Abstract

We present a formal treatment of the modification of spontaneous emission rate by a cavity (Purcell effect) in sub-wavelength semiconductor lasers. To explicitly express the assumptions upon which our formalism builds, we summarize the results of non-relativistic quantum electrodynamics (QED) and the emitter-field-reservoir model in the quantum theory of damping. Within this model, the emitter-field interaction is modified to the extent that the field mode is modified by its environment. We show that the Purcell factor expressions frequently encountered in the literature are recovered only in the hypothetical condition when the gain medium is replaced by a transparent medium. Further, we argue that to accurately evaluate the Purcell effect, both the passive cavity boundary and the collective effect of all emitters must be included as part of the mode environment.

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2012

K. Ding and C. Ning, “Metallic subwavelength-cavity semiconductor nanolasers,” Light: Sci. Appl.1(7), e20 (2012).
[CrossRef]

C. A. Ni and S. L. Chuang, “Theory of high-speed nanolasers and nanoLEDs,” Opt. Express20(15), 16450–16470 (2012).
[CrossRef]

M. Glauser, G. Rossbach, G. Cosendey, J. Levrat, M. Cobet, J. Carlin, J. Besbas, M. Gallart, P. Gilliot, R. Butté, and N. Grandjean, “Investigation of InGaN/GaN quantum wells for polariton laser diodes,” Phys. Status Solidi C9(5), 1325–1329 (2012).
[CrossRef]

M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Thresholdless nanoscale coaxial lasers,” Nature482(7384), 204–207 (2012).
[CrossRef] [PubMed]

2011

D. Martín-Cano, A. González-Tudela, L. Martín-Moreno, F. J. García-Vidal, C. Tejedor, and E. Moreno, “Dissipation-driven generation of two-qubit entanglement mediated by plasmonic waveguides,” Phys. Rev. B84(23), 235306 (2011).
[CrossRef]

2010

2009

S. W. Chang and S. L. Chuang, “Normal modes for plasmonic nanolasers with dispersive and inhomogeneous media,” Opt. Lett.34(1), 91–93 (2009).
[CrossRef] [PubMed]

S. W. Chang and S. L. Chuang, “Fundamental formulation for plasmonic nanolasers,” IEEE J. Quantum Electron.45(8), 1014–1023 (2009).
[CrossRef]

2007

M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. De Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Nötzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics1(10), 589–594 (2007).
[CrossRef]

K. Srinivasan and O. Painter, “Linear and nonlinear optical spectroscopy of a strongly coupled microdisk-quantum dot system,” Nature450(7171), 862–865 (2007).
[CrossRef] [PubMed]

2006

G. Khitrova, H. Gibbs, M. Kira, S. Koch, and A. Scherer, “Vacuum Rabi splitting in semiconductors,” Nat. Phys.2(2), 81–90 (2006).
[CrossRef]

H. Altug, D. Englund, and J. Vučković, “Ultrafast photonic crystal nanocavity laser,” Nat. Phys.2(7), 484–488 (2006).
[CrossRef]

2004

T. Baba, D. Sano, K. Nozaki, K. Inoshita, Y. Kuroki, and F. Koyama, “Observation of fast spontaneous emission decay in GaInAsP photonic crystal point defect nanocavity at room temperature,” Appl. Phys. Lett.85(18), 3989–3991 (2004).
[CrossRef]

2003

P. Yu, P. Bhattacharya, and J. Cheng, “Enhanced spontaneous emission from InAs/GaAs self-organized quantum dots in a GaAs photonic-crystal-based microcavity,” J. Appl. Phys.93(10), 6173–6176 (2003).
[CrossRef]

T. Baba and D. Sano, “Low-threshold lasing and Purcell effect in microdisk lasers at room temperature,” IEEE J. Sel. Top. Quantum Electron.9(5), 1340–1346 (2003).
[CrossRef]

