Abstract

Dissipative Kerr cavity solitons experience a so-called self-frequency shift (SFS) as a result of Raman interactions. The frequency shift has been observed in several microcavity systems. The Raman process has also been shown numerically to influence the soliton pumping efficiency. Here, a perturbed Lagrangian approach is used to derive simple analytical expressions for the SFS and the soliton efficiency. The predicted dependences of these quantities on soliton pulse width are compared with measurements in a high-Q silica microcavity. The Raman time constant in silica is also inferred. Analytical expressions for the Raman SFS and soliton efficiency greatly simplify the prediction of soliton behavior over a wide range of microcavity platforms.

© 2016 Optical Society of America

Full Article  |  PDF Article

Corrections

19 July 2016: A correction was made to the text on p. 3420, six lines above Eq. (1), and p. 3421, in the line below Eq. (15).


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References

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2016 (5)

V. Brasch, M. Geiselmann, T. Herr, G. Lihachev, M. Pfeiffer, M. Gorodetsky, and T. Kippenberg, Science 351, 357 (2016).
[Crossref]

M. Karpov, H. Guo, A. Kordts, V. Brasch, M. H. Pfeiffer, M. Zervas, M. Geiselmann, and T. J. Kippenberg, Phys. Rev. Lett. 116, 103902 (2016).
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[Crossref]

2015 (2)

X. Yi, Q.-F. Yang, K. Y. Yang, M.-G. Suh, and K. Vahala, Optica 2, 1078 (2015).
[Crossref]

C. Milián, A. V. Gorbach, M. Taki, A. V. Yulin, and D. V. Skryabin, Phys. Rev. A 92, 033851 (2015).
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2014 (3)

T. Herr, V. Brasch, J. Jost, C. Wang, N. Kondratiev, M. Gorodetsky, and T. Kippenberg, Nat. Photonics 8, 145 (2014).
[Crossref]

T. Herr, V. Brasch, J. Jost, I. Mirgorodskiy, G. Lihachev, M. Gorodetsky, and T. Kippenberg, Phys. Rev. Lett. 113, 123901 (2014).
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C. Bao, L. Zhang, A. Matsko, Y. Yan, Z. Zhao, G. Xie, A. M. Agarwal, L. C. Kimerling, J. Michel, L. Maleki, and A. E. Willner, Opt. Lett. 39, 6126 (2014).
[Crossref]

2013 (1)

2011 (2)

2010 (1)

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit, and M. Haelterman, Nat. Photonics 4, 471 (2010).
[Crossref]

2008 (1)

2007 (1)

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. Kippenberg, Nature 450, 1214 (2007).
[Crossref]

2006 (1)

J. M. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys. 78, 1135 (2006).
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2000 (1)

A. Hasegawa, IEEE J. Sel. Top. Quantum Electron. 6, 1161 (2000).
[Crossref]

1999 (1)

1996 (1)

I. Barashenkov and Y. S. Smirnov, Phys. Rev. E 54, 5707 (1996).
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1987 (1)

L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).
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1986 (2)

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Bao, C.

Barashenkov, I.

I. Barashenkov and Y. S. Smirnov, Phys. Rev. E 54, 5707 (1996).
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Brasch, V.

V. Brasch, M. Geiselmann, T. Herr, G. Lihachev, M. Pfeiffer, M. Gorodetsky, and T. Kippenberg, Science 351, 357 (2016).
[Crossref]

M. Karpov, H. Guo, A. Kordts, V. Brasch, M. H. Pfeiffer, M. Zervas, M. Geiselmann, and T. J. Kippenberg, Phys. Rev. Lett. 116, 103902 (2016).
[Crossref]

T. Herr, V. Brasch, J. Jost, I. Mirgorodskiy, G. Lihachev, M. Gorodetsky, and T. Kippenberg, Phys. Rev. Lett. 113, 123901 (2014).
[Crossref]

T. Herr, V. Brasch, J. Jost, C. Wang, N. Kondratiev, M. Gorodetsky, and T. Kippenberg, Nat. Photonics 8, 145 (2014).
[Crossref]

Chrostowski, J.

Coen, S.

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit, and M. Haelterman, Nat. Photonics 4, 471 (2010).
[Crossref]

J. M. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys. 78, 1135 (2006).
[Crossref]

Del’Haye, P.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. Kippenberg, Nature 450, 1214 (2007).
[Crossref]

Diddams, S.

T. J. Kippenberg, R. Holzwarth, and S. Diddams, Science 332, 555 (2011).
[Crossref]

Dudley, J. M.

