Media 14, 61–74 (2004).
    [CrossRef]
  18. K. Edee, G. Granet, R. Dusséaux, and C. Baudier, “A hybrid method for the study of plane waves scattering by rough surfaces,” J. Electromagn. Waves Appl. 18, 1001–1015 (2004).
    [CrossRef]
  19. K. Edee and G. Granet, “Improvement of the curvilinear coordinate method for scattering from rough surfaces: reduction of the eigenvalue equation by using eigenvalue degenerscence,” J. Electromagn. Waves Appl. 18, 763–768 (2004).
    [CrossRef]
  20. K. Edee, B. Guizal, G. Granet, and A. Moreau, “Beam implementation in a nonorthogonal coordinate system : application to the scattering from random rough surfaces,” J. Opt. Soc. Am. A 25, 796–804 (2008).
    [CrossRef]
  21. E. Silberstein, P. Lalanne, J.-P. Hugonin, and Q. Cao, “Use of gratings in integrated optics,” J. Opt. Soc. Am. A 18, 2865–2875 (2001).
    [CrossRef]
  22. J. P. Hugonin and P. Lalanne, “Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization,” J. Opt. Soc. Am. A 22, 1844–1849 (2005).
    [CrossRef]
  23. J.-P. Plumey, K. Edee, and G. Granet, “Modal expansion for 2D Green’s function in a non-orthogonal coordinates system,” Progress Electromagn. Res. 59, 101–112 (2006).
    [CrossRef]
  24. K. Edee, G. Granet, and J.-P. Plumey, “Complex coordinate implementation in the curvilinear coordinate method : application to plane-wave diffraction by nonperiodic rough surfaces,” J. Opt. Soc. Am. A 24, 1097–1102 (2007).
    [CrossRef]
  25. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).

2013 (1)

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: weithing function, convergence and stability,” Progress Electromagn. Res. 133, 17–35 (2013).

2011 (1)

2010 (2)

A. M. Armeanu, K. Edee, G. Granet, and P. Schiavone, “Modal method based on spline expansion for the electromagnetic analysis of the lamellar grating,” Progress Electromagn. Res. 106, 243–261 (2010).
[CrossRef]

G. Granet, L. B. Andriamanampisoa, K. Raniriharinosy, A. M. Armeanu, and K. Edee, “Modal analysis of lamellar gratings using the moment method with subsectional basis and adaptive spatial resolution,” J. Opt. Soc. Am. A 27, 1303–1310(2010).
[CrossRef]

2008 (1)

2007 (1)

2006 (1)

J.-P. Plumey, K. Edee, and G. Granet, “Modal expansion for 2D Green’s function in a non-orthogonal coordinates system,” Progress Electromagn. Res. 59, 101–112 (2006).
[CrossRef]

2005 (2)

J. P. Hugonin and P. Lalanne, “Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization,” J. Opt. Soc. Am. A 22, 1844–1849 (2005).
[CrossRef]

K. Edee, P. Schiavone, and G. Granet, “Analysis of defect in E.U.V lithography mask using a modal method by nodal B-spline expansion,” Jpn. J. Appl. Phys. 44, 6458–6462 (2005).
[CrossRef]

2004 (3)

C. Baudier, R. Dusséeaux, K. Edee, and G. Granet, “Scattering of a plane wave by one-dimensional dielectric random surfaces. Study with the curvilinear method,” Waves Random Media 14, 61–74 (2004).
[CrossRef]

K. Edee, G. Granet, R. Dusséaux, and C. Baudier, “A hybrid method for the study of plane waves scattering by rough surfaces,” J. Electromagn. Waves Appl. 18, 1001–1015 (2004).
[CrossRef]

K. Edee and G. Granet, “Improvement of the curvilinear coordinate method for scattering from rough surfaces: reduction of the eigenvalue equation by using eigenvalue degenerscence,” J. Electromagn. Waves Appl. 18, 763–768 (2004).
[CrossRef]

2003 (2)

B. Guizal, D. Barchiesi, and D. Felbacq, “Electromagnetic beam diffraction by a finite lamellar structure: an aperiodic coupled-wave method,” J. Opt. Soc. Am. A 20, 2274–2280 (2003).

