Abstract

In photonics, Dispersive Quasi-Normal Modes (DQNMs) refer to optical resonant modes, solutions of spectral problems associated with Maxwell’s equations for open photonic structures involving dispersive media. Since these DQNMs are the constituents determining optical responses, studying DQNM expansion formalisms is the key to model the physical properties of a considered system. In this paper, we emphasize the non-uniqueness of the expansions related to the over-completeness of the set of modes and discuss a family of DQNM expansions depending on continuous parameters that can be freely chosen. These expansions can be applied to dispersive, anisotropic, and even non-reciprocal materials. As an example, we particularly demonstrate the modal analysis on a 2-D scattering model where the permittivity of a silicon object is drawn directly from actual measurement data.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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  2. Q. Bai, M. Perrin, C. Sauvan, J.-P. Hugonin, and P. Lalanne, “Efficient and intuitive method for the analysis of light scattering by a resonant nanostructure,” Opt. Express 21(22), 27371–27382 (2013).
    [Crossref]
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    [Crossref]
  4. W. Yan, R. Faggiani, and P. Lalanne, “Rigorous modal analysis of plasmonic nanoresonators,” Phys. Rev. B 97(20), 205422 (2018).
    [Crossref]
  5. B. Vial, F. Zolla, A. Nicolet, and M. Commandré, “Quasimodal expansion of electromagnetic fields in open two-dimensional structures,” Phys. Rev. A 89(2), 023829 (2014).
    [Crossref]
  6. E. A. Muljarov and T. Weiss, “Resonant-state expansion for open optical systems: generalization to magnetic, chiral, and bi-anisotropic materials,” Opt. Lett. 43(9), 1978 (2018).
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  7. A. Gras, P. Lalanne, and M. Duruflé, “Non-uniqueness of the quasinormal mode expansion of electromagnetic lorentz dispersive materials,” J. Opt. Soc. Am. (A 37), 1219–1228 (2020).
  8. C. Engström, H. Langer, and C. Tretter, “Non-linear eigenvalue problems and applications to photonic crystals,” J. Math. Analysis Appl. (445), 240–279 (2017).
  9. V. Hernandez, J. E. Roman, and V. Vidal, “SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems,” ACM Trans. Math. Softw. 31(3), 351–362 (2005).
    [Crossref]
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    [Crossref]
  11. W.-J. Beyn, “An integral method for solving nonlinear eigenvalue problems,” Linear Algebr. its Appl. 436(10), 3839–3863 (2012).
    [Crossref]
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    [Crossref]
  13. M. Garcia-Vergara, G. Demésy, and F. Zolla, “Extracting an accurate model for permittivity from experimental data: hunting complex poles from the real line,” Opt. Lett. 42(6), 1145–1148 (2017).
    [Crossref]
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    [Crossref]
  16. P. Lalanne, W. Yan, A. Gras, C. Sauvan, J.-P. Hugonin, M. Besbes, G. Demésy, M. D. Truong, B. Gralak, F. Zolla, A. Nicolet, F. Binkowski, L. Zschiedrich, S. Burger, J. Zimmerling, R. Remis, P. Urbach, H. T. Liu, and T. Weiss, “Quasinormal mode solvers for resonators with dispersive materials,” J. Opt. Soc. Am. A 36(4), 686–704 (2019).
    [Crossref]
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  18. M. D. Truong, G. Demésy, F. Zolla, and A. Nicolet, “The exact dispersive quasi-normal mode (DQNM) expansion for photonic structures with highly dispersive media in unbounded geometries,” in Metamaterials XII, vol. 11025 (International Society for Optics and Photonics, 2019), p. 110250Q.
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    [Crossref]
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    [Crossref]
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    [Crossref]
  26. B. Vial, F. Zolla, A. Nicolet, M. Commandré, and S. Tisserand, “Adaptive Perfectly Matched Layer for Wood’s anomalies in diffraction gratings,” Opt. Express 20(27), 28094–105 (2012).
    [Crossref]
  27. K. Nguyen, F. Treyssède, and C. Hazard, “Numerical modeling of three-dimensional open elastic waveguides combining semi-analytical finite element and perfectly matched layer methods,” J. Sound Vib. 344, 158–178 (2015).
    [Crossref]
  28. P. Hislop and I. Sigal, Introduction to Spectral Theory: With Applications to Schrödinger Operators, Applied Mathematical Sciences (Springer New York, 2012).
  29. L. N. Trefethen, “Pseudospectra of linear operators,” SIAM Rev. 39(3), 383–406 (1997).
    [Crossref]