J. M. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys. 78, 1135 (2006).
[Crossref]

Emplit, P.

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit, and M. Haelterman, Nat. Photonics 4, 471 (2010).
[Crossref]

Gaeta, A. L.

Galko, P.

Gambetta, A.

Geiselmann, M.

V. Brasch, M. Geiselmann, T. Herr, G. Lihachev, M. Pfeiffer, M. Gorodetsky, and T. Kippenberg, Science 351, 357 (2016).
[Crossref]

M. Karpov, H. Guo, A. Kordts, V. Brasch, M. H. Pfeiffer, M. Zervas, M. Geiselmann, and T. J. Kippenberg, Phys. Rev. Lett. 116, 103902 (2016).
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Genty, G.

J. M. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys. 78, 1135 (2006).
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C. Milián, A. V. Gorbach, M. Taki, A. V. Yulin, and D. V. Skryabin, Phys. Rev. A 92, 033851 (2015).
[Crossref]

Gordon, J. P.

Gorodetsky, M.

V. Brasch, M. Geiselmann, T. Herr, G. Lihachev, M. Pfeiffer, M. Gorodetsky, and T. Kippenberg, Science 351, 357 (2016).
[Crossref]

T. Herr, V. Brasch, J. Jost, C. Wang, N. Kondratiev, M. Gorodetsky, and T. Kippenberg, Nat. Photonics 8, 145 (2014).
[Crossref]

T. Herr, V. Brasch, J. Jost, I. Mirgorodskiy, G. Lihachev, M. Gorodetsky, and T. Kippenberg, Phys. Rev. Lett. 113, 123901 (2014).
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Gorodetsky, M. L.

T. Herr, M. L. Gorodetsky, and T. J. Kippenberg, “Dissipative Kerr solitons in optical microresonators,” arXiv:1508.04989 (2015).

Gorza, S.-P.

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit, and M. Haelterman, Nat. Photonics 4, 471 (2010).
[Crossref]

Grelu, P.

P. Grelu, Nonlinear Optical Cavity Dynamics: from Microresonators to Fiber Lasers (Wiley, 2015).

Guo, H.

M. Karpov, H. Guo, A. Kordts, V. Brasch, M. H. Pfeiffer, M. Zervas, M. Geiselmann, and T. J. Kippenberg, Phys. Rev. Lett. 116, 103902 (2016).
[Crossref]

Haelterman, M.

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit, and M. Haelterman, Nat. Photonics 4, 471 (2010).
[Crossref]

Hasegawa, A.

A. Hasegawa, IEEE J. Sel. Top. Quantum Electron. 6, 1161 (2000).
[Crossref]

Herr, T.

V. Brasch, M. Geiselmann, T. Herr, G. Lihachev, M. Pfeiffer, M. Gorodetsky, and T. Kippenberg, Science 351, 357 (2016).
[Crossref]

T. Herr, V. Brasch, J. Jost, C. Wang, N. Kondratiev, M. Gorodetsky, and T. Kippenberg, Nat. Photonics 8, 145 (2014).
[Crossref]

T. Herr, V. Brasch, J. Jost, I. Mirgorodskiy, G. Lihachev, M. Gorodetsky, and T. Kippenberg, Phys. Rev. Lett. 113, 123901 (2014).
[Crossref]

T. Herr, M. L. Gorodetsky, and T. J. Kippenberg, “Dissipative Kerr solitons in optical microresonators,” arXiv:1508.04989 (2015).

Holzwarth, R.

T. J. Kippenberg, R. Holzwarth, and S. Diddams, Science 332, 555 (2011).
[Crossref]

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. Kippenberg, Nature 450, 1214 (2007).
[Crossref]

Ilchenko, V.

Jang, J. K.

Jaramillo-Villegas, J. A.

Ji, X.

Joshi, C.

Jost, J.

T. Herr, V. Brasch, J. Jost, I. Mirgorodskiy, G. Lihachev, M. Gorodetsky, and T. Kippenberg, Phys. Rev. Lett. 113, 123901 (2014).
[Crossref]

T. Herr, V. Brasch, J. Jost, C. Wang, N. Kondratiev, M. Gorodetsky, and T. Kippenberg, Nat. Photonics 8, 145 (2014).
[Crossref]

Karpov, M.

M. Karpov, H. Guo, A. Kordts, V. Brasch, M. H. Pfeiffer, M. Zervas, M. Geiselmann, and T. J. Kippenberg, Phys. Rev. Lett. 116, 103902 (2016).
[Crossref]

Kimerling, L. C.

