Abstract

A fundamental feature of the quasi-normal modes (QNMs), which describe light interaction with open (leaky) systems like nanoparticles, lies in the question of the completeness of the QNMs representation and in the divergence of their field profile due to their leaky behavior and complex eigenfrequency. In this article, the QNMs expansion is obtained by taking into consideration the frequency dispersion and the causality principle. The derivation based on the complex analysis ensures the completeness of the QNMs expansion and prevents from any divergence of the field profile. The general derivation is tested in the case of a one-dimensional open resonator made of a homogeneous absorptive medium with frequency dispersion given by the Lorentz model. For a harmonic excitation, the result of the QNMs expansion perfectly matches the exact formula for the field distribution outside as well as inside the resonator.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref] [PubMed]
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    [Crossref]
  4. X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband light bending with plasmonic nanoantennas,” Science 335, 427 (2012).
    [Crossref]
  5. P. Leung, S. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3057 (1994).
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  22. B. Gralak, “Analytic properties of the electromagnetic green’s function,” J. Math. Phys. 58, 071501 (2017).
    [Crossref]
  23. J. Makialo, M. Kauranen, and S. Suuriniemi, “Modes and resonances of plasmonic scatterers,” Phys. Rev. B 89, 165429 (2014).
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2018 (3)

M. Kamandar Dezfouli and S. Hughes, “Regularized quasinormal modes for plasmonic resonators and open cavities,” Phys. Rev. B 97, 115302 (2018).
[Crossref]

P. Lalanne, W. Yan, K. Vynck, C. Sauvan, and J. Hugonin, “Light interaction with photonic and plasmonic resonances,” Laser Photon. Rev. 12, 1700113 (2018).
[Crossref]

M. I. Abdelrahman and B. Gralak, “Modal analysis of wave propagation in dispersive media,” Phys. Rev. A 97, 013824 (2018).
[Crossref]

2017 (3)

B. Gralak, “Analytic properties of the electromagnetic green’s function,” J. Math. Phys. 58, 071501 (2017).
[Crossref]

D. A. Powell, “Interference between the modes of an all-dielectric meta-atom,” Phys. Rev. Appl. 7, 034006 (2017).
[Crossref]

F. Alpeggiani, N. Parappurath, E. Verhagen, and L. Kuipers, “Quasinormal-mode expansion of the scattering matrix,” Phys. Rev. X 7, 021035 (2017).

2015 (1)

2014 (5)

R.-C. Ge, P. T. Kristensen, J. F. Young, and S. Hughes, “Quasinormal mode approach to modelling light-emission and propagation in nanoplasmonics,” New J. Phys. 16, 113048 (2014).
[Crossref]

R.-C. Ge and S. Hughes, “Design of an efficient single photon source from a metallic nanorod dimer: a quasi-normal mode finite-difference time-domain approach,” Opt. Lett. 39, 4235–4238 (2014).
[Crossref] [PubMed]

B. Vial, G. Demésy, F. Zolla, A. Nicolet, M. Commandré, C. Hecquet, T. Begou, S. Tisserand, S. Gautier, and V. Sauget, “Resonant metamaterial absorbers for infrared spectral filtering: quasimodal analysis, design, fabrication, and characterization,” J. Opt. Soc. Am. B 31, 1339–1346 (2014).
[Crossref]

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

J. Makialo, M. Kauranen, and S. Suuriniemi, “Modes and resonances of plasmonic scatterers,” Phys. Rev. B 89, 165429 (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, 27371–27382 (2013).
[Crossref] [PubMed]

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

2012 (2)

P. T. Kristensen, C. Van Vlack, and S. Hughes, “Generalized effective mode volume for leaky optical cavities,” Opt. Lett. 37, 1649–1651 (2012).
[Crossref] [PubMed]

X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband light bending with plasmonic nanoantennas,” Science 335, 427 (2012).
[Crossref]

2008 (1)

A. Grigorenko, N. Roberts, M. Dickinson, and Y. Zhang, “Nanometric optical tweezers based on nanostructured substrates,” Nat. Photonics 2, 365 (2008).
[Crossref]

