Abstract

Determining the electromagnetic field response of photonic and plasmonic resonators is a formidable task in general. Field expansions in terms of quasi-normal modes (QNMs) are often used, since only a few of these modes are typically required for an accurate field description. We show that by exploiting the structure of Maxwell’s equations, conjugate-symmetric frequency-domain field expansions can be efficiently computed via a Lanczos-type algorithm. Dominant QNMs can be identified a posteriori with error control and without a priori mode selection. Discrete QNM approximations of resonating nanostructures are presented and the spontaneous decay rate of a quantum emitter is also considered.

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

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  1. F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80(21), 4057–4059 (2002).
    [Crossref]
  2. B. Rolly, B. Stout, and N. Bonod, “Boosting the directivity of optical antennas with magnetic and electric dipolar resonant particles,” Opt. Express 20(18), 20376–20386 (2012).
    [Crossref]
  3. S. Peng and G. M. Morris, “Efficient implementation of rigorous coupled-wave analysis for surface-relief gratings,” J. Opt. Soc. Am. A 12(5), 1087–1096 (1995).
    [Crossref]
  4. R. Faggiani, A. Losquin, J. Yang, E. Marsell, A. Mikkelsen, and P. Lalanne, “Modal analysis of the ultrafast dynamics of optical nanoresonators,” ACS Photonics 4(4), 897–904 (2017).
    [Crossref]
  5. L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd Ed. (Cambridge University, 2012).
  6. 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]
  7. 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]
  8. L. Zschiedrich, F. Binkowski, N. Nikolay, O. Benson, G. Kewes, and S. Burger, “Riesz-projection-based theory of light-matter interaction in dispersive nanoresonators,” Phys. Rev. A 98(4), 043806 (2018).
    [Crossref]
  9. A. Gras, W. Yan, and P. Lalanne, “Quasinormal-mode analysis of grating spectra at fixed incidence angles,” Opt. Lett. 44(14), 3494–3497 (2019).
    [Crossref]
  10. 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]
  11. F. Zolla, A. Nicolet, and G. Demésy, “Photonics in highly dispersive media: the exact modal expansion,” Opt. Lett. 43(23), 5813–5816 (2018).
    [Crossref]
  12. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd Ed. (Artech House, 2000).
  13. J. Zimmerling, L. Wei, P. Urbach, and R. Remis, “A Lanczos model-order reduction technique to efficiently simulate electromagnetic wave propagation in dispersive media,” J. Comput. Phys. 315, 348–362 (2016).
    [Crossref]
  14. W. Chew, “Electromagnetic theory on a lattice,” J. Appl. Phys. 75(10), 4843–4850 (1994).
    [Crossref]
  15. J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994).
    [Crossref]
  16. W. Chew and W. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7(13), 599–604 (1994).
    [Crossref]
  17. V. Druskin and R. Remis, “A Krylov stability-corrected coordinate-stretching method to simulate wave propagation in unbounded domains,” SIAM J. on Sci. Comput. 35(2), B376–B400 (2013).
    [Crossref]
  18. V. Druskin, S. Güttel, and L. Knizhnerman, “Near-optimal perfectly matched layers for indefinite Helmholtz problems,” SIAM Rev. 58(1), 90–116 (2016).
    [Crossref]
  19. J. Zimmerling, L. Wei, P. Urbach, and R. Remis, “Efficient computation of the spontaneous decay rate of arbitrarily shaped 3D nanosized resonators: a Krylov model-order reduction approach,” Appl. Phys. A 122(3), 158 (2016).
    [Crossref]
  20. A. N. Krylov, “On the numerical solution of the equation by which in technical questions frequencies of small oscillations of material systems are determined,” Otdelenie Matematicheskikh i Estestvennykh Nauk 7, 491–539 (1931).
  21. J. Liesen and Z. Strakos, Krylov Subspace Methods (Oxford University, 2013).
  22. G. Golub and C. V. Loan, Matrix Computations, 4th Ed. (Johns Hopkins University, 2013).