H. Y. Ryu and M. Notomi, “Enhancement of spontaneous emission from the resonant modes of a photonic crystal slab single-defect cavity,” Opt. Lett.28(23), 2390–2392 (2003).
[CrossRef] [PubMed]

1999

1992

R. H. Groeneveld, R. Sprik, and A. Lagendijk, “Effect of a nonthermal electron distribution on the electron-phonon energy relaxation process in noble metals,” Phys. Rev. B Condens. Matter45(9), 5079–5082 (1992).
[CrossRef] [PubMed]

W. S. Fann, R. Storz, H. W. Tom, and J. Bokor, “Electron thermalization in gold,” Phys. Rev. B Condens. Matter46(20), 13592–13595 (1992).
[CrossRef] [PubMed]

H. Walther, “Experiments on cavity quantum electrodynamics,” Phys. Rep.219(3-6), 263–281 (1992).
[CrossRef]

1991

R. J. Glauber and M. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A43(1), 467–491 (1991).
[CrossRef] [PubMed]

G. Björk, S. Machida, Y. Yamamoto, and K. Igeta, “Modification of spontaneous emission rate in planar dielectric microcavity structures,” Phys. Rev. A44(1), 669–681 (1991).
[CrossRef] [PubMed]

B. Deveaud, F. Clérot, N. Roy, K. Satzke, B. Sermage, and D. S. Katzer, “Enhanced radiative recombination of free excitons in GaAs quantum wells,” Phys. Rev. Lett.67(17), 2355–2358 (1991).
[CrossRef] [PubMed]

1989

M. Asada, “Intraband relaxation time in quantum-well lasers,” IEEE J. Quantum Electron.25(9), 2019–2026 (1989).
[CrossRef]

G. P. Agrawal and N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron.25(11), 2297–2306 (1989).
[CrossRef]

1988

S. R. Chinn, P. Zory, and A. R. Reisinger, “A model for GRIN-SCH-SQW diode lasers,” IEEE J. Quantum Electron.24(11), 2191–2214 (1988).
[CrossRef]

1987

M. Yamanishi and Y. Lee, “Phase dampings of optical dipole moments and gain spectra in semiconductor lasers,” IEEE J. Quantum Electron.23(4), 367–370 (1987).
[CrossRef]

1981

W. Kowalsky, A. Schlachetzki, and F. Fiedler, “Near‐band‐gap absorption of InGaAsP at 1.3 μm wavelength,” Phys. Status Solidi A68(1), 153–158 (1981).
[CrossRef]

M. Yamada and Y. Suematsu, “Analysis of gain suppression in undoped injection lasers,” J. Appl. Phys.52(4), 2653–2664 (1981).
[CrossRef]

1946

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev.69, 681 (1946).

1930

V. Weisskopf and E. Wigner, “Calculation of the natural brightness of spectral lines on the basis of Dirac’s theory,” Z. Phys.63, 54–73 (1930).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal and N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron.25(11), 2297–2306 (1989).
[CrossRef]

Altug, H.

H. Altug, D. Englund, and J. Vučković, “Ultrafast photonic crystal nanocavity laser,” Nat. Phys.2(7), 484–488 (2006).
[CrossRef]

Asada, M.

M. Asada, “Intraband relaxation time in quantum-well lasers,” IEEE J. Quantum Electron.25(9), 2019–2026 (1989).
[CrossRef]

Baba, T.

T. Baba, D. Sano, K. Nozaki, K. Inoshita, Y. Kuroki, and F. Koyama, “Observation of fast spontaneous emission decay in GaInAsP photonic crystal point defect nanocavity at room temperature,” Appl. Phys. Lett.85(18), 3989–3991 (2004).
[CrossRef]

T. Baba and D. Sano, “Low-threshold lasing and Purcell effect in microdisk lasers at room temperature,” IEEE J. Sel. Top. Quantum Electron.9(5), 1340–1346 (2003).
[CrossRef]

M. Fujita, A. Sakai, and T. Baba, “Ultrasmall and ultralow threshold GaInAsP-InP microdisk injection lasers: design, fabrication, lasing characteristics, and spontaneous emission factor,” IEEE J. Sel. Top. Quantum Electron.5(3), 673–681 (1999).
[CrossRef]

Besbas, J.

