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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 23 — Nov. 18, 2013
  • pp: 28862–28876
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On timing jitter of mode locked Kerr frequency combs

Andrey B. Matsko and Lute Maleki  »View Author Affiliations


Optics Express, Vol. 21, Issue 23, pp. 28862-28876 (2013)
http://dx.doi.org/10.1364/OE.21.028862


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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

The recently emerged microcavity-based Kerr frequency combs [1

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]

24

24. P. Del’Haye, S. B. Papp, and S. A. Diddams, “Self-injection locking and phase-locked states in microresonator-based optical frequency combs,” arXiv:1307.4091 (2013).

] have a great deal of promise for applications in science and technology [25

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

] and their properties are by now well understood. Generation of short optical pulses [21

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]

, 22

22. T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Mode-locking in an optical microresonator via soliton formation,” arXiv:1211.0733v2 (2013).

] as well as stable and spectrally pure radio frequency (RF) signals [3

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

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

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]

] from chip-scale devices are among the attractive practical applications of the frequency combs.

The usefulness of LL approach for description of Kerr frequency combs has been identified recently [55

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 Proceedings of 7th Symposium on Frequency Standards and Metrology, L. Maleki, ed., (World Scientific, 2009), pp. 539–558.

]. It was revealed that the equation correctly describes modulation instability for light confined in a pair of resonant optical modes located at lower and higher frequency sides of the optically pumped mode. It was argued that LL equation is also suitable for description of optical pulses generated in the resonator. It was further theoretically elaborated [22

22. T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Mode-locking in an optical microresonator via soliton formation,” arXiv:1211.0733v2 (2013).

, 35

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

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

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

, 42

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]

], and also experimentally demonstrated [21

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]

, 22

22. T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Mode-locking in an optical microresonator via soliton formation,” arXiv:1211.0733v2 (2013).

], that there is a mode locked regime in the formation of a microcavity-based Kerr frequency comb. In this regime, optical pulses can be generated directly in the nonlinear microresonator pumped with cw light. These stable autosoliton pulses are located on top of a cw background. In this system, the loss of background power is compensated by the external cw pump, while the loss of pulse energy is compensated by the nonlinear interaction of pulses with the background.

An important application of Kerr frequency combs is production of stable radio frequency (RF) signals by demodulating the optical comb on a fast photodiode [3

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

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

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 CaF2resonator,” Phys. Rev. Lett. 93, 243905 (2004). [CrossRef]

, 57

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]

]. The RF frequency produced is the repetition rate of the optical pulses leaving the resonator. Signals generated in this manner are characterized with both high stability and high spectral purity. So, it is important to understand restrictions on the timing jitter of pulses, and find the optimal conditions for reduction of phase noise of the RF signals generated with the frequency comb to determine the fundamental performance limitations of RF photonic oscillators based on Kerr comb generation.

In this paper we utilize the soliton variational approach [58

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

, 59

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

] to find an analytical formula describing pulse shape as well as fundamental timing jitter of the repetition rate of Kerr frequency combs. We show analytically that the peak power of the optical pulse leaving the resonator is nearly the same as the power of the cw pump. The pulse is bright if it is retrieved from the resonator using an output coupler different from the coupler used for pumping; and it is dark if the pulse is retrieved from the same input coupler used for introducing the pumping light into the resonator.

We also show that a Kerr frequency comb operating in a mode locked regime produces a signal with superior spectral purity when compared to a three-mode hyper parametric oscillator [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]

]. Furthermore, we show that there is an optimal pump power for achieving the highest stability of the repetition rate of the comb oscillator. Finally, it is shown that resonators with larger mode volume generate lower phase noise signals when compared with resonators characterized with smaller mode volume.

