Abstract

The correct numerical calculation of the resonance characteristics and, principally, the quality factor Q of contemporary photonic and plasmonic resonant systems is of utmost importance, since Q defines the bandwidth and affects nonlinear and spontaneous emission processes. Here, we comparatively assess the commonly used methods for calculating Q using spectral simulations with commercially available, general-purpose software. We study the applicability range of these methods through judiciously selected examples covering different material systems and frequency regimes from the far-infrared to the visible. We take care in highlighting the underlying physical and numerical reasons limiting the applicability of each one. Our findings demonstrate that in contemporary systems (plasmonics, 2D materials) Q calculation is not trivial, mainly due to the physical complication of strong material dispersion and light leakage. Our work can act as a reference for the mindful and accurate calculation of the quality factor and can serve as a handbook for its evaluation in guided-wave and free-space photonic and plasmonic resonant systems.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (1)

A. Abrashuly and C. Valagiannopoulos, “Limits for absorption and scattering by core-shell nanowires in the visible spectrum,” Phys. Rev. Appl. 11, 014051 (2019).
[Crossref]

2018 (4)

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

O. Tsilipakos, A. C. Tasolamprou, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Pairing toroidal and magnetic dipole resonances in elliptic dielectric rod metasurfaces for reconfigurable wavefront manipulation in reflection,” Adv. Opt. Mater. 6, 1800633 (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, 1700113 (2018).
[Crossref]

V. Stojanović, R. J. Ram, M. Popović, S. Lin, S. Moazeni, M. Wade, C. Sun, L. Alloatti, A. Atabaki, F. Pavanello, N. Mehta, and P. Bhargava, “Monolithic silicon-photonic platforms in state-of-the-art CMOS SOI processes [invited],” Opt. Express 26, 13106 (2018).
[Crossref]

2017 (1)

2016 (3)

O. Tsilipakos, T. Christopoulos, and E. E. Kriezis, “Long-range hybrid plasmonic disk resonators for mw bistability and self-pulsation,” J. Lightw. Technol. 34, 1333 (2016).
[Crossref]

T. Christopoulos, O. Tsilipakos, N. Grivas, and E. E. Kriezis, “Coupled-mode-theory framework for nonlinear resonators comprising graphene,” Phys. Rev. E 94, 062219 (2016).
[Crossref]

A. C. Tasolamprou, O. Tsilipakos, M. Kafesaki, C. M. Soukoulis, and E. N. Economou, “Toroidal eigenmodes in all-dielectric metamolecules,” Phys. Rev. B 94, 205433 (2016).
[Crossref]

2015 (1)

I. S. Lamata, P. Alonso-González, R. Hillenbrand, and A. Y. Nikitin, “Plasmons in cylindrical 2D materials as a platform for nanophotonic circuits,” ACS Photonics 2, 280 (2015).
[Crossref]

2014 (1)

L. Li, H. Lin, S. Qiao, Y. Zou, S. Danto, K. Richardson, J. D. Musgraves, N. Lu, and J. Hu, “Integrated flexible chalcogenide glass photonic devices,” Nat. Photon. 8, 643 (2014).
[Crossref]

2013 (3)

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, 237401 (2013).
[Crossref] [PubMed]

P. T. Kristensen and S. Hughes, “Modes and mode volumes of leaky optical cavities and plasmonic nanoresonators,” ACS Photonics 1, 2 (2013).
[Crossref]

P. Ma, D.-Y. Choi, Y. Yu, X. Gai, Z. Yang, S. Debbarma, S. Madden, and B. Luther-Davies, “Low-loss chalcogenide waveguides for chemical sensing in the mid-infrared,” Opt. Express 21, 29927 (2013).
[Crossref]

2012 (2)

2011 (2)

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics 5, 523–530 (2011).
[Crossref]

A. Raman and S. Fan, “Perturbation theory for plasmonic modulation and sensing,” Phys. Rev. B 83, 205131 (2011).
[Crossref]

2010 (4)

P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82, 053825 (2010).
[Crossref]

M. Notomi, “Manipulating light with strongly modulated photonic crystals,” Rep. Prog. Phys. 73, 096501 (2010).
[Crossref]

Q. Li, T. Wang, Y. Su, M. Yan, and M. Qiu, “Coupled mode theory analysis of mode-splitting in coupled cavity system,” Opt. Express 18, 8367 (2010).
[Crossref] [PubMed]

Z. Ruan and S. Fan, “Superscattering of light from subwavelength nanostructures,” Phys. Rev. Lett. 105, 013901 (2010).
[Crossref] [PubMed]

2009 (1)

P. Bharadwaj, B. Deutsch, and L. Novotny, “Optical antennas,” Adv. Opt. Photonics 1, 438 (2009).
[Crossref]

2008 (2)

K. R. Catchpole and A. Polman, “Plasmonic solar cells,” Opt. Express 16, 21793 (2008).
[Crossref] [PubMed]

G. W. Hanson, “Dyadic green’s functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103, 064302 (2008).
[Crossref]

2007 (2)