M. Glauser, G. Rossbach, G. Cosendey, J. Levrat, M. Cobet, J. Carlin, J. Besbas, M. Gallart, P. Gilliot, R. Butté, and N. Grandjean, “Investigation of InGaN/GaN quantum wells for polariton laser diodes,” Phys. Status Solidi C9(5), 1325–1329 (2012).
[CrossRef]

Bhattacharya, P.

P. Yu, P. Bhattacharya, and J. Cheng, “Enhanced spontaneous emission from InAs/GaAs self-organized quantum dots in a GaAs photonic-crystal-based microcavity,” J. Appl. Phys.93(10), 6173–6176 (2003).
[CrossRef]

Bianucci, P.

Björk, G.

G. Björk, S. Machida, Y. Yamamoto, and K. Igeta, “Modification of spontaneous emission rate in planar dielectric microcavity structures,” Phys. Rev. A44(1), 669–681 (1991).
[CrossRef] [PubMed]

Bokor, J.

W. S. Fann, R. Storz, H. W. Tom, and J. Bokor, “Electron thermalization in gold,” Phys. Rev. B Condens. Matter46(20), 13592–13595 (1992).
[CrossRef] [PubMed]

Bondarenko, O.

M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics4(6), 395–399 (2010).
[CrossRef]

Butté, R.

M. Glauser, G. Rossbach, G. Cosendey, J. Levrat, M. Cobet, J. Carlin, J. Besbas, M. Gallart, P. Gilliot, R. Butté, and N. Grandjean, “Investigation of InGaN/GaN quantum wells for polariton laser diodes,” Phys. Status Solidi C9(5), 1325–1329 (2012).
[CrossRef]

Carlin, J.

M. Glauser, G. Rossbach, G. Cosendey, J. Levrat, M. Cobet, J. Carlin, J. Besbas, M. Gallart, P. Gilliot, R. Butté, and N. Grandjean, “Investigation of InGaN/GaN quantum wells for polariton laser diodes,” Phys. Status Solidi C9(5), 1325–1329 (2012).
[CrossRef]

Chang, S. W.

S. W. Chang and S. L. Chuang, “Normal modes for plasmonic nanolasers with dispersive and inhomogeneous media,” Opt. Lett.34(1), 91–93 (2009).
[CrossRef] [PubMed]

S. W. Chang and S. L. Chuang, “Fundamental formulation for plasmonic nanolasers,” IEEE J. Quantum Electron.45(8), 1014–1023 (2009).
[CrossRef]

Cheng, J.

P. Yu, P. Bhattacharya, and J. Cheng, “Enhanced spontaneous emission from InAs/GaAs self-organized quantum dots in a GaAs photonic-crystal-based microcavity,” J. Appl. Phys.93(10), 6173–6176 (2003).
[CrossRef]

Chinn, S. R.

S. R. Chinn, P. Zory, and A. R. Reisinger, “A model for GRIN-SCH-SQW diode lasers,” IEEE J. Quantum Electron.24(11), 2191–2214 (1988).
[CrossRef]

Chuang, S. L.

Clérot, F.

B. Deveaud, F. Clérot, N. Roy, K. Satzke, B. Sermage, and D. S. Katzer, “Enhanced radiative recombination of free excitons in GaAs quantum wells,” Phys. Rev. Lett.67(17), 2355–2358 (1991).
[CrossRef] [PubMed]

Cobet, M.

M. Glauser, G. Rossbach, G. Cosendey, J. Levrat, M. Cobet, J. Carlin, J. Besbas, M. Gallart, P. Gilliot, R. Butté, and N. Grandjean, “Investigation of InGaN/GaN quantum wells for polariton laser diodes,” Phys. Status Solidi C9(5), 1325–1329 (2012).
[CrossRef]

Cosendey, G.

