Abstract

We reply to the comment on our recent article [Optica 6, 104 (2019) [CrossRef]  ], in which we demonstrated that nonreciprocal cavities comply with the resonance time-bandwidth limit. We fully stand by our original claims and further elucidate how breaking of reciprocity is not required to achieve the large field enhancements and time-bandwidth products observed in the comment and our article. We also further clarify that these hotspots do not overcome the conventional time-bandwidth limit.

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

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2020 (3)

S. Buddhiraju, Y. Shi, A. Song, C. Wojcik, M. Minkov, I. A. D. Williamson, A. Dutt, and S. Fan, “Absence of unidirectionally propagating surface plasmon-polaritons at nonreciprocal metal-dielectric interfaces,” Nat. Commun. 11, 674 (2020).
[Crossref]

K. L. Tsakmakidis, Y. You, T. Stefanski, and L. Shen, “Nonreciprocal cavities and the time–bandwidth limit: comment,” Optica 7, 1097–1101 (2020).
[Crossref]

M. A. Green, E. D. Dunlop, J. Hohl-Ebinger, M. Yoshita, N. Kopidakis, and A. W. Y. Ho-Baillie, “Solar cell efficiency tables (version 55),” Prog. Photovoltaics 28, 3–15 (2020).
[Crossref]

2019 (7)

P. Lalanne, S. Coudert, G. Duchateau, S. Dilhaire, and K. Vynck, “Structural slow waves: parallels between photonic crystals and plasmonic waveguides,” ACS Photon. 6, 4–17 (2019).
[Crossref]

D. E. Fernandes and M. G. Silveirinha, “Topological origin of electromagnetic energy sinks,” Phys. Rev. Appl. 12, 014021 (2019).
[Crossref]

H. Li, A. Mekawy, and A. Alù, “Beyond Chu’s limit with floquet impedance matching,” Phys. Rev. Lett. 123, 164102 (2019).
[Crossref]

S. A. Hassani Gangaraj and F. Monticone, “Do truly unidirectional surface plasmon-polaritons exist?” Optica 6, 1158–1165 (2019).
[Crossref]

S. A. Mann, D. L. Sounas, and A. Alù, “Nonreciprocal cavities and the time–bandwidth limit,” Optica 6, 104–110 (2019).
[Crossref]

M. Tymchenko, D. Sounas, A. Nagulu, H. Krishnaswamy, and A. Alù, “Quasielectrostatic wave propagation beyond the delay-bandwidth limit in switched networks,” Phys. Rev. X 9, 031015 (2019).
[Crossref]

S. A. Mann, D. L. Sounas, and A. Alù, “Broadband delay lines and nonreciprocal resonances in unidirectional waveguides,” Phys. Rev. B 100, 020303 (2019).
[Crossref]

2018 (3)

M. Tsang, “Quantum limits on the time-bandwidth product of an optical resonator,” Opt. Lett. 43, 150–153 (2018).
[Crossref]

C. Caloz, A. Alù, S. Tretyakov, D. Sounas, K. Achouri, and Z.-L. Deck-Léger, “Electromagnetic nonreciprocity,” Phys. Rev. Appl. 10, 047001 (2018).
[Crossref]

A. Shlivinski and Y. Hadad, “Beyond the bode-Fano bound: wideband impedance matching for short pulses using temporal switching of transmission-line parameters,” Phys. Rev. Lett. 121, 204301 (2018).
[Crossref]

2017 (3)

K. L. Tsakmakidis, L. Shen, S. A. Schulz, X. Zheng, J. Upham, X. Deng, H. Altug, A. F. Vakakis, and R. W. Boyd, “Breaking Lorentz reciprocity to overcome the time-bandwidth limit in physics and engineering,” Science 356, 1260–1264 (2017).
[Crossref]

M. Marvasti and B. Rejaei, “Formation of hotspots in partially filled ferrite-loaded rectangular waveguides,” J. Appl. Phys. 122, 233901 (2017).
[Crossref]

D. A. B. Miller, L. Zhu, and S. Fan, “Universal modal radiation laws for all thermal emitters,” Proc. Natl. Acad. Sci. USA 114, 4336–4341 (2017).
[Crossref]

2016 (2)