2020 (1)

G. Demésy, A. Nicolet, B. Gralak, C. Geuzaine, C. Campos, and J. E. Roman, “Non-linear eigenvalue problems with GetDP and SLEPc: Eigenmode computations of frequency-dispersive photonic open structures,” Comput. Phys. Commun. 257, 107509 (2020).
[Crossref]

2019 (1)

2018 (5)

A.-S. Bonnet-Ben Dhia, C. Carvalho, and P. Ciarlet, “Mesh requirements for the finite element approximation of problems with sign-changing coefficients,” Numer. Math. 138(4), 801–838 (2018).
[Crossref]

P. Lalanne, W. Yan, K. Vynck, C. Sauvan, and J.-P. Hugonin, “Light Interaction with Photonic and Plasmonic Resonances,” Laser Photonics Rev. 12(5), 1700113 (2018).
[Crossref]

W. Yan, R. Faggiani, and P. Lalanne, “Rigorous modal analysis of plasmonic nanoresonators,” Phys. Rev. B 97(20), 205422 (2018).
[Crossref]

E. A. Muljarov and T. Weiss, “Resonant-state expansion for open optical systems: generalization to magnetic, chiral, and bi-anisotropic materials,” Opt. Lett. 43(9), 1978 (2018).
[Crossref]

F. Zolla, A. Nicolet, and G. Demésy, “Photonics in highly dispersive media: the exact modal expansion,” Opt. Lett. 43(23), 5813 (2018).
[Crossref]

2017 (1)

2016 (1)

M. Van Barel and P. Kravanja, “Nonlinear eigenvalue problems and contour integrals,” J. Comput. Appl. Math. 292, 526–540 (2016).
[Crossref]

2015 (1)

K. Nguyen, F. Treyssède, and C. Hazard, “Numerical modeling of three-dimensional open elastic waveguides combining semi-analytical finite element and perfectly matched layer methods,” J. Sound Vib. 344, 158–178 (2015).
[Crossref]

2014 (1)

B. Vial, F. Zolla, A. Nicolet, and M. Commandré, “Quasimodal expansion of electromagnetic fields in open two-dimensional structures,” Phys. Rev. A 89(2), 023829 (2014).
[Crossref]

2013 (2)

Q. Bai, M. Perrin, C. Sauvan, J.-P. Hugonin, and P. Lalanne, “Efficient and intuitive method for the analysis of light scattering by a resonant nanostructure,” Opt. Express 21(22), 27371–27382 (2013).
[Crossref]

C. Sauvan, J. P. Hugonin, I. S. Maksymov, and P. Lalanne, “Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators,” Phys. Rev. Lett. 110(23), 237401 (2013).
[Crossref]

2012 (2)

2005 (1)

V. Hernandez, J. E. Roman, and V. Vidal, “SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems,” ACM Trans. Math. Softw. 31(3), 351–362 (2005).
[Crossref]

1997 (1)

L. N. Trefethen, “Pseudospectra of linear operators,” SIAM Rev. 39(3), 383–406 (1997).
[Crossref]

1995 (1)

M. A. Green and M. J. Keevers, “Optical properties of intrinsic silicon at 300 k,” Prog. Photovolt: Res. Appl. 3(3), 189–192 (1995).
[Crossref]

1994 (1)

J.-P. Berenger, “A Perfectly Matched Layer for the Absorption of Electromagnetic Waves,” J. Comput. Phys. 114(2), 185–200 (1994).
[Crossref]

1976 (1)

1960 (1)

P. Lancaster, “Inversion of lambda-matrices and application to the theory of linear vibrations,” Arch. Ration. Mech. Anal. 6(1), 105–114 (1960).
[Crossref]

Bai, Q.

Berenger, J.-P.

J.-P. Berenger, “A Perfectly Matched Layer for the Absorption of Electromagnetic Waves,” J. Comput. Phys. 114(2), 185–200 (1994).
[Crossref]

Besbes, M.

Beyn, W.-J.

W.-J. Beyn, “An integral method for solving nonlinear eigenvalue problems,” Linear Algebr. its Appl. 436(10), 3839–3863 (2012).
[Crossref]

Binkowski, F.

