Self-locking of the frequency comb repetition rate in microring resonators with higher order dispersions

D.V. Skryabin; Y.V. Kartashov

doi:10.1364/OE.25.027442

1. Introduction

Whispering-gallery-mode microresonators attract nowadays considerable attention in photonics community [1]. These compact devices generating equidistant frequency combs through the Kerr mode locking have a host of diverse applications, e.g., in high-precision spectroscopy and optical clocks [2–4], and in classical and quantum information processing [4–6]. The process of the comb generation in these devices has revealed a plethora of nonlinear effects related to formation of solitons, see, e.g., [7–12]. These solitons have the broad comb spectrum (soliton combs) and orbit around the microring resonator indefinitely sustaining their temporal and spectral shapes due the perfect balance between the Kerr nonlinearity, dispersion, pump and losses. Microresonators and frequency combs also link to various breaking fundamental and applied ideas from across many branches of photonics, e.g., photonic topological insulators [13, 14] and synthetic dimensions [15, 16] have been proposed and demonstrated using arrays of microrings and frequency combs.

There are some important similarities between the supercontinuum generation in photonic crystal fibers and generation of the broad microresonator frequency combs. In particular, emission of the resonant dispersive radiation by quasi-soliton pulses (soliton Cherenkov radiation) is one of the key phenomenon in broadening of both the fiber supercontinua [17] and the microresonator combs [18–23]. These ultra broad spectra allow to self-reference frequency combs by measuring the beat between the frequency-doubled lower-frequency end of the comb with its higher-frequency end [22, 23]. Unlike in a fiber, the Cherenkov radiation in a ring resonator propagates along the loop and therefore it has a chance to rich the other end of the soliton and thus to provide a feedback mechanism. In particular, the work on the octave wide combs was conducted in relatively short microrings with strong Cherenkov radiation [22, 23], thereby creating favourable conditions for the above effect to occur. Formation of soliton crystals, or multi-soliton bound states, in the presence of various radiation sources was considered in [19, 24–26]. Effects of the higher order dispersions on solitons were also studied in fiber loops, see, e.g., [27, 28] and references therein.

Since the radiation carries some momentum away from a soliton, the soliton recoils and changes its velocity as a result [29]. In micoresonators the recoil effect [19] directly translates to the soliton repetition rate and hence modifies the comb free spectral range (FSR), i.e., the distance between the comb teeth. It was shown that the recoil induced change of the soliton velocity as a result of the 3rd dispersion and Raman effect in microresonators is a smooth function of the pump frequency [30]. Here we report that, this is true only until the length of the Cherenkov radiation tails is shorter than the resonator, so that there is no feedback loop through the Cherenkov radiation closing onto the parent soliton. As soon as the radiation tail is long enough to fill the entire resonator, the soliton velocity starts changing in steps of the variable height and length. This ’quasi-quantisation’ of the velocity directly provides the pump frequency intervals, where the FSR is stabilised around some quasi-constant values and hence is expected to be more robust against fluctuations of any origin. Thus, soliton interaction with the Cherenkov radiation provides a potentially viable method for self-stabilisation of frequency combs, extending a range of already known comb stabilisation techniques, see, e.g., [31, 32] and references therein.

Below, we first describe a mathematical model and then demonstrate that the octave wide soliton combs similar to ones in [22, 23] are readily excitable from the cw input and strong Cherenkov radiation tails can fill the entire resonator circumference. We compute numerically examples of the soliton families embedded into the Cherenkov radiation background and demonstrate existence of a sequence of the pump frequency intervals, where FSR becomes quasi-insensitive to the pump frequency (FSR self-locking). We conclude by deriving the Adler equation explaining the FSR self-locking and indicate possible connections of the FSR quasi-quantisation diagram to other staircase-like structures known in the synchronisation of oscillators, see, e.g., [33–35] and references therein.