2019 (2)

2018 (3)

F. Zolla, A. Nicolet, and G. Demésy, “Photonics in highly dispersive media: the exact modal expansion,” Opt. Lett. 43(23), 5813–5816 (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]

L. Zschiedrich, F. Binkowski, N. Nikolay, O. Benson, G. Kewes, and S. Burger, “Riesz-projection-based theory of light-matter interaction in dispersive nanoresonators,” Phys. Rev. A 98(4), 043806 (2018).
[Crossref]

2017 (1)

R. Faggiani, A. Losquin, J. Yang, E. Marsell, A. Mikkelsen, and P. Lalanne, “Modal analysis of the ultrafast dynamics of optical nanoresonators,” ACS Photonics 4(4), 897–904 (2017).
[Crossref]

2016 (3)

J. Zimmerling, L. Wei, P. Urbach, and R. Remis, “A Lanczos model-order reduction technique to efficiently simulate electromagnetic wave propagation in dispersive media,” J. Comput. Phys. 315, 348–362 (2016).
[Crossref]

V. Druskin, S. Güttel, and L. Knizhnerman, “Near-optimal perfectly matched layers for indefinite Helmholtz problems,” SIAM Rev. 58(1), 90–116 (2016).
[Crossref]

J. Zimmerling, L. Wei, P. Urbach, and R. Remis, “Efficient computation of the spontaneous decay rate of arbitrarily shaped 3D nanosized resonators: a Krylov model-order reduction approach,” Appl. Phys. A 122(3), 158 (2016).
[Crossref]

2013 (2)

V. Druskin and R. Remis, “A Krylov stability-corrected coordinate-stretching method to simulate wave propagation in unbounded domains,” SIAM J. on Sci. Comput. 35(2), B376–B400 (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 (1)

2002 (1)

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80(21), 4057–4059 (2002).
[Crossref]

1995 (1)

1994 (3)

W. Chew, “Electromagnetic theory on a lattice,” J. Appl. Phys. 75(10), 4843–4850 (1994).
[Crossref]

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994).
[Crossref]

W. Chew and W. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7(13), 599–604 (1994).
[Crossref]

1931 (1)

A. N. Krylov, “On the numerical solution of the equation by which in technical questions frequencies of small oscillations of material systems are determined,” Otdelenie Matematicheskikh i Estestvennykh Nauk 7, 491–539 (1931).

Arnold, S.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80(21), 4057–4059 (2002).
[Crossref]

Benson, O.

L. Zschiedrich, F. Binkowski, N. Nikolay, O. Benson, G. Kewes, and S. Burger, “Riesz-projection-based theory of light-matter interaction in dispersive nanoresonators,” Phys. Rev. A 98(4), 043806 (2018).
[Crossref]

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.

Binkowski, F.

Bonod, N.

Braun, D.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80(21), 4057–4059 (2002).
[Crossref]

Burger, S.

Chew, W.

W. Chew and W. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7(13), 599–604 (1994).
[Crossref]

W. Chew, “Electromagnetic theory on a lattice,” J. Appl. Phys. 75(10), 4843–4850 (1994).
[Crossref]

Demésy, G.

Druskin, V.

V. Druskin, S. Güttel, and L. Knizhnerman, “Near-optimal perfectly matched layers for indefinite Helmholtz problems,” SIAM Rev. 58(1), 90–116 (2016).
[Crossref]

V. Druskin and R. Remis, “A Krylov stability-corrected coordinate-stretching method to simulate wave propagation in unbounded domains,” SIAM J. on Sci. Comput. 35(2), B376–B400 (2013).
[Crossref]

Faggiani, R.

R. Faggiani, A. Losquin, J. Yang, E. Marsell, A. Mikkelsen, and P. Lalanne, “Modal analysis of the ultrafast dynamics of optical nanoresonators,” ACS Photonics 4(4), 897–904 (2017).
[Crossref]

Golub, G.

G. Golub and C. V. Loan, Matrix Computations, 4th Ed. (Johns Hopkins University, 2013).

Gralak, B.

Gras, A.

Güttel, S.

V. Druskin, S. Güttel, and L. Knizhnerman, “Near-optimal perfectly matched layers for indefinite Helmholtz problems,” SIAM Rev. 58(1), 90–116 (2016).
[Crossref]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd Ed. (Artech House, 2000).

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd Ed. (Cambridge University, 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.

Kewes, G.

L. Zschiedrich, F. Binkowski, N. Nikolay, O. Benson, G. Kewes, and S. Burger, “Riesz-projection-based theory of light-matter interaction in dispersive nanoresonators,” Phys. Rev. A 98(4), 043806 (2018).
[Crossref]

Khoshsima, M.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80(21), 4057–4059 (2002).
[Crossref]

Knizhnerman, L.