R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljačić, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A 75, 053801 (2007).
[Crossref]

P. K. Jain, X. Huang, I. H. El-Sayed, and M. A. El-Sayed, “Review of some interesting surface plasmon resonance-enhanced properties of noble metal nanoparticles and their applications to biosystems,” Plasmonics 2, 107 (2007).
[Crossref]

2006 (2)

K.-S. Lee and M. A. El-Sayed, “Gold and silver nanoparticles in sensing and imaging: sensitivity of plasmon response to size, shape, and metal composition,” J. Phys. Chem. B 110, 19220 (2006).
[Crossref] [PubMed]

V. Ilchenko and A. Matsko, “Optical resonators with whispering-gallery modes-part II: applications,” IEEE J. Sel. Top. Quant. 12, 15 (2006).
[Crossref]

2004 (2)

M. Soljačić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3, 211 (2004).
[Crossref]

K. C. Huang, M. L. Povinelli, and J. D. Joannopoulos, “Negative effective permeability in polaritonic photonic crystals,” Appl. Phys. Lett. 85, 543 (2004).
[Crossref]

2003 (1)

K. J. Vahala, “Optical microcavities,” Nature 424, 839846 (2003).
[Crossref]

2002 (2)

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E 66, 055601 (2002).
[Crossref]

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).
[Crossref]

1997 (1)

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightw. Technol. 15, 998 (1997).
[Crossref]

1996 (1)

1990 (1)

M. Kawachi, “Silica waveguides on silicon and their application to integrated-optic components,” Opt. Quant. Electron. 22, 391 (1990).
[Crossref]

1984 (1)

1965 (1)

Abrashuly, A.

A. Abrashuly and C. Valagiannopoulos, “Limits for absorption and scattering by core-shell nanowires in the visible spectrum,” Phys. Rev. Appl. 11, 014051 (2019).
[Crossref]

Alloatti, L.

Alonso-González, P.

I. S. Lamata, P. Alonso-González, R. Hillenbrand, and A. Y. Nikitin, “Plasmons in cylindrical 2D materials as a platform for nanophotonic circuits,” ACS Photonics 2, 280 (2015).
[Crossref]

Asatryan, A. A.

P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82, 053825 (2010).
[Crossref]

Atabaki, A.

Bharadwaj, P.

P. Bharadwaj, B. Deutsch, and L. Novotny, “Optical antennas,” Adv. Opt. Photonics 1, 438 (2009).
[Crossref]

Bhargava, P.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999), 7 ed.
[Crossref]

Botten, L. C.

P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82, 053825 (2010).
[Crossref]

Campillo, A. J.

R. K. Chang and A. J. Campillo, Optical Processes in Microcavities (World Scientific, 1996).
[Crossref]

Catchpole, K. R.

Chang, R. K.

R. K. Chang and A. J. Campillo, Optical Processes in Microcavities (World Scientific, 1996).
[Crossref]

Chen, P. Y.

P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82, 053825 (2010).
[Crossref]

Choi, D.-Y.

Christopoulos, T.

O. Tsilipakos, T. Christopoulos, and E. E. Kriezis, “Long-range hybrid plasmonic disk resonators for mw bistability and self-pulsation,” J. Lightw. Technol. 34, 1333 (2016).
[Crossref]

T. Christopoulos, O. Tsilipakos, N. Grivas, and E. E. Kriezis, “Coupled-mode-theory framework for nonlinear resonators comprising graphene,” Phys. Rev. E 94, 062219 (2016).
[Crossref]

Chu, S. T.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightw. Technol. 15, 998 (1997).
[Crossref]

Combrié, S.

Danto, S.

L. Li, H. Lin, S. Qiao, Y. Zou, S. Danto, K. Richardson, J. D. Musgraves, N. Lu, and J. Hu, “Integrated flexible chalcogenide glass photonic devices,” Nat. Photon. 8, 643 (2014).
[Crossref]

de Sterke, C. M.

P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82, 053825 (2010).
[Crossref]

Debbarma, S.

Deutsch, B.

P. Bharadwaj, B. Deutsch, and L. Novotny, “Optical antennas,” Adv. Opt. Photonics 1, 438 (2009).
[Crossref]

Economou, E. N.

O. Tsilipakos, A. C. Tasolamprou, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Pairing toroidal and magnetic dipole resonances in elliptic dielectric rod metasurfaces for reconfigurable wavefront manipulation in reflection,” Adv. Opt. Mater. 6, 1800633 (2018).
[Crossref]

A. C. Tasolamprou, O. Tsilipakos, M. Kafesaki, C. M. Soukoulis, and E. N. Economou, “Toroidal eigenmodes in all-dielectric metamolecules,” Phys. Rev. B 94, 205433 (2016).
[Crossref]

El-Sayed, I. H.

