## On timing jitter of mode locked Kerr frequency combs |

Optics Express, Vol. 21, Issue 23, pp. 28862-28876 (2013)

http://dx.doi.org/10.1364/OE.21.028862

Acrobat PDF (822 KB)

### Abstract

We study fundamental timing jitter in repetition rate of a mode locked Kerr frequency comb generated in an externally pumped nonlinear ring resonator. We show that the increase in the integrated power of the comb harmonics, and the corresponding decrease of the duration of the associated pulse, results in the increase of low frequency noise, and a decrease in high frequency noise.

© 2013 OSA

## 1. Introduction

1. P. Del-Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature **450**, 1214–1217 (2007). [CrossRef]

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

21. K. Saha, Y. Okawachi, B. Shim, J. S. Levy, R. Salem, A. R. Johnson, M. A. Foster, M. R. E. Lamont, M. Lipson, and A. L. Gaeta, “Modelocking and femtosecond pulse generation in chip-based frequency combs,” Opt. Express **21**, 1335–1343 (2013). [CrossRef] [PubMed]

3. A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, “Tunable optical frequency comb with a crystalline whispering gallery mode resonator,” Phys. Rev. Lett. **101**, 093902 (2008). [CrossRef] [PubMed]

8. S. B. Papp and S. A. Diddams, “Spectral and temporal characterization of a fused quartz microresonator optical frequency comb,” Phys. Rev. A **84**, 053833 (2011). [CrossRef]

9. P. Del-Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett. **107**, 063901 (2011). [CrossRef]

26. M. Nakazawa, K. Suzuki, and H. A. Haus, “The modulational instability laser-Part I: Experiment,” IEEE J. Quantum Electron. **25**, 2036–2044 (1989). [CrossRef]

29. S. Coen and M. Haelterman, “Continuous-wave ultrahigh-repetition-rate pulse-train generation through modulational instability in a passive fiber cavity,” Opt. Lett. **26**, 39–41 (2001). [CrossRef]

30. D. K. Serkland and P. Kumar, “Tunable fiber-optic parametric oscillator,” Opt. Lett. **24**, 92–94 (1999). [CrossRef]

31. L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett. **58**, 2209–2211 (1987). [CrossRef] [PubMed]

32. Y. K. Chembo, D. V. Strekalov, and N. Yu, “Spectrum and dynamics of optical frequency combs generated with monolithic whispering gallery mode resonators,” Phys. Rev. Lett. **104**, 103902 (2010). [CrossRef] [PubMed]

47. S. Wabnitz, “Suppression of interactions in a phase-locked soliton optical memory,” Opt. Lett. **18**, 601–603 (1993). [CrossRef] [PubMed]

48. I. V. Barashenkov and Y. S. Smirnov, “Existence and stability chart for the ac-driven, damped nonlinear Schrödinger solitons,” Phys. Rev. E **54**, 5707–5725 (1996). [CrossRef]

53. Q.-H. Park and H. J. Shin, “Parametric control of soliton light traffic by cw traffic light,” Phys. Rev. Lett. **82**, 4432–4435 (1999). [CrossRef]

54. S. Li, L. Li, Z. Li, and G. Zhou, “Properties of soliton solutions on a cw background in optical fibers with higher-order effects,” J. Opt. Soc. Am. B **21**, 2089–2094 (2004). [CrossRef]

35. A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Mode-locked Kerr frequency combs,” Opt. Lett. **36**, 2845–2847 (2011). [CrossRef] [PubMed]

36. A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, “Hard and soft excitation regimes of Kerr frequency combs,” Phys. Rev. A **85**, 023830 (2012). [CrossRef]

41. S. Coen and M. Erkintalo, “Universal scaling laws of Kerr frequency combs,” Opt. Lett. **38**, 1790–1792 (2013). [CrossRef] [PubMed]

42. A. B. Matsko, W. Liang, A. A. Savchenkov, and L. Maleki, “Chaotic dynamics of frequency combs generated with continuously pumped nonlinear microresonators,” Opt. Lett. **38**, 525–527 (2013). [CrossRef] [PubMed]

21. K. Saha, Y. Okawachi, B. Shim, J. S. Levy, R. Salem, A. R. Johnson, M. A. Foster, M. R. E. Lamont, M. Lipson, and A. L. Gaeta, “Modelocking and femtosecond pulse generation in chip-based frequency combs,” Opt. Express **21**, 1335–1343 (2013). [CrossRef] [PubMed]