V. Druskin, S. Güttel, and L. Knizhnerman, “Near-optimal perfectly matched layers for indefinite Helmholtz problems,” SIAM Rev. 58(1), 90–116 (2016).
[Crossref]

Krylov, A. N.

A. N. Krylov, “On the numerical solution of the equation by which in technical questions frequencies of small oscillations of material systems are determined,” Otdelenie Matematicheskikh i Estestvennykh Nauk 7, 491–539 (1931).

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]

A. Gras, W. Yan, and P. Lalanne, “Quasinormal-mode analysis of grating spectra at fixed incidence angles,” Opt. Lett. 44(14), 3494–3497 (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]

R. Faggiani, A. Losquin, J. Yang, E. Marsell, A. Mikkelsen, and P. Lalanne, “Modal analysis of the ultrafast dynamics of optical nanoresonators,” ACS Photonics 4(4), 897–904 (2017).
[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]

Libchaber, A.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80(21), 4057–4059 (2002).
[Crossref]

Liesen, J.

J. Liesen and Z. Strakos, Krylov Subspace Methods (Oxford University, 2013).

Liu, H. T.

Loan, C. V.

G. Golub and C. V. Loan, Matrix Computations, 4th Ed. (Johns Hopkins University, 2013).

Losquin, A.

R. Faggiani, A. Losquin, J. Yang, E. Marsell, A. Mikkelsen, and P. Lalanne, “Modal analysis of the ultrafast dynamics of optical nanoresonators,” ACS Photonics 4(4), 897–904 (2017).
[Crossref]

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]

Marsell, E.

R. Faggiani, A. Losquin, J. Yang, E. Marsell, A. Mikkelsen, and P. Lalanne, “Modal analysis of the ultrafast dynamics of optical nanoresonators,” ACS Photonics 4(4), 897–904 (2017).
[Crossref]

Mikkelsen, A.

R. Faggiani, A. Losquin, J. Yang, E. Marsell, A. Mikkelsen, and P. Lalanne, “Modal analysis of the ultrafast dynamics of optical nanoresonators,” ACS Photonics 4(4), 897–904 (2017).
[Crossref]

Morris, G. M.

Nicolet, A.

Nikolay, N.

L. Zschiedrich, F. Binkowski, N. Nikolay, O. Benson, G. Kewes, and S. Burger, “Riesz-projection-based theory of light-matter interaction in dispersive nanoresonators,” Phys. Rev. A 98(4), 043806 (2018).
[Crossref]

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd Ed. (Cambridge University, 2012).

Peng, S.

Remis, R.

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]

J. Zimmerling, L. Wei, P. Urbach, and R. Remis, “Efficient computation of the spontaneous decay rate of arbitrarily shaped 3D nanosized resonators: a Krylov model-order reduction approach,” Appl. Phys. A 122(3), 158 (2016).
[Crossref]

J. Zimmerling, L. Wei, P. Urbach, and R. Remis, “A Lanczos model-order reduction technique to efficiently simulate electromagnetic wave propagation in dispersive media,” J. Comput. Phys. 315, 348–362 (2016).
[Crossref]

V. Druskin and R. Remis, “A Krylov stability-corrected coordinate-stretching method to simulate wave propagation in unbounded domains,” SIAM J. on Sci. Comput. 35(2), B376–B400 (2013).
[Crossref]

Rolly, B.

Sauvan, C.

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]

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]

Stout, B.

Strakos, Z.

J. Liesen and Z. Strakos, Krylov Subspace Methods (Oxford University, 2013).

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd Ed. (Artech House, 2000).

Teraoka, I.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80(21), 4057–4059 (2002).
[Crossref]

Truong, M. D.

Urbach, 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]

J. Zimmerling, L. Wei, P. Urbach, and R. Remis, “Efficient computation of the spontaneous decay rate of arbitrarily shaped 3D nanosized resonators: a Krylov model-order reduction approach,” Appl. Phys. A 122(3), 158 (2016).
[Crossref]

J. Zimmerling, L. Wei, P. Urbach, and R. Remis, “A Lanczos model-order reduction technique to efficiently simulate electromagnetic wave propagation in dispersive media,” J. Comput. Phys. 315, 348–362 (2016).
[Crossref]

Vollmer, F.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80(21), 4057–4059 (2002).
[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]

Weedon, W.

W. Chew and W. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7(13), 599–604 (1994).
[Crossref]

Wei, L.