A. Polman, M. Knight, E. C. Garnett, B. Ehrler, and W. C. Sinke, “Photovoltaic materials: Present efficiencies and future challenges,” Science 352, aad4424 (2016).
[Crossref]

S. A. Mann, R. R. Grote, R. M. Osgood, A. Alù, and E. C. Garnett, “Opportunities and limitations for nanophotonic structures to exceed the Shockley-Queisser limit,” ACS Nano 10, 8620–8631 (2016).
[Crossref]

2015 (1)

2014 (3)

U. K. Chettiar, A. R. Davoyan, and N. Engheta, “Hotspots from nonreciprocal surface waves,” Opt. Lett. 39, 1760–1763 (2014).
[Crossref]

E. D. Kosten, B. M. Kayes, and H. A. Atwater, “Experimental demonstration of enhanced photon recycling in angle-restricted GaAs solar cells,” Energy Environ. Sci. 7, 1907 (2014).
[Crossref]

L. Zhu and S. Fan, “Near-complete violation of detailed balance in thermal radiation,” Phys. Rev. B 90, 220301 (2014).
[Crossref]

2013 (4)

L. Zhu, S. Sandhu, C. Otey, S. Fan, M. B. Sinclair, and T. Shan Luk, “Temporal coupled mode theory for thermal emission from a single thermal emitter supporting either a single mode or an orthogonal set of modes,” Appl. Phys. Lett. 102, 103104 (2013).
[Crossref]

E. D. E. Kosten, J. H. J. Atwater, J. Parsons, A. Polman, and H. A. Atwater, “Highly efficient GaAs solar cells by limiting light emission angle,” Light Sci. Appl. 2, e45 (2013).
[Crossref]

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is—and what is not—an optical isolator,” Nat. Photonics 7, 579–582 (2013).
[Crossref]

A. A. Maznev, A. G. Every, and O. B. Wright, “Reciprocity in reflection and transmission: What is a ‘phonon diode’?” Wave Motion 50, 776–784 (2013).
[Crossref]

2012 (1)

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on “nonreciprocal light propagation in a silicon photonic circuit”,” Science 335, 38 (2012).
[Crossref]

2009 (1)

Ş. E. Kocabaş, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B 79, 035120 (2009).
[Crossref]

2008 (1)

2007 (4)

Q. Xu, P. Dong, and M. Lipson, “Breaking the delay-bandwidth limit in a photonic structure,” Nat. Phys. 3, 406–410 (2007).
[Crossref]

D. A. B. Miller, “Fundamental limit for optical components,” J. Opt. Soc. Am. B 24, A1 (2007).
[Crossref]

J. B. Khurgin, “Dispersion and loss limitations on the performance of optical delay lines based on coupled resonant structures,” Opt. Lett. 32, 133 (2007).
[Crossref]

D. A. B. Miller, “Fundamental limit to linear one-dimensional slow light structures,” Phys. Rev. Lett. 99, 203903 (2007).
[Crossref]

2006 (1)

D. F. P. Pile and D. K. Gramotnev, “Adiabatic and nonadiabatic nanofocusing of plasmons by tapered gap plasmon waveguides,” Appl. Phys. Lett. 89, 2004–2007 (2006).
[Crossref]

2005 (1)

R. S. Tucker, P. C. Ku, and C. J. Chang-Hasnain, “Delay–bandwidth product and storage density in slow-light optical buffers,” Electron. Lett. 41, 208 (2005).
[Crossref]

2004 (4)

M. F. Yanik and S. Fan, “Stopping light all optically,” Phys. Rev. Lett. 92, 083901 (2004).
[Crossref]

M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. 93, 137404 (2004).
[Crossref]

S. Wonjoo, W. Zheng, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40, 1511–1518 (2004).
[Crossref]

M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. 93, 233903 (2004).
[Crossref]

2000 (1)

K. N. Rozanov, “Ultimate thickness to bandwidth ratio of radar absorbers,” IEEE Trans. Antennas Propag. 48, 1230–1234 (2000).
[Crossref]

1966 (1)

G. Barzilai and G. Gerosa, “Rectangular waveguides loaded with magnetised ferrite, and the so-called thermodynamic paradox,” Proc. Inst. Electr. Eng. 113, 285 (1966).
[Crossref]

1961 (1)

W. Shockley and H. J. Queisser, “Detailed balance limit of efficiency of p-n junction solar cells,” J. Appl. Phys. 32, 510–519 (1961).
[Crossref]

1950 (1)

R. M. Fano, “Theoretical limitations on the broad-band matching of arbitrary impedances,” J. Franklin Inst. 249, 57–83 (1950).
[Crossref]

1948 (1)

L. J. Chu, “Physical limitations of omni-directional antennas,” J. Appl. Phys. 19, 1163–1175 (1948).
[Crossref]

Achouri, K.