Bonnet-Ben Dhia, A.-S.

A.-S. Bonnet-Ben Dhia, C. Carvalho, and P. Ciarlet, “Mesh requirements for the finite element approximation of problems with sign-changing coefficients,” Numer. Math. 138(4), 801–838 (2018).
[Crossref]

Burger, S.

Campos, C.

G. Demésy, A. Nicolet, B. Gralak, C. Geuzaine, C. Campos, and J. E. Roman, “Non-linear eigenvalue problems with GetDP and SLEPc: Eigenmode computations of frequency-dispersive photonic open structures,” Comput. Phys. Commun. 257, 107509 (2020).
[Crossref]

Carvalho, C.

A.-S. Bonnet-Ben Dhia, C. Carvalho, and P. Ciarlet, “Mesh requirements for the finite element approximation of problems with sign-changing coefficients,” Numer. Math. 138(4), 801–838 (2018).
[Crossref]

Ciarlet, P.

A.-S. Bonnet-Ben Dhia, C. Carvalho, and P. Ciarlet, “Mesh requirements for the finite element approximation of problems with sign-changing coefficients,” Numer. Math. 138(4), 801–838 (2018).
[Crossref]

Commandré, M.

B. Vial, F. Zolla, A. Nicolet, and M. Commandré, “Quasimodal expansion of electromagnetic fields in open two-dimensional structures,” Phys. Rev. A 89(2), 023829 (2014).
[Crossref]

B. Vial, F. Zolla, A. Nicolet, M. Commandré, and S. Tisserand, “Adaptive Perfectly Matched Layer for Wood’s anomalies in diffraction gratings,” Opt. Express 20(27), 28094–105 (2012).
[Crossref]

Demésy, G.

G. Demésy, A. Nicolet, B. Gralak, C. Geuzaine, C. Campos, and J. E. Roman, “Non-linear eigenvalue problems with GetDP and SLEPc: Eigenmode computations of frequency-dispersive photonic open structures,” Comput. Phys. Commun. 257, 107509 (2020).
[Crossref]

P. Lalanne, W. Yan, A. Gras, C. Sauvan, J.-P. Hugonin, M. Besbes, G. Demésy, M. D. Truong, B. Gralak, F. Zolla, A. Nicolet, F. Binkowski, L. Zschiedrich, S. Burger, J. Zimmerling, R. Remis, P. Urbach, H. T. Liu, and T. Weiss, “Quasinormal mode solvers for resonators with dispersive materials,” J. Opt. Soc. Am. A 36(4), 686–704 (2019).
[Crossref]

F. Zolla, A. Nicolet, and G. Demésy, “Photonics in highly dispersive media: the exact modal expansion,” Opt. Lett. 43(23), 5813 (2018).
[Crossref]

M. Garcia-Vergara, G. Demésy, and F. Zolla, “Extracting an accurate model for permittivity from experimental data: hunting complex poles from the real line,” Opt. Lett. 42(6), 1145–1148 (2017).
[Crossref]

M. D. Truong, G. Demésy, F. Zolla, and A. Nicolet, “Dispersive Quasi-Normal Mode (DQNM) Expansion in Open and Periodic Nanophotonic Structures,” in 2019 22nd International Conference on the Computation of Electromagnetic Fields (COMPUMAG), (2019), pp. 1–4.

M. D. Truong, G. Demésy, F. Zolla, and A. Nicolet, “The exact dispersive quasi-normal mode (DQNM) expansion for photonic structures with highly dispersive media in unbounded geometries,” in Metamaterials XII, vol. 11025 (International Society for Optics and Photonics, 2019), p. 110250Q.

Dular, P.

P. Dular and C. Geuzaine, GetDP reference manual: the documentation for GetDP, a general environment for the treatment of discrete problems, http://getdp.info .

Duruflé, M.

A. Gras, P. Lalanne, and M. Duruflé, “Non-uniqueness of the quasinormal mode expansion of electromagnetic lorentz dispersive materials,” J. Opt. Soc. Am. (A 37), 1219–1228 (2020).

Engström, C.

C. Engström, H. Langer, and C. Tretter, “Non-linear eigenvalue problems and applications to photonic crystals,” J. Math. Analysis Appl. (445), 240–279 (2017).