P. K. Jain, X. Huang, I. H. El-Sayed, and M. A. El-Sayed, “Review of some interesting surface plasmon resonance-enhanced properties of noble metal nanoparticles and their applications to biosystems,” Plasmonics 2, 107 (2007).
[Crossref]

El-Sayed, M. A.

P. K. Jain, X. Huang, I. H. El-Sayed, and M. A. El-Sayed, “Review of some interesting surface plasmon resonance-enhanced properties of noble metal nanoparticles and their applications to biosystems,” Plasmonics 2, 107 (2007).
[Crossref]

K.-S. Lee and M. A. El-Sayed, “Gold and silver nanoparticles in sensing and imaging: sensitivity of plasmon response to size, shape, and metal composition,” J. Phys. Chem. B 110, 19220 (2006).
[Crossref] [PubMed]

Faggiani, R.

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

Fan, S.

A. Raman and S. Fan, “Perturbation theory for plasmonic modulation and sensing,” Phys. Rev. B 83, 205131 (2011).
[Crossref]

Z. Ruan and S. Fan, “Superscattering of light from subwavelength nanostructures,” Phys. Rev. Lett. 105, 013901 (2010).
[Crossref] [PubMed]

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).
[Crossref]

Favero, I.

Fink, Y.

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E 66, 055601 (2002).
[Crossref]

Foresi, J.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightw. Technol. 15, 998 (1997).
[Crossref]

Gai, X.

Gorodetsky, M. L.

Grivas, N.

T. Christopoulos, O. Tsilipakos, N. Grivas, and E. E. Kriezis, “Coupled-mode-theory framework for nonlinear resonators comprising graphene,” Phys. Rev. E 94, 062219 (2016).
[Crossref]

Guha, B.

Hamam, R. E.

R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljačić, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A 75, 053801 (2007).
[Crossref]

Hanson, G. W.

G. W. Hanson, “Dyadic green’s functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103, 064302 (2008).
[Crossref]

Haus, H. A.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightw. Technol. 15, 998 (1997).
[Crossref]

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

Hillenbrand, R.

I. S. Lamata, P. Alonso-González, R. Hillenbrand, and A. Y. Nikitin, “Plasmons in cylindrical 2D materials as a platform for nanophotonic circuits,” ACS Photonics 2, 280 (2015).
[Crossref]

Hu, J.

L. Li, H. Lin, S. Qiao, Y. Zou, S. Danto, K. Richardson, J. D. Musgraves, N. Lu, and J. Hu, “Integrated flexible chalcogenide glass photonic devices,” Nat. Photon. 8, 643 (2014).
[Crossref]

Huang, K. C.

K. C. Huang, M. L. Povinelli, and J. D. Joannopoulos, “Negative effective permeability in polaritonic photonic crystals,” Appl. Phys. Lett. 85, 543 (2004).
[Crossref]

Huang, X.

P. K. Jain, X. Huang, I. H. El-Sayed, and M. A. El-Sayed, “Review of some interesting surface plasmon resonance-enhanced properties of noble metal nanoparticles and their applications to biosystems,” Plasmonics 2, 107 (2007).
[Crossref]

Hughes, S.

P. T. Kristensen and S. Hughes, “Modes and mode volumes of leaky optical cavities and plasmonic nanoresonators,” ACS Photonics 1, 2 (2013).
[Crossref]

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

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, 237401 (2013).
[Crossref] [PubMed]

Hugonin, J.-P.

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

Ibanescu, M.

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E 66, 055601 (2002).
[Crossref]

Ilchenko, V.

V. Ilchenko and A. Matsko, “Optical resonators with whispering-gallery modes-part II: applications,” IEEE J. Sel. Top. Quant. 12, 15 (2006).
[Crossref]

Ilchenko, V. S.

Jain, P. K.

P. K. Jain, X. Huang, I. H. El-Sayed, and M. A. El-Sayed, “Review of some interesting surface plasmon resonance-enhanced properties of noble metal nanoparticles and their applications to biosystems,” Plasmonics 2, 107 (2007).
[Crossref]

Joannopoulos, J. D.

R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljačić, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A 75, 053801 (2007).
[Crossref]

K. C. Huang, M. L. Povinelli, and J. D. Joannopoulos, “Negative effective permeability in polaritonic photonic crystals,” Appl. Phys. Lett. 85, 543 (2004).
[Crossref]

M. Soljačić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3, 211 (2004).
[Crossref]

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E 66, 055601 (2002).
[Crossref]

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).
[Crossref]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008), 2nd ed.

Johnson, S. G.

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E 66, 055601 (2002).
[Crossref]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008), 2nd ed.

Kafesaki, M.