C. Caloz, A. Alù, S. Tretyakov, D. Sounas, K. Achouri, and Z.-L. Deck-Léger, “Electromagnetic nonreciprocity,” Phys. Rev. Appl. 10, 047001 (2018).
[Crossref]

Altug, H.

K. L. Tsakmakidis, L. Shen, S. A. Schulz, X. Zheng, J. Upham, X. Deng, H. Altug, A. F. Vakakis, and R. W. Boyd, “Breaking Lorentz reciprocity to overcome the time-bandwidth limit in physics and engineering,” Science 356, 1260–1264 (2017).
[Crossref]

Alù, A.

S. A. Mann, D. L. Sounas, and A. Alù, “Nonreciprocal cavities and the time–bandwidth limit,” Optica 6, 104–110 (2019).
[Crossref]

H. Li, A. Mekawy, and A. Alù, “Beyond Chu’s limit with floquet impedance matching,” Phys. Rev. Lett. 123, 164102 (2019).
[Crossref]

M. Tymchenko, D. Sounas, A. Nagulu, H. Krishnaswamy, and A. Alù, “Quasielectrostatic wave propagation beyond the delay-bandwidth limit in switched networks,” Phys. Rev. X 9, 031015 (2019).
[Crossref]

S. A. Mann, D. L. Sounas, and A. Alù, “Broadband delay lines and nonreciprocal resonances in unidirectional waveguides,” Phys. Rev. B 100, 020303 (2019).
[Crossref]

C. Caloz, A. Alù, S. Tretyakov, D. Sounas, K. Achouri, and Z.-L. Deck-Léger, “Electromagnetic nonreciprocity,” Phys. Rev. Appl. 10, 047001 (2018).
[Crossref]

S. A. Mann, R. R. Grote, R. M. Osgood, A. Alù, and E. C. Garnett, “Opportunities and limitations for nanophotonic structures to exceed the Shockley-Queisser limit,” ACS Nano 10, 8620–8631 (2016).
[Crossref]

Asadchy, V.

V. Asadchy, M. S. Mirmoosa, A. Díaz-Rubio, S. Fan, and S. A. Tretyakov, “Tutorial on electromagnetic nonreciprocity and its origins,” arXiv:2001.04848 (2020).

Atwater, H. A.

E. D. Kosten, B. M. Kayes, and H. A. Atwater, “Experimental demonstration of enhanced photon recycling in angle-restricted GaAs solar cells,” Energy Environ. Sci. 7, 1907 (2014).
[Crossref]

E. D. E. Kosten, J. H. J. Atwater, J. Parsons, A. Polman, and H. A. Atwater, “Highly efficient GaAs solar cells by limiting light emission angle,” Light Sci. Appl. 2, e45 (2013).
[Crossref]

Atwater, J. H. J.

E. D. E. Kosten, J. H. J. Atwater, J. Parsons, A. Polman, and H. A. Atwater, “Highly efficient GaAs solar cells by limiting light emission angle,” Light Sci. Appl. 2, e45 (2013).
[Crossref]

Baets, R.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is—and what is not—an optical isolator,” Nat. Photonics 7, 579–582 (2013).
[Crossref]

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on “nonreciprocal light propagation in a silicon photonic circuit”,” Science 335, 38 (2012).
[Crossref]

Barzilai, G.

G. Barzilai and G. Gerosa, “Rectangular waveguides loaded with magnetised ferrite, and the so-called thermodynamic paradox,” Proc. Inst. Electr. Eng. 113, 285 (1966).
[Crossref]

Boyd, R. W.

K. L. Tsakmakidis, L. Shen, S. A. Schulz, X. Zheng, J. Upham, X. Deng, H. Altug, A. F. Vakakis, and R. W. Boyd, “Breaking Lorentz reciprocity to overcome the time-bandwidth limit in physics and engineering,” Science 356, 1260–1264 (2017).
[Crossref]

Brinkmeyer, E.