M. Glauser, G. Rossbach, G. Cosendey, J. Levrat, M. Cobet, J. Carlin, J. Besbas, M. Gallart, P. Gilliot, R. Butté, and N. Grandjean, “Investigation of InGaN/GaN quantum wells for polariton laser diodes,” Phys. Status Solidi C9(5), 1325–1329 (2012).
[CrossRef]

De Vries, T.

M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. De Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Nötzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics1(10), 589–594 (2007).
[CrossRef]

de Waardt, H.

M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. De Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Nötzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics1(10), 589–594 (2007).
[CrossRef]

Deveaud, B.

B. Deveaud, F. Clérot, N. Roy, K. Satzke, B. Sermage, and D. S. Katzer, “Enhanced radiative recombination of free excitons in GaAs quantum wells,” Phys. Rev. Lett.67(17), 2355–2358 (1991).
[CrossRef] [PubMed]

Ding, K.

K. Ding and C. Ning, “Metallic subwavelength-cavity semiconductor nanolasers,” Light: Sci. Appl.1(7), e20 (2012).
[CrossRef]

Eijkemans, T. J.

M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. De Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Nötzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics1(10), 589–594 (2007).
[CrossRef]

Englund, D.

Fainman, Y.

M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Thresholdless nanoscale coaxial lasers,” Nature482(7384), 204–207 (2012).
[CrossRef] [PubMed]

M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics4(6), 395–399 (2010).
[CrossRef]

Fann, W. S.

W. S. Fann, R. Storz, H. W. Tom, and J. Bokor, “Electron thermalization in gold,” Phys. Rev. B Condens. Matter46(20), 13592–13595 (1992).
[CrossRef] [PubMed]

Feng, L.

M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics4(6), 395–399 (2010).
[CrossRef]

Fiedler, F.

W. Kowalsky, A. Schlachetzki, and F. Fiedler, “Near‐band‐gap absorption of InGaAsP at 1.3 μm wavelength,” Phys. Status Solidi A68(1), 153–158 (1981).
[CrossRef]

Fujita, M.

M. Fujita, A. Sakai, and T. Baba, “Ultrasmall and ultralow threshold GaInAsP-InP microdisk injection lasers: design, fabrication, lasing characteristics, and spontaneous emission factor,” IEEE J. Sel. Top. Quantum Electron.5(3), 673–681 (1999).
[CrossRef]

Gallart, M.

M. Glauser, G. Rossbach, G. Cosendey, J. Levrat, M. Cobet, J. Carlin, J. Besbas, M. Gallart, P. Gilliot, R. Butté, and N. Grandjean, “Investigation of InGaN/GaN quantum wells for polariton laser diodes,” Phys. Status Solidi C9(5), 1325–1329 (2012).
[CrossRef]

García-Vidal, F. J.

D. Martín-Cano, A. González-Tudela, L. Martín-Moreno, F. J. García-Vidal, C. Tejedor, and E. Moreno, “Dissipation-driven generation of two-qubit entanglement mediated by plasmonic waveguides,” Phys. Rev. B84(23), 235306 (2011).
[CrossRef]

Gayral, B.

Geluk, E. J.

M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. De Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Nötzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics1(10), 589–594 (2007).
[CrossRef]

Gérard, J. M.

Gibbs, H.