J. Zimmerling, L. Wei, P. Urbach, and R. Remis, “A Lanczos model-order reduction technique to efficiently simulate electromagnetic wave propagation in dispersive media,” J. Comput. Phys. 315, 348–362 (2016).
[Crossref]

J. Zimmerling, L. Wei, P. Urbach, and R. Remis, “Efficient computation of the spontaneous decay rate of arbitrarily shaped 3D nanosized resonators: a Krylov model-order reduction approach,” Appl. Phys. A 122(3), 158 (2016).
[Crossref]

Weiss, T.

Yan, W.

Yang, J.

R. Faggiani, A. Losquin, J. Yang, E. Marsell, A. Mikkelsen, and P. Lalanne, “Modal analysis of the ultrafast dynamics of optical nanoresonators,” ACS Photonics 4(4), 897–904 (2017).
[Crossref]

Zimmerling, J.

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]

J. Zimmerling, L. Wei, P. Urbach, and R. Remis, “Efficient computation of the spontaneous decay rate of arbitrarily shaped 3D nanosized resonators: a Krylov model-order reduction approach,” Appl. Phys. A 122(3), 158 (2016).
[Crossref]

J. Zimmerling, L. Wei, P. Urbach, and R. Remis, “A Lanczos model-order reduction technique to efficiently simulate electromagnetic wave propagation in dispersive media,” J. Comput. Phys. 315, 348–362 (2016).
[Crossref]

Zolla, F.

Zschiedrich, L.

ACS Photonics (1)

R. Faggiani, A. Losquin, J. Yang, E. Marsell, A. Mikkelsen, and P. Lalanne, “Modal analysis of the ultrafast dynamics of optical nanoresonators,” ACS Photonics 4(4), 897–904 (2017).
[Crossref]

Appl. Phys. A (1)

J. Zimmerling, L. Wei, P. Urbach, and R. Remis, “Efficient computation of the spontaneous decay rate of arbitrarily shaped 3D nanosized resonators: a Krylov model-order reduction approach,” Appl. Phys. A 122(3), 158 (2016).
[Crossref]

Appl. Phys. Lett. (1)

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80(21), 4057–4059 (2002).
[Crossref]

J. Appl. Phys. (1)

W. Chew, “Electromagnetic theory on a lattice,” J. Appl. Phys. 75(10), 4843–4850 (1994).
[Crossref]

J. Comput. Phys. (2)

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994).
[Crossref]

J. Zimmerling, L. Wei, P. Urbach, and R. Remis, “A Lanczos model-order reduction technique to efficiently simulate electromagnetic wave propagation in dispersive media,” J. Comput. Phys. 315, 348–362 (2016).
[Crossref]

J. Opt. Soc. Am. A (2)

Laser Photonics Rev. (1)

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]

Microw. Opt. Technol. Lett. (1)

W. Chew and W. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7(13), 599–604 (1994).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Otdelenie Matematicheskikh i Estestvennykh Nauk (1)

A. N. Krylov, “On the numerical solution of the equation by which in technical questions frequencies of small oscillations of material systems are determined,” Otdelenie Matematicheskikh i Estestvennykh Nauk 7, 491–539 (1931).

Phys. Rev. A (1)

L. Zschiedrich, F. Binkowski, N. Nikolay, O. Benson, G. Kewes, and S. Burger, “Riesz-projection-based theory of light-matter interaction in dispersive nanoresonators,” Phys. Rev. A 98(4), 043806 (2018).
[Crossref]

Phys. Rev. Lett. (1)

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]

SIAM J. on Sci. Comput. (1)

V. Druskin and R. Remis, “A Krylov stability-corrected coordinate-stretching method to simulate wave propagation in unbounded domains,” SIAM J. on Sci. Comput. 35(2), B376–B400 (2013).
[Crossref]

SIAM Rev. (1)

V. Druskin, S. Güttel, and L. Knizhnerman, “Near-optimal perfectly matched layers for indefinite Helmholtz problems,” SIAM Rev. 58(1), 90–116 (2016).
[Crossref]

Other (4)

J. Liesen and Z. Strakos, Krylov Subspace Methods (Oxford University, 2013).

G. Golub and C. V. Loan, Matrix Computations, 4th Ed. (Johns Hopkins University, 2013).

L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd Ed. (Cambridge University, 2012).

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd Ed. (Artech House, 2000).