Kippenberg, T.

V. Brasch, M. Geiselmann, T. Herr, G. Lihachev, M. Pfeiffer, M. Gorodetsky, and T. Kippenberg, Science 351, 357 (2016).
[Crossref]

T. Herr, V. Brasch, J. Jost, C. Wang, N. Kondratiev, M. Gorodetsky, and T. Kippenberg, Nat. Photonics 8, 145 (2014).
[Crossref]

T. Herr, V. Brasch, J. Jost, I. Mirgorodskiy, G. Lihachev, M. Gorodetsky, and T. Kippenberg, Phys. Rev. Lett. 113, 123901 (2014).
[Crossref]

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. Kippenberg, Nature 450, 1214 (2007).
[Crossref]

Kippenberg, T. J.

M. Karpov, H. Guo, A. Kordts, V. Brasch, M. H. Pfeiffer, M. Zervas, M. Geiselmann, and T. J. Kippenberg, Phys. Rev. Lett. 116, 103902 (2016).
[Crossref]

T. J. Kippenberg, R. Holzwarth, and S. Diddams, Science 332, 555 (2011).
[Crossref]

T. Herr, M. L. Gorodetsky, and T. J. Kippenberg, “Dissipative Kerr solitons in optical microresonators,” arXiv:1508.04989 (2015).

Klenner, A.

Kockaert, P.

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit, and M. Haelterman, Nat. Photonics 4, 471 (2010).
[Crossref]

Kondratiev, N.

T. Herr, V. Brasch, J. Jost, C. Wang, N. Kondratiev, M. Gorodetsky, and T. Kippenberg, Nat. Photonics 8, 145 (2014).
[Crossref]

Kordts, A.

M. Karpov, H. Guo, A. Kordts, V. Brasch, M. H. Pfeiffer, M. Zervas, M. Geiselmann, and T. J. Kippenberg, Phys. Rev. Lett. 116, 103902 (2016).
[Crossref]

Leaird, D. E.

Lefever, R.

L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).
[Crossref]

Leo, F.

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit, and M. Haelterman, Nat. Photonics 4, 471 (2010).
[Crossref]

Liang, W.

Lihachev, G.

V. Brasch, M. Geiselmann, T. Herr, G. Lihachev, M. Pfeiffer, M. Gorodetsky, and T. Kippenberg, Science 351, 357 (2016).
[Crossref]

T. Herr, V. Brasch, J. Jost, I. Mirgorodskiy, G. Lihachev, M. Gorodetsky, and T. Kippenberg, Phys. Rev. Lett. 113, 123901 (2014).
[Crossref]

Lipson, M.

Lugiato, L. A.

L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).
[Crossref]

Luke, K.

Maleki, L.

Marangoni, M.

Matsko, A.

Matsko, A. B.

Michel, J.

Milián, C.

C. Milián, A. V. Gorbach, M. Taki, A. V. Yulin, and D. V. Skryabin, Phys. Rev. A 92, 033851 (2015).
[Crossref]

Miller, S. A.

Mirgorodskiy, I.

T. Herr, V. Brasch, J. Jost, I. Mirgorodskiy, G. Lihachev, M. Gorodetsky, and T. Kippenberg, Phys. Rev. Lett. 113, 123901 (2014).
[Crossref]

Mitschke, F. M.

Mollenauer, L. F.

Myslinski, P.

Okawachi, Y.

Pfeiffer, M.

V. Brasch, M. Geiselmann, T. Herr, G. Lihachev, M. Pfeiffer, M. Gorodetsky, and T. Kippenberg, Science 351, 357 (2016).
[Crossref]

Pfeiffer, M. H.

M. Karpov, H. Guo, A. Kordts, V. Brasch, M. H. Pfeiffer, M. Zervas, M. Geiselmann, and T. J. Kippenberg, Phys. Rev. Lett. 116, 103902 (2016).
[Crossref]

Qi, M.

Ramponi, R.

Savchenkov, A.

Schliesser, A.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. Kippenberg, Nature 450, 1214 (2007).
[Crossref]

Seidel, D.

Skryabin, D. V.

C. Milián, A. V. Gorbach, M. Taki, A. V. Yulin, and D. V. Skryabin, Phys. Rev. A 92, 033851 (2015).
[Crossref]

Smirnov, Y. S.

I. Barashenkov and Y. S. Smirnov, Phys. Rev. E 54, 5707 (1996).
[Crossref]

Suh, M.-G.

