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

Corrections

Xu Yi, Qi-Fan Yang, Ki Youl Yang, and Kerry Vahala, "Theory and measurement of the soliton self-frequency shift and efficiency in optical microcavities: publisher’s note," Opt. Lett. 41, 3722-3722 (2016)
https://www.osapublishing.org/ol/abstract.cfm?uri=ol-41-16-3722

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).

Recently, a new type of temporal soliton, the dissipative Kerr cavity soliton, has been observed in both fiber [1] and microcavity resonators [26]. In microcavities, these solitons are being explored as possible routes to create a chip-based frequency comb system [7,8]. Like all optical solitons, dissipative Kerr cavity (DKC) solitons balance waveguide dispersion with the Kerr effect (intensity dependence of refractive index). Beyond this, however, DKC solitons also use the Kerr effect to compensate cavity loss through parametric amplification [2].

While the Kerr effect is essential to soliton formation, other nonlinearities can alter soliton properties. Soliton interaction with the Raman nonlinearity causes the so-called self-frequency shift (SFS). In optical fiber transmission, this effect causes the soliton’s central frequency to experience increasing redshifting with propagation distance [9]. Intuitively, this is understood as a continuous energy transfer from the blue to red side of the soliton spectrum that is mediated by the Raman interaction [10]. The effect is of practical importance in supercontinuum generation using optical fibers [11]. It has also been used in difference frequency generation of mid-IR frequency combs [12]. Recently, the Raman SFS has been numerically modeled [13] and observed to influence DKC solitons [3,14]. Here, rather than producing a continuously increasing redshift, the Raman effect produces a constant frequency offset between the optical pump and the soliton spectral maximum (i.e., a frequency locked Raman soliton [13]). Without the Raman SFS, the optical pump would be centered on the soliton comb spectrum (at the spectral maximum). Instead, the soliton spectral maximum is red-shifted away from the pump. The amount of shift increases with soliton peak power and, so far, has been numerically calculated [3,13,14] by solving the Lugiato–Lefever equation [15].

Here, we present an analysis of the SFS for DKC solitons using the perturbed Lagrangian formalism and develop an analytical expression for the frequency shift in terms of soliton and cavity properties. In addition, an analytical expression for the soliton efficiency, including the Raman effect, is found and compared with new measurements, as well as those reported earlier [3]. The Raman SFS and soliton efficiency expressions are first compared with measurements to confirm the predicted behavior; then the derivation of these expressions is outlined.

To form DKC solitons, a microresonator must have a transverse mode family with dispersion that is both anomalous and free of avoided mode crossings [16]. Pumping of a single mode belonging to this mode family will generate solitons when the pumping frequency is red detuned relative to the mode resonance. Further details on the excitation of DKC solitons are provided in [26]. DKC solitons were generated in a 3 mm silica resonator (free spectral range 22GHz). General information on the resonators and the techniques used to measure the solitons are described in [3,17]. Soliton spectra measured in a silica microresonator at two different operating points are shown in Fig. 1(a). Soliton generation is confirmed by time domain intensity autocorrelation and frequency resolved optical gating (FROG) [3]. Additional confirmation is provided by fitting to the theoretically predicted hyperbolic secant-squared spectrum [see the spectral envelopes in Fig. 1(a)]. The Raman SFS offset of the pump line relative to the soliton spectral center is also indicated in the figure. It can be seen that the broader soliton spectrum (narrower pulse width) features a larger Raman SFS. Finally, soliton pulse width, power, and Raman SFS depend on a single operating point parameter, the frequency detuning of the resonator mode being pumped relative to the pump frequency, δω=ω0ωp. (ωp is the pump frequency, and ω0 is the frequency of the resonator mode that is pumped.) To set and hold this parameter (and, in turn, other soliton properties), soliton power was used to servo control the pump laser frequency as described elsewhere [3,17]. In the measurements, the protocol was to hold frequency detuning close to the soliton existence maximum detuning [2,3] and to then vary this maximum detuning by adjusting the pumping power.

 figure: 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 sech2 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/τs4 for two devices. The red line is a linear fit according to Eq. (1).