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M. Glauser, G. Rossbach, G. Cosendey, J. Levrat, M. Cobet, J. Carlin, J. Besbas, M. Gallart, P. Gilliot, R. Butté, and N. Grandjean, “Investigation of InGaN/GaN quantum wells for polariton laser diodes,” Phys. Status Solidi C9(5), 1325–1329 (2012).
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G. Khitrova, H. Gibbs, M. Kira, S. Koch, and A. Scherer, “Vacuum Rabi splitting in semiconductors,” Nat. Phys.2(2), 81–90 (2006).
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T. Baba, D. Sano, K. Nozaki, K. Inoshita, Y. Kuroki, and F. Koyama, “Observation of fast spontaneous emission decay in GaInAsP photonic crystal point defect nanocavity at room temperature,” Appl. Phys. Lett.85(18), 3989–3991 (2004).
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T. Baba, D. Sano, K. Nozaki, K. Inoshita, Y. Kuroki, and F. Koyama, “Observation of fast spontaneous emission decay in GaInAsP photonic crystal point defect nanocavity at room temperature,” Appl. Phys. Lett.85(18), 3989–3991 (2004).
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M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. De Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Nötzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics1(10), 589–594 (2007).
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M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Thresholdless nanoscale coaxial lasers,” Nature482(7384), 204–207 (2012).
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Figures (3)

Fig. 1
Fig. 1

Schematic of the subwavelength metallo-dielectric laser in Fig. 4 of Nezhad et al. [8]. Key geometrical parameters are rmajor = 245nm, rminor = 210nm, Δ = 200nm, h1, h2, h3 and hcore are 200nm, 550nm, 250nm and 480nm, respectively.

Fig. 2
Fig. 2

(a) The lasing mode’s electric field profile and the three spectra in the evaluation of the Purcell factor: (b) cavity lineshape, (c) homogeneous broadening lineshape and (d) PL spectra. Dashed red: measured at low pump powers, and solid blue: datasheet provided by OEpic Inc.

Fig. 3
Fig. 3

Simulated mode distribution of all modes that falls within the spectral window of PL and have cavity Q>20. Also shown are Purcell factors for each mode, Fcav, calculated using two different sources of PL spectra.

Tables (1)

Tables Icon

Table 1 Evaluation of the Purcell factor F cav ( TE012 ) using different methods

Equations (19)