Faggiani, R.

W. Yan, R. Faggiani, and P. Lalanne, “Rigorous modal analysis of plasmonic nanoresonators,” Phys. Rev. B 97(20), 205422 (2018).
[Crossref]

Frederick, W.

W. Frederick, Optical properties of solids (Academic press, New York London, 1972).

Garcia-Vergara, M.

Geuzaine, C.

G. Demésy, A. Nicolet, B. Gralak, C. Geuzaine, C. Campos, and J. E. Roman, “Non-linear eigenvalue problems with GetDP and SLEPc: Eigenmode computations of frequency-dispersive photonic open structures,” Comput. Phys. Commun. 257, 107509 (2020).
[Crossref]

P. Dular and C. Geuzaine, GetDP reference manual: the documentation for GetDP, a general environment for the treatment of discrete problems, http://getdp.info .

C. Geuzaine, “GetDP: a general finite-element solver for the de Rham complex,” in PAMM Volume 7 Issue 1. Special Issue: Sixth International Congress on Industrial Applied Mathematics (ICIAM07) and GAMM Annual Meeting, Zürich 2007, vol. 7 (Wiley, 2008), pp. 1010603–1010604.

Gralak, B.

G. Demésy, A. Nicolet, B. Gralak, C. Geuzaine, C. Campos, and J. E. Roman, “Non-linear eigenvalue problems with GetDP and SLEPc: Eigenmode computations of frequency-dispersive photonic open structures,” Comput. Phys. Commun. 257, 107509 (2020).
[Crossref]

P. Lalanne, W. Yan, A. Gras, C. Sauvan, J.-P. Hugonin, M. Besbes, G. Demésy, M. D. Truong, B. Gralak, F. Zolla, A. Nicolet, F. Binkowski, L. Zschiedrich, S. Burger, J. Zimmerling, R. Remis, P. Urbach, H. T. Liu, and T. Weiss, “Quasinormal mode solvers for resonators with dispersive materials,” J. Opt. Soc. Am. A 36(4), 686–704 (2019).
[Crossref]

Gras, A.

Green, M. A.

M. A. Green and M. J. Keevers, “Optical properties of intrinsic silicon at 300 k,” Prog. Photovolt: Res. Appl. 3(3), 189–192 (1995).
[Crossref]

Hazard, C.

K. Nguyen, F. Treyssède, and C. Hazard, “Numerical modeling of three-dimensional open elastic waveguides combining semi-analytical finite element and perfectly matched layer methods,” J. Sound Vib. 344, 158–178 (2015).
[Crossref]

Hernandez, V.

V. Hernandez, J. E. Roman, and V. Vidal, “SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems,” ACM Trans. Math. Softw. 31(3), 351–362 (2005).
[Crossref]

Hislop, P.

P. Hislop and I. Sigal, Introduction to Spectral Theory: With Applications to Schrödinger Operators, Applied Mathematical Sciences (Springer New York, 2012).

Hugonin, J. P.

C. Sauvan, J. P. Hugonin, I. S. Maksymov, and P. Lalanne, “Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators,” Phys. Rev. Lett. 110(23), 237401 (2013).
[Crossref]

Hugonin, J.-P.

Keevers, M. J.

M. A. Green and M. J. Keevers, “Optical properties of intrinsic silicon at 300 k,” Prog. Photovolt: Res. Appl. 3(3), 189–192 (1995).
[Crossref]

Kravanja, P.

M. Van Barel and P. Kravanja, “Nonlinear eigenvalue problems and contour integrals,” J. Comput. Appl. Math. 292, 526–540 (2016).
[Crossref]

Lalanne, P.

P. Lalanne, W. Yan, A. Gras, C. Sauvan, J.-P. Hugonin, M. Besbes, G. Demésy, M. D. Truong, B. Gralak, F. Zolla, A. Nicolet, F. Binkowski, L. Zschiedrich, S. Burger, J. Zimmerling, R. Remis, P. Urbach, H. T. Liu, and T. Weiss, “Quasinormal mode solvers for resonators with dispersive materials,” J. Opt. Soc. Am. A 36(4), 686–704 (2019).
[Crossref]

P. Lalanne, W. Yan, K. Vynck, C. Sauvan, and J.-P. Hugonin, “Light Interaction with Photonic and Plasmonic Resonances,” Laser Photonics Rev. 12(5), 1700113 (2018).
[Crossref]