Taki, M.

C. Milián, A. V. Gorbach, M. Taki, A. V. Yulin, and D. V. Skryabin, Phys. Rev. A 92, 033851 (2015).
[Crossref]

Vahala, K.

X. Yi, Q.-F. Yang, K. Youl, and K. Vahala, Opt. Lett. 41, 2037 (2016).
[Crossref]

X. Yi, Q.-F. Yang, K. Y. Yang, M.-G. Suh, and K. Vahala, Optica 2, 1078 (2015).
[Crossref]

Q.-F. Yang, X. Yi, K. Y. Yang, and K. Vahala, “Spatial-mode-interaction-induced dispersive-waves and their active tuning in microresonators,” arXiv:1606.00954 (2016).

Wang, C.

T. Herr, V. Brasch, J. Jost, C. Wang, N. Kondratiev, M. Gorodetsky, and T. Kippenberg, Nat. Photonics 8, 145 (2014).
[Crossref]

Wang, P.-H.

Weiner, A. M.

Wilken, T.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. Kippenberg, Nature 450, 1214 (2007).
[Crossref]

Willner, A. E.

Xie, G.

Xuan, Y.

Xue, X.

Yan, Y.

Yang, K. Y.

X. Yi, Q.-F. Yang, K. Y. Yang, M.-G. Suh, and K. Vahala, Optica 2, 1078 (2015).
[Crossref]

Q.-F. Yang, X. Yi, K. Y. Yang, and K. Vahala, “Spatial-mode-interaction-induced dispersive-waves and their active tuning in microresonators,” arXiv:1606.00954 (2016).

Yang, Q.-F.

X. Yi, Q.-F. Yang, K. Youl, and K. Vahala, Opt. Lett. 41, 2037 (2016).
[Crossref]

X. Yi, Q.-F. Yang, K. Y. Yang, M.-G. Suh, and K. Vahala, Optica 2, 1078 (2015).
[Crossref]

Q.-F. Yang, X. Yi, K. Y. Yang, and K. Vahala, “Spatial-mode-interaction-induced dispersive-waves and their active tuning in microresonators,” arXiv:1606.00954 (2016).

Yi, X.

X. Yi, Q.-F. Yang, K. Youl, and K. Vahala, Opt. Lett. 41, 2037 (2016).
[Crossref]

X. Yi, Q.-F. Yang, K. Y. Yang, M.-G. Suh, and K. Vahala, Optica 2, 1078 (2015).
[Crossref]

Q.-F. Yang, X. Yi, K. Y. Yang, and K. Vahala, “Spatial-mode-interaction-induced dispersive-waves and their active tuning in microresonators,” arXiv:1606.00954 (2016).

Youl, K.

Yulin, A. V.

C. Milián, A. V. Gorbach, M. Taki, A. V. Yulin, and D. V. Skryabin, Phys. Rev. A 92, 033851 (2015).
[Crossref]

Zervas, M.

M. Karpov, H. Guo, A. Kordts, V. Brasch, M. H. Pfeiffer, M. Zervas, M. Geiselmann, and T. J. Kippenberg, Phys. Rev. Lett. 116, 103902 (2016).
[Crossref]

Zhang, L.

Zhao, Z.

IEEE J. Sel. Top. Quantum Electron. (1)

A. Hasegawa, IEEE J. Sel. Top. Quantum Electron. 6, 1161 (2000).
[Crossref]

J. Lightwave Technol. (1)

Nat. Photonics (2)

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit, and M. Haelterman, Nat. Photonics 4, 471 (2010).
[Crossref]

T. Herr, V. Brasch, J. Jost, C. Wang, N. Kondratiev, M. Gorodetsky, and T. Kippenberg, Nat. Photonics 8, 145 (2014).
[Crossref]

Nature (1)

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. Kippenberg, Nature 450, 1214 (2007).
[Crossref]

Opt. Express (2)

Opt. Lett. (7)

Optica (1)

Phys. Rev. A (1)

C. Milián, A. V. Gorbach, M. Taki, A. V. Yulin, and D. V. Skryabin, Phys. Rev. A 92, 033851 (2015).
[Crossref]

Phys. Rev. E (1)

I. Barashenkov and Y. S. Smirnov, Phys. Rev. E 54, 5707 (1996).
[Crossref]

Phys. Rev. Lett. (3)

M. Karpov, H. Guo, A. Kordts, V. Brasch, M. H. Pfeiffer, M. Zervas, M. Geiselmann, and T. J. Kippenberg, Phys. Rev. Lett. 116, 103902 (2016).
[Crossref]