Download Full Size | PPT Slide | PDF

The key theoretical results of this Letter are the following expressions for the Raman SFS, Ω; the minimum input pump power required to generate a soliton, Pmin; and the soliton pumping efficiency, Γ=Psol/Pmin [18] (Psol is the average soliton power):

Ω=8D2QτR15ω0D121τs4=8cτRQβ215n0ω01τs4,
Pmin=P0cosh2πΩτs2,
Γ=Γ0sech2(πΩτs/2),
where c, n0, n2 and β2 are the speed of light, refractive index, Kerr nonlinear refractive index coefficient, and group velocity dispersion, respectively. D1/2π is the cavity-free spectral range in Hertz units at the pumping mode (i.e., soliton repetition frequency). Q (QE) is the total (external) optical Q factor and η=Q/QE is the coupling factor. In addition, τs is the soliton pulse width, τR is the Raman time constant [19], and D2/2π gives the second-order dispersion in units of change in free-spectral range per mode (D2=cnoD12β2). In addition, P0 and Γ0 are the soliton minimum pumping power and soliton efficiency in the absence of the Raman SFS (i.e., Ω=0) and are given by [3]:
P0=2cπAeffβ2ηn2QD11τs2,
Γ0=πη2D1τs,
where Aeff is the effective nonlinear mode area. Before deriving Eqs. (1)–(3), they are compared with measurements.

To test the theory, a series of soliton spectra of increasing spectral width were obtained using the operating point locking method [17]. The Raman SFS, Ω, and the pulse width, τs, were measured by least-squares fitting of the soliton optical spectrum [see Fig. 1(a)] to the theoretical soliton spectral envelope P(ω)=Pcsech2[πτs(ωωpΩ)/2], where Pc is the spectral maximum power. Figure 1(b) plots the measured Ω versus 1/τs4 in two devices. For reference, device I is a device characterized previously [3]. A linear fitting to the two sets of data is also provided, confirming the predicted theoretical dependence in Eq. (1). Furthermore, with measured parameters (device I: Q=200 million, D1/2π=22GHz, D2/2π=17kHz; and device II: Q=235 million, D1/2π=22GHz, D2/2π=17kHz), a value of τR=1.8fs, both for device I and for device II, is inferred from the linear fitting, which is in a reasonable agreement with the value of 2–4 fs reported in silica optical fibers [19]. The small non-zero intercept of the linear fitting is in the range of 1 free-spectral range (22 GHz) of the two resonators.

To measure the soliton efficiency, Γ, the operating point is fixed (i.e., laser-cavity detuning δω is fixed) while the pump power is decreased until the soliton disappears. Near the disappearance point, the soliton average power, Psol, and the minimum pump power, Pmin, are recorded and used to obtain efficiency as Γ=Psol/Pmin. The pulse width is obtained from the optical spectrum as before. Measured efficiency is plotted versus the pulse width in Fig. 2 for devices I and II. The prediction given by Eq. (3) is shown as the solid line. The value of τR used in the plots is that inferred from fitting in Fig. 1(b). Also, η=0.29, 0.37 for devices I and II. Device II has a higher efficiency as a result of a larger coupling coefficient. The agreement between the theory and measurement in Fig. 2(b) is very good, especially considering that there are no free parameters. The dashed lines in the plots give the uncorrected efficiency prediction of Eq. (5) (i.e., Γ0 versus τs).

 figure: 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].

Download Full Size | PPT Slide | PDF

The derivation of Eqs. (1)–(3) is now presented. For transverse mode families that are relatively free of avoided mode crossings, the slowly varying internal cavity field A(T,t) can be described by the Lugiato–Lefever equation [15] augmented with an additional Raman effect term [3,13,14]:

iATcβ22no2At2+g|A|2A=(iκ2δω)A+iκηPinωo+gτRA|A|2t,
where T and t are the slow and fast times, g=ωo2n2D1/2πnoAeff denotes the normalized Kerr effect nonlinear coefficient, κ=ω0/Q is the power dissipation rate of the soliton mode family, and Pin is the pump power. Exact solutions of Eq. (6) are only known in the absence of dissipation and the Raman effect [20,21]. However, an approximate solution, including the dissipation term, has been calculated using the Lagrangian perturbation approach [22,23]. Here, we extend this method by including the Raman term as a perturbation.