The basic result of the study presented in this paper can be summarized by the formula
Lϕ=π2PthPin(D)gγ2[1+π296Pth(D)Pinγ2π2f2+124(1+π2f2γ2)1γ2π2f2Pin(D)Pth],
(1)
derived for single sideband phase noise of the radio frequency signal generated by a mode locked Kerr frequency comb on a fast photodiode. Here γ is the half width at half maximum of the resonator mode (critical coupling is assumed), Pth is the threshold input power for start of the modulation instability in a microresonator characterized with anomalous group velocity dispersion [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]

], Pin > Pth is the power of the external continuous wave optical pumping source,
D=2ω0ω+ωγ,
(2)
is a dimensionless parameter describing group velocity dispersion of resonator modes (ω0 is the frequency of the optically pumped mode, ω, ω0, and ω+ are frequencies of consecutive resonator modes belonging to the same mode family, so that ω0ω and ω+ω0 represent the free spectral range of the resonator),
g=n2n0h¯ω02c𝒱n0
(3)
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]

] (n2 is an optical cubic nonlinearity coefficient, n0 is the linear refractive index of the material, 𝒱 is the effective mode volume, and c is the speed of light in the vacuum).

Equation (1) is valid for an idealized case of a Kerr frequency comb generator when only fundamental quantum noise sources are present and the resonator has spectrum of identical modes characterized with anomalous group velocity dispersion. Any disruptions in the resonator spectrum as well as asymmetry of the distribution of quality factor of the modes can lead to noise increase. The phase noise of a real oscillator will be influenced by the technical noises of the pump laser as well as the resonator. For instance, change of the frequency detuning of the cw pump light from the pumped mode will result in change of the power circulating in the resonator, that will change the temperature of the resonator host material and change of the mode frequency and comb repetition rate and the phase noise of the associated signal. Change of the coupling efficiency of the light to the mode will also result in shift of the mode frequency due to circulating power change as well as change of the effective refractive index of the mode (the last one is valid if an open dielectric resonator is used for comb generation). Changes of ambient temperature and pressure will impact the timing jitter and phase noise. Reduction of the resonator size will make fundamental thermodynamic fluctuations of the resonator spectrum more pronounced. All these and many other noise sources must be taken into account to evaluate the performance of a realistic Kerr frequency comb oscillator.

This paper is organize as follows. In Section 2 we present the master (LL) equation. The steady state solution is found in Section 3 using the variational approach. The timing jitter of pulses associated with the Kerr frequency comb generation is studied in Section 4 using the Langevin formalism.

2. Master equation

To study Kerr frequency comb generation in a ring resonator possessing a fast cubic nonlinearity one can use the LL master equation describing the optical field inside the resonator, A, supplied with the equation for the output field, Aout,
TRAT+i2β2Σ2At2iγΣ|A|2A=(αΣ+Tc2+iδ0)A+iTcPineiϕin,
(4)
Aout=Pineiϕin+iTcA,
(5)
where A(T, t) is the slowly varying envelope of the electric field, T = z/Vg, z is the is distance traveled by the photon around the resonator circumference (T is ”slow time”), Vg = 1/β1 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, tz/Vg, where is the physical time). By definition, time scale T is much longer than the pulse round trip time TR = 2πR/Vg, 2πR is the length of the resonator circumference, αΣ is the amplitude attenuation per round trip, Tc/2 is the coupling loss per one round trip (both αΣ and Tc are much less than unity), δ0 = (ω0ω)TR is the detuning between the pump light and the pumped mode, ω0 is the eigenfrequency of the optically pumped mode, ω is the carrier frequency of the pumping light, Pinexp(iϕin) stands for the external pump, γΣ is the cubic nonlinearity paramater, β2Σ is the group velocity dispersion parameter.

It is worth noting that the equation for the output field, Eq. (5), is valid for a single coupler only. In the case of usage of another coupler for retrieving the pulses, it is possible to get rid of the cw background term. For the case of two couplers with coupling loss Tc1 and Tc2 per round trip, we have to replace αΣ + Tc1/2 + Tc2/2 in Eq. (4) and also replace Eq. (5) with
Aout1=Pineiϕin+iTc1A,
(6)
Aout2=iTc2A.
(7)
Equation (4) is written for an infinite time t interval, while the interval is physically restricted by the round trip time of the pulse, TR. 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 (−TR/2, TR/2) taken over time t in the following sections of the paper, to infinite boundaries.