O. Tsilipakos, A. C. Tasolamprou, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Pairing toroidal and magnetic dipole resonances in elliptic dielectric rod metasurfaces for reconfigurable wavefront manipulation in reflection,” Adv. Opt. Mater. 6, 1800633 (2018).
[Crossref]

A. C. Tasolamprou, O. Tsilipakos, M. Kafesaki, C. M. Soukoulis, and E. N. Economou, “Toroidal eigenmodes in all-dielectric metamolecules,” Phys. Rev. B 94, 205433 (2016).
[Crossref]

Karalis, A.

R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljačić, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A 75, 053801 (2007).
[Crossref]

Kawachi, M.

M. Kawachi, “Silica waveguides on silicon and their application to integrated-optic components,” Opt. Quant. Electron. 22, 391 (1990).
[Crossref]

Kivshar, Y. S.

N. I. Zheludev and Y. S. Kivshar, “From metamaterials to metadevices,” Nature Mater. 11, 917 (2012).
[Crossref]

Koschny, T.

O. Tsilipakos, A. C. Tasolamprou, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Pairing toroidal and magnetic dipole resonances in elliptic dielectric rod metasurfaces for reconfigurable wavefront manipulation in reflection,” Adv. Opt. Mater. 6, 1800633 (2018).
[Crossref]

Kriezis, E. E.

T. Christopoulos, O. Tsilipakos, N. Grivas, and E. E. Kriezis, “Coupled-mode-theory framework for nonlinear resonators comprising graphene,” Phys. Rev. E 94, 062219 (2016).
[Crossref]

O. Tsilipakos, T. Christopoulos, and E. E. Kriezis, “Long-range hybrid plasmonic disk resonators for mw bistability and self-pulsation,” J. Lightw. Technol. 34, 1333 (2016).
[Crossref]

Kristensen, P. T.

P. T. Kristensen and S. Hughes, “Modes and mode volumes of leaky optical cavities and plasmonic nanoresonators,” ACS Photonics 1, 2 (2013).
[Crossref]

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

Laine, J.-P.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightw. Technol. 15, 998 (1997).
[Crossref]

Lalanne, P.

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

W. Yan, R. Faggiani, and P. Lalanne, “Rigorous modal analysis of plasmonic nanoresonators,” Phys. Rev. B 97, 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, 237401 (2013).
[Crossref] [PubMed]

Lamata, I. S.

I. S. Lamata, P. Alonso-González, R. Hillenbrand, and A. Y. Nikitin, “Plasmons in cylindrical 2D materials as a platform for nanophotonic circuits,” ACS Photonics 2, 280 (2015).
[Crossref]

Landau, L. D.

L. D. Landau, L. P. Pitaevskii, and E. M. Lifshitz, Electrodynamics of Continuous Media (Elsevier Butterworth-Heinemann, 1984), 2nd ed.

Lee, K.-S.

K.-S. Lee and M. A. El-Sayed, “Gold and silver nanoparticles in sensing and imaging: sensitivity of plasmon response to size, shape, and metal composition,” J. Phys. Chem. B 110, 19220 (2006).
[Crossref] [PubMed]

Lemaître, A.

Leo, G.

Li, L.

L. Li, H. Lin, S. Qiao, Y. Zou, S. Danto, K. Richardson, J. D. Musgraves, N. Lu, and J. Hu, “Integrated flexible chalcogenide glass photonic devices,” Nat. Photon. 8, 643 (2014).
[Crossref]

Li, Q.

Lifshitz, E. M.

L. D. Landau, L. P. Pitaevskii, and E. M. Lifshitz, Electrodynamics of Continuous Media (Elsevier Butterworth-Heinemann, 1984), 2nd ed.

Lin, H.

L. Li, H. Lin, S. Qiao, Y. Zou, S. Danto, K. Richardson, J. D. Musgraves, N. Lu, and J. Hu, “Integrated flexible chalcogenide glass photonic devices,” Nat. Photon. 8, 643 (2014).
[Crossref]

Lin, S.

Little, B. E.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightw. Technol. 15, 998 (1997).
[Crossref]

Lu, N.

L. Li, H. Lin, S. Qiao, Y. Zou, S. Danto, K. Richardson, J. D. Musgraves, N. Lu, and J. Hu, “Integrated flexible chalcogenide glass photonic devices,” Nat. Photon. 8, 643 (2014).
[Crossref]

Luther-Davies, B.

Ma, P.

Madden, S.

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, 237401 (2013).
[Crossref] [PubMed]

Malitson, I. H.

Mariani, S.

Matsko, A.

V. Ilchenko and A. Matsko, “Optical resonators with whispering-gallery modes-part II: applications,” IEEE J. Sel. Top. Quant. 12, 15 (2006).
[Crossref]

McPhedran, R. C.

P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82, 053825 (2010).
[Crossref]

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008), 2nd ed.

Mehta, N.

Moazeni, S.

Musgraves, J. D.

L. Li, H. Lin, S. Qiao, Y. Zou, S. Danto, K. Richardson, J. D. Musgraves, N. Lu, and J. Hu, “Integrated flexible chalcogenide glass photonic devices,” Nat. Photon. 8, 643 (2014).
[Crossref]

Nikitin, A. Y.