3. A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, “Tunable optical frequency comb with a crystalline whispering gallery mode resonator,” Phys. Rev. Lett. **101**, 093902 (2008). [CrossRef] [PubMed]

8. S. B. Papp and S. A. Diddams, “Spectral and temporal characterization of a fused quartz microresonator optical frequency comb,” Phys. Rev. A **84**, 053833 (2011). [CrossRef]

56. A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, “Low threshold optical oscillations in a whispering gallery mode CaF_{2}resonator,” Phys. Rev. Lett. **93**, 243905 (2004). [CrossRef]

57. A. A. Savchenkov, E. Rubiola, A. B. Matsko, V. S. Ilchenko, and L. Maleki, “Phase noise of whispering gallery photonic hyper-parametric microwave oscillators,” Opt. Express **16**, 4130–4144 (2008). [CrossRef] [PubMed]

58. H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. **29**, 983–996 (1993). [CrossRef]

59. A. Hasegawa, “Soliton-based optical communications: An overwiew,” IEEE J. Sel. Top. Quantum Electron. **6**, 1161–1172 (2000). [CrossRef]

61. A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Optical hyperparametric oscillations in a whispering-gallery-mode resonator: Threshold and phase diffusion,” Phys. Rev. A **71**, 033804 (2005). [CrossRef]

*γ*is the half width at half maximum of the resonator mode (critical coupling is assumed),

*P*is the threshold input power for start of the modulation instability in a microresonator characterized with anomalous group velocity dispersion [61

_{th}61. A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Optical hyperparametric oscillations in a whispering-gallery-mode resonator: Threshold and phase diffusion,” Phys. Rev. A **71**, 033804 (2005). [CrossRef]

*P*>

_{in}*P*is the power of the external continuous wave optical pumping source, is a dimensionless parameter describing group velocity dispersion of resonator modes (

_{th}*ω*

_{0}is the frequency of the optically pumped mode,

*ω*

_{−},

*ω*

_{0}, and

*ω*

_{+}are frequencies of consecutive resonator modes belonging to the same mode family, so that

*ω*

_{0}−

*ω*

_{−}and

*ω*

_{+}−

*ω*

_{0}represent the free spectral range of the resonator), is the nonlinearity parameter characterizing frequency shift of a resonator mode per photon [61

61. A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Optical hyperparametric oscillations in a whispering-gallery-mode resonator: Threshold and phase diffusion,” Phys. Rev. A **71**, 033804 (2005). [CrossRef]

*n*

_{2}is an optical cubic nonlinearity coefficient,

*n*

_{0}is the linear refractive index of the material,

*𝒱*is the effective mode volume, and

*c*is the speed of light in the vacuum).

_{2}whispering gallery mode resonators 2

*γ*= 2

*π*× 10

^{5}rad/s and

*g*≈ 2 × 10

^{−3}rad/s (35 GHz resonator) [10

10. W. Liang, A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, “Generation of near-infrared frequency combs from a MgF_{2}whispering gallery mode resonator,” Opt. Lett. **36**, 2290–2292 (2011). [CrossRef] [PubMed]

*dBc/Hz*for

*P*(−

_{in}*D*)/

*P*≃ 1. Moreover, since

_{th}*P*∼

_{th}*γ*

^{2}/

*g*, the noise floor is lower in resonators with higher

*P*, i.e. in resonators with smaller quality factor and larger mode volume. Since reduction of the quality factor leads to the increase of the close-in phase noise (noise at small Fourier frequency

_{th}*f*), a reasonable way for reduction of the noise floor is increasing the volume of the resonator modes while keeping the quality factor essentially unchanged. Note that the timing jitter of the pulses generated by the resonator is the integrated phase noise of the generated RF signal.

## 2. Master equation

*A*, supplied with the equation for the output field,

*A*, where

_{out}*A*(

*T*,

*t*) is the slowly varying envelope of the electric field,

*T*=

*z/V*,

_{g}*z*is the is distance traveled by the photon around the resonator circumference (

*T*is ”slow time”),

*V*= 1/

_{g}*β*

_{1}is the group velocity, and

*t*is the retarded time (the time in the frame of reference travelling along with the pulse with the group velocity,

*t*≡

*t̃*−

*z/V*, where

_{g}*t̃*is the physical time). By definition, time scale

*T*is much longer than the pulse round trip time

*T*= 2

_{R}*πR/V*, 2

_{g}*πR*is the length of the resonator circumference,

*α*

_{Σ}is the amplitude attenuation per round trip,

*T*/2 is the coupling loss per one round trip (both

_{c}*α*

_{Σ}and

*T*are much less than unity),

_{c}*δ*

_{0}= (

*ω*

_{0}−

*ω*)

*T*is the detuning between the pump light and the pumped mode,

_{R}*ω*

_{0}is the eigenfrequency of the optically pumped mode,

*ω*is the carrier frequency of the pumping light,

*γ*

_{Σ}is the cubic nonlinearity paramater,

*β*

_{2}

_{Σ}is the group velocity dispersion parameter.