G. Granet, K. Edee, and D. Felbacq, “Scattering of a plane wave by rough surfaces: a new curvilinear coordinate system based approach,” Progress Electromagn. Res. 41, 235–250 (2003).
[CrossRef]

2001 (1)

2000 (1)

J. C. Weeber, A. Dereux, C. Girard, G. Colas des Francs, J. R. Krenn, and J. P. Goudonnet, “Optical addressing at the subwavelength scale,” Phys. Rev. E. 62, 7381–7388 (2000).
[CrossRef]

1999 (1)

F. L. Teixeira and W. C. Chew, “Differential forms, metrics, and the reflectionless absorption of electromagnetic waves,” J. Electromagn. Waves Appl. 13, 665–686 (1999).
[CrossRef]

1997 (2)

W. C. Chew, J. M. Jin, and E. Michielssen, “Complex coordinate stretching as a generalized absorbing boundary condition,” Microw. Opt. Technol. Lett. 15, 363–369 (1997).
[CrossRef]

L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997).
[CrossRef]

1995 (2)

R. H. Morf, “Exponentially convergent and numerically efficient solution of Maxwell’s equations for lamellar gratings,” J. Opt. Soc. Am. A 12, 1043–1056 (1995).
[CrossRef]

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

1994 (2)

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

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

1978 (1)

Andriamanampisoa, L. B.

Armeanu, A. M.

G. Granet, L. B. Andriamanampisoa, K. Raniriharinosy, A. M. Armeanu, and K. Edee, “Modal analysis of lamellar gratings using the moment method with subsectional basis and adaptive spatial resolution,” J. Opt. Soc. Am. A 27, 1303–1310(2010).
[CrossRef]

A. M. Armeanu, K. Edee, G. Granet, and P. Schiavone, “Modal method based on spline expansion for the electromagnetic analysis of the lamellar grating,” Progress Electromagn. Res. 106, 243–261 (2010).
[CrossRef]

Barchiesi, D.

Baudier, C.

C. Baudier, R. Dusséeaux, K. Edee, and G. Granet, “Scattering of a plane wave by one-dimensional dielectric random surfaces. Study with the curvilinear method,” Waves Random Media 14, 61–74 (2004).
[CrossRef]

K. Edee, G. Granet, R. Dusséaux, and C. Baudier, “A hybrid method for the study of plane waves scattering by rough surfaces,” J. Electromagn. Waves Appl. 18, 1001–1015 (2004).
[CrossRef]

Bérenger, J.

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

Cao, Q.

Chew, W. C.

F. L. Teixeira and W. C. Chew, “Differential forms, metrics, and the reflectionless absorption of electromagnetic waves,” J. Electromagn. Waves Appl. 13, 665–686 (1999).
[CrossRef]

W. C. Chew, J. M. Jin, and E. Michielssen, “Complex coordinate stretching as a generalized absorbing boundary condition,” Microw. Opt. Technol. Lett. 15, 363–369 (1997).
[CrossRef]

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]

Dereux, A.

J. C. Weeber, A. Dereux, C. Girard, G. Colas des Francs, J. R. Krenn, and J. P. Goudonnet, “Optical addressing at the subwavelength scale,” Phys. Rev. E. 62, 7381–7388 (2000).
[CrossRef]

des Francs, G. Colas

J. C. Weeber, A. Dereux, C. Girard, G. Colas des Francs, J. R. Krenn, and J. P. Goudonnet, “Optical addressing at the subwavelength scale,” Phys. Rev. E. 62, 7381–7388 (2000).
[CrossRef]

Dusséaux, R.

