## Abstract

We investigate the impact of stimulated Raman scattering (SRS) and self-steepening (SS) on breather soliton dynamics in octave-spanning Kerr frequency comb generation. SRS and SS can transform chaotic fluctuations in cavity solitons into periodic breathing. Furthermore, with SRS and SS considered, bandwidth of the soliton breathes more than two times stronger. The simultaneous presence of SRS and SS also make the soliton breathe slower and degrades the coherence of the soliton.

© 2015 Optical Society of America

## 1. Introduction

Kerr frequency combs generated from the parametric process in high-Q microresonators pave a way to obtain on-chip coherent multi-wavelength source [1–4]. It has been demonstrated theoretically and experimentally that the generated comb lines can be phase locked under appropriate conditions, in which a cw pump is converted into an ultrashort pulse train with high repetition rate [5–9]. The mode-locked intra-cavity pulses, often termed as cavity solitons (CSs), can be quite short through dispersion engineering of the microresonators [10–12], exhibiting rich nonlinear phenomena, such as breather solitons [13–15], optical rogue waves [16], and soliton self-frequency shift [17]. Among them, the breather soliton is an important dynamics that affects the formation and noise property of Kerr frequency combs [13, 18]. The excitation of breather solitons [13, 14, 19], and its influence on coherence property for the generated Kerr frequency comb [20] have been extensively explored. However, only a few works have focused on the impact of stimulated Raman scattering (SRS) and self-steepening (SS) on Kerr frequency comb generation [21]. Most of the work on breather solitons excluded SRS and SS, which play a critical role in various nonlinear optical phenomena, especially when the spectrum approaches octave-spanning or the pulse peak power increases, e.g., supercontinuum generation [22] and rogue wave generation in mode-locked fiber lasers [23].

Here, we numerically analyze the impact of SRS and SS on breather soliton dynamics in octave-spanning Kerr frequency comb generation. We show that SRS and SS will induce or enhance breather soliton dynamics. The coherence property of breather solitons is also degraded due to SS and SRS.

## 2. Modeling of octave-spanning comb generation

We use a generalized Lugiato-Lefever equation with SRS and SS to investigate the intra-cavity pulse dynamics in the octave-spanning Kerr frequency comb generation [16],

*exp(-iωt)*in deriving Eq. (1). In Eq. (1), E(t,τ) is the intra-cavity electric field and

*τ*(5 ps, a resonator with radius of 115 μm) is the roundtrip time, t and

_{0}*τ*are the slow and fast time respectively, L is the cavity length,

*α*are the intrinsic cavity loss and coupling loss respectively,

_{i}, θ*γ*is the Kerr nonlinear coefficient of the cavity. It is noteworthy we drop the contribution from the direct current component of Raman spectrum to Kerr effect when calculating

*γ*.

*β*is the nth-order dispersion coefficient, and

_{n}*τ*is Kerr shock time.

_{Kshock}*δ*is the phase detuning of the pump, and the pump power equals to |E

_{0}_{in}|

^{2}. For simplicity of expression, the pump power and detuning are normalized as X = 8|E

_{in}|

^{2}

*γθL/(θ + α*, Δ =

_{i})^{3}*2δ*respectively. The last term of Eq. (1) is the effect of SRS, where γ

_{0}/(θ + α_{i})_{R}is the Raman nonlinear parameter (0.11 (Wm)

^{−1}),

*h*is the Raman response function [21],

_{R}*τ*is the Raman shock time (1.8 fs), and SRS is calculated in the frequency domain with a Lorentzian gain spectrum whose bandwidth is 3.44 THz.

_{Rshock}To obtain an octave-spanning comb, we consider a low-dispersion microring resonator, with the dispersion coefficients at 1.55 μm as *β _{2 =}* -12 ps

^{2}/km,

*β*-0.033 ps

_{3 =}^{3}/km,

*β*-0.0004 ps

_{4 =}^{4}/km (higher order dispersion terms not given here), and the dispersion curve is shown in Fig. 1. The Kerr nonlinear coefficient

*γ*and Kerr shock time

*τ*are 0.76 (Wm)

_{Kshock}^{−1}and 2.0 fs, respectively. The intrinsic loss is 0.2 dB/cm and the coupling ratio is 0.4%. These parameters can be obtained in a low dispersion SiN resonator. We first exclude the effