Abstract

We predict that the free spectral range (FSR) of the soliton combs in microring resonators can self-lock through the back-action of the Cherenkov dispersive radiation on its parent soliton under the conditions typical for recent experiments on the generation of the octave wide combs. The comb FSR in the self-locked state remains quasi-constant over sufficiently broad intervals of the pump frequencies, implying that this effect can be potentially used as the comb self-stabilisation technique. The intervals of self-locking form a sequence of the discrete plateaus reminiscent to other staircase-like structures known in the oscillator synchronisation research. We derive a version of the Adler equation for the self-locking regime and confirm that it is favoured by the strong overlap between the soliton and the dispersive radiation parts of the comb signal.

© 2017 Optical Society of America

Full Article  |  PDF Article

Corrections

26 October 2017: A typographical correction was made to the article title.


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2017 (9)

K. Beha, D. C. Cole, P. D. Haye, A. Coillet, S. A. Diddams, and S. B. Papp, “Electronic synthesis of light,” Optica 4, 406–411 (2017).
[Crossref]

H. Guo, M. Karpov, E. Lucas, A. Kordts, M. H. P. Pfeiffer, V. Brasch, G. Lihachev, V. E. Lobanov, M. L. Gorodetsky, and T. J. Kippenberg, “Universal dynamics and deterministic switching of dissipative Kerr solitons in optical microresonators,” Nature Phys. 13, 94 (2017).
[Crossref]

Q. Li, T. C. Briles, D. A. Westly, T. E. Drake, J. R. Stone, B. R. Ilic, S. A. Diddams, S. B. Papp, and K. Srinivasan, “Stably accessing octave-spanning microresonator frequency combs in the soliton regime,” Optica 4, 193–203 (2017).
[Crossref] [PubMed]

M. H. P. Pfeiffer, C. Herkommer, J. Liu, H. Guo, M. Karpov, E. Lucas, M. Zervas, and T. J. Kippenberg, “Octave-spanning dissipative Kerr soliton frequency combs in Si3N4 microresonators,” Optica 4, 684–691 (2017).
[Crossref]

D. C. Cole, E. S. Lamb, P. D. Haye, S. A. Diddams, and S. B. Papp, “Soliton crystals in Kerr resonators,” Nature Photon. 11, 671–676 (2017).
[Crossref]

H. Taheri, A. B. Matsko, and L. Maleki, “Optical lattice trap for Kerr solitons,” Eur. Phys. Jour. D 71, 153 (2017).
[Crossref]

Y. V. Kartashov, O. Alexander, and D. V. Skryabin, “Multistability and coexisting soliton combs in ring resonators: the Lugiato-Lefever approach,” Opt. Express 25, 11550–11555 (2017).
[Crossref] [PubMed]

H. Guo, M. Karpov, E. Lucas, A. Kordts, M. H. P. Pfeiffer, V. Brasch, G. Lihachev, V. E. Lobanov, M. L. Gorodetsky, and T. J. Kippenberg, “Universal dynamics and deterministic switching of dissipative Kerr solitons in optical microresonators,” Nat. Phys. 13, 94–102 (2017).
[Crossref]

M. Yu, J. K. Jang, Y. Okawachi, A. G. Griffith, K. Luke, S. A. Miller, X. Ji, M. Lipson, and A. L. Gaeta, “Breather soliton dynamics in microresonators,” Nat. Comm. 8, 14569 (2017).
[Crossref]

2016 (8)

Y. H. Wen, M. R. E. Lamont, S. H. Strogatz, and A. L. Gaeta, “Self-organization in Kerr-cavity-soliton formation in parametric frequency combs,” Phys. Rev. A 94, 063843 (2016).
[Crossref]

V. Brasch, M. Geiselmann, T. Herr, G. Lihachev, M. H. P. Pfeiffer, M. L. Gorodetsky, and T. J. Kippenberg, “Photonic chip-based optical frequency comb using soliton Cherenkov radiation,” Science 351, 357–360 (2016).
[Crossref] [PubMed]

M. Yu, Y. Okawachi, A. G. Griffith, M. Lipson, and A. L. Gaeta, “Mode-locked mid-infrared frequency combs in a silicon microresonator,” Optica 3, 854–860 (2016).
[Crossref]

C. Joshi, J. K. Jang, K. Luke, X. Ji, S. A. Miller, A. Klenner, Y. Okawachi, M. Lipson, and A. L. Gaeta, “Thermally controlled comb generation and soliton mode locking in microresonators,” Opt. Lett. 41, 2565–2568 (2016).
[Crossref] [PubMed]

V. Peano, M. Houde, F. Marquardt, and A. A. Clerk, “Topological quantum fluctuations and travelling wave amplifiers,” Phys. Rev. X 6, 041026 (2016).

T. Ozawa, H. M. Price, N. Goldman, O. Zilberberg, and I. Carusotto, “Synthetic dimensions in integrated photonics: From optical isolation to four-dimensional quantum Hall physics,” Phys. Rev. A 93, 043827 (2016).
[Crossref]

M. G. Suh, Q. F. Yang, K. Y. Yang, X. Yi, and K. J. Vahala, “Microresonator soliton dual-comb spectroscopy,” Science 354, 600–603 (2016).
[Crossref] [PubMed]

C. Reimer, M. Kues, P. Roztocki, B. Wetzel, F. Grazioso, B. E. Little, S. T. Chu, T. Johnston, Y. Bromberg, L. Caspan, D. J. Moss, and R. Morandotti, “Generation of multiphoton entangled quantum states by means of integrated frequency combs,” Science 351, 1176–1180 (2016).
[Crossref] [PubMed]

2015 (4)

X. Yi, Q.F. Yang, K.Y. Yang, M.G. Suh, and K. Vahala, “Soliton frequency comb at microwave rates in a high-Q silica microresonator,” Optica 2, 1078–1085 (2015).
[Crossref]

J. D. Jost, T. Herr, C. Lecaplain, V. Brasch, M. H. P. Pfeiffer, and T. J. Kippenberg, “Counting the cycles of light using a self-referenced optical microresonator,” Optica 2, 706–711 (2015).
[Crossref]

X. Xue, Y. Xuan, Y. Liu, P. H. Wang, S. Chen, J. Wang, D. E. Leaird, M. Qi, and A. M. Weiner, “Mode-locked dark pulse Kerr combs in normal-dispersion microresonators,” Nature Photon. 9, 594 (2015).
[Crossref]

C. Milian, A. V. Gorbach, M. Taki, A. V. Yulin, and D. V. Skryabin, “Solitons and frequency combs in silica microring resonators: Interplay of the Raman and higher-order dispersion effects,” Phys. Rev. A 92, 033851 (2015).
[Crossref]

2014 (8)

P. D. Haye, K. Beha, S. B. Papp, and S. A. Diddams, “Self-injection locking and phase-locked states in microresonator-based optical frequency combs,” Phys. Rev. Lett. 112, 043905 (2014).
[Crossref]

C. Milian and D. V. Skryabin, “Soliton families and resonant radiation in a micro-ring resonator near zero group-velocity dispersion,” Opt. Express 22, 3732 (2014).
[Crossref] [PubMed]

S. Wang, H. Guo, X. Bai, and X. Zeng, “Broadband Kerr frequency combs and intracavity soliton dynamics influenced by high-order cavity dispersion,” Opt. Lett. 39, 2880–2883 (2014).
[Crossref] [PubMed]

J. K. Jang, M. Erkintalo, S. G. Murdoch, and S. Coen, “Observation of dispersive wave emission by temporal cavity solitons,” Opt. Lett. 39, 5503–5506 (2014).
[Crossref] [PubMed]

P. Parra-Rivas, D. Gomila, F. Leo, S. Coen, and L. Gelens, “Third-order chromatic dispersion stabilizes Kerr frequency combs,” Opt. Lett. 39, 2971–2974 (2014).
[Crossref] [PubMed]

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J.S. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T.J. Kippenberg, and C. Koos, “Coherent terabit communications with microresonator Kerr frequency combs,” Nat. Photon. 8, 375–380 (2014).
[Crossref]

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nature Photon. 8, 145 (2014).
[Crossref]

T. Hansson and S. Wabnitz, “Bichromatically pumped microresonator frequency combs,” Phys. Rev. A 90, 013811 (2014).
[Crossref]

2013 (3)

2011 (3)

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nature Phys. 7, 907–912 (2011).
[Crossref]

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams,“Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
[Crossref] [PubMed]

I. V. Barashenkov and E. V. Zemlyanaya, “Travelling solitons in the externally driven nonlinear Schrödinger equation,” J. Phys. A: Math. Theor. 44, 1–23 (2011).
[Crossref]

2010 (1)

D. V. Skryabin and A. V. Gorbach, “Colloquium: Looking at a soliton through the prism of optical supercontinuum,” Rev. Mod. Phys. 82, 1287 (2010).
[Crossref]

2008 (1)

P. D. Haye, O. Arcizet, A. Schliesser, R. Holzwarth, and T. J. Kippenberg, “Full stabilization of a microresonator based optical frequency comb,” Phys. Rev. Lett. 101, 053903 (2008).
[Crossref]

2007 (1)

2003 (2)

F. Biancalana, D. V. Skryabin, and P. S. J. Russell, “Four-wave mixing instabilities in photonic-crystal and tapered fibers,” Phys. Rev. E 68, 046603 (2003).
[Crossref]

D. V. Skryabin, F. Luan, J. C. Knight, and P. S. J. Russell, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science 301, 1705–1708 (2003).
[Crossref] [PubMed]

2002 (1)

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knödl, M. Miller, and R. Jäger, “Cavity solitons as pixels in semiconductor microcavities,” Nature 419, 699–702 (2002).
[Crossref] [PubMed]

1996 (1)

W. J. Firth and A. J. Scroggie, “Optical Bullet Holes: Robust Controllable Localized States of a Nonlinear Cavity,” Phys. Rev. Lett. 76, 1623 (1996).
[Crossref] [PubMed]

1994 (1)

D. Cai, A. R. Bishop, N. Gronbech-Jensen, and B. A. Malomed, “Bound solitons in the ac-driven, damped nonlinear Schrödinger equation,” Phys. Rev. E 49, 1677 (1994).
[Crossref]

1991 (1)

B. A. Malomed, “Bound solitons in the nonlinear Schrödinger-Ginzburg-Landau equation,” Phys. Rev. A 44, 6954 (1991).
[Crossref] [PubMed]

1989 (1)

P. Laurent, A. Clairon, and C. Breant, “Frequency noise analysis of optically self-locked diode lasers,” IEEE J. of Quant. Electr. 25, 1131 (1989).
[Crossref]

Alexander, O.

Arcizet, O.

P. D. Haye, O. Arcizet, A. Schliesser, R. Holzwarth, and T. J. Kippenberg, “Full stabilization of a microresonator based optical frequency comb,” Phys. Rev. Lett. 101, 053903 (2008).
[Crossref]

Bai, X.

Balle, S.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knödl, M. Miller, and R. Jäger, “Cavity solitons as pixels in semiconductor microcavities,” Nature 419, 699–702 (2002).
[Crossref] [PubMed]

Barashenkov, I. V.

I. V. Barashenkov and E. V. Zemlyanaya, “Travelling solitons in the externally driven nonlinear Schrödinger equation,” J. Phys. A: Math. Theor. 44, 1–23 (2011).
[Crossref]

Barland, S.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knödl, M. Miller, and R. Jäger, “Cavity solitons as pixels in semiconductor microcavities,” Nature 419, 699–702 (2002).
[Crossref] [PubMed]

Beha, K.

K. Beha, D. C. Cole, P. D. Haye, A. Coillet, S. A. Diddams, and S. B. Papp, “Electronic synthesis of light,” Optica 4, 406–411 (2017).
[Crossref]

P. D. Haye, K. Beha, S. B. Papp, and S. A. Diddams, “Self-injection locking and phase-locked states in microresonator-based optical frequency combs,” Phys. Rev. Lett. 112, 043905 (2014).
[Crossref]

Biancalana, F.

F. Biancalana, D. V. Skryabin, and P. S. J. Russell, “Four-wave mixing instabilities in photonic-crystal and tapered fibers,” Phys. Rev. E 68, 046603 (2003).
[Crossref]

Bishop, A. R.