I. S. Lamata, P. Alonso-González, R. Hillenbrand, and A. Y. Nikitin, “Plasmons in cylindrical 2D materials as a platform for nanophotonic circuits,” ACS Photonics 2, 280 (2015).
[Crossref]

Notomi, M.

M. Notomi, “Manipulating light with strongly modulated photonic crystals,” Rep. Prog. Phys. 73, 096501 (2010).
[Crossref]

Novotny, L.

P. Bharadwaj, B. Deutsch, and L. Novotny, “Optical antennas,” Adv. Opt. Photonics 1, 438 (2009).
[Crossref]

Pavanello, F.

Pitaevskii, L. P.

L. D. Landau, L. P. Pitaevskii, and E. M. Lifshitz, Electrodynamics of Continuous Media (Elsevier Butterworth-Heinemann, 1984), 2nd ed.

Polman, A.

Popovic, M.

Poulton, C. G.

P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82, 053825 (2010).
[Crossref]

Povinelli, M. L.

K. C. Huang, M. L. Povinelli, and J. D. Joannopoulos, “Negative effective permeability in polaritonic photonic crystals,” Appl. Phys. Lett. 85, 543 (2004).
[Crossref]

Pozar, D. M.

D. M. Pozar, Microwave Engineering (John Wiley & Sons, 2005), 3rd ed.

Qiao, S.

L. Li, H. Lin, S. Qiao, Y. Zou, S. Danto, K. Richardson, J. D. Musgraves, N. Lu, and J. Hu, “Integrated flexible chalcogenide glass photonic devices,” Nat. Photon. 8, 643 (2014).
[Crossref]

Qiu, M.

Ram, R. J.

Raman, A.

A. Raman and S. Fan, “Perturbation theory for plasmonic modulation and sensing,” Phys. Rev. B 83, 205131 (2011).
[Crossref]

Richardson, K.

L. Li, H. Lin, S. Qiao, Y. Zou, S. Danto, K. Richardson, J. D. Musgraves, N. Lu, and J. Hu, “Integrated flexible chalcogenide glass photonic devices,” Nat. Photon. 8, 643 (2014).
[Crossref]

Ruan, Z.

Z. Ruan and S. Fan, “Superscattering of light from subwavelength nanostructures,” Phys. Rev. Lett. 105, 013901 (2010).
[Crossref] [PubMed]

Sauvan, C.

P. Lalanne, W. Yan, K. Vynck, C. Sauvan, and J.-P. Hugonin, “Light interaction with photonic and plasmonic resonances,” Laser Photonics Rev. 12, 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, 237401 (2013).
[Crossref] [PubMed]

Savchenkov, A. A.

Soljacic, M.

R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljačić, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A 75, 053801 (2007).
[Crossref]

M. Soljačić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3, 211 (2004).
[Crossref]

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E 66, 055601 (2002).
[Crossref]

Soukoulis, C. M.

O. Tsilipakos, A. C. Tasolamprou, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Pairing toroidal and magnetic dipole resonances in elliptic dielectric rod metasurfaces for reconfigurable wavefront manipulation in reflection,” Adv. Opt. Mater. 6, 1800633 (2018).
[Crossref]

A. C. Tasolamprou, O. Tsilipakos, M. Kafesaki, C. M. Soukoulis, and E. N. Economou, “Toroidal eigenmodes in all-dielectric metamolecules,” Phys. Rev. B 94, 205433 (2016).
[Crossref]

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics 5, 523–530 (2011).
[Crossref]

Steel, M. J.

P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82, 053825 (2010).
[Crossref]

Stojanovic, V.

Su, Y.

Sun, C.

Tasolamprou, A. C.

O. Tsilipakos, A. C. Tasolamprou, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Pairing toroidal and magnetic dipole resonances in elliptic dielectric rod metasurfaces for reconfigurable wavefront manipulation in reflection,” Adv. Opt. Mater. 6, 1800633 (2018).
[Crossref]

A. C. Tasolamprou, O. Tsilipakos, M. Kafesaki, C. M. Soukoulis, and E. N. Economou, “Toroidal eigenmodes in all-dielectric metamolecules,” Phys. Rev. B 94, 205433 (2016).
[Crossref]

Tatian, B.

Tsilipakos, O.

O. Tsilipakos, A. C. Tasolamprou, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Pairing toroidal and magnetic dipole resonances in elliptic dielectric rod metasurfaces for reconfigurable wavefront manipulation in reflection,” Adv. Opt. Mater. 6, 1800633 (2018).
[Crossref]

A. C. Tasolamprou, O. Tsilipakos, M. Kafesaki, C. M. Soukoulis, and E. N. Economou, “Toroidal eigenmodes in all-dielectric metamolecules,” Phys. Rev. B 94, 205433 (2016).
[Crossref]

T. Christopoulos, O. Tsilipakos, N. Grivas, and E. E. Kriezis, “Coupled-mode-theory framework for nonlinear resonators comprising graphene,” Phys. Rev. E 94, 062219 (2016).
[Crossref]

O. Tsilipakos, T. Christopoulos, and E. E. Kriezis, “Long-range hybrid plasmonic disk resonators for mw bistability and self-pulsation,” J. Lightw. Technol. 34, 1333 (2016).
[Crossref]

Vahala, K. J.