2006 (1)

X. Huang, I. H. El-Sayed, W. Qian, and M. A. El-Sayed, “Cancer cell imaging and photothermal therapy in the near-infrared region by using gold nanorods,” J. Am. Chem. Soc. 128, 2115–2120 (2006).
[Crossref] [PubMed]

1998 (1)

E. Ching, P. Leung, A. M. van den Brink, W. Suen, S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545 (1998).
[Crossref]

1997 (1)

S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced raman scattering,” Science 275, 1102–1106 (1997).
[Crossref] [PubMed]

1994 (2)

P. Leung, S. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3057 (1994).
[Crossref] [PubMed]

P. Leung, S. Liu, and K. Young, “Completeness and time-independent perturbation of the quasinormal modes of an absorptive and leaky cavity,” Phys. Rev. A 49, 3982 (1994).
[Crossref] [PubMed]

1992 (1)

H.-P. Nollert and B. G. Schmidt, “Quasinormal modes of schwarzschild black holes: Defined and calculated via laplace transformation,” Phys. Rev. D 45, 2617 (1992).
[Crossref]

Abdelrahman, M. I.

M. I. Abdelrahman and B. Gralak, “Modal analysis of wave propagation in dispersive media,” Phys. Rev. A 97, 013824 (2018).
[Crossref]

Alpeggiani, F.

F. Alpeggiani, N. Parappurath, E. Verhagen, and L. Kuipers, “Quasinormal-mode expansion of the scattering matrix,” Phys. Rev. X 7, 021035 (2017).

Bai, Q.

Begou, T.

Boltasseva, A.

X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband light bending with plasmonic nanoantennas,” Science 335, 427 (2012).
[Crossref]

Ching, E.

E. Ching, P. Leung, A. M. van den Brink, W. Suen, S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545 (1998).
[Crossref]

Commandré, M.

de Lasson, J. R.

Demésy, G.

Dickinson, M.

A. Grigorenko, N. Roberts, M. Dickinson, and Y. Zhang, “Nanometric optical tweezers based on nanostructured substrates,” Nat. Photonics 2, 365 (2008).
[Crossref]

El-Sayed, I. H.

X. Huang, I. H. El-Sayed, W. Qian, and M. A. El-Sayed, “Cancer cell imaging and photothermal therapy in the near-infrared region by using gold nanorods,” J. Am. Chem. Soc. 128, 2115–2120 (2006).
[Crossref] [PubMed]

El-Sayed, M. A.

X. Huang, I. H. El-Sayed, W. Qian, and M. A. El-Sayed, “Cancer cell imaging and photothermal therapy in the near-infrared region by using gold nanorods,” J. Am. Chem. Soc. 128, 2115–2120 (2006).
[Crossref] [PubMed]

Emani, N. K.

X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband light bending with plasmonic nanoantennas,” Science 335, 427 (2012).
[Crossref]

Emory, S. R.

S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced raman scattering,” Science 275, 1102–1106 (1997).
[Crossref] [PubMed]

Gautier, S.

Ge, R.-C.

R.-C. Ge and S. Hughes, “Design of an efficient single photon source from a metallic nanorod dimer: a quasi-normal mode finite-difference time-domain approach,” Opt. Lett. 39, 4235–4238 (2014).
[Crossref] [PubMed]

R.-C. Ge, P. T. Kristensen, J. F. Young, and S. Hughes, “Quasinormal mode approach to modelling light-emission and propagation in nanoplasmonics,” New J. Phys. 16, 113048 (2014).
[Crossref]

Gralak, B.

M. I. Abdelrahman and B. Gralak, “Modal analysis of wave propagation in dispersive media,” Phys. Rev. A 97, 013824 (2018).
[Crossref]

B. Gralak, “Analytic properties of the electromagnetic green’s function,” J. Math. Phys. 58, 071501 (2017).
[Crossref]

Gregersen, N.

Grigorenko, A.