W. Yan, R. Faggiani, and P. Lalanne, “Rigorous modal analysis of plasmonic nanoresonators,” Phys. Rev. B 97(20), 205422 (2018).
[Crossref]

C. Sauvan, J. P. Hugonin, I. S. Maksymov, and P. Lalanne, “Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators,” Phys. Rev. Lett. 110(23), 237401 (2013).
[Crossref]

Q. Bai, M. Perrin, C. Sauvan, J.-P. Hugonin, and P. Lalanne, “Efficient and intuitive method for the analysis of light scattering by a resonant nanostructure,” Opt. Express 21(22), 27371–27382 (2013).
[Crossref]

A. Gras, P. Lalanne, and M. Duruflé, “Non-uniqueness of the quasinormal mode expansion of electromagnetic lorentz dispersive materials,” J. Opt. Soc. Am. (A 37), 1219–1228 (2020).

Lancaster, P.

P. Lancaster, “Inversion of lambda-matrices and application to the theory of linear vibrations,” Arch. Ration. Mech. Anal. 6(1), 105–114 (1960).
[Crossref]

Langer, H.

C. Engström, H. Langer, and C. Tretter, “Non-linear eigenvalue problems and applications to photonic crystals,” J. Math. Analysis Appl. (445), 240–279 (2017).

Liu, H. T.

Maksymov, I. S.

C. Sauvan, J. P. Hugonin, I. S. Maksymov, and P. Lalanne, “Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators,” Phys. Rev. Lett. 110(23), 237401 (2013).
[Crossref]

Muljarov, E. A.

Nguyen, K.

K. Nguyen, F. Treyssède, and C. Hazard, “Numerical modeling of three-dimensional open elastic waveguides combining semi-analytical finite element and perfectly matched layer methods,” J. Sound Vib. 344, 158–178 (2015).
[Crossref]

Nicolet, A.

G. Demésy, A. Nicolet, B. Gralak, C. Geuzaine, C. Campos, and J. E. Roman, “Non-linear eigenvalue problems with GetDP and SLEPc: Eigenmode computations of frequency-dispersive photonic open structures,” Comput. Phys. Commun. 257, 107509 (2020).
[Crossref]

P. Lalanne, W. Yan, A. Gras, C. Sauvan, J.-P. Hugonin, M. Besbes, G. Demésy, M. D. Truong, B. Gralak, F. Zolla, A. Nicolet, F. Binkowski, L. Zschiedrich, S. Burger, J. Zimmerling, R. Remis, P. Urbach, H. T. Liu, and T. Weiss, “Quasinormal mode solvers for resonators with dispersive materials,” J. Opt. Soc. Am. A 36(4), 686–704 (2019).
[Crossref]

F. Zolla, A. Nicolet, and G. Demésy, “Photonics in highly dispersive media: the exact modal expansion,” Opt. Lett. 43(23), 5813 (2018).
[Crossref]

B. Vial, F. Zolla, A. Nicolet, and M. Commandré, “Quasimodal expansion of electromagnetic fields in open two-dimensional structures,” Phys. Rev. A 89(2), 023829 (2014).
[Crossref]

B. Vial, F. Zolla, A. Nicolet, M. Commandré, and S. Tisserand, “Adaptive Perfectly Matched Layer for Wood’s anomalies in diffraction gratings,” Opt. Express 20(27), 28094–105 (2012).
[Crossref]

M. D. Truong, G. Demésy, F. Zolla, and A. Nicolet, “The exact dispersive quasi-normal mode (DQNM) expansion for photonic structures with highly dispersive media in unbounded geometries,” in Metamaterials XII, vol. 11025 (International Society for Optics and Photonics, 2019), p. 110250Q.

M. D. Truong, G. Demésy, F. Zolla, and A. Nicolet, “Dispersive Quasi-Normal Mode (DQNM) Expansion in Open and Periodic Nanophotonic Structures,” in 2019 22nd International Conference on the Computation of Electromagnetic Fields (COMPUMAG), (2019), pp. 1–4.

Perrin, M.

Remis, R.

Roman, J. E.