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Figures (3)

Fig. 1.
Fig. 1. Purcell factor of a quantum emitter (arrow) centered 10 nm above a $30~\textrm {nm} \times 100~\textrm {nm}$ nanorod computed using the Fourier modal method of [7] (solid line) and the Lanczos ROM (dashed line). (a) Simulated configuration, (b) isosurface plots of $\textrm {Re}(\hat {E}_z)$ (b) and $\textrm {Re}(\hat {E}_x)$ (c) of the dominant QNM with a wavelength $\lambda =926 + 47\textrm {i}$ nm. The small shift between the two responses is due to the use of a 2 nm staggered finite-difference grid as opposed to the method in [7], which exploits the cylindrical symmetry of the configuration.
Fig. 2.
Fig. 2. (Bottom:) Purcell factor of a quantum emitter (arrow) located 10 nm above a $102~\textrm {nm} \times 40~\textrm {nm} \times 20~\textrm {nm}$ nanoplate. The Purcell factor is computed using Lanczos reduction (Eq. (16)) and an expansion in the three most dominant QNMs. The real part of the $\hat {E}_x$ field of the three dominant QNMs is depicted along with their individual contribution to the SD rate. (a) $\textrm {Re}(\hat {E}_x)$ of the QNM with $\lambda =542.4+10.8\textrm {i}$ nm. (b) $\textrm {Re}(\hat {E}_x)$ of the QNM with $\lambda =599.5+13.5\textrm {i}$ nm. (c) Simulated configuration. (d) $\textrm {Re}(\hat {E}_x)$ of the QNM with $\lambda =942.7+50.5\textrm {i}$ nm. (Top:) Pointwise relative error of the QNM expansion with respect to the Lanczos solution.
Fig. 3.
Fig. 3. Electric field distributions of QNMs in a coupled parallel plate configuration, as a function of the wavelength in vacuum. (a) $\hat {E}_x$-field of the fundamental symmetric QNM ($\lambda =891+68\textrm {i}$ nm). (b$)\hat {E}_x$-field of a higher harmonic anti-symmetric QNM ($\lambda =622+14\textrm {i}$ nm). (c) – (e$)\hat {E}_x$, $\hat {E}_y$, and $\hat {E}_z$-fields of the fundamental anti-symmetric QNM ($\lambda =1034 +34\textrm {i}$ nm).

Equations (21)

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P ( ω ) = ω 2 Im [ p ^ ( ω ) E ^ ( x S , ω ) n s ] ,
ω 2 P ^ i ω β 2 P ^ + β 1 P ^ = β 0 E ^ ,
[ i ω ε 1 × i ω 1 β 0 β 1 β 2 i ω × i ω μ ] [ E ^ P ^ U ^ H ^ ] = [ J ^ ext 0 0 0 ] ,
( D + S i ω M ) f ^ cs = p ^ ( ω ) q ,
f ^ ( ω ) = p ^ ( ω ) G ^ ( A , ω ) q ,
G ^ ( A , ω ) = R ^ ( A , ω ) + R ^ ( A , ω ) ,
R ^ ( A , ω ) = χ ( A ) ( A i ω I ) 1 ,
( A x ) T W M y = x T W M A y x , y C n .
L = Ω ε E ^ 2 + β 1 β 0 1 P ^ 2 β 0 1 U ^ 2 μ H ^ 2 d V = Ω E ^ ω ε c ( ω ) ω E ^ μ H ^ H ^ d V ,
A V m = V m T m + β m + 1 v m + 1 e m T ,
v i T W M v j = δ i j ,
v 1 = [ q T W M q ] 1 / 2 q ,
y j = V m z j [ m ] ,
f ^ m ( ω ) = i ω p ^ ( ω ) [ q T W M q ] 1 / 2 [ V m R ^ ( T m , ω ) e 1 + V m R ^ ( T m , ω ) e 1 ] ,
E ^ ( x S , ω ) n s f ^ m T ( ω ) W M q ,
P m ( ω ) = P a Re [ e 1 T G ^ ( T m , ω ) e 1 ] ,
P m ( ω ) = P a Re [ k = 1 m w k 2 R ^ ( θ k [ m ] , ω ) + ( w k ) 2 R ^ ( θ k [ m ] , ω ) ] ,
χ ^ k ( k , i ω ) = a k ( k ) b k ( k ) i ω ,
[ D ( i ω ) + S i ω M ] f = 0 ,
[ D ( i ω 0 ) + S i ω M ] f = 0 .
φ m ( i ω ) = j = 1 m r j i ω θ j .