D. Cai, A. R. Bishop, N. Gronbech-Jensen, and B. A. Malomed, “Bound solitons in the ac-driven, damped nonlinear Schrödinger equation,” Phys. Rev. E 49, 1677 (1994).
[Crossref]

Brambilla, M.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knödl, M. Miller, and R. Jäger, “Cavity solitons as pixels in semiconductor microcavities,” Nature 419, 699–702 (2002).
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Figures (4)

Fig. 1
Fig. 1 (a) The full line shows the integrated dispersion D m used in our modelling. The dashed line is the resonator GVD, D″ (m). Anomalous GVD, D″ (m) >0, favours the soliton formation. Cherenkov radiation is shed into the intervals of normal GVD, D″ (m) >0. (b) The thick and thin lines show the left-hand sides of Eq. (2) with ± , respectively. Intersections with zero mark modal indexes of the Cherenkov resonances. Arrows indicate the dominant pair of resonances with large amplitudes.
Fig. 2
Fig. 2 Numerical simulation of Eq. (1) with the initial conditions taken as the perturbed upper branch of the cw bistability loop. (a) Shows the temporal evolution of the modal spectrum, |ψm(T)|, ψ m = 0 2 π Ψ ( T , θ ) e i m θ d θ . The vertical axis t = T/T0 counts the round trips. δ = 0.05, γ = 0.005, h = 0.0015. (b) is the spectral snapshot at t = 2000. m = 0 is the modal index of the pump. Two dominant side bands correspond to the two Cherenkov radiation peaks at m = −60 and m = 84. Positions of these peaks are exactly predicted by Eq. (2).
Fig. 3
Fig. 3 (a) An angular profile of the soliton amplitude for parameters as in Fig. 2, but with δ = 0.036. (b) An angular profile of the soliton in the five times longer cavity for δ as in (a).
Fig. 4
Fig. 4 (a) The soliton parameter v characterising the comb FSR vs the detuning parameter δ. The full line shows the staircase dependence of v vs δ, that gives the FSR self-locking providing δ is tuned within the quasi-flat plateaus. Corresponding Cherenkov radiation forms a modulated background extending over the entire resonator circumference. m values indicate m = m (the negative m Cherenkov resonance) corresponding to every plateau, while m = m+ (positive m resonance) remains fixed at +84. The smooth dotted line shows the case of the five time longer resonator with the comb solitons having Cherenkov tails that decay to zero over distances shorter than half of the resonator length. (b) The soliton amplitude across the same δ interval as in (a). (c) The soliton amplitude same as in (b) (marked here as |ψs|) plotted together with the amplitude of the bistable cw solution (marked here as |ψcw|). The black/red part of the curve corresponds to the stable/unstable cw. The soliton background ψ0 corresponds to the black part of the curve located within the bistability interval.

Equations (11)

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i T Ψ = ( ω 0 i D 1 θ 1 2 ! D 2 θ 2 + i 3 ! D 3 θ 3 + i 4 ! D 4 θ 4 ) Ψ i κ Ψ 1 T 0 | Ψ | 2 Ψ 1 T 0 h e i ω p t .
± ( π [ D ( m ) + D ( m ) ] + D 1 δ 2 D 1 | ψ 0 | 2 ) 2 D 1 2 | ψ 0 | 4 v D 1 m + π ( D ( m ) D ( m ) ) = 0 .
Ψ ( T , θ ) = ψ ( x , t ) e i ω p T , x = ( θ 2 π t ) v t , t = T T 0 .
i t ψ = ( δ + i v x β 2 x 2 + i β 3 x 3 + β 4 x 4 ) ψ i γ ψ | ψ | 2 ψ h , ψ ( t , x + 2 π ) = ψ ( t , x ) .
M = 1 2 i π π ( ψ * x ψ c . c . ) d x , t M = 2 γ M .
( δ + i v x β 2 x 2 ) ϕ s | ϕ s | 2 ϕ s = 0 , ϕ s = 2 a sech ( ( x x s ) a β 2 ) exp [ i v ˜ x ] ,
ψ r a d = ε + ( x x s ) e i m + x + ε ( x x s ) e i m x ,
M v ˜ Q s + m + Q + + m Q + 1 2 i ( i m + e i ( m + v ˜ x s ) + + i m e i ( m v ˜ x s c . c . )
± π 2 β 2 ε ± sech ( π m ± 2 β 2 a ) exp [ i α ± ] ,
Q s 2 β 2 ( v v i m b ) | m | | | cos ( m x s + α ) | m + | | + | cos ( m + x s + α + ) .
t x s v m Δ m 2 β 2 Q s | m | | | cos ( m x s + α ) | m + | | + | cos ( m + x s + α + ) .

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