K. Edee, G. Granet, R. Dusséaux, and C. Baudier, “A hybrid method for the study of plane waves scattering by rough surfaces,” J. Electromagn. Waves Appl. 18, 1001–1015 (2004).
[CrossRef]

Dusséeaux, R.

C. Baudier, R. Dusséeaux, K. Edee, and G. Granet, “Scattering of a plane wave by one-dimensional dielectric random surfaces. Study with the curvilinear method,” Waves Random Media 14, 61–74 (2004).
[CrossRef]

Edee, K.

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: weithing function, convergence and stability,” Progress Electromagn. Res. 133, 17–35 (2013).

K. Edee, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings,” J. Opt. Soc. Am. A 28, 2006–2013 (2011).
[CrossRef]

A. M. Armeanu, K. Edee, G. Granet, and P. Schiavone, “Modal method based on spline expansion for the electromagnetic analysis of the lamellar grating,” Progress Electromagn. Res. 106, 243–261 (2010).
[CrossRef]

G. Granet, L. B. Andriamanampisoa, K. Raniriharinosy, A. M. Armeanu, and K. Edee, “Modal analysis of lamellar gratings using the moment method with subsectional basis and adaptive spatial resolution,” J. Opt. Soc. Am. A 27, 1303–1310(2010).
[CrossRef]

K. Edee, B. Guizal, G. Granet, and A. Moreau, “Beam implementation in a nonorthogonal coordinate system : application to the scattering from random rough surfaces,” J. Opt. Soc. Am. A 25, 796–804 (2008).
[CrossRef]

K. Edee, G. Granet, and J.-P. Plumey, “Complex coordinate implementation in the curvilinear coordinate method : application to plane-wave diffraction by nonperiodic rough surfaces,” J. Opt. Soc. Am. A 24, 1097–1102 (2007).
[CrossRef]

J.-P. Plumey, K. Edee, and G. Granet, “Modal expansion for 2D Green’s function in a non-orthogonal coordinates system,” Progress Electromagn. Res. 59, 101–112 (2006).
[CrossRef]

K. Edee, P. Schiavone, and G. Granet, “Analysis of defect in E.U.V lithography mask using a modal method by nodal B-spline expansion,” Jpn. J. Appl. Phys. 44, 6458–6462 (2005).
[CrossRef]

C. Baudier, R. Dusséeaux, K. Edee, and G. Granet, “Scattering of a plane wave by one-dimensional dielectric random surfaces. Study with the curvilinear method,” Waves Random Media 14, 61–74 (2004).
[CrossRef]

K. Edee and G. Granet, “Improvement of the curvilinear coordinate method for scattering from rough surfaces: reduction of the eigenvalue equation by using eigenvalue degenerscence,” J. Electromagn. Waves Appl. 18, 763–768 (2004).
[CrossRef]

K. Edee, G. Granet, R. Dusséaux, and C. Baudier, “A hybrid method for the study of plane waves scattering by rough surfaces,” J. Electromagn. Waves Appl. 18, 1001–1015 (2004).
[CrossRef]

G. Granet, K. Edee, and D. Felbacq, “Scattering of a plane wave by rough surfaces: a new curvilinear coordinate system based approach,” Progress Electromagn. Res. 41, 235–250 (2003).
[CrossRef]

Felbacq, D.

G. Granet, K. Edee, and D. Felbacq, “Scattering of a plane wave by rough surfaces: a new curvilinear coordinate system based approach,” Progress Electromagn. Res. 41, 235–250 (2003).
[CrossRef]

B. Guizal, D. Barchiesi, and D. Felbacq, “Electromagnetic beam diffraction by a finite lamellar structure: an aperiodic coupled-wave method,” J. Opt. Soc. Am. A 20, 2274–2280 (2003).

Fenniche, I.

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: weithing function, convergence and stability,” Progress Electromagn. Res. 133, 17–35 (2013).

Girard, C.