3. Steady state solution

The nonzero cw background which is always present in the resonator is the basic difference between the cw-pumped ring resonator-based four wave mixing oscillator and a passively mode locked laser. This background interacts with the optical pulses generated in the resonator if the oscillator operates in the mode locked regime. The interaction is responsible for the compensation of the energy lost by the optical pulse per round trip. There exists an analytical solution of the lossless master Eq. (4) with anomalous group velocity dispersion (β2Σ < 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

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]

]. This solution can be roughly approximated by an autosoliton pulse and a cw background.

We look for the autosoliton solution of Eq. (4) located at cw background and use the variational method to find parameters of the solution [59

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

]. The method is based on the assumption that there is a quasi-steady state regime when the parameters determining duration and shape of the pulse do not change much per single round trip.
A(T,t)=Ac+Ap(T,t),Ac=Pceiϕc,Ap(T,t)=[Pp2]1/2[sech(tξτ)]1+iqeiΩ(tξ)+iϕp,
(8)
where Ac and Ap(T, t) are the amplitudes of the background and the autosoliton pulse, respectively; Pc is the power of the DC background, ϕc is the phase of the background wave, Pp is the pulse peak power (E = Ppτ is the pulse energy), ξ is the temporal shift (timing), Ω is the frequency shift, q is the chirp, and τ is the pulse duration, and ϕp is the phase of the pulse envelope. It is also convenient to introduce averaged anergy of the pulse per round trip time, Pave = Pp(τ/TR).

3.1. DC background

We start from separation of the equations describing DC background and the pulse formed in the resonator. To do it we substitute Eq. (8) into Eq. (4) and separate time dependent terms from time independent term. To find the time independent terms we average the nonlinear part of the equation over the pulse round trip in the resonator
Pc(αΣ+Tc2+iδ0)iξDCeiϕc=iTcPinei(ϕinϕc),
(9)
where the nonlinear term is given by
ξDC=γΣPceiϕc[Pc+PpτTR(2+e2i(ϕcϕp)+π(Pp+8Pc)42PpPcei(ϕcϕp)+πPc2Ppei(ϕcϕp))]
(10)

To simplify the expression we assumed that 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.

To solve Eq. (9) we assume that the nonlinear loss of the pump field is much smaller than the linear loss: 1 ≫ (Pp/Pc)(2τ/TR). 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: PpPc and the pulse duration is much smaller than the round-trip time.

The equation for the DC background parameters transforms to [22

22. T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Mode-locking in an optical microresonator via soliton formation,” arXiv:1211.0733v2 (2013).

]
Pc(αΣ+Tc2+iδ0iγΣPc)=iTcPinei(ϕinϕc).
(11)
which results in an approximate solution
ϕcϕinα+Tc/2δ0,
(12)
PcTcPinδ02(1+2TcγΣPinδ03).
(13)

Equation (11) is identical to the exact cw solution, suggesting that the background is unaffected by the presence of the pulse. This was noted previously with respect to modulation instability and Kerr frequency comb generation [22

22. T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Mode-locking in an optical microresonator via soliton formation,” arXiv:1211.0733v2 (2013).

, 35

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

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

, 48

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

The equations describing the parameters of the autosoliton solution, Eq. (8), can be found using variational approach [59

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

]. We introduce a Lagrangian density in form
=TR2(Ap*ApTApAp*T)i2(β2Σ|Apt|2+γΣ|Ap|4)
(14)
Variation of the Lagrangian density results in the unperturbed nonlinear Schrödinger equation (NLSE)
δδA*=A*T(A*/T)t(A*/t)=TRApT+i2β2Σ2Apt2iγΣ|Ap|2Ap=0.
(15)

To study evolution of the autosoliton parameters we introduce a Lagrangian
L=dt,
(16)
which is a function of Pp(T), ϕp(T), ξ(T), Ω(T), q(T), and τ(T).