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E ^ ( r,t )= k,ε ω k 2 ε 0 L 3 i( a ^ k,ε ( t ) e ikr a ^ k,ε ( t ) e ikr )ε
a ^ k,ε ( t )= a ^ k,ε ( 0 ) e i ω k t ; a ^ k,ε ( t )= a ^ k,ε ( 0 ) e i ω k t
E ^ ( r,t )= E ^ + ( r,t )+ E ^ ( r,t ), E ^ + ( r,t )= k,ε ω k 2 ε 0 L 3 i a ^ k,ε ( t ) e ikr ε , E ^ ( r,t )= ( E ^ + ( r,t ) ) = k,ε ω k 2 ε 0 L 3 i a ^ k,ε ( t ) e ikr ε
E ^ u ( r,t )= E ^ + u ( r,t )+ E ^ u ( r,t )= ω k >0 i ω k ( a ^ k ( t ) a ^ k ( t ) ) e k ( r )
e k ( r )= E k ( r ) N k , N k V [ ε( r ) E k 2 ( r )+μ( r ) H k 2 ( r ) ] d 3 r
N k = V [ ( ω ε R ( r, ω ) ) ω | ω = ω k E k 2 ( r )+μ( r ) H k 2 ( r ) ] d 3 r = V [ ( ( ω ε R ( r, ω ) ) ω | ω = ω k + ε R ( r, ω k ) ) E k 2 ( r ) ] d 3 r
d dt [ a ^ k ( t ) a ^ k ( t+τ ) ] R = C k [ a ^ k ( t ) a ^ k ( t+τ ) ] R + C k n ¯ ( ω k ) e 1 2 C k | τ | e i ω k τ d dt [ a ^ k ( t ) a ^ k ( t+τ ) ] R = C k [ a ^ k ( t ) a ^ k ( t+τ ) ] R + C k ( n ¯ ( ω k )+1 ) e 1 2 C k | τ | e i ω k τ
[ a ^ k (t) a ^ k (t+τ) ] R = n ¯ ( ω k ) e 1 2 C k | τ | e i ω k τ [ a ^ k (t) a ^ k (t+τ) ] R =( n ¯ ( ω k )+1 ) e 1 2 C k | τ | e i ω k τ
P 21,i ( t )= 1 2 t 0 t 0 + τ coll t 0 t 0 + τ coll e i ω 21 ( t t ) i|( * 12 ( ω 21 ) E ^ + ( r, t ) )( 12 ( ω 21 ) E ^ ( r, t ) )|i D( ω 21 )d ω 21 d t d t
P 21,| 0...0 free = 1 2 t 0 t 0 + τ coll t 0 t 0 + τ coll e i ω 21 ( t t ) 00 |( 12 * ( ω 21 ) k , ε ω k 2 ε 0 L 3 ε a ^ k , ε ( t ) e i k r e )× ( 12 ( ω 21 ) k , ε ω k 2 ε 0 L 3 ε a ^ k , ε ( t ) e i k r e )| 00 D( ω 21 )d ω 21 d t d t = k,ε ω k 2 ε 0 L 3 | 12 ( ω 21 )ε | 2 D( ω 21 )R( ω k ω 21 , τ coll )d ω 21 ω 21 3 3π ε 0 c 3 τ coll | 12 ( ω 21 ) | 2 D( ω 21 )d ω 21
P 21,| 0...0 cav = k ω k | 12 ( ω 21 ) e k ( r e ) | 2 D( ω 21 )R( ω k ω 21 , τ coll )d ω 21
P 21,equilibrium cav = k ω k ( n ¯ ( ω k )+1 ) | 12 ( ω 21 ) e k ( r e ) | 2 D( ω 21 ) L k ( ω ω k ) R( ω ω 21 , τ coll )dωd ω 21
L k ( ω ω k ) 1 π 1 2 C k ( 1 2 C k ) 2 + ( ω ω k ) 2 = 2 π Q ω k ( 1 2 Δ ω k ) 2 ( 1 2 Δ ω k ) 2 + ( ω ω k ) 2 , where C k =Δ ω k
H ^ = H ^ A + H ^ F + H ^ AF + H ^ R + H ^ FR
P 21,| 0...0 material ω 21 3 3π ε r ( c/ n r ) 3 τ coll | 12 ( ω 21 ) | 2 D( ω 21 )d ω 21 ω ¯ 21 3 3π ε r ( c/ n r ) 3 τ coll | 12 ( ω ¯ 21 ) | 2
F cav P 21,equilibrium cav P 21,| 0...0 material = k 3π ε r ( c/ n r ) 3 τ coll ω k ω ¯ 21 3 1 | 12 ( ω ¯ 21 ) | 2 | 12 ( ω 21 ) e k ( r e ) | 2 D( ω 21 ) L k ( ω ω k )R( ω ω 21 , τ coll ) dωd ω 21 k 3π ε r ( c/ n r ) 3 τ coll ω k ω ¯ 21 3 | 12 ( ω ¯ 21 ) e k ( r e ) | 2 | 12 ( ω ¯ 21 ) | 2 D( ω 21 ) L k ( ω ω k )R( ω ω 21 , τ coll ) dωd ω 21
| 12 ( ω ¯ 21 ) e k ( r e ) | 2 1 3 | 12 ( ω ¯ 21 ) | 2 1 V a V a | e k ( r ) | 2 d 3 r
F cav = k π ( c/ n r ) 3 τ coll ω k ω ¯ 21 3 1 V a { ε r V a | E k ( r ) | 2 d 3 r [ ( ( ω ε R ( r, ω ) ) ω | ω = ω k + ε R ( r, ω k ) ) E k 2 ( r ) ] d 3 r }× D( ω 21 ) L k ( ω ω k )R( ω ω 21 , τ coll ) dωd ω 21 = k π ( c/ n r ) 3 τ coll ω k ω ¯ 21 3 1 V a { Γ k } D( ω 21 ) L k ( ω ω k )R( ω ω 21 , τ coll ) dωd ω 21 = k F cav ( k )
β= F cav ( 1 ) k F cav ( k )

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