J. C. Weeber, A. Dereux, C. Girard, G. Colas des Francs, J. R. Krenn, and J. P. Goudonnet, “Optical addressing at the subwavelength scale,” Phys. Rev. E. 62, 7381–7388 (2000).
[CrossRef]

Goudonnet, J. P.

J. C. Weeber, A. Dereux, C. Girard, G. Colas des Francs, J. R. Krenn, and J. P. Goudonnet, “Optical addressing at the subwavelength scale,” Phys. Rev. E. 62, 7381–7388 (2000).
[CrossRef]

Granet, G.

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: weithing function, convergence and stability,” Progress Electromagn. Res. 133, 17–35 (2013).

A. M. Armeanu, K. Edee, G. Granet, and P. Schiavone, “Modal method based on spline expansion for the electromagnetic analysis of the lamellar grating,” Progress Electromagn. Res. 106, 243–261 (2010).
[CrossRef]

G. Granet, L. B. Andriamanampisoa, K. Raniriharinosy, A. M. Armeanu, and K. Edee, “Modal analysis of lamellar gratings using the moment method with subsectional basis and adaptive spatial resolution,” J. Opt. Soc. Am. A 27, 1303–1310(2010).
[CrossRef]

K. Edee, B. Guizal, G. Granet, and A. Moreau, “Beam implementation in a nonorthogonal coordinate system : application to the scattering from random rough surfaces,” J. Opt. Soc. Am. A 25, 796–804 (2008).
[CrossRef]

K. Edee, G. Granet, and J.-P. Plumey, “Complex coordinate implementation in the curvilinear coordinate method : application to plane-wave diffraction by nonperiodic rough surfaces,” J. Opt. Soc. Am. A 24, 1097–1102 (2007).
[CrossRef]

J.-P. Plumey, K. Edee, and G. Granet, “Modal expansion for 2D Green’s function in a non-orthogonal coordinates system,” Progress Electromagn. Res. 59, 101–112 (2006).
[CrossRef]

K. Edee, P. Schiavone, and G. Granet, “Analysis of defect in E.U.V lithography mask using a modal method by nodal B-spline expansion,” Jpn. J. Appl. Phys. 44, 6458–6462 (2005).
[CrossRef]

K. Edee, G. Granet, R. Dusséaux, and C. Baudier, “A hybrid method for the study of plane waves scattering by rough surfaces,” J. Electromagn. Waves Appl. 18, 1001–1015 (2004).
[CrossRef]

K. Edee and G. Granet, “Improvement of the curvilinear coordinate method for scattering from rough surfaces: reduction of the eigenvalue equation by using eigenvalue degenerscence,” J. Electromagn. Waves Appl. 18, 763–768 (2004).
[CrossRef]

C. Baudier, R. Dusséeaux, K. Edee, and G. Granet, “Scattering of a plane wave by one-dimensional dielectric random surfaces. Study with the curvilinear method,” Waves Random Media 14, 61–74 (2004).
[CrossRef]

G. Granet, K. Edee, and D. Felbacq, “Scattering of a plane wave by rough surfaces: a new curvilinear coordinate system based approach,” Progress Electromagn. Res. 41, 235–250 (2003).
[CrossRef]

Guizal, B.

Hugonin, J. P.

Hugonin, J.-P.

Jin, J. M.

W. C. Chew, J. M. Jin, and E. Michielssen, “Complex coordinate stretching as a generalized absorbing boundary condition,” Microw. Opt. Technol. Lett. 15, 363–369 (1997).
[CrossRef]

Kingsland, D. M.

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

Knop, K.

Krenn, J. R.

J. C. Weeber, A. Dereux, C. Girard, G. Colas des Francs, J. R. Krenn, and J. P. Goudonnet, “Optical addressing at the subwavelength scale,” Phys. Rev. E. 62, 7381–7388 (2000).
[CrossRef]

Lalanne, P.

Lee, J.-F.

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

Li, L.

Michielssen, E.