2. Model

We describe the dimensionless electric field amplitude Ψ in a microring resonator using the well established generalized Lugiato-Lefever model, see, e.g., [7, 19, 21]. We consider a dispersion operator that includes terms up to the fourth order, which is the minimal requirement allowing to qualitatively approximate dispersion of the short resonators used to experimentally observe the octave wide combs [22, 23]

\begin{array}{l} i \partial_{T} Ψ = (ω_{0} - i D_{1} \partial_{θ} - \frac{1}{2!} D_{2} \partial_{θ}^{2} + \frac{i}{3!} D_{3} \partial_{θ}^{3} + \frac{i}{4!} D_{4} \partial_{θ}^{4}) Ψ \\ - i κ Ψ - \frac{1}{T_{0}} {| Ψ |}^{2} Ψ - \frac{1}{T_{0}} h e^{- i ω_{p} t} . \end{array}

Here T is time and θ is the polar angle varying around the resonator perimeter. The resonator dispersion coefficients D_j_≥1 are defined following [7, 21] and have dimensions of frequency. All the parameters and variables in the above equation, apart from h and Ψ, are measured in physical units. Ψ obeys an obvious periodicity condition Ψ(T, θ) = Ψ(T, θ +2π). κ is the loss rate, h and ω_p are the pump amplitude and its frequency,

T_{0}^{- 1} {| Ψ |}^{2}

is the nonlinear shift of the resonances. ω₀ is the frequency of the m = 0 (reference) resonator mode. The coefficient before Ψ ² has the dimension of the inverse time, but its value can be scaled to any convenient number, through the scaling of h and Ψ only. Our choice was to make it equal to the inverse round trip time, 1/T₀ = D₁/(2π), of the pulse propagating with the group velocity at the frequency ω₀. We are interested here in large THz FSRs, therefore h can be safely assumed θ-independent, cf. [36]. Disregarding the pump, loss and nonlinearity and assuming

Ψ = e^{i m θ - i ω_{m} T}

, where |m| = 0, 1, 2, 3,…, one finds for the frequency of the m’th mode ω_m = ω₀ + D₁m + D₂m² / 2! + D₃m³ / 3! + D₄m⁴ / 4!. FSR of 1THz [5, 22] corresponds to D₁ = 2π/T₀ = 2π THz. Assuming the central resonance at 1.55 µm gives ω₀ = 2π × 193.5THz, thus the self-referenced combs with the width Δ = 2ω₀/3, estimated from 2(ω₀ − Δ/2) = ω₀ + Δ/2, should contain ~ 130 or more of the detectable modes. Other dispersion coefficients are chosen by us so that the frequencies of the red and blue detuned dispersive waves, see below, span over the required range: D₂ = 2π × 25MHz, D₃ = 2π × 700kHz, D₄ = −2π × 71kHz. We define the so-called integrated dispersion as D (m) = ω_m − (ω₀ + D₁m) = D₂m² / 2! + D₃m³ / 3! + D₄m⁴ / 4! [22], which is plotted with the full line in Fig. 1(a). The resonator length L enters the dispersion coefficients as D_j ~ 1 L^j [19, 30]. The above set of D_j’s gives an approximate dispersion for the SiN microring with ∼ 23µm radius used in [22]. Group velocity dispersion (GVD) is characterized by D″ (m) = D₂ + mD₃ + m² D₄/2 (dashed line in Fig. 1(a)), which is anomalous around m = 0 (D₂, (D″ > 0) and turns normal (D″ < 0) for m < −20 and m > 40.

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 $\pm \sqrt{\dots}$ , respectively. Intersections with zero mark modal indexes of the Cherenkov resonances. Arrows indicate the dominant pair of resonances with large amplitudes.