Taking into account that A = A(t, Pp, ϕp, ξ, Ω, q, τ) does not depend on T directly, we write
ApT=ApPpPpT+ApϕpϕpT+ApξξT+ApΩΩT+ApqqT+ApττT.
(17)

Substituting Eq. (8) into Eq. (17), substituting Eq. (17) into Eq. (14), and Eq. (14) into Eq. (16) we get the Lagrangian for the unperturbed nonlinear Schrödinger equation, if the solution is selected in the autosoliton form, per Eq. (8):
L=iβ2ΣPp6τ(1+q2+3Ω2τ2)i6γΣPp2τ+i2PpTR[qτT+2τ(qT(ln(2)1)ΩξT+ϕpT)].
(18)
The Lagrangian generates equations for the pulse parameters in accordance with
ddT(Lr˙j)Lrj=0,
(19)
where j = {∂Pp/∂T, ∂ϕp/∂T, ∂ξ/∂T, Ω/∂T, ∂q/∂T, ∂τ/∂T}, and rj = {Pp, ϕp, ξ, Ω, q, τ}.

To take the loss as well as interaction of the autosoliton pulse with the DC background into account we introduce perturbation of the NLSE in form
R=[αΣ+Tc2+iδ0]Ap+i[γΣ(|Ac+Ap|2(Ac+Ap)|Ap|2Ap)ξDC]
(20)
and modify Eq. (19) as
ddT(Lr˙j)Lrj=(R*AprjRAp*rj)dt,
(21)
Those equations can be rewritten after the integration is performed
TRdEdT=E[Tc+2αΣ+π22γΣPcsin(ϕcϕp)(PpPc+82πcos(ϕcϕp))],
(22)
TRdξdT=β2ΣΩ,
(23)
TRdΩdT=TREdEdTΩ(Tc+2αΣ)Ω,
(24)
TRdqdT=TREdEdTq+2β2Σ3τ2(1+q2+3Ω2τ2)+13γΣPp+(Tc+2αΣ)q+π22γΣPc[PpPccos(ϕcϕp)+π2τTR(8+4cos2(ϕcϕp)+PpPc)],
(25)
TRτdτdT=TR2EdEdTτ2+6π2β2Σqτ22[Tc+2αΣ+32(π28)4πγΣPcsin(ϕcϕp)(PpPc+16π32(π28)cos(ϕcϕp))],
(26)
TRdϕpdT=TRdqdT(1ln2)+TRdξdTΩTR2τdτdTq+β2Σ6τ2(1+q2+3Ω2τ2)+13γΣPpδ0+γΣPc(2+cos2(ϕcϕp)+3π42PpPccos(ϕcϕp)).
(27)

In the case of no loss and no pump the set of Eqs. (22)(27) results in a standard autosoliton solution with all parameters equal to zero except
τ2=2β2ΣγΣPp,
(28)
ϕp=14TTRγΣPp.
(29)
As the soliton travels in the ring cavity, the phase has to be a periodic function with respect to the round trip which results in condition γΣPp = 8, where N is a natural number. The duration of the pulse depends on the peak power Pp, which means that the soliton duration should be quantized in a lossless cavity.

In the case of nonzero loss we find from Eqs. (23) and (24) that steady state for the temporal shift and frequency is 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]

].

We find that 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:
2β2Σ3τ2+13γΣPp0,
(30)
β2Σ6τ2+13γΣPpδ00,
(31)
we find pulse duration, Eq. (28), as well as pulse peak power
Pp=4δ0γΣ.
(32)
Equation (32) introduces requirement δ0 > 0. From the analysis of the modulation instability [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]

] we also can find more strict restriction on the detuning δ0>3. This condition tells that the autosolitons are generated when the optical pump exceeds the bistability threshold.