K. J. Vahala, “Optical microcavities,” Nature 424, 839846 (2003).
[Crossref]

Valagiannopoulos, C.

A. Abrashuly and C. Valagiannopoulos, “Limits for absorption and scattering by core-shell nanowires in the visible spectrum,” Phys. Rev. Appl. 11, 014051 (2019).
[Crossref]

Vlack, C. V.

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, 1700113 (2018).
[Crossref]

Wade, M.

Wang, T.

Wegener, M.

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics 5, 523–530 (2011).
[Crossref]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008), 2nd ed.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999), 7 ed.
[Crossref]

Yan, M.

Yan, W.

W. Yan, R. Faggiani, and P. Lalanne, “Rigorous modal analysis of plasmonic nanoresonators,” Phys. Rev. B 97, 205422 (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, 1700113 (2018).
[Crossref]

Yang, Z.

Yu, Y.

Zheludev, N. I.

N. I. Zheludev and Y. S. Kivshar, “From metamaterials to metadevices,” Nature Mater. 11, 917 (2012).
[Crossref]

Zou, Y.

L. Li, H. Lin, S. Qiao, Y. Zou, S. Danto, K. Richardson, J. D. Musgraves, N. Lu, and J. Hu, “Integrated flexible chalcogenide glass photonic devices,” Nat. Photon. 8, 643 (2014).
[Crossref]

ACS Photonics (2)

I. S. Lamata, P. Alonso-González, R. Hillenbrand, and A. Y. Nikitin, “Plasmons in cylindrical 2D materials as a platform for nanophotonic circuits,” ACS Photonics 2, 280 (2015).
[Crossref]

P. T. Kristensen and S. Hughes, “Modes and mode volumes of leaky optical cavities and plasmonic nanoresonators,” ACS Photonics 1, 2 (2013).
[Crossref]

Adv. Opt. Mater. (1)

O. Tsilipakos, A. C. Tasolamprou, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Pairing toroidal and magnetic dipole resonances in elliptic dielectric rod metasurfaces for reconfigurable wavefront manipulation in reflection,” Adv. Opt. Mater. 6, 1800633 (2018).
[Crossref]

Adv. Opt. Photonics (1)

P. Bharadwaj, B. Deutsch, and L. Novotny, “Optical antennas,” Adv. Opt. Photonics 1, 438 (2009).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

K. C. Huang, M. L. Povinelli, and J. D. Joannopoulos, “Negative effective permeability in polaritonic photonic crystals,” Appl. Phys. Lett. 85, 543 (2004).
[Crossref]

IEEE J. Sel. Top. Quant. (1)

V. Ilchenko and A. Matsko, “Optical resonators with whispering-gallery modes-part II: applications,” IEEE J. Sel. Top. Quant. 12, 15 (2006).
[Crossref]

J. Appl. Phys. (1)

G. W. Hanson, “Dyadic green’s functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103, 064302 (2008).
[Crossref]

J. Lightw. Technol. (2)

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightw. Technol. 15, 998 (1997).
[Crossref]

O. Tsilipakos, T. Christopoulos, and E. E. Kriezis, “Long-range hybrid plasmonic disk resonators for mw bistability and self-pulsation,” J. Lightw. Technol. 34, 1333 (2016).
[Crossref]

J. Opt. Soc. Am. (1)

J. Phys. Chem. B (1)

K.-S. Lee and M. A. El-Sayed, “Gold and silver nanoparticles in sensing and imaging: sensitivity of plasmon response to size, shape, and metal composition,” J. Phys. Chem. B 110, 19220 (2006).
[Crossref] [PubMed]

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, 1700113 (2018).
[Crossref]

Nat. Mater. (1)

M. Soljačić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3, 211 (2004).
[Crossref]

Nat. Photon. (1)

L. Li, H. Lin, S. Qiao, Y. Zou, S. Danto, K. Richardson, J. D. Musgraves, N. Lu, and J. Hu, “Integrated flexible chalcogenide glass photonic devices,” Nat. Photon. 8, 643 (2014).
[Crossref]

Nat. Photonics (1)

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics 5, 523–530 (2011).
[Crossref]

Nature (1)

K. J. Vahala, “Optical microcavities,” Nature 424, 839846 (2003).
[Crossref]

Nature Mater. (1)

N. I. Zheludev and Y. S. Kivshar, “From metamaterials to metadevices,” Nature Mater. 11, 917 (2012).
[Crossref]

Opt. Express (5)