L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).
[Crossref]

T. Herr, V. Brasch, J. Jost, I. Mirgorodskiy, G. Lihachev, M. Gorodetsky, and T. Kippenberg, Phys. Rev. Lett. 113, 123901 (2014).
[Crossref]

Rev. Mod. Phys. (1)

J. M. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys. 78, 1135 (2006).
[Crossref]

Science (2)

V. Brasch, M. Geiselmann, T. Herr, G. Lihachev, M. Pfeiffer, M. Gorodetsky, and T. Kippenberg, Science 351, 357 (2016).
[Crossref]

T. J. Kippenberg, R. Holzwarth, and S. Diddams, Science 332, 555 (2011).
[Crossref]

Other (4)

T. Herr, M. L. Gorodetsky, and T. J. Kippenberg, “Dissipative Kerr solitons in optical microresonators,” arXiv:1508.04989 (2015).

P. Grelu, Nonlinear Optical Cavity Dynamics: from Microresonators to Fiber Lasers (Wiley, 2015).

Q.-F. Yang, X. Yi, K. Y. Yang, and K. Vahala, “Spatial-mode-interaction-induced dispersive-waves and their active tuning in microresonators,” arXiv:1606.00954 (2016).

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2007).

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Figures (2)

Fig. 1.
Fig. 1.

(a) Optical spectra measured for a dissipative Kerr cavity soliton at two operating points. The pump power is suppressed using a fiber grating filter. A sech 2 fit is shown (orange curves), and pulse widths inferred from the fitting are shown in the legend. The location of the pump line is indicated as the black line. The centers of the spectra are indicated by the green lines. (b) Measured Raman SFS plotted versus 1 / τ s 4 for two devices. The red line is a linear fit according to Eq. (1).

Fig. 2.
Fig. 2.

Measured efficiency versus soliton pulse width is plotted (blue points) for two devices and compared with theory. A theory comparison with Raman (solid blue lines) and without Raman (dashed blue lines) is presented. There are no free parameters in the comparison. The small deviations between the measurement and the theory could result from the presence of weak avoided mode crossings in the dispersion spectrum [16].

Equations (20)

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Ω = 8 D 2 Q τ R 15 ω 0 D 1 2 1 τ s 4 = 8 c τ R Q β 2 15 n 0 ω 0 1 τ s 4 ,
P min = P 0 cosh 2 π Ω τ s 2 ,
Γ = Γ 0 sech 2 ( π Ω τ s / 2 ) ,
P 0 = 2 c π A eff β 2 η n 2 Q D 1 1 τ s 2 ,
Γ 0 = π η 2 D 1 τ s ,
i A T c β 2 2 n o 2 A t 2 + g | A | 2 A = ( i κ 2 δ ω ) A + i κ η P in ω o + g τ R A | A | 2 t ,
L = i 2 ( A * A T A A * T ) + c β 2 2 n o | A t | 2 + 1 2 g | A | 4 δ ω | A | 2 ,
R = i κ 2 A + i κ η P in ω o + g τ R A | A | 2 t .
A = B sech [ ( t t o ) / τ s ] e i Ω ( t t o ) e i φ ,
L r i d d T L r ˙ i = ( R A * r i + R * A r i ) d t ,
L = L d t = 2 B c β 2 g n o ( g B 2 6 + c β 2 Ω 2 2 n o δ ω Ω t 0 T φ T ) .
d B d T = κ B + π κ η P in ω o cos φ sech ( c β 2 g n 0 π Ω 2 B ) ,
d φ d T = g 2 B 2 δ ω + c β 2 2 n 0 Ω 2 t o T Ω ,
d B Ω d T = κ B Ω + 8 n 0 τ R g 2 15 c β 2 B 5 ,
d t o d T = c β 2 n 0 Ω ,
B = c β 2 g n o 1 τ s = 2 g ( δ ω + c β 2 2 n o Ω 2 ) ,
cos φ = 1 π τ s c β 2 κ ω o η g n o P in cosh π Ω τ s 2 .
δ ω = 15 c β 2 ω 0 32 n o Q Ω τ R c β 2 2 n o Ω 2 ,
δ ω = c β 2 2 n o 1 τ s 2 ( 1 + 64 c 2 τ R 2 Q 2 β 2 2 225 n o 2 ω 0 2 1 τ s 6 ) .
P sol = 2 c η A eff β 2 n 2 Q 1 τ s .

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