Finally, writing the steady state form of the equation for the pulse energy, Eq. (22)
αΣ+Tc2+π42γΣPcsin(ϕcϕp)(PpPc+82πcos(ϕcϕp))=0,
(33)
and substituting into it the expression for the power of the DC background as well as pulse peak power, Eqs. (13) and (32), we derive expression for the phase difference between the background and the pulse envelope
sin(ϕcϕp)=8δ0(αΣ+Tc/2)2π2TcγΣPin.
(34)
From this equation we find that
δ0π2TcγΣPin8(αΣ+Tc/2)2,
(35)
which places another boundary on the peak power of the autosoliton pulse [22

22. T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Mode-locking in an optical microresonator via soliton formation,” arXiv:1211.0733v2 (2013).

].

Thus we have found a complete approximate solution of the LL equation. It is easy to verify that the solution coincides with solution found in [22

22. T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Mode-locking in an optical microresonator via soliton formation,” arXiv:1211.0733v2 (2013).

, 47

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

]. The analysis presented here also allows for studying the behavior of the system in more detail. For example, Eq. (33) allows to calculate the phase of the pulse with better accuracy, if needed.

3.3. Output pulses

Taking into account parameters of the packground as well as of the autosoliton pulse and substituting those parameters either to Eq. (5) or to Eq. (7) we can calculate expressions for the power of the output light for the case of a single coupler
PoutPin[1π2sin2(ϕpϕc)sech(tτ)]2,
(36)
or two couplers
Pout2Pin=Tc2δ02+π24sin4(ϕpϕc)sech2(tτ).
(37)
Deriving Eq. (36) we assumed 2αΣ = Tc (critical coupling); and deriving Eq. (37) we assumed that αΣ = 0, Tc1 = Tc2 = Tc (an overcoupled resonator with two identical couplers).

One can see that Eq. (36) shows that dark pulses leave the resonator, while Eq. (37) shows that bright pulses leave the resonator. The peak power of the bright pulses is nearly identical to the power of the cw pump light. The maximum possible peak power is π2/4 compared with the pump power.

3.4. Dimensionless form of LL equation

The values of the coupling, intrinsic, and the total half width at the half maximum of the resonator modes, as well as optical detuning and normalized group velocity dispersion are determined as
γc=Tc2TR,γ0=αΣTR,γ=γc+γ0,Δ0=δ0TR,D=β2ΣωR2cγn0.
(40)

It is possible to transfer Eq. (4) to the dimensionless form used in [22

22. T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Mode-locking in an optical microresonator via soliton formation,” arXiv:1211.0733v2 (2013).

]
iΨτ0+122Ψθ2+|Ψ|2Ψ=(i+ζ0)Ψ+if,
(41)
by substitution
τ0=T(αΣ+Tc/2)TR,ζ0=δ0αΣ+Tc/2,f=TcγΣPin(αΣ+Tc/2)3/2,θ=t/(β2Σ)αΣ+Tc/2,Ψ=AγΣαΣ+Tc/2(ieiϕin).

4. Langevin formalism

An advantage of the variational method described above is its applicability to calculation of the noise of the mode locked system. To find the timing jitter of the pulses corresponding to a base band mode locked Kerr frequency comb (only one pulse is excited in the cavity) we start from the NLSE and add a Langevin force (F(t, T)) to it
TRAT+i2β2Σ2At2iγΣ|A|2A=(αΣ+Tc2+iδ0)A+iTcPineiϕin+TRF(t,T).
(42)

The Langevin force has the following properties
F(t,T)F(t,T)=DFF+δ(tt)δ(TT),
(43)
F(t,T)F(t,T)=DF+Fδ(tt)δ(TT),
(44)
DFF+=h¯ω0TR[2αΣ+Tc],
(45)
DF+F=0.
(46)
Here we consider shot noise only. Since there are no elements in the system having frequencies dependent on the pulse spectrum, the Langevin force does not depend on the pulse shape.