Opt. Lett. (2)

Opt. Quant. Electron. (1)

M. Kawachi, “Silica waveguides on silicon and their application to integrated-optic components,” Opt. Quant. Electron. 22, 391 (1990).
[Crossref]

Phys. Rev. A (2)

R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljačić, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A 75, 053801 (2007).
[Crossref]

P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82, 053825 (2010).
[Crossref]

Phys. Rev. Appl. (1)

A. Abrashuly and C. Valagiannopoulos, “Limits for absorption and scattering by core-shell nanowires in the visible spectrum,” Phys. Rev. Appl. 11, 014051 (2019).
[Crossref]

Phys. Rev. B (4)

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).
[Crossref]

A. Raman and S. Fan, “Perturbation theory for plasmonic modulation and sensing,” Phys. Rev. B 83, 205131 (2011).
[Crossref]

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

A. C. Tasolamprou, O. Tsilipakos, M. Kafesaki, C. M. Soukoulis, and E. N. Economou, “Toroidal eigenmodes in all-dielectric metamolecules,” Phys. Rev. B 94, 205433 (2016).
[Crossref]

Phys. Rev. E (2)

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E 66, 055601 (2002).
[Crossref]

T. Christopoulos, O. Tsilipakos, N. Grivas, and E. E. Kriezis, “Coupled-mode-theory framework for nonlinear resonators comprising graphene,” Phys. Rev. E 94, 062219 (2016).
[Crossref]

Phys. Rev. Lett. (2)

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, 237401 (2013).
[Crossref] [PubMed]

Z. Ruan and S. Fan, “Superscattering of light from subwavelength nanostructures,” Phys. Rev. Lett. 105, 013901 (2010).
[Crossref] [PubMed]

Plasmonics (1)

P. K. Jain, X. Huang, I. H. El-Sayed, and M. A. El-Sayed, “Review of some interesting surface plasmon resonance-enhanced properties of noble metal nanoparticles and their applications to biosystems,” Plasmonics 2, 107 (2007).
[Crossref]

Rep. Prog. Phys. (1)

M. Notomi, “Manipulating light with strongly modulated photonic crystals,” Rep. Prog. Phys. 73, 096501 (2010).
[Crossref]

Other (6)

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008), 2nd ed.

R. K. Chang and A. J. Campillo, Optical Processes in Microcavities (World Scientific, 1996).
[Crossref]

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999), 7 ed.
[Crossref]

D. M. Pozar, Microwave Engineering (John Wiley & Sons, 2005), 3rd ed.

L. D. Landau, L. P. Pitaevskii, and E. M. Lifshitz, Electrodynamics of Continuous Media (Elsevier Butterworth-Heinemann, 1984), 2nd ed.

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Figures (15)