G. Demésy, A. Nicolet, B. Gralak, C. Geuzaine, C. Campos, and J. E. Roman, “Non-linear eigenvalue problems with GetDP and SLEPc: Eigenmode computations of frequency-dispersive photonic open structures,” Comput. Phys. Commun. 257, 107509 (2020).
[Crossref]

V. Hernandez, J. E. Roman, and V. Vidal, “SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems,” ACM Trans. Math. Softw. 31(3), 351–362 (2005).
[Crossref]

Sammut, R.

Sauvan, C.

Sigal, I.

P. Hislop and I. Sigal, Introduction to Spectral Theory: With Applications to Schrödinger Operators, Applied Mathematical Sciences (Springer New York, 2012).

Snyder, A. W.

Tisserand, S.

Trefethen, L. N.

L. N. Trefethen, “Pseudospectra of linear operators,” SIAM Rev. 39(3), 383–406 (1997).
[Crossref]

Tretter, C.

C. Engström, H. Langer, and C. Tretter, “Non-linear eigenvalue problems and applications to photonic crystals,” J. Math. Analysis Appl. (445), 240–279 (2017).

Treyssède, F.

K. Nguyen, F. Treyssède, and C. Hazard, “Numerical modeling of three-dimensional open elastic waveguides combining semi-analytical finite element and perfectly matched layer methods,” J. Sound Vib. 344, 158–178 (2015).
[Crossref]

Truong, M. D.

P. Lalanne, W. Yan, A. Gras, C. Sauvan, J.-P. Hugonin, M. Besbes, G. Demésy, M. D. Truong, B. Gralak, F. Zolla, A. Nicolet, F. Binkowski, L. Zschiedrich, S. Burger, J. Zimmerling, R. Remis, P. Urbach, H. T. Liu, and T. Weiss, “Quasinormal mode solvers for resonators with dispersive materials,” J. Opt. Soc. Am. A 36(4), 686–704 (2019).
[Crossref]

M. D. Truong, G. Demésy, F. Zolla, and A. Nicolet, “The exact dispersive quasi-normal mode (DQNM) expansion for photonic structures with highly dispersive media in unbounded geometries,” in Metamaterials XII, vol. 11025 (International Society for Optics and Photonics, 2019), p. 110250Q.

M. D. Truong, G. Demésy, F. Zolla, and A. Nicolet, “Dispersive Quasi-Normal Mode (DQNM) Expansion in Open and Periodic Nanophotonic Structures,” in 2019 22nd International Conference on the Computation of Electromagnetic Fields (COMPUMAG), (2019), pp. 1–4.

Urbach, P.

Van Barel, M.

M. Van Barel and P. Kravanja, “Nonlinear eigenvalue problems and contour integrals,” J. Comput. Appl. Math. 292, 526–540 (2016).
[Crossref]

Vial, B.

B. Vial, F. Zolla, A. Nicolet, and M. Commandré, “Quasimodal expansion of electromagnetic fields in open two-dimensional structures,” Phys. Rev. A 89(2), 023829 (2014).
[Crossref]

B. Vial, F. Zolla, A. Nicolet, M. Commandré, and S. Tisserand, “Adaptive Perfectly Matched Layer for Wood’s anomalies in diffraction gratings,” Opt. Express 20(27), 28094–105 (2012).
[Crossref]

Vidal, V.

V. Hernandez, J. E. Roman, and V. Vidal, “SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems,” ACM Trans. Math. Softw. 31(3), 351–362 (2005).
[Crossref]

Vynck, K.

P. Lalanne, W. Yan, K. Vynck, C. Sauvan, and J.-P. Hugonin, “Light Interaction with Photonic and Plasmonic Resonances,” Laser Photonics Rev. 12(5), 1700113 (2018).
[Crossref]

Weiss, T.

Yan, W.

P. Lalanne, W. Yan, A. Gras, C. Sauvan, J.-P. Hugonin, M. Besbes, G. Demésy, M. D. Truong, B. Gralak, F. Zolla, A. Nicolet, F. Binkowski, L. Zschiedrich, S. Burger, J. Zimmerling, R. Remis, P. Urbach, H. T. Liu, and T. Weiss, “Quasinormal mode solvers for resonators with dispersive materials,” J. Opt. Soc. Am. A 36(4), 686–704 (2019).
[Crossref]

P. Lalanne, W. Yan, K. Vynck, C. Sauvan, and J.-P. Hugonin, “Light Interaction with Photonic and Plasmonic Resonances,” Laser Photonics Rev. 12(5), 1700113 (2018).
[Crossref]