A. Grigorenko, N. Roberts, M. Dickinson, and Y. Zhang, “Nanometric optical tweezers based on nanostructured substrates,” Nat. Photonics 2, 365 (2008).
[Crossref]

Hecquet, C.

Huang, X.

X. Huang, I. H. El-Sayed, W. Qian, and M. A. El-Sayed, “Cancer cell imaging and photothermal therapy in the near-infrared region by using gold nanorods,” J. Am. Chem. Soc. 128, 2115–2120 (2006).
[Crossref] [PubMed]

Hughes, S.

M. Kamandar Dezfouli and S. Hughes, “Regularized quasinormal modes for plasmonic resonators and open cavities,” Phys. Rev. B 97, 115302 (2018).
[Crossref]

R.-C. Ge, P. T. Kristensen, J. F. Young, and S. Hughes, “Quasinormal mode approach to modelling light-emission and propagation in nanoplasmonics,” New J. Phys. 16, 113048 (2014).
[Crossref]

R.-C. Ge and S. Hughes, “Design of an efficient single photon source from a metallic nanorod dimer: a quasi-normal mode finite-difference time-domain approach,” Opt. Lett. 39, 4235–4238 (2014).
[Crossref] [PubMed]

P. T. Kristensen, C. Van Vlack, and S. Hughes, “Generalized effective mode volume for leaky optical cavities,” Opt. Lett. 37, 1649–1651 (2012).
[Crossref] [PubMed]

Hugonin, J.

P. Lalanne, W. Yan, K. Vynck, C. Sauvan, and J. Hugonin, “Light interaction with photonic and plasmonic resonances,” Laser Photon. Rev. 12, 1700113 (2018).
[Crossref]

Hugonin, J.-P.

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, 27371–27382 (2013).
[Crossref] [PubMed]

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

Kamandar Dezfouli, M.

M. Kamandar Dezfouli and S. Hughes, “Regularized quasinormal modes for plasmonic resonators and open cavities,” Phys. Rev. B 97, 115302 (2018).
[Crossref]

Kauranen, M.

J. Makialo, M. Kauranen, and S. Suuriniemi, “Modes and resonances of plasmonic scatterers,” Phys. Rev. B 89, 165429 (2014).
[Crossref]

Kildishev, A. V.

X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband light bending with plasmonic nanoantennas,” Science 335, 427 (2012).
[Crossref]

Kristensen, P. T.

Kuipers, L.

F. Alpeggiani, N. Parappurath, E. Verhagen, and L. Kuipers, “Quasinormal-mode expansion of the scattering matrix,” Phys. Rev. X 7, 021035 (2017).

Lalanne, P.

P. Lalanne, W. Yan, K. Vynck, C. Sauvan, and J. Hugonin, “Light interaction with photonic and plasmonic resonances,” Laser Photon. Rev. 12, 1700113 (2018).
[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, 27371–27382 (2013).
[Crossref] [PubMed]

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

Leung, P.

E. Ching, P. Leung, A. M. van den Brink, W. Suen, S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545 (1998).
[Crossref]

P. Leung, S. Liu, and K. Young, “Completeness and time-independent perturbation of the quasinormal modes of an absorptive and leaky cavity,” Phys. Rev. A 49, 3982 (1994).
[Crossref] [PubMed]

P. Leung, S. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3057 (1994).
[Crossref] [PubMed]

Liu, S.

P. Leung, S. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3057 (1994).
[Crossref] [PubMed]

P. Leung, S. Liu, and K. Young, “Completeness and time-independent perturbation of the quasinormal modes of an absorptive and leaky cavity,” Phys. Rev. A 49, 3982 (1994).
[Crossref] [PubMed]

Makialo, J.

J. Makialo, M. Kauranen, and S. Suuriniemi, “Modes and resonances of plasmonic scatterers,” Phys. Rev. B 89, 165429 (2014).
[Crossref]

Maksymov, I.

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

Mørk, J.

Ni, X.