Download Full Size | PDF

3. Cherenkov radiation and generation of the octave-wide soliton combs

Results of [19] concerning predictions of the Cherenkov radiation emission by soliton combs in microresonators with the 3rd order dispersion can be readily extended onto the present system. In the approximation that neglects loss, we obtain that the soliton frequency resonates with the frequencies of the modes having indices m satisfying the following condition [19]:

\begin{array}{l} \pm \sqrt{{(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 . \end{array}

Here v is the dimensionless soliton velocity shift, such that the soliton angular velocity is D₁(1 + v/(2π)). v ≠ 0 due to spectral recoil on the soliton from the Cherenkov radiation [19]. |ψ₀| ² is the constant intensity of the soliton background determined by h ≠ 0 and δ = (ω₀ – ω_p) 2π/D₁ is the normalized detuning between the cavity resonance and the pump frequencies. Figure 1(b) shows plots of the left-hand sides of Eq. (2), revealing four real roots. Our numerical simulations, demonstrated that two of the predicted roots associate with the Cherenkov waves having negligible amplitudes, while the other two (indicated by the arrows in Fig. 1(b)) are responsible for generation of very strong radiation. Let us note, that Eq. (2) accounts for the modification of the dispersion of linear waves due to presence of the soliton background ψ₀ [19], and hence predictions of Eq. (2) are noticeably different from using D (m) = 0 as an estimate for the Cherenkov resonances [5, 18, 22].

To confirm that the soliton combs spectrally centered inside the anomalous GVD range shed Cherenkov radiation as predicted by Eq. (2), we have conducted a series of numerical simulations of Eq. (1) for δ > 0, where the cw (∂_θ |Ψ| = ∂_t |Ψ| = 0) state of the microcavity field is bistable, see Section 4 below. The aforementioned soliton background ψ₀ is the lower state of the bistability loop, while simulations where initialised with the upper state perturbed by noise. We used the relatively large value of the cavity linewidth κ = 2π * 800MHz, which corresponds to the quality factor Q = ω₀/κ ≃ 2.5 ∗ 10⁵, like in the resonator used in [22].

We have found that upto δ ≃ 0.03 the generated signal is dominated by the octave broad noisy spectrum typical for the modulational instability, which spectral range was extended into the normal GVD domain through the 4th order dispersion effects [37, 38]. However, for δ above ≃ 0.03 and upto ≃ 0.055, the initially generated noisy spectrum becomes suppressed after few hundreds of round trips and it is replaced by the coherent spectra typical for the soliton combs, see Figs. 2(a), 2(b) and 3. These spectra have the octave width and their edges are shaped by the pronounced spectral peaks, which are the direct consequence of the two Cherenkov waves emitted by the soliton core. The soliton core spectrally recoils towards positive ’m’ since the red detuned (m < 0, ω_m < ω₀) Cherenkov peak in the range of m ≃ −60 is stronger than the blue detuned one in the range of m ~ 80.

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, |ψ_m(T)|, $ψ_{m} = \int_{0}^{2 π} Ψ (T, θ) e^{i m θ} d θ$ . 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).

Download Full Size | PDF

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).

Download Full Size | PDF

Positions of the Cherenkov maxima is precisely given by Eq. (2), providing the velocity shift v is extracted from simulations. Real roots of Eq. (2) should be taken upto the nearest integer, since we are dealing with a periodic system. We have found that within the soliton existence interval of detunings the red Cherenkov peaks varies between m = −56 and m = −60, see Fig. 4(a), while the blue Cherenkov peak stays fixed at m = +84.

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 |ψ_s|) plotted together with the amplitude of the bistable cw solution (marked here as |ψ_cw|). 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.