*T*

_{c}_{1}and

*T*

_{c}_{2}per round trip, we have to replace

*α*

_{Σ}+

*T*

_{c}_{1}/2 +

*T*

_{c}_{2}/2 in Eq. (4) and also replace Eq. (5) with Equation (4) is written for an infinite time

*t*interval, while the interval is physically restricted by the round trip time of the pulse,

*T*. Thence, the considered model is valid for the case of a short optical pulse which basically decays at the boundaries of the time interval. Same approximation leads to change of the boundaries of integration (−

_{R}*T*/2,

_{R}*T*/2) taken over time

_{R}*t*in the following sections of the paper, to infinite boundaries.

## 3. Steady state solution

*β*

_{2}

_{Σ}< 0) showing coexistence of the optical pulse and a cw background [53

53. Q.-H. Park and H. J. Shin, “Parametric control of soliton light traffic by cw traffic light,” Phys. Rev. Lett. **82**, 4432–4435 (1999). [CrossRef]

54. S. Li, L. Li, Z. Li, and G. Zhou, “Properties of soliton solutions on a cw background in optical fibers with higher-order effects,” J. Opt. Soc. Am. B **21**, 2089–2094 (2004). [CrossRef]

59. A. Hasegawa, “Soliton-based optical communications: An overwiew,” IEEE J. Sel. Top. Quantum Electron. **6**, 1161–1172 (2000). [CrossRef]

*A*and

_{c}*A*(

_{p}*T*,

*t*) are the amplitudes of the background and the autosoliton pulse, respectively;

*P*is the power of the DC background,

_{c}*ϕ*is the phase of the background wave,

_{c}*P*is the pulse peak power (

_{p}*E*=

*P*is the pulse energy),

_{p}τ*ξ*is the temporal shift (timing), Ω is the frequency shift,

*q*is the chirp, and

*τ*is the pulse duration, and

*ϕ*is the phase of the pulse envelope. It is also convenient to introduce averaged anergy of the pulse per round trip time,

_{p}*P*=

_{ave}*P*(

_{p}*τ/T*).

_{R}### 3.1. DC background

*q*= 0 and Ω = 0. This assumption is reasonable when the center of the pulse does not deviate much in frequency from the DC background. Its validity is proven in what follows.

*P*)(2

_{p}/P_{c}*τ/T*). In some way this condition shows that the power conversion into the soliton mode is inefficient. This assumption is backed by both experiments and numerical simulations showing that the amount of light reflected from the resonator increases with increase of the Kerr frequency comb spectral width. We also assume that the intracavity peak power of the pulse is much larger as compared to the power of the DC background:

_{R}*P*≫

_{p}*P*and the pulse duration is much smaller than the round-trip time.

_{c}35. A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Mode-locked Kerr frequency combs,” Opt. Lett. **36**, 2845–2847 (2011). [CrossRef] [PubMed]

41. S. Coen and M. Erkintalo, “Universal scaling laws of Kerr frequency combs,” Opt. Lett. **38**, 1790–1792 (2013). [CrossRef] [PubMed]

48. I. V. Barashenkov and Y. S. Smirnov, “Existence and stability chart for the ac-driven, damped nonlinear Schrödinger solitons,” Phys. Rev. E **54**, 5707–5725 (1996). [CrossRef]

### 3.2. Pulse parameters

59. A. Hasegawa, “Soliton-based optical communications: An overwiew,” IEEE J. Sel. Top. Quantum Electron. **6**, 1161–1172 (2000). [CrossRef]

*P*(

_{p}*T*),

*ϕ*(

_{p}*T*),

*ξ*(

*T*), Ω(

*T*),

*q*(

*T*), and

*τ*(

*T*).