Fig. 1
Fig. 1 Characteristic contemporary guided-wave and free-space resonant photonic/plasmonic systems. (a) Silicon microring resonator coupled to an access waveguide. (b) Carbon tube resonator accessed through a graphene sheet. (c) Free-space dielectric rod metasurface. (d) Free-space plasmonic core-shell nanoparticle.
Fig. 2
Fig. 2 (a) Lorentzian lineshape and (b) inverse Lorentzian lineshape. Minimum/maximum values and the FWHM bandwidth are clearly marked.
Fig. 3
Fig. 3 Schematic of (a) an uncoupled and (b) a coupled, silica-clad, silicon ring resonator of radius R with a coupling gap g. Energy integration domains and the respective power flux integration boundaries for the correct application of Eq. (1) are clearly highlighted.
Fig. 4
Fig. 4 (a) Uncoupled intrinsic Q-factor calculated using the eigenfrequency (black circles) and the respective eigenmode (yellow and red crosses). (b) Coupled intrinsic Q-factor (yellow stars and blue triangles). External coupling results in higher Q i ' since a non-negligible part of the radiation is coupled to the bus waveguide (inset). All modes have resonance wavelengths around 1.55 μm.
Fig. 5
Fig. 5 Loaded Q-factor calculated for the m = 11 mode using the eigenfrequency (black solid line), the eigenmode (red dashed line), and the spectral response (blue dash-dot line), revealing very good agreement. Q i ' and Qe are also included, equated at critical coupling.
Fig. 6
Fig. 6 FEM-based spectral response (black dots) compared to CMT transmission using the coupled Q i ' (blue thick curve) and the uncoupled Qi (red thin curve). When Q i ' is used, FEM and CMT coincide.
Fig. 7
Fig. 7 Schematic of (a) an uncoupled and (b) a coupled graphene tube resonator of radius R with coupling gap g. Energy integration domains and the respective power flux integration boundaries for the correct application of Eq. (1) are denoted as well.
Fig. 8
Fig. 8 (a) Intrinsic Q-factor calculated after including (upper markers) or neglecting (lower markers) graphene dispersion. The strong impact of dispersion is seen. (b) Resistive and (c) radiation Q for the same conditions. All modes have resonance frequencies around 10 THz.
Fig. 9
Fig. 9 (a) Loaded Q-factor obtained from the eigenmode (red dashed line) and the spectral response (blue dashed-dotted line) when graphene dispersion is included. (b) Loaded Q-factor obtained from the eigenfrequency (black solid line) and the eigenmode (red dashed line) when graphene dispersion is erroneously neglected. Qi and Qe are also included to identify critical coupling conditions, actually being comparable in both considerations.
Fig. 10
Fig. 10 FEM results (black dots) compared to CMT, with the Q-factors calculated including (blue thick curve) or ignoring (red thin curve) graphene dispersion. The correct spectral response is recovered when dispersion is taken intoaccount.
Fig. 11
Fig. 11 Schematics of (a) a LiTaO3 microrod of radius R, (b) a LiTaO3 metasurface with lattice constant a, and (c) a plasmonic core-shell nanoparticle of outer radius R with a silica core and a gold shielding of width w. A typical integration domain and the respective power integration limit for applying Eq. (1) are sketched in all structures.
Fig. 12
Fig. 12 (a) Intrinsic Q-factor for low order Mie resonances (TM polarization, H H z z ^ ) of the dielectric meta-atom for resonance frequencies around 2 THz. All the Q-factor calculation methods coincide. Absorption cross-section calculated using FEM simulations (black dots) and CMT (blue solid curves) for (b) the TM00 mode and (c) the TM10 mode. FEM-CMT agreement is excellent.
Fig. 13
Fig. 13 Dielectric metasurface absorption using FEM simulations (black dots) and CMT (blue solid curve), revealing excellent agreement for all four modes.
Fig. 14
Fig. 14 (a) Intrinsic Q-factor for various core-shell modes. “Eigenfrequency” (black circles) and “eigenmode” (yellow crosses) methods coincide but both fail to capture material dispersion. Field distribution (red crosses) and spectral response (blue squares) correctly incorporate dispersion. All modes have resonance wavelengths around 500 nm. Qi may be carefully decomposed in its (b) Qres and (c) Qrad contributions. (d) FEM (black dots) and CMT results using the calculated Q-factors including (blue thick curve) or neglecting (red thin curve) material dispersion. The correct absorption cross-section spectral response is recovered when dispersion is correctly taken into account.
Fig. 15
Fig. 15 Applicability of Q-factor calculation methods, schematically imprinted in the Leakage-Loss “plane”, when dispersion is (a) weak and (b) strong.

Tables (4)

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Table 1 Q-factor Calculation Methods in Guided-Wave Systems with Weak Dispersion, Negligible Resistive, and Potentially Significant Radiation Loss. n/a stands for “not applicable.”

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Table 2 Q-factor Calculation Methods in Guided-Wave Systems with Strong Material Dispersion, Significant Resistive, and Negligible Radiation Loss. n/a stands for “not applicable.”

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Table 3 Q-factor Calculation Methods in Strongly Leaky and Lossy Scatterers with Negligible Dispersion. n/a stands for “not applicable.”

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Table 4 Q-factor Calculation Methods in Leaky and Lossy Scatterers with High Dispersion. n/a stands for “not applicable.”

Equations (14)

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Q = ω 0 W P loss ,
1 Q l = 1 Q i + 1 Q e = 1 Q res + 1 Q rad + 1 Q e .
Q = ω 2 ω .
V Re { S c } d V = V Re { S c } n ^   d S = 2 ω V 1 4 ( ε 0 ε r ' | E | 2 + μ 0 | H | 2 ) d V V 1 2 ω ε 0 ε r ' ' | E | 2 d V V 1 2 σ | E | 2 d V ,
P rad + P e = 2 ω W P res .
d a ( t ) d t = ( j ω 0 1 τ ) a ( t ) ,
Q = ω 0 Δ ω ,
W = 1 4 A ε 0 { ω ε r ( ω ) } ω | ω = ω 0 | E | 2 d S + 1 4 A μ 0 | H | 2 d S ,
P rad = A Re { S c } n ^   d l = 1 2 A Re { E × H * } n ^   d l .
T = δ 2 + ( 1 r Q ) 2 δ 2 + ( 1 + r Q ) 2 ,
P res = 1 2 gr Re { E J * } d l = 1 2 gr Re { σ gr } | E | 2 d l ,
W = 1 4 A ε 0 { ω ε r ( ω ) } ω | ω = ω 0 | E | 2 d S + 1 4 A μ 0 | H | 2 d S + 1 4 gr Im { σ gr ( ω ) } ω | ω = ω 0 | E | 2 d l .
C abs = 1 2 rod ω ε 0 ε r '' | E | 2 d S I 0 ,
C abs = 2 ( m + 1 ) λ π r Q δ 2 + ( 1 + r Q ) 2 ,

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