Download Full Size | PDF

A typical angular profile of the soliton comb circling around the resonator is shown in Fig. 3(a). The positive/negative m Cherenkov radiation tail spreads to the right/left from the soliton core. The radiation tails fill the entire resonator and form a strongly modulated soliton background. The spatial modulation of the background has its origin in the interference between the pump field at m = 0 and the Cherenkov radiation at m ≠ 0 and also between the counter-propagating Cherenkov tails. Contrary, typical intensity profiles of the Cherenkov tails emitted by solitons in photonic crystal fibers are not modulated, since these tails spread over the zero background. Note, that for δ above 0.06 the initial modulational instability pattern switches into the stable small amplitude cw state after few hundreds of round trips and that close to δ = 0.03, the solitons typically demonstrate breathing dynamics [39, 40]. Losses and length of the resonator are the factors that control if the modulated non-decaying Cherenkov background fills the entire resonator or not. An example of the soliton in the five times longer cavity with the same losses is shown in Fig. 3(b). This soliton still has very pronounced radiation tails, which decay to the constant flat background over the distance that is shorter than the half of the cavity length, cf. Figs. 3(a) and 3(b).

4. Staircase-like changes of the soliton FSR

In order to understand properties of the soliton states embedded into and interacting with their own Cherenkov radiation, we sought the soliton solution of Eq. (1) in the reference frame moving with the velocity D₁ (1 + v/(2π)):

Ψ (T, θ) = ψ (x, t) e^{- i ω_{p} T}, x = (θ - 2 π t) - v t, t = \frac{T}{T_{0}} .

One unit of the dimensionless time t corresponds to one round trip. Introducing β_j>₂ = 2πD_j/(D₁ j!) and γ = κ2π/D₁, we find that ψ obeys

\begin{array}{l} i \partial_{t} ψ = (δ + i v \partial_{x} - β_{2} \partial_{x}^{2} + i β_{3} \partial_{x}^{3} + β_{4} \partial_{x}^{4}) ψ \\ - i γ ψ - {| ψ |}^{2} ψ - h, ψ (t, x + 2 π) = ψ (t, x) . \end{array}

The exact soliton solution ψ_s satisfies ∂_tψ_s = 0 and therefore can be found numerically by solving an ordinary differential equation. v ≠ 0 and the proportional shift of the soliton spectral center of mass, see Fig. 2(b), are triggered by the 3rd order dispersion responsible for the power and momentum imbalances of the positive and negative Cherenkov radiation peaks. Thus, the net FSR for the soliton comb, D₁ (1 + v / (2π)), is different from the FSR of the GVD free cavity D₁. For the cavity and pump parameters fixed, there is a unique value of v that allows for the soliton solution to exist. Soliton profiles and v were found self-consistently by solving Eq. (4) with ∂_t = 0 using a variant of the Newton method. Plots of v vs δ in Fig. 4(a) below show the FSR change with δ.

Figure 4(a) shows how v varies with the detuning δ. The staircase-like line in Fig. 4(a) corresponds to the soliton family found in the resonator with the D_j values specified in Section 2, see the corresponding resonator dispersion in Fig. 1(a) and a typical soliton profile in Fig. 3(a). The dotted smoothly varying line in Fig. 4(a) corresponds to the five times longer resonator, see Fig. 3(b) for the typical soliton profile with the background that decays to a practical zero on the distances shorter than the half of the resonance length. In the former case the soliton is efficiently trapped within the standing wave pattern formed by the radiation filling the entire resonator, while in the latter case any remnants of the soliton-radiation interaction forces are irrelevant. Note here a connection of this effect with trapping of spatial cavity solitons by external beams creating two-dimensional optically induced potentials in broad-area microcavities [41, 42] and lattice potentials created in microrings with a polychromatic pump [26, 43]. We have verified by extensive numerical modelling that the v vs δ dependence forms a sequence of steps connecting quasi-flat plateaus in any situation, when the tails of the radiation emitted by the soliton form a modulated background extending over the entire cavity length. Thus, the staircase shape of v (δ) is associated with the feedback action of the radiation on its parent soliton.