X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband light bending with plasmonic nanoantennas,” Science 335, 427 (2012).
[Crossref]

Nicolet, A.

Nie, S.

S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced raman scattering,” Science 275, 1102–1106 (1997).
[Crossref] [PubMed]

Nollert, H.-P.

H.-P. Nollert and B. G. Schmidt, “Quasinormal modes of schwarzschild black holes: Defined and calculated via laplace transformation,” Phys. Rev. D 45, 2617 (1992).
[Crossref]

Parappurath, N.

F. Alpeggiani, N. Parappurath, E. Verhagen, and L. Kuipers, “Quasinormal-mode expansion of the scattering matrix,” Phys. Rev. X 7, 021035 (2017).

Perrin, M.

Powell, D. A.

D. A. Powell, “Interference between the modes of an all-dielectric meta-atom,” Phys. Rev. Appl. 7, 034006 (2017).
[Crossref]

Qian, W.

X. Huang, I. H. El-Sayed, W. Qian, and M. A. El-Sayed, “Cancer cell imaging and photothermal therapy in the near-infrared region by using gold nanorods,” J. Am. Chem. Soc. 128, 2115–2120 (2006).
[Crossref] [PubMed]

Reed, M.

M. Reed and B. Simon, Methods Of Mathematical Physics. Vol. 3: Scattering Theory (Academic Press, 1979).

Roberts, N.

A. Grigorenko, N. Roberts, M. Dickinson, and Y. Zhang, “Nanometric optical tweezers based on nanostructured substrates,” Nat. Photonics 2, 365 (2008).
[Crossref]

Sauget, V.

Sauvan, C.

P. Lalanne, W. Yan, K. Vynck, C. Sauvan, and J. Hugonin, “Light interaction with photonic and plasmonic resonances,” Laser Photon. Rev. 12, 1700113 (2018).
[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, 27371–27382 (2013).
[Crossref] [PubMed]

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

Schmidt, B. G.

H.-P. Nollert and B. G. Schmidt, “Quasinormal modes of schwarzschild black holes: Defined and calculated via laplace transformation,” Phys. Rev. D 45, 2617 (1992).
[Crossref]

Shalaev, V. M.

X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband light bending with plasmonic nanoantennas,” Science 335, 427 (2012).
[Crossref]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

Simon, B.

M. Reed and B. Simon, Methods Of Mathematical Physics. Vol. 3: Scattering Theory (Academic Press, 1979).

Suen, W.

E. Ching, P. Leung, A. M. van den Brink, W. Suen, S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545 (1998).
[Crossref]

Suuriniemi, S.

J. Makialo, M. Kauranen, and S. Suuriniemi, “Modes and resonances of plasmonic scatterers,” Phys. Rev. B 89, 165429 (2014).
[Crossref]

Tisserand, S.

Tong, S.

E. Ching, P. Leung, A. M. van den Brink, W. Suen, S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545 (1998).
[Crossref]

van den Brink, A. M.

E. Ching, P. Leung, A. M. van den Brink, W. Suen, S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545 (1998).
[Crossref]

Van Vlack, C.

Verhagen, E.

F. Alpeggiani, N. Parappurath, E. Verhagen, and L. Kuipers, “Quasinormal-mode expansion of the scattering matrix,” Phys. Rev. X 7, 021035 (2017).

Vial, B.

Vynck, K.

P. Lalanne, W. Yan, K. Vynck, C. Sauvan, and J. Hugonin, “Light interaction with photonic and plasmonic resonances,” Laser Photon. Rev. 12, 1700113 (2018).
[Crossref]

Yan, W.

P. Lalanne, W. Yan, K. Vynck, C. Sauvan, and J. Hugonin, “Light interaction with photonic and plasmonic resonances,” Laser Photon. Rev. 12, 1700113 (2018).
[Crossref]

Young, J. F.

R.-C. Ge, P. T. Kristensen, J. F. Young, and S. Hughes, “Quasinormal mode approach to modelling light-emission and propagation in nanoplasmonics,” New J. Phys. 16, 113048 (2014).
[Crossref]

Young, K.