*A*=

*A*(

*t*,

*P*,

_{p}*ϕ*,

_{p}*ξ*, Ω,

*q*,

*τ*) does not depend on

*T*directly, we write

*ṙ*= {

_{j}*∂P*,

_{p}/∂T*∂ϕ*,

_{p}/∂T*∂ξ/∂T*,

*∂*Ω

*/∂T*,

*∂q/∂T*,

*∂τ/∂T*}, and

*r*= {

_{j}*P*,

_{p}*ϕ*,

_{p}*ξ*, Ω,

*q*,

*τ*}.

*γ*

_{Σ}

*P*= 8

_{p}*Nπ*, where

*N*is a natural number. The duration of the pulse depends on the peak power

*P*, which means that the soliton duration should be quantized in a lossless cavity.

_{p}*ξ*=

*const*and Ω = 0, respectively, if

*dE/dT*= 0. It is interesting to note that the value of frequency Ω converges to zero with the rate determined by the linear loss of the cavity modes. In other words, the existence of the parametric gain mode locks the system. In conventional mode locked lasers Ω usually stabilizes due to presence of an optical filter or a gain media within the loop of the mode locked laser characterized with a finite spectral width [58

58. H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. **29**, 983–996 (1993). [CrossRef]

*q*≃ 0 from the steady state approximation of equation for the pulse duration, Eq. (26). Rewriting equations for the chirp, Eq. (25), and phase, Eq. (27), in steady state and omitting small terms: we find pulse duration, Eq. (28), as well as pulse peak power Equation (32) introduces requirement

*δ*

_{0}> 0. From the analysis of the modulation instability [61

**71**, 033804 (2005). [CrossRef]

47. S. Wabnitz, “Suppression of interactions in a phase-locked soliton optical memory,” Opt. Lett. **18**, 601–603 (1993). [CrossRef] [PubMed]

### 3.3. Output pulses

*α*

_{Σ}=

*T*(critical coupling); and deriving Eq. (37) we assumed that

_{c}*α*

_{Σ}= 0,

*T*

_{c}_{1}=

*T*

_{c}_{2}=

*T*(an overcoupled resonator with two identical couplers).

_{c}### 3.4. Dimensionless form of LL equation

## 4. Langevin formalism

*F*(

*t*,

*T*)) to it

*dE/dT*from Eq. (48).

### 4.1. Solution

*F*(

_{mw}*ω*) results from the measurement setup. The Langevin force describing the noise,

*F*(

_{mw}*ω*), represents the classical white thermal noise as well as shot noise where the thermal noise term increases proportionally to the resonator temperature, Θ, and the shot noise term increases proportionally to the averaged power at the photodiode,

*P̄*;

_{D}*q*is the charge of an electron,

*R*is responsivity of the photodiode,

*ρ*is the load resistance

*F*is the noise figure of the RF circuit,

*k*is the Boltzmann constant,

_{B}*ω*= 2

_{R}*π/T*, is the demodulated RF power leaving the photodiode (in the case of two-coupler system), is the average optical power at the photodiode.

_{R}*α*

_{Σ}and there is no linear loss in the resonator.

*S*is the spectral density for the phase noise of the RF signal, and arrive at the simplified expression for the RF phase noise of the photonic generator based on the frequency comb The timing jitter of the pulses is determined by the first term in the expression. By construction, Eq. (66) is valid for small enough spectral frequencies (

_{ϕ}*ωT*≪ 1). The second term in the square brackets exceeds unity for frequencies below the resonator mode bandwidth.

_{R}*η*is the quantum efficiency of the detector. We also assumed, per our discussion above, that the resonator is either critically coupled or it has no loss and two identical couplers. In both cases it results in

*γ*=

_{c}*γ*/2. Parameters

*D*and Δ

_{0}are given by Eq. (40).

**71**, 033804 (2005). [CrossRef]

*ν*≃

_{RF}*ω*/(2

_{R}*π*) is the linear repetition frequency of the comb. To make the transition from the phase noise to Allan deviation we used a general formula which transforms to if

*S*∼

_{ϕ}*f*

^{−2}.

**71**, 033804 (2005). [CrossRef]

*D*)

^{−1/4}> 1 times better as compared with the hyper-parametric oscillation. This makes sense since the fundamental hyper-parametric oscillation (the harmonics are separated by a single free spectral range of the resonator) is possible when −

*D*≃ 2. In other words, the formula derived for the phase noise generated by the comb is also valid for estimation of the hyperparametric oscillator phase noise if we take into account that |

*D*| > 2 is required for the oscillator operation.