In accordance with Eq. (21) we obtain two coupled equations
dξdT=β2ΣΩTR+Fξ,
(47)
dΩdT=Tc+2ασTRΩ+FΩ.
(48)
It is easy to see that the equations are uncoupled from the fluctuations of the power since the expectation value of the frequency, Ω, approaches zero, efficiently removing term proportional to dE/dT from Eq. (48).

The noise forces pertain to the parameters of the pulse are given by
Fξ(T)=1Et[A*F(t,T)+F(t,T)A]dt,
(49)
FΩ(T)=iE[A*tF(t,T)F(t,T)At]dt,
(50)
where we used assumption that
[A*F(t,T)t+A*tF(t,T)]dt=A*F(t,T)|=0.
(51)
The corresponding diffusion coefficients for the forces are
Fξ(T)Fξ(T)=Dξξδ(TT)=π2τ212EDFF+δ(TT),
(52)
FΩ(T)FΩ(T)=DΩΩδ(TT)=13Eτ2DFF+δ(TT),
(53)
Fξ(T)FΩ(T)=DξΩδ(TT)=i2EDFF+δ(TT),
(54)
FΩ(T)Fξ(T)=DΩξδ(TT)=i2EDFF+δ(TT).
(55)

4.1. Solution

The solution of the set of Eqs. (47) and (48) with respect to the intracavity timing jitter can be found using Fourier transformation
ξ(ω)=iω[β2ΣTRFΩ(ω)iω+(Tc+2α)/TR+Fξ(ω)].
(56)
where
ξ(T)=ξ(ω)eiωTdω2π,
(57)
ξ(ω)=ξ(T)eiωTdT.
(58)

To calculate the timing jitter tm(T) of the output pulse train measured by means of demodulation of the optical pulses on a fast photodiode we assume that the noise that is added to the output light is uncorrelated with the noise of the input light. This is not entirely true if a single coupler is used and TcαΣ; however, in the case of a critically coupled resonator or a resonator that uses two couplers, the correlation is limited.

It is convenient to write the equation for the measured timing jitter in form
tm(ω)=ξ(ω)+Fmw(ω),
(59)
The RF noise Fmw(ω) results from the measurement setup. The Langevin force describing the noise, Fmw(ω), represents the classical white thermal noise as well as shot noise
Fmw(T)Fmw(T)=Dmwδ(TT),
(60)
Dmw=1ωR22qρP¯D+FkBΘPmw|D,
(61)
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, 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, kB is the Boltzmann constant, ωR = 2π/TR,
Pmw|D=ρ2Tc2|TR/2TR/2|A(T,t)|2exp[i2π(t1t)TR]dt1TR|2ρ2Tc2Pave2
(62)
is the demodulated RF power leaving the photodiode (in the case of two-coupler system),
P¯D=TcPave
(63)
is the average optical power at the photodiode.

Writing Eq. (62) we neglected the DC signal added at the coupler. This is possible to do either using an optical filter or assuming that the resonator has two couplers, where the coupling coefficient of the second coupler is equal to αΣ and there is no linear loss in the resonator.

The noise spectrum
Stm(ω)=tm(T)tm(T+T)eiωTdT
(64)
is then given by
Stm(ω)1ω2[Dξξ+(β2ΣTR)2DΩΩω2+[(Tc+2αΣ)/TR]2β2ΣTRDξΩiω+(Tc+2αΣ)/TRβ2ΣTRDΩξiω+(Tc+2αΣ)/TR]+Dmw.
(65)

We take into account the relationship Sϕ=ωR2Stm between phase noise of the generated RF signal and timing RF noise, where Sϕ 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
Sϕ=π2ω2h¯ω0Pave(αΣ+Tc/2)ωR2τ26TR2[1+(2β2Σπτ2TR)21ω2+4(αΣ+Tc/2)2/TR2]+2qρP¯D+FkBΘPmw|D
(66)
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 (ωTR ≪ 1). The second term in the square brackets exceeds unity for frequencies below the resonator mode bandwidth.