Graphically, the phenomenon appears to be similar to the frequency self-locking effect in a laser with feedback, when the laser frequency forms a sequence of plateaus as a function of the external cavity length, where the frequency becomes insensitive to the change of parameters leading to the linewidth narrowing [33]. If the pump frequency is tuned into the middle of the v(δ) plateau, then the comb FSR is in the self-locking regime. Indeed, in this case any fluctuations of the pump frequency or of the cavity resonance will result in the FSR changes that are much smaller than the same fluctuations would give for the five times longer cavity, cf. the full and dotted lines in Fig. 4(a). We found, that while changing from one plateau to another, the index m = m₋ of the stronger (negative) Cherenkov resonance changes in steps of one, while the weaker resonance stays fixed at m = m₊ = 84. The corresponding values of m found in the simulations are shown in Fig. 4(a) and are reproduced through Eq. (2). Thus our system features a sequence of the quasi-discrete FSR locking intervals associated with a sequence of the Cherenkov resonances, which by the nature of the cavity periodicity can not change continuously as a system parameter is varied.

5. Adler equation for the FSR self-locking

The above results connect not only to the theory of lasers with feedback [33], but also to a bulk of research on the oscillator synchronisation, where discrete sequences of the frequency locking intervals are well known in a variety of contexts, see, e.g., [35] and references therein. In order to make this connection obvious we present here a conservation law based approach capturing the main qualitative aspects of the FSR locking regime and leading to the Adler equation, which is a paradigm model in the oscillator synchronisation theory [35]. Note, that the comb formation itself was also linked to this theory [44].

The angular momentum integral M (an average modal number of the field) and its time derivative are given by

M = \frac{1}{2 i} \int_{- π}^{π} (ψ^{*} \partial_{x} ψ - c . c .) d x, \partial_{t} M = - 2 γ M .

We assume that D₃_,₄, γ and h are relatively small, and that the soliton is much narrower than the resonator length. Then, the soliton and radiation solution ψ_s can be represented as a superposition of the soliton ϕ_s localised on the zero background over the infinite interval, plus perturbations in the form of the Cherenkov radiation tails ψ_rad and of the constant background ψ₀: ψ_s(x) ≃ ϕ_s(x) + ψ_rad(x) + ψ₀ [45, 46]. ϕ_s(x) solves

\begin{array}{l} (δ + i v \partial_{x} - β_{2} \partial_{x}^{2}) ϕ_{s} - {| ϕ_{s} |}^{2} ϕ_{s} = 0, \\ ϕ_{s} = \sqrt{2 a} sech ((x - x_{s}) \frac{\sqrt{a}}{\sqrt{β_{2}}}) \exp [i \tilde{v} x], \end{array}

see [47], and

ψ_{r a d} = ε_{+} (x - x_{s}) e^{i m_{+} x} + ε_{-} (x - x_{s}) e^{i m_{-} x},

where a = δ − v²/(4β₂) > 0,

\tilde{v} \equiv v / (2 β_{2})

is the shift of the soliton spectral maximum away from the zero (pump) momentum, x_s is the soliton angular coordinate in the frame that moves with the velocity v, m_± are the modal numbers (angular momenta) of the positive and negative Cherenkov radiation peaks. ε_± are the radiation envelopes. The comb solutions found above are stationary in the x -frame and hence the momentum balance condition is M = 0. Noting

| \tilde{v} | ≪ | m_{\pm} |

, find that the leading terms in M are

\begin{array}{l} M ≃ \tilde{v} Q_{s} + m_{+} Q_{+} + m_{-} Q_{-} \\ + \frac{1}{2 i} (i m_{+} e^{i (m_{+} - \tilde{v} x_{s}}) ℐ_{+} + i m_{-} e^{i (m_{-} - \tilde{v} x_{s}} ℐ_{-} - c . c .) \end{array}

where

Q_{\pm} = \int_{- π}^{π} {| ε_{\pm} (x) |}^{2} d x, Q_{s} = \int_{- π}^{π} {| ϕ_{s} (x) |}^{2} d x, ≃ 2 \sqrt{4 β_{2} δ - v^{2}}

and

ℐ_{\pm} = \int_{- π}^{π} ϕ_{s} (x) ε_{\pm} (x) e^{i (m_{\pm} - \tilde{v}) x} d x

. Thus, naturally, the net momentum of the intracavity field splits into the independent soliton and radiation momenta and the term responsible for their interaction.