W. C. Chew, J. M. Jin, and E. Michielssen, “Complex coordinate stretching as a generalized absorbing boundary condition,” Microw. Opt. Technol. Lett. 15, 363–369 (1997).
[CrossRef]

Moreau, A.

Morf, R. H.

Plumey, J.-P.

K. Edee, G. Granet, and J.-P. Plumey, “Complex coordinate implementation in the curvilinear coordinate method : application to plane-wave diffraction by nonperiodic rough surfaces,” J. Opt. Soc. Am. A 24, 1097–1102 (2007).
[CrossRef]

J.-P. Plumey, K. Edee, and G. Granet, “Modal expansion for 2D Green’s function in a non-orthogonal coordinates system,” Progress Electromagn. Res. 59, 101–112 (2006).
[CrossRef]

Raether, H.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).

Raniriharinosy, K.

Sacks, Z. S.

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

Schiavone, P.

A. M. Armeanu, K. Edee, G. Granet, and P. Schiavone, “Modal method based on spline expansion for the electromagnetic analysis of the lamellar grating,” Progress Electromagn. Res. 106, 243–261 (2010).
[CrossRef]

K. Edee, P. Schiavone, and G. Granet, “Analysis of defect in E.U.V lithography mask using a modal method by nodal B-spline expansion,” Jpn. J. Appl. Phys. 44, 6458–6462 (2005).
[CrossRef]

Silberstein, E.

Teixeira, F. L.

F. L. Teixeira and W. C. Chew, “Differential forms, metrics, and the reflectionless absorption of electromagnetic waves,” J. Electromagn. Waves Appl. 13, 665–686 (1999).
[CrossRef]

Weeber, J. C.

J. C. Weeber, A. Dereux, C. Girard, G. Colas des Francs, J. R. Krenn, and J. P. Goudonnet, “Optical addressing at the subwavelength scale,” Phys. Rev. E. 62, 7381–7388 (2000).
[CrossRef]

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]

IEEE. Trans. Antennas Propag. (1)

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

J. Comput. Phys. (1)

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

J. Electromagn. Waves Appl. (3)

K. Edee, G. Granet, R. Dusséaux, and C. Baudier, “A hybrid method for the study of plane waves scattering by rough surfaces,” J. Electromagn. Waves Appl. 18, 1001–1015 (2004).
[CrossRef]

K. Edee and G. Granet, “Improvement of the curvilinear coordinate method for scattering from rough surfaces: reduction of the eigenvalue equation by using eigenvalue degenerscence,” J. Electromagn. Waves Appl. 18, 763–768 (2004).
[CrossRef]

F. L. Teixeira and W. C. Chew, “Differential forms, metrics, and the reflectionless absorption of electromagnetic waves,” J. Electromagn. Waves Appl. 13, 665–686 (1999).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (9)

L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997).
[CrossRef]

R. H. Morf, “Exponentially convergent and numerically efficient solution of Maxwell’s equations for lamellar gratings,” J. Opt. Soc. Am. A 12, 1043–1056 (1995).
[CrossRef]

K. Edee, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings,” J. Opt. Soc. Am. A 28, 2006–2013 (2011).
[CrossRef]

G. Granet, L. B. Andriamanampisoa, K. Raniriharinosy, A. M. Armeanu, and K. Edee, “Modal analysis of lamellar gratings using the moment method with subsectional basis and adaptive spatial resolution,” J. Opt. Soc. Am. A 27, 1303–1310(2010).
[CrossRef]

K. Edee, B. Guizal, G. Granet, and A. Moreau, “Beam implementation in a nonorthogonal coordinate system : application to the scattering from random rough surfaces,” J. Opt. Soc. Am. A 25, 796–804 (2008).
[CrossRef]

E. Silberstein, P. Lalanne, J.-P. Hugonin, and Q. Cao, “Use of gratings in integrated optics,” J. Opt. Soc. Am. A 18, 2865–2875 (2001).
[CrossRef]

J. P. Hugonin and P. Lalanne, “Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization,” J. Opt. Soc. Am. A 22, 1844–1849 (2005).
[CrossRef]

B. Guizal, D. Barchiesi, and D. Felbacq, “Electromagnetic beam diffraction by a finite lamellar structure: an aperiodic coupled-wave method,” J. Opt. Soc. Am. A 20, 2274–2280 (2003).