Assuming that shot noise dominates, we get
Dmw=1ωR22h¯ω0ηTcPave=2πωR2γΔ0(D)gηγ2,
(67)
where 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 γc = γ/2. Parameters D and Δ0 are given by Eq. (40).

Equation (66) can be simplified if we substitute pulse parameters
Sϕ=2πγΔ0(D)gηγ2[1+196γ(D)Δ0ηγ2f2+124(1+π2f2γ2)1ηγ2π2f2Δ0(D)γ]
(68)
which becomes
Sϕ|f0=gf21122π((D)Δ0γ)1/2
(69)
for small frequencies. It shows that the detuning increase results in phase noise increase. Since the detuning is proportional to the pulse peak power, in accordance with Eq. (32), the closing phase noise increases with the pulse power increase and pulse duration decrease. It means that a broad Kerr frequency comb has higher timing jitter and associated phase noise as compared to the narrow frequency comb.

The smallest detuning for which Kerr comb can be generated is Δ0=3γ [22

22. T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Mode-locking in an optical microresonator via soliton formation,” arXiv:1211.0733v2 (2013).

, 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]

]. For this detuning
Sϕmin|f0gf231/4122πD
(70)
which corresponds to Allan deviation
σmin2g1/2(D)1/45πνRF1τ,
(71)
where νRFωR/(2π) is the linear repetition frequency of the comb. To make the transition from the phase noise to Allan deviation we used a general formula
σ2=20Sϕf2νRF2sin4(πfτ)(πfτ)2df,
(72)
which transforms to
σ=Sϕf22νRF2τ,
(73)
if Sϕf−2.

To compare, the minimum Allan deviation of the hyper-parametric oscillation that involves only two optical sidebands, found in [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]

], is
σminhpg1/222πνRF1τ.
(74)
It means that the frequency comb has Allan deviation (−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.

Finally, it is useful to present the phase noise of the RF signal generated by the comb on a fast photodiode using values easily measurable in an experiment. We introduce threshold power, Pth, of hyper-parametric oscillator defined as the power when oscillation with optical sidebands separated by multiple free spectral ranges starts in a resonator with −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]

]
Pth(αΣ+Tc/2)3TcγΣ,
(75)
so that the equation for the phase of the comb envelope can be written as
Δ0γ=π28sin2(ϕcϕp)PinPthPinPth.
(76)
The approximation is valid in the case of well developed comb. for the sake of simplicity of the resultant formula we also assume η = 1, replace phase noise by a single sideband phase noise Sϕ = 2Lϕ, and obtain the resultant phase noise in the form presented at the beginning of the paper by Eq. (1).

5. Conclusion

Expectation values as well as phase fluctuations of the mode locked Kerr frequency comb and associated optical pulses generated in a nonlinear ring cavity pumped with continuous wave light are studied. Mode locking mechanism of the frequency comb is explained. An analytical expression for the timing jitter of the pulses is found. It is shown that low spectral frequency frequency timing jitter of the generated pulses (close-in phase noise of the repetition rate of the comb) is optimized for the low pump power. The increase of the pump power results in decrease of the fundamental stability of the comb repetition rate, and also results in improvement of the high frequency phase noise floor.

Acknowledgments

The authors acknowledge support from Defense Sciences Office of Defense Advanced Research Projects Agency under contract No. W911QX-12-C-0067 as well as support from Air Force Office of Scientific Research under contract No. FA9550-12-C-0068.

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F. Ferdous, H. X. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics 5, 770–776 (2011). [CrossRef]

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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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References

  1. P. Del-Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450, 1214–1217 (2007). [CrossRef]
  2. P. Del-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]
  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]
  4. I. S. Grudinin, N. Yu, and L. Maleki, “Generation of optical frequency combs with a CaF2resonator,” Opt. Lett.34, 878–880 (2009). [CrossRef] [PubMed]
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