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on “nonreciprocal light propagation in a silicon photonic circuit”,” Science 335, 38 (2012).
[Crossref]

Buddhiraju, S.

S. Buddhiraju, Y. Shi, A. Song, C. Wojcik, M. Minkov, I. A. D. Williamson, A. Dutt, and S. Fan, “Absence of unidirectionally propagating surface plasmon-polaritons at nonreciprocal metal-dielectric interfaces,” Nat. Commun. 11, 674 (2020).
[Crossref]

Caloz, C.

C. Caloz, A. Alù, S. Tretyakov, D. Sounas, K. Achouri, and Z.-L. Deck-Léger, “Electromagnetic nonreciprocity,” Phys. Rev. Appl. 10, 047001 (2018).
[Crossref]

Chang-Hasnain, C. J.

R. S. Tucker, P. C. Ku, and C. J. Chang-Hasnain, “Delay–bandwidth product and storage density in slow-light optical buffers,” Electron. Lett. 41, 208 (2005).
[Crossref]

Chettiar, U. K.

Chu, L. J.

L. J. Chu, “Physical limitations of omni-directional antennas,” J. Appl. Phys. 19, 1163–1175 (1948).
[Crossref]

Coudert, S.

P. Lalanne, S. Coudert, G. Duchateau, S. Dilhaire, and K. Vynck, “Structural slow waves: parallels between photonic crystals and plasmonic waveguides,” ACS Photon. 6, 4–17 (2019).
[Crossref]

Davoyan, A. R.

Deck-Léger, Z.-L.

C. Caloz, A. Alù, S. Tretyakov, D. Sounas, K. Achouri, and Z.-L. Deck-Léger, “Electromagnetic nonreciprocity,” Phys. Rev. Appl. 10, 047001 (2018).
[Crossref]

Deng, X.

K. L. Tsakmakidis, L. Shen, S. A. Schulz, X. Zheng, J. Upham, X. Deng, H. Altug, A. F. Vakakis, and R. W. Boyd, “Breaking Lorentz reciprocity to overcome the time-bandwidth limit in physics and engineering,” Science 356, 1260–1264 (2017).
[Crossref]

L. Shen, X. Zheng, and X. Deng, “Stopping terahertz radiation without backscattering over a broad band,” Opt. Express 23, 11790–11798 (2015).
[Crossref]

Díaz-Rubio, A.

V. Asadchy, M. S. Mirmoosa, A. Díaz-Rubio, S. Fan, and S. A. Tretyakov, “Tutorial on electromagnetic nonreciprocity and its origins,” arXiv:2001.04848 (2020).

Dilhaire, S.

P. Lalanne, S. Coudert, G. Duchateau, S. Dilhaire, and K. Vynck, “Structural slow waves: parallels between photonic crystals and plasmonic waveguides,” ACS Photon. 6, 4–17 (2019).
[Crossref]

Doerr, C. R.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is—and what is not—an optical isolator,” Nat. Photonics 7, 579–582 (2013).
[Crossref]

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K. L. Tsakmakidis, L. Shen, S. A. Schulz, X. Zheng, J. Upham, X. Deng, H. Altug, A. F. Vakakis, and R. W. Boyd, “Breaking Lorentz reciprocity to overcome the time-bandwidth limit in physics and engineering,” Science 356, 1260–1264 (2017).
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Electron. Lett. (1)

R. S. Tucker, P. C. Ku, and C. J. Chang-Hasnain, “Delay–bandwidth product and storage density in slow-light optical buffers,” Electron. Lett. 41, 208 (2005).
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S. Buddhiraju, Y. Shi, A. Song, C. Wojcik, M. Minkov, I. A. D. Williamson, A. Dutt, and S. Fan, “Absence of unidirectionally propagating surface plasmon-polaritons at nonreciprocal metal-dielectric interfaces,” Nat. Commun. 11, 674 (2020).
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Nat. Photonics (1)

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is—and what is not—an optical isolator,” Nat. Photonics 7, 579–582 (2013).
[Crossref]

Nat. Phys. (1)

Q. Xu, P. Dong, and M. Lipson, “Breaking the delay-bandwidth limit in a photonic structure,” Nat. Phys. 3, 406–410 (2007).
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Opt. Express (2)