K. Edee, G. Granet, and J.-P. Plumey, “Complex coordinate implementation in the curvilinear coordinate method : application to plane-wave diffraction by nonperiodic rough surfaces,” J. Opt. Soc. Am. A 24, 1097–1102 (2007).
[CrossRef]

Jpn. J. Appl. Phys. (1)

K. Edee, P. Schiavone, and G. Granet, “Analysis of defect in E.U.V lithography mask using a modal method by nodal B-spline expansion,” Jpn. J. Appl. Phys. 44, 6458–6462 (2005).
[CrossRef]

Microw. Opt. Technol. Lett. (2)

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Figures (10)

Fig. 1.
Fig. 1.

Configuration of the structure under study: one period is depicted.

Fig. 2.
Fig. 2.

Modulus of the wave function A0 describing the plasmon mode for (a) different values of epml while (χ,η)=(1,0) and (b) different values of χ while (epml=λ and η=0).

Fig. 3.
Fig. 3.

Convergence of the (a) real part and (b) imaginary part of the effective index β of the plasmon mode with respect to Nmax, for η=0 and different values of χ. Used numerical parameters: epml=λ.

Fig. 4.
Fig. 4.

Results presented are the same as those of Figs. 3(a) and 3(b), except η=1.

Fig. 5.
Fig. 5.

Corrugated photonic waveguide is inserted in the basic cell of a grating equipped with PMLs.

Fig. 6.
Fig. 6.

Convergence of the effective indices of the fundamental modes TE0 and TM0 with respect to Nmax and for different values of χ (η=0).

Fig. 7.
Fig. 7.

Convergence of the real and imaginary parts of the effective indices of the fundamental modes (a) TE0 and (b) TM0 with respect to Nmax and for different values of η (while χ=5).

Fig. 8.
Fig. 8.

Computed imaginary part of the effective indices of the fundamental modes with the FMM (a) TE0 and (b) TM0 with respect to Nmax=2M+1 (where M is the truncation order) for different values of η (while χ=5).

Fig. 9.
Fig. 9.

Nonconvergence of the reflectivity of modes TE0 and TM0 in the case of real PMLs; i.e., η=0 no matter the values of χ.

Fig. 10.
Fig. 10.

Convergence of the reflectivity of modes TE0 and TM0 in the case of complex PMLs, i.e., η0, and for two different values of χ: (a) χ=5 and (b) χ=10.

Equations (13)

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([εlm],[μlm])=(ε,μ)[1χiη000χiη000χiη],(χ,η)R2.
{x˜(x)=S(x)=0xs(x)dxy˜=yandz˜=z.
{ξlmnmHn=iωεlmEmξlmnmEn=iωμlmHm(l,m,n)=(1,2,3)=(x,y,z),
([εnm],[μnm])=(ε,μ)[1s(x)000s(x)000s(x)].
x˜(x)={(χiη)xinside the PMLsxoutside the PMLs.
[1k2d2dx2+νi2]|Xpi=βp2|Xpi;
u,v[a,b]=ab(1f(x)2)Λ.5u(x)v(x)dx,
f(x)=ξ=2baxb+aba.
CmΛ,CnΛ[a,b]=dxdξCmΛ,CnΛ[1,1]
CmΛ,dCnΛdx[a,b]=CmΛ,dCnΛdξ[1,1].
{x˜(x)=S(x)=0xs(x)dxy˜=yz˜=z,
[1k21s(x)ddx1s(x)ddx+νi2]|Xpi=βp2|Xpi.
K(ω)=kε1ε2ε1+ε2,k=ωc,

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