E. Ching, P. Leung, A. M. van den Brink, W. Suen, S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545 (1998).
[Crossref]

P. Leung, S. Liu, and K. Young, “Completeness and time-independent perturbation of the quasinormal modes of an absorptive and leaky cavity,” Phys. Rev. A 49, 3982 (1994).
[Crossref] [PubMed]

P. Leung, S. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3057 (1994).
[Crossref] [PubMed]

Zhang, Y.

A. Grigorenko, N. Roberts, M. Dickinson, and Y. Zhang, “Nanometric optical tweezers based on nanostructured substrates,” Nat. Photonics 2, 365 (2008).
[Crossref]

Zolla, F.

J. Am. Chem. Soc. (1)

X. Huang, I. H. El-Sayed, W. Qian, and M. A. El-Sayed, “Cancer cell imaging and photothermal therapy in the near-infrared region by using gold nanorods,” J. Am. Chem. Soc. 128, 2115–2120 (2006).
[Crossref] [PubMed]

J. Math. Phys. (1)

B. Gralak, “Analytic properties of the electromagnetic green’s function,” J. Math. Phys. 58, 071501 (2017).
[Crossref]

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

Laser Photon. Rev. (1)

P. Lalanne, W. Yan, K. Vynck, C. Sauvan, and J. Hugonin, “Light interaction with photonic and plasmonic resonances,” Laser Photon. Rev. 12, 1700113 (2018).
[Crossref]

Nat. Photonics (1)

A. Grigorenko, N. Roberts, M. Dickinson, and Y. Zhang, “Nanometric optical tweezers based on nanostructured substrates,” Nat. Photonics 2, 365 (2008).
[Crossref]

New J. Phys. (1)

R.-C. Ge, P. T. Kristensen, J. F. Young, and S. Hughes, “Quasinormal mode approach to modelling light-emission and propagation in nanoplasmonics,” New J. Phys. 16, 113048 (2014).
[Crossref]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. A (4)

P. Leung, S. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3057 (1994).
[Crossref] [PubMed]

P. Leung, S. Liu, and K. Young, “Completeness and time-independent perturbation of the quasinormal modes of an absorptive and leaky cavity,” Phys. Rev. A 49, 3982 (1994).
[Crossref] [PubMed]

M. I. Abdelrahman and B. Gralak, “Modal analysis of wave propagation in dispersive media,” Phys. Rev. A 97, 013824 (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, 023829 (2014).
[Crossref]

Phys. Rev. Appl. (1)

D. A. Powell, “Interference between the modes of an all-dielectric meta-atom,” Phys. Rev. Appl. 7, 034006 (2017).
[Crossref]

Phys. Rev. B (2)

M. Kamandar Dezfouli and S. Hughes, “Regularized quasinormal modes for plasmonic resonators and open cavities,” Phys. Rev. B 97, 115302 (2018).
[Crossref]

J. Makialo, M. Kauranen, and S. Suuriniemi, “Modes and resonances of plasmonic scatterers,” Phys. Rev. B 89, 165429 (2014).
[Crossref]

Phys. Rev. D (1)

H.-P. Nollert and B. G. Schmidt, “Quasinormal modes of schwarzschild black holes: Defined and calculated via laplace transformation,” Phys. Rev. D 45, 2617 (1992).
[Crossref]

Phys. Rev. Lett. (1)

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

Phys. Rev. X (1)

F. Alpeggiani, N. Parappurath, E. Verhagen, and L. Kuipers, “Quasinormal-mode expansion of the scattering matrix,” Phys. Rev. X 7, 021035 (2017).

Rev. Mod. Phys. (1)

E. Ching, P. Leung, A. M. van den Brink, W. Suen, S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545 (1998).
[Crossref]

Science (2)

X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband light bending with plasmonic nanoantennas,” Science 335, 427 (2012).
[Crossref]

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Other (2)

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A. E. Siegman, Lasers (University Science Books, 1986).