The Lagrangian density for this system is given by [2224]

L=i2(A*ATAA*T)+cβ22no|At|2+12g|A|4δω|A|2,
and the dissipation, pumping, and Raman terms are combined in a perturbation R, where
R=iκ2A+iκηPinωo+gτRA|A|2t.

Equation (6) is recovered by taking δL/δA*=R [24]. The form for the slowly varying field envelope of the solitons wherein the central frequency is shifted by Ω is given by

A=Bsech[(tto)/τs]eiΩ(tto)eiφ,
which is an exact solution for the case with no perturbation (R=0). This form is also consistent with measurement and numerical modeling of the DKC solitons in the presence of Raman interactions [3,13,14]. For the proceeding analysis, the soliton phase (φ), temporal position (to), amplitude (B), pulse width (τs), and shift frequency (Ω) are considered functions of T, the slow time variable. Finally, Eq. (9) assumes δωκ (i.e., large pump detuning). In this case, the background field associated with DKC solitons is much weaker than the soliton peak power and is not included in Eq. (9). However, it can be retrieved from Eq. (6) for |(tt0)/τs|1, where the soliton pulse is no longer dominant [2,23,25].

It is observed experimentally that the central soliton comb line power is a constant (independent of the operating point) and in good agreement with the Raman-free Lugiato–Lefever prediction [3]. By Fourier transform of a train of periodic pulses (period=2π/D1) of the form in Eq. (9), the power per comb line at the central maximum of the soliton spectrum is given by ω02D12/4Qext×B2τs2. Accordingly, to be consistent with experimental observations, the product Bτs is assumed to be given by the Raman-free result, Bτs=cβ2/gno. This is used to eliminate τs in the calculation below. It is also possible to show that this result follows directly from variation of the Raman-perturbed system [Eqs. (7) and (8)] with B and τs treated as independent variables.

The equations of motion for the perturbed Lagrangian have the form [2325],

LriddTLr˙i=(RA*ri+R*Ari)dt,
where the coordinates ri in the Lagrangian are taken as the parameters B, Ω, to, and φ. Using Eq. (9) in the Lagrangian density, Eq. (7) gives the following result:
L=Ldt=2Bcβ2gno(gB26+cβ2Ω22noδωΩt0TφT).

Inserting Eqs. (9) and (11) into Eq. (10) yields the equations of motion for parameters B, to, Ω, and φ:

dBdT=κB+πκηPinωocosφsech(cβ2gn0πΩ2B),
dφdT=g2B2δω+cβ22n0Ω2toTΩ,
dBΩdT=κBΩ+8n0τRg215cβ2B5,
dtodT=cβ2n0Ω,
where δωκ and (to be consistent with the measurement protocol) operation near the soliton maximum existence detuning are assumed. The latter is equivalent to assuming φ is near zero in the analysis. Equation (15) corresponds to the change of soliton repetition rate due to the SFS [22] and has recently been observed [26]. The steady-state solution of Eq. (14) gives Eq. (1) for the SFS Ω (also using Bτs=cβ2/gno), while Eqs. (12) and (13) give the following steady-state results:
B=cβ2gno1τs=2g(δω+cβ22noΩ2),
cosφ=1πτscβ2κωoηgnoPincoshπΩτs2.

Combining Eqs. (1) and (16), the SFS and soliton pulse width can be related to the laser-cavity detuning,

δω=15cβ2ω032noQΩτRcβ22noΩ2,
δω=cβ22no1τs2(1+64c2τR2Q2β22225no2ω021τs6).
It is important to remember that despite the form of these equations, detuning (δω) is the parameter that is controlled in a measurement. In addition, in Eq. (18), note that in the limit of τR0, Ω0 for finite δω.