Opt. Lett. (3)

Optica (3)

Phys. Rev. Appl. (2)

C. Caloz, A. Alù, S. Tretyakov, D. Sounas, K. Achouri, and Z.-L. Deck-Léger, “Electromagnetic nonreciprocity,” Phys. Rev. Appl. 10, 047001 (2018).
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Ş. E. Kocabaş, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B 79, 035120 (2009).
[Crossref]

L. Zhu and S. Fan, “Near-complete violation of detailed balance in thermal radiation,” Phys. Rev. B 90, 220301 (2014).
[Crossref]

S. A. Mann, D. L. Sounas, and A. Alù, “Broadband delay lines and nonreciprocal resonances in unidirectional waveguides,” Phys. Rev. B 100, 020303 (2019).
[Crossref]

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M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. 93, 233903 (2004).
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M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. 93, 137404 (2004).
[Crossref]

M. F. Yanik and S. Fan, “Stopping light all optically,” Phys. Rev. Lett. 92, 083901 (2004).
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D. A. B. Miller, L. Zhu, and S. Fan, “Universal modal radiation laws for all thermal emitters,” Proc. Natl. Acad. Sci. USA 114, 4336–4341 (2017).
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M. A. Green, E. D. Dunlop, J. Hohl-Ebinger, M. Yoshita, N. Kopidakis, and A. W. Y. Ho-Baillie, “Solar cell efficiency tables (version 55),” Prog. Photovoltaics 28, 3–15 (2020).
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S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on “nonreciprocal light propagation in a silicon photonic circuit”,” Science 335, 38 (2012).
[Crossref]

K. L. Tsakmakidis, L. Shen, S. A. Schulz, X. Zheng, J. Upham, X. Deng, H. Altug, A. F. Vakakis, and R. W. Boyd, “Breaking Lorentz reciprocity to overcome the time-bandwidth limit in physics and engineering,” Science 356, 1260–1264 (2017).
[Crossref]

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A. A. Maznev, A. G. Every, and O. B. Wright, “Reciprocity in reflection and transmission: What is a ‘phonon diode’?” Wave Motion 50, 776–784 (2013).
[Crossref]

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V. Asadchy, M. S. Mirmoosa, A. Díaz-Rubio, S. Fan, and S. A. Tretyakov, “Tutorial on electromagnetic nonreciprocity and its origins,” arXiv:2001.04848 (2020).

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Figures (2)

Fig. 1.
Fig. 1. (a) Adiabatically tapered MIM waveguide formed by silicon and gold. A pulse is launched towards the apex and progressively slows down. We record the fields inside the purple square (the area surrounding the apex). (b) The stored energy in the purple square quickly rises, as the broadband pulse approaches the termination relatively quickly but slowly decays as it is not reflected but only absorbed. (c) As the pulse approaches the termination, the pulse is compressed as the group velocity is continuously reduced.
Fig. 2.
Fig. 2. (a) Unidirectional delay line of length $L$. The dashed lines are input and output planes, and the section in between has low group velocity due to the narrow dielectric region. (b) A one-dimensional representation of the geometry in (a), where the effective index of each region is given by the plasmon wavenumber. In this case, ${k_s} \gg {k_0}$.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

d d t a = ( i ω 0 γ r γ i ) a + k s + .
Δ ω Δ t L k c 2 3 max [ k s 2 ( x , ω ) k 0 2 ( ω ) k 0 2 ( ω ) ] ,
d d t a = ( i ω 0 γ r γ i ) a + k r n r ( t ) + k i n i ( t ) .
a ( ω ) = k r n r ( ω ) + k i n i ( ω ) i ( ω 0 ω ) γ r γ i .
n i ( ω ) n i ( ω ) = 1 2 π k B T i δ ( ω ω ) , n r ( ω ) n r ( ω ) = 1 2 π k B T e δ ( ω ω ) , n r ( ω ) n i ( ω ) = 0 ,
a ( ω ) a ( ω ) d ω d ω = ( | k i | 2 + | k r | 2 ) k B T 2 π × δ ( ω ω ) ( ω 0 ω ) 2 + ( γ r + γ i ) 2 d ω d ω ,
a ( ω ) a ( ω ) d ω d ω = ( 2 γ i + | k r | 2 ) k B T 2 ( γ i + γ r ) .