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Figures (5)

Fig. 1
Fig. 1 A one-dimensional resonator made of a homogeneous dispersive material is excited by a harmonic source located at x0. This interaction results in diffracted fields both in the forward (transmission) and backward (reflection) directions.
Fig. 2
Fig. 2 A comparison between the results of the exact formulas (solid lines) of T(z) and R(z) and their corresponding QNMs expansion (dashed lines) for below-resonance excitation ωs < ω0. The results are shown for different numbers of summation terms of the expansion.
Fig. 3
Fig. 3 Top) The field distribution of a Lorentz-dispersive resonator of parameters ω0L/c = 10, γL/c = 0.2, ΩL/c = 20, and for a near-resonance harmonic excitation at frequency ωsL/c = 8. The result of the QNMs expansion using 31 modes perfectly matches the exact formula. Bottom) The field distribution of the five nearest QNMs to the excitation frequency.
Fig. 4
Fig. 4 A comparison between the field distribution of QNMs formulation with the additional causality-related factor (solid lines) that shows no divergence outside the resonator, and the conventional QNMs formulation (dashed lines) that exhibits a divergence behavior. The results are shown for the two nearest QNMs to the excitation frequency of the example in Fig. 3.
Fig. 5
Fig. 5 The exact formula for T(z) and its corresponding QNMs expansion for different excitation regions. The results are shown for different numbers of summation terms in dashed lines.

Equations (16)

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× × G ( r , r ; z ) z 2 μ 0 ε ( r , z ) G ( r , r ; z ) = i z μ 0 δ ( r r ) .
G d ( r , r ; z ) = G ( r , r ; z ) G ref ( r , r ; z ) .
G ( r , r ; z ) G ref ( r , r ; z ) G 0 ( r r ; z ) G d ( r , r ; z ) 0 ,
G ^ d ( r , r ; t ) = Γ η d z e i z t G d ( r , r ; z ) ,
G d ( r , r ; z ) × e i z τ e i z | r r | / c z 0 .
G ^ d ( r , r , t ) = 2 i π { poles z q } e i z q t G d q ( r , r , z q ) ,
G ˜ τ , d ( r , r ; z ) = 1 2 π τ + | r r | / c d t e i z t G ^ d ( r , r , t ) .
G d ( r , r ; z ) = lim τ 0 { poles z q } G d q ( r , r , z q ) z z q e i ( z z q ) ( τ + | r r | / c ) .
E R ( x , z ) = E s ( x , z ) + R ( z ) e i z ( x + L / 2 ) / c x L / 2 , E i n ( x , z ) = A ( z ) e i z ε ( z ) x / c + B ( z ) e i z ε ( z ) x / c | x | L / 2 , E T ( x , z ) = T ( z ) e i z ( x L / 2 ) / c x L / 2 ,
T ( z ) = [ 1 r 0 2 ( z ) ] e i z ε ( z ) L / c 1 r 0 2 ( z ) e 2 i z ε ( z ) L / c , r 0 ( z ) = 1 ε ( z ) 1 + ε ( z ) .
G T ( x , z ) = T ( z ) z e i z ( x L / 2 ) / c x L / 2 .
E T ( x , z ) = z z q T q ( z q ) e i z q ( x L / 2 ) / c z q ( z z q ) e i ( z z q ) ( x + L / 2 ) / c + T 0 e i z ( x + L / 2 ) / c ,
E T ( x , z ) e i z ( x + L / 2 ) / c = z z q T q ( z q ) e i z q L / c z q ( z z q ) e i z ( x + L / 2 ) / c .
R ( z ) = r 0 ( z ) 1 e 2 i z ε ( z ) L / c 1 r 0 2 ( z ) e 2 i z ε ( z ) L / c .
G R ( x , z ) = R ( z ) z e i z ( x + L / 2 ) / c e i z τ x L / 2 .
E R ( x , z ) = E s ( x , z ) + z z q R q ( z q ) z q ( z z q ) e i z ( x + L / 2 ) / c e i ( z z q ) τ ,

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