The minimum input power for soliton existence given by Eq. (2) can be obtained by requiring |cosφ|1 in Eq. (17). The average soliton power is given by [2,3]

Psol=2cηAeffβ2n2Q1τs.
This expression follows from Eq. (9) and the relation between B and τs noted above. Using Eqs. (2) and (20), the soliton efficiency in Eq. (3) can be obtained from Γ=Psol/Pmin.

By numerical simulation of Eq. (6), the relations among Ω, τs, and δω given by Eqs. (1), (18), and (19) were confirmed for both the silica resonator in this Letter and for silicon nitride resonators (parameters as in [14]: D1/2π=100GHz, D2/2π=2MHz, κ/2π=350MHz, and τR=0.2×20/2π=0.64fs). The form of the Raman interaction used here assumes that higher order terms beyond the shock term are negligible. This imposes a restriction on pulse times to be much longer than the damping time of vibrations [27]. This assumption is satisfied here. Finally, even though the behavior of the Raman SFS in microcavities differs in comparison to conventional soliton propagation in optical fiber (i.e., frequency locking behavior in microcavities [13]), it is interesting to note that both systems exhibit an inverse quartic dependence on soliton pulse width [10].

In summary, closed-form expressions for the Raman self-frequency-shift and the efficiency of dissipative Kerr cavity solitons have been derived using the Lagrangian perturbation approach. The results are in good agreement with measurements in silica microcavities. The expressions should be applicable to predict soliton behavior in any microcavity system.

Funding

Defense Advanced Research Projects Agency (DARPA) under the QuASAR and PULSE programs; National Aeronautics and Space Administration (NASA); Kavli Nanoscience Institute; National Science Foundation (NSF) Institute for Quantum Information and Matter, a NSF Physics Frontiers Center; Gordon and Betty Moore Foundation.

Acknowledgment

The authors thank Xueyue Zhang for checking the derivations in this paper and for confirming that the relation between B and τs follows from variation of the Raman-perturbed system.

REFERENCES

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

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

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

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

5. P.-H. Wang, J. A. Jaramillo-Villegas, Y. Xuan, X. Xue, C. Bao, D. E. Leaird, M. Qi, and A. M. Weiner, Opt. Express 24, 10890 (2016). [CrossRef]  

6. C. Joshi, J. K. Jang, K. Luke, X. Ji, S. A. Miller, A. Klenner, Y. Okawachi, M. Lipson, and A. L. Gaeta, Opt. Lett. 41, 2565 (2016). [CrossRef]  

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

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

9. F. M. Mitschke and L. F. Mollenauer, Opt. Lett. 11, 659 (1986). [CrossRef]  

10. J. P. Gordon, Opt. Lett. 11, 662 (1986). [CrossRef]  

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

12. A. Gambetta, R. Ramponi, and M. Marangoni, Opt. Lett. 33, 2671 (2008). [CrossRef]  

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

14. 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]  

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

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

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

18. 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]  

19. A. Atieh, P. Myslinski, J. Chrostowski, and P. Galko, J. Lightwave Technol. 17, 216 (1999). [CrossRef]  

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

21. A. Matsko, A. Savchenkov, W. Liang, V. Ilchenko, D. Seidel, and L. Maleki, Opt. Lett. 36, 2845 (2011). [CrossRef]  

22. A. B. Matsko and L. Maleki, Opt. Express 21, 28862 (2013). [CrossRef]  

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

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

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

26. 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).