*P*, of hyper-parametric oscillator defined as the power when oscillation with optical sidebands separated by multiple free spectral ranges starts in a resonator with −

_{th}*D*≫ 1 [36

36. A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, “Hard and soft excitation regimes of Kerr frequency combs,” Phys. Rev. A **85**, 023830 (2012). [CrossRef]

*η*= 1, replace phase noise by a single sideband phase noise

*S*= 2L

_{ϕ}*, and obtain the resultant phase noise in the form presented at the beginning of the paper by Eq. (1).*

_{ϕ}## 5. Conclusion

## Acknowledgments

## References and links

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40. | S. Coen, H. G. Randle, T. Sylvestre, and M. Erkintalo, “Modeling of octave-spanning Kerr frequency combs using a generalized mean-field Lugiato-Lefever model,” Opt. Lett. |

41. | S. Coen and M. Erkintalo, “Universal scaling laws of Kerr frequency combs,” Opt. Lett. |

42. | A. B. Matsko, W. Liang, A. A. Savchenkov, and L. Maleki, “Chaotic dynamics of frequency combs generated with continuously pumped nonlinear microresonators,” Opt. Lett. |

43. | T. Hansson, D. Modotto, and S. Wabnitz, “Dynamics of the modulational instability in microresonator frequency combs,” Phys. Rev. A |

44. | C. Godey, I. Balakireva, A. Coillet, and Y. K. Chembo, “Stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs. Part I: Case of normal dispersion,” arXiv:1308.2539 (2013). |

45. | I. Balakireva, A. Coillet, C. Godey, and Y. K. Chembo, “Stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs. Part II: Case of anomalous dispersion,” arXiv:1308.2542 (2013). |

46. | M. Lamont, Y. Okawachi, and A. L. Gaeta, “Route to stabilized ultrabroadband microresonator-based frequency combs,” arXiv:1305.4921 (2013). |

47. | S. Wabnitz, “Suppression of interactions in a phase-locked soliton optical memory,” Opt. Lett. |

48. | I. V. Barashenkov and Y. S. Smirnov, “Existence and stability chart for the ac-driven, damped nonlinear Schrödinger solitons,” Phys. Rev. E |

49. | J.-M. Ghidaglia, “Finite dimensional behavior for weakly damped driven Schrödinger equation,” Ann. Inst. Henri Poincare |

50. | N. I. Karachalios and N. M. Stavrakakis, “Global attractor for the weakly damped driven Schrodinger equation in H2(R),” Nonlinear Diff. Eq. Appl. |

51. | C. Zhu, “Attractor of the nonlinear Schrodinger equation,” Commun. Math. Anal. |

52. | K. J. Blow and N. J. Doran, “Global and local chaos in the pumped nonlinear Schrödinger equation,” Phys. Rev. Lett. |

53. | Q.-H. Park and H. J. Shin, “Parametric control of soliton light traffic by cw traffic light,” Phys. Rev. Lett. |

54. | S. Li, L. Li, Z. Li, and G. Zhou, “Properties of soliton solutions on a cw background in optical fibers with higher-order effects,” J. Opt. Soc. Am. B |

55. | A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Whispering gallery mode oscillators and optical comb generators,” in |

56. | A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, “Low threshold optical oscillations in a whispering gallery mode CaF |

57. | A. A. Savchenkov, E. Rubiola, A. B. Matsko, V. S. Ilchenko, and L. Maleki, “Phase noise of whispering gallery photonic hyper-parametric microwave oscillators,” Opt. Express |

58. | H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. |

59. | A. Hasegawa, “Soliton-based optical communications: An overwiew,” IEEE J. Sel. Top. Quantum Electron. |

60. | A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. M. Dudley, C. R. Menyuk, and Y. K. Chembo, “Azimuthal Turing patterns, bright and dark cavity solitons in Kerr combs generated with whispering-gallery-mode resonators,” IEEE Photonics J. |

61. | A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Optical hyperparametric oscillations in a whispering-gallery-mode resonator: Threshold and phase diffusion,” Phys. Rev. A |

**OCIS Codes**

(190.4360) Nonlinear optics : Nonlinear optics, devices

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

(230.4910) Optical devices : Oscillators

(270.2500) Quantum optics : Fluctuations, relaxations, and noise

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: September 17, 2013

Revised Manuscript: October 14, 2013

Manuscript Accepted: October 15, 2013

Published: November 15, 2013

**Virtual Issues**

Nonlinear Optics (2013) *Optics Express*

**Citation**

Andrey B. Matsko and Lute Maleki, "On timing jitter of mode locked Kerr frequency combs," Opt. Express **21**, 28862-28876 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-23-28862

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