W. Yan, R. Faggiani, and P. Lalanne, “Rigorous modal analysis of plasmonic nanoresonators,” Phys. Rev. B 97(20), 205422 (2018).
[Crossref]

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Figures (11)

Fig. 1.
Fig. 1. (Left) A 2-D square box containing an elliptical scatterer $\Omega _1$: The major and minor radius of the ellipse are 2 and 1.2 respectively. The length of the side of a square is 5.6 (All lengths are measured in $(\times 10^{-1} \mu m)$). (Right) The Finite Element Mesh.
Fig. 2.
Fig. 2. The real part (top) and imaginary part (bottom) of the permittivity of silicon computed with different numbers of poles. The blue dots represent the actual measurement data of silicon.
Fig. 3.
Fig. 3. Spectrum of complex eigenfrequencies (bottom left) corresponding to the 1-pole relative permittivity (top left). Three eigenfields (real part) are depicted at the right (the blue color of the field maps indicates the minimum value and the red is the maximum).
Fig. 4.
Fig. 4. On the left: Scattered field $\mathbf {E}$ obtained by different expansion formulas Eq. (9) or by solving a direct problem classically (green dots) corresponding to the 1-pole permittivity. (Top-left) Integral over $\Omega _1$ of the norm of the electric field $\int _{\Omega _1}\vert \mathbf {E}\vert d\Omega$. (Middle-left and bottom-left) The real and imaginary part of the electric field calculated at the detector point. On the right: (Top-right) Scattered field $\mathbf {E}$ reconstructed by Eq. (9) with $f_\rho (\lambda )=\lambda -\lambda _0$ and $f_\rho (\lambda )=\lambda ^2$. The orange vertical line indicates the value $\Re (\omega _0)$. (Bottom-right) Scattered field $\mathbf {E}$ rebuilt by Eq. (7).
Fig. 5.
Fig. 5. Spectrum of complex eigenfrequencies (bottom left) corresponding to the 4-pole permittivity (top left). Four eigenfields (real part) are depicted at the right.
Fig. 6.
Fig. 6. Scattered field $\mathbf {E}$ obtained by expansion for different functions of $f_\rho$ (blue, red and purple curves) or by solving a direct problem classically (green dots) corresponding to the 4-pole permittivity. The orange vertical line indicates the value $\Re (\omega _0)$.
Fig. 7.
Fig. 7. Spectrum of complex eigenfrequencies (bottom left) of magnetic field $\mathbf {H}$ corresponding to the 4-pole permittivity (top left). The green crosses refer to the plasmons, solutions of $\varepsilon (\omega _2)=-1$ (There should be 4 green crosses but the forth one is out of our domain of interest). Four eigenfields (real part) are also depicted at the bottom.
Fig. 8.
Fig. 8. Scattered field $\mathbf {H}$ obtained by expansion for different functions $f_\rho =1$ (blue curves) and $f_\rho =\lambda$ (red curves) or by solving a direct problem classically (green dots) corresponding to the 4-pole permittivity. The purple vertical lines indicate the positions of $\Re (\omega _2)$.
Fig. 9.
Fig. 9. The upper half of the geometry for the unbounded structure.
Fig. 10.
Fig. 10. Spectrum of complex eigenfrequencies (bottom left) of electric field in the unbounded structure. The eigenfield of PML mode is depicted on the right. The purple line illustrates the slope $\theta \approx -0.20456$.
Fig. 11.
Fig. 11. Electric field obtained by expansion for different functions $f_\rho =1$ (blue curves) and $f_\rho =\lambda$(red curves) or by solving a direct problem classically (green dots) for the unbounded structure.