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

References

  • View by:

  1. F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit, and M. Haelterman, Nat. Photonics 4, 471 (2010).
    [Crossref]
  2. T. Herr, V. Brasch, J. Jost, C. Wang, N. Kondratiev, M. Gorodetsky, and T. Kippenberg, Nat. Photonics 8, 145 (2014).
    [Crossref]
  3. X. Yi, Q.-F. Yang, K. Y. Yang, M.-G. Suh, and K. Vahala, Optica 2, 1078 (2015).
    [Crossref]
  4. V. Brasch, M. Geiselmann, T. Herr, G. Lihachev, M. Pfeiffer, M. Gorodetsky, and T. Kippenberg, Science 351, 357 (2016).
    [Crossref]
  5. P.-H. Wang, J. A. Jaramillo-Villegas, Y. Xuan, X. Xue, C. Bao, D. E. Leaird, M. Qi, and A. M. Weiner, Opt. Express 24, 10890 (2016).
    [Crossref]
  6. C. Joshi, J. K. Jang, K. Luke, X. Ji, S. A. Miller, A. Klenner, Y. Okawachi, M. Lipson, and A. L. Gaeta, Opt. Lett. 41, 2565 (2016).
    [Crossref]
  7. P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. Kippenberg, Nature 450, 1214 (2007).
    [Crossref]
  8. T. J. Kippenberg, R. Holzwarth, and S. Diddams, Science 332, 555 (2011).
    [Crossref]
  9. F. M. Mitschke and L. F. Mollenauer, Opt. Lett. 11, 659 (1986).
    [Crossref]
  10. J. P. Gordon, Opt. Lett. 11, 662 (1986).
    [Crossref]
  11. J. M. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys. 78, 1135 (2006).
    [Crossref]
  12. A. Gambetta, R. Ramponi, and M. Marangoni, Opt. Lett. 33, 2671 (2008).
    [Crossref]
  13. C. Milián, A. V. Gorbach, M. Taki, A. V. Yulin, and D. V. Skryabin, Phys. Rev. A 92, 033851 (2015).
    [Crossref]
  14. 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]
  15. L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).
    [Crossref]
  16. T. Herr, V. Brasch, J. Jost, I. Mirgorodskiy, G. Lihachev, M. Gorodetsky, and T. Kippenberg, Phys. Rev. Lett. 113, 123901 (2014).
    [Crossref]
  17. X. Yi, Q.-F. Yang, K. Youl, and K. Vahala, Opt. Lett. 41, 2037 (2016).
    [Crossref]
  18. 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]
  19. A. Atieh, P. Myslinski, J. Chrostowski, and P. Galko, J. Lightwave Technol. 17, 216 (1999).
    [Crossref]
  20. I. Barashenkov and Y. S. Smirnov, Phys. Rev. E 54, 5707 (1996).
    [Crossref]
  21. A. Matsko, A. Savchenkov, W. Liang, V. Ilchenko, D. Seidel, and L. Maleki, Opt. Lett. 36, 2845 (2011).
    [Crossref]
  22. A. B. Matsko and L. Maleki, Opt. Express 21, 28862 (2013).
    [Crossref]
  23. T. Herr, M. L. Gorodetsky, and T. J. Kippenberg, “Dissipative Kerr solitons in optical microresonators,” arXiv:1508.04989 (2015).
  24. A. Hasegawa, IEEE J. Sel. Top. Quantum Electron. 6, 1161 (2000).
    [Crossref]
  25. P. Grelu, Nonlinear Optical Cavity Dynamics: from Microresonators to Fiber Lasers (Wiley, 2015).
  26. 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).
  27. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2007).

2016 (5)

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

P.-H. Wang, J. A. Jaramillo-Villegas, Y. Xuan, X. Xue, C. Bao, D. E. Leaird, M. Qi, and A. M. Weiner, Opt. Express 24, 10890 (2016).
[Crossref]

C. Joshi, J. K. Jang, K. Luke, X. Ji, S. A. Miller, A. Klenner, Y. Okawachi, M. Lipson, and A. L. Gaeta, Opt. Lett. 41, 2565 (2016).
[Crossref]

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

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).
[Crossref]

2014 (3)

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]

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]

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).
[Crossref]

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).
[Crossref]

1987 (1)

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

1986 (2)

Agarwal, A. M.

Agrawal, G. P.

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

Arcizet, O.

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

Atieh, A.

Bao, C.

Barashenkov, I.

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

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, 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]

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.

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]

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

Genty, G.

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

Gorbach, A. V.

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).
[Crossref]

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, 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]

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).

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


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)

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

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 .

Metrics