ℐ_{\pm}

are the complex overlap integrals making up the dominant part of the radiation interaction with its the parent soliton, which is critical for description of the FSR self-locking.

In the long cavity case ε₋ (ε₊) falls abruptly to zero for x < x_s (x > x_s) and decays exponentially for x > x_s (x < x_s). Figure 3(b) clearly shows such behaviour. In this case, $ℐ_{\pm} ≃ 0$ and the momentum balance equation gives: v ≃ 2β₂ (−m₊|Q₊ + |m₋|Q₋)/Q_s, so that v ≠ 0 is due to the imbalance between the net momenta of the two Cherenkov waves, v = v_imb. In the short cavity case, see Fig. 3(a), the radiation envelopes can be approximately replaced by the complex constants $ε_{\pm} (x) ≃ ε_{\pm} e^{i α_{\pm}}$ , so that v_imb = 4πβ₂(|m₋| ℇ ₋² − |m₊ |ℇ₊ ²)/Q_s, while the limits in the overlap integrals can be extended to the infinities to give

ℐ_{\pm} ≃ π \sqrt{2 β_{2}} ε_{\pm} sech (\frac{π m_{\pm}}{2} \sqrt{\frac{β_{2}}{a}}) \exp [i α_{\pm}],

and the momentum balance condition is now given by

\begin{array}{l} \frac{Q_{s}}{2 β_{2}} (v - v_{i m b}) ≃ | m_{-} | | ℐ_{-} | \cos (m_{-} x_{s} + α_{-}) \\ - | m_{+} | | ℐ_{+} | \cos (m_{+} x_{s} + α_{+}) . \end{array}

Introducing

δ_{m_{+}, m_{-}}

corresponding to the middle of one of the self-locking plateaus and denoting the corresponding value of v_imb as v_m, we assume that

δ - δ_{m_{+}, m_{-}} \equiv Δ_{m}

is small. This gives

v_{i m b} = v_{m} + v_{m}^{'} Δ_{m} + \dots

, where

v_{m}^{'} = \partial_{δ} v_{i m b}

for

δ = δ_{m_{+}, m_{-}}

and

m \equiv (m_{+}, m_{-})

. If x_s is a slow function of time, such that γ∂_t M is still an approximate zero with the required accuracy, we have v = v_m + ∂_t x_s. Now it becomes obvious that Eq. (10) is an example of the Adler equation with the two frequency driving:

\begin{array}{l} \partial_{t} x_{s} - v_{m}^{'} Δ_{m} ≃ \frac{2 β_{2}}{Q_{s}} | m_{-} | | ℐ_{-} | \cos (m_{-} x_{s} + α_{-}) \\ - | m_{+} | | ℐ_{+} | \cos (m_{+} x_{s} + α_{+}) . \end{array}

The soliton coordinate x_s has the meaning of the phase of the oscillations at the radio frequency. If the right hand side in Eq. (11) representing the effective self-driving force, can be neglected, then the phase becomes

v_{m}^{'} Δ_{m} t

. This corresponds to the FSR in the unlocked regime, meaning the soliton is freely drifting with the velocity shift

v = v_{m} + v_{m}^{'} Δ_{m}

. If the effective force dominates over the

v_{m}^{'} Δ_{m}

, then the phase of the oscillator self-locks to the phase of the driving, meaning that the radiation lattice traps the solitons, i.e., there exists a stationary, ∂_t x_s = 0, real valued x_s, and hence v = v_m. As soon as δ changes enough to shift either m₋ or m₊ by 1, then the oscillator finds its new locked frequency around another v_m. The maximal possible value of the driving force is

\frac{4 β_{2}}{Q_{s}} \sqrt{m_{-}^{2} {| ℐ_{-} |}^{2} + m_{+}^{2} {| ℐ_{+} |}^{2}}