Equations (26)

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× H ( r ) + i ω ε ε ( r , i ω ) E ( r ) = J × E ( r ) i ω μ μ ( r , i ω ) H ( r ) = 0
M ξ ξ , χ χ ( λ ) u = S ,
u = E , ξ ξ = μ μ , χ χ = ε ε , S = λ J for electric fields u = H , ξ ξ = ε ε , χ χ = μ μ , S = × ( ε ε 1 J ) for magnetic fields .
M ξ ξ , χ χ ( λ n ) v n = 0 ,
R L ( λ ) := i = 0 N R i ( λ ) L i ,
R i ( λ ) = n i ( λ ) d i ( λ ) .
w n | R L ( λ n ) = 0 and R L ( λ n ) | v n = 0 ,
N N = sup i ( deg ( n i ) + h i deg ( d h ) ) and N D = i deg ( d i ) ,
u = n g σ ( λ n ) g σ ( λ ) 1 λ λ n w n , D ( λ ) S w n , N L ( λ n ) v n v n
u = n g σ ( λ n ) D ( λ ) g σ ( λ ) D ( λ n ) 1 λ λ n w n , S w n , R L ( λ n ) v n v n = n f ρ ( λ n ) f ρ ( λ ) 1 λ λ n w n , S w n , R L ( λ n ) v n v n
E = n f ρ ( λ n ) f ρ ( λ ) 1 λ λ n E l n , S E l n , M μ μ , ε ε ( λ n ) E r n E r n ,
E l n , M μ μ , ε ε ( λ n ) E r n = Ω [ E l n ¯ ( × ( ( μ μ 1 ( λ n ) ) × E r n ) ) + E l n ¯ ( ( λ n 2 ε ε ( λ n ) ) E r n ) ] d Ω .
N N N D = 2.
M μ μ , ε ε ( λ ) = μ 1 ( λ ) n 0 / d 0 × ( × ) L 0 + λ 2 ε ( λ ) n 1 / d 1 I L 1 ,
E = n λ n λ ( λ λ n ) Ω E l n ¯ S d Ω Ω [ E l n ¯ ( × ( ( μ μ 1 ( λ n ) ) × E r n ) ) + E l n ¯ ( ( λ n 2 ε ε ( λ n ) ) E r n ) ] d Ω E r n ,
E ( r , ω ) n α n ( ω ) E r n ( r ) α n ( ω ) = ω p E r n ( r 0 ) ( ω ω ~ n ) Ω [ E r n ( ω ε ) ω E r n H r n ( ω μ ) ω H r n ] d Ω + f n ( ω )
H = n f ρ ( λ n ) f ρ ( λ ) 1 λ λ n H l n , S H l n , M ε ε , μ μ ( λ n ) H r n H r n ,
H l n , M ε ε , μ μ ( λ n ) H r n = Ω [ H l n ¯ ( × ( ( ε ε 1 ( λ n ) ) × H r n ) ) + H l n ¯ ( ( λ n 2 μ μ ( λ n ) ) H r n ) ] d Ω .
ε Si ( r , ω ) = 1 + i = 1 N p ( A i ( r ) ω ω i ε A i ( r ) ¯ ω + ω i ε ¯ ) ,
δ δ s := J s 1 δ δ J s det ( J s ) for δ δ = { ε ε , μ μ } ,
J s = { diag ( s x , 1 , 1 ) in Ω PML x diag ( 1 , s y , 1 ) in Ω PML y diag ( s x , s y , 1 ) in Ω PML x y ,
y , M ξ ξ , χ χ ( λ ) x = Ω y ¯ ( × ( ξ ξ 1 ( λ ) × x ) + λ 2 χ χ ( λ ) x ) d Ω = Ω ( × ( ξ ξ ( λ ) × y ¯ ) + λ 2 χ χ ( λ ) y ¯ ) x d Ω + Ω [ ( ξ ξ ( λ ) × y ¯ ) ( n × x ) ( n × y ¯ ) ( ξ ξ 1 ( λ ) × x ) ] d S
y , M ξ ξ , χ χ ( λ ) x = Ω ( × ( ξ ξ ( λ ) × y ¯ ) + λ ¯ 2 χ χ ( λ ) y ¯ ¯ ) x d Ω = M ξ ξ , χ χ ( λ ) y , x
M ξ ξ , χ χ ( λ ) = × ( ξ ξ ( λ ¯ ) × ) + λ ¯ 2 χ χ ( λ ¯ ) = M ξ ξ , χ χ ( λ ¯ ) .
M ξ ξ , χ χ ( λ n ) w n ¯ = × ( ξ ξ 1 ( λ n ) × w n ¯ ) + λ 2 χ χ ( λ n ) w n ¯ = 0 ,
w n κ κ ( r ) = w # n κ κ ( r ) exp ( i κ κ r ) v n κ κ ( r ) = v # n κ κ ( r ) exp ( i κ κ r )