. Therefore, we can approximate the locking condition as

Q_{s} | v_{m}^{'} Δ_{m} | / (4 β_{2}) < \sqrt{m_{-}^{2} {| ℐ_{-} |}^{2} + m_{+}^{2} {| ℐ_{+} |}^{2}}

. The above shows that, maximizing values of the soliton-radiaton overlap integrals promotes a broader parameter range for observation of the FSR self-locking. The above calculation also shows explicitly that v determines not only the FSR, but also the shift of the carrier mode number of the soliton away from the pump reference. Therefore the above self-locking may also stabilise the absolute position of the comb lines and it can be one of the possible future directions of the theory development.

6. Summary

We have predicted that, under the conditions typical for the recent experiments on generation of the octave wide combs shaped by the solitons and two Cherenkov waves [22, 23], the comb FSR can self-lock through the back-action of the Cherenkov radiation on its parent soliton. We have derived the Adler equations for the soliton velocity and position, which is a well known model describing the frequency locking effects in a variety of dynamical systems. We have demonstrated that the comb FSR remains quasi-constant over sufficiently broad intervals of the pump frequency, implying that the phenomenon we have described can be potentially used as the self-stabilisation technique for the octave-wide combs in the relatively short microring resonators.

Funding

The Leverhulme Trust (RPG-2015-456); H2020 (691011, Soliring); ITMO University Visiting Professorship via the Government of Russia Grant 074-U01; Russian Foundation for Basic Research (17-02-00081); Russian Science Foundation (17-12-01413).

References and links

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

2. 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]

3. 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]

4. 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]

5. 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]

6. 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]

7. 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]

8. 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]

9. 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]

10. 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]

11. 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]

12. 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]

13. 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]

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

15. 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]

16. A. Schwartz and B. Fischer, “Laser mode hyper-combs,” Opt. Express 21, 6196–6204 (2013). [CrossRef] [PubMed]

17. 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]

18. 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. 38, 37–39 (2013). [CrossRef] [PubMed]

19. 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]

20. 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]

21. 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]

22. 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]

23. 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]

24. 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]

25. A. G. Vladimirov, S. V. Gurevich, and M. Tlidi, “Effect of Cherenkov radiation on the interaction of temporal dissipative solitons in a driven cavity with high order dispersion,” in 2017 European Conference on Lasers and Electro-Optics - European Quantum Electronics Conference, paper EF-1.2.

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

27. 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]

28. 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]

29. 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]

30. 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]

31. 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]

32. A. A. Savchenkov, A. Eliyahu, W. Liang, V. S. Ilchenko, J. Byrd, A. B. Matsko, D. Seidel, and L. Maleki, “Stabilization of a Kerr frequency comb oscillator,” Opt. Lett. 38, 2636 (2013). [CrossRef] [PubMed]

33. 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]

34. 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]

35. A. Pikovsky, M. Rosenblum, and J. Kurths, “Synchronization. A Universal Concept in Nonlinear Sciences” (Cambridge University, 2001). [CrossRef]

36. 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]

37. 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]

38. M. Tlidi, A. Mussot, E. Louvergneaux, G. Kozyreff, A. G. Vladimirov, and M. Taki, “Control and removal of modulational instabilities in low-dispersion photonic crystal fiber cavities,” Opt. Lett. 32, 662–664 (2007). [CrossRef] [PubMed]

39. 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]

40. 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]

41. 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]

42. 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]

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

44. 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]

45. 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]

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

47. 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]

Self-locking of the frequency comb repetition rate in microring resonators with higher order dispersions

Abstract

Corrections

1. Introduction

2. Model

3. Cherenkov radiation and generation of the octave-wide soliton combs

4. Staircase-like changes of the soliton FSR

5. Adler equation for the FSR self-locking

6. Summary

Funding

References and links

Cited By

Figures (4)

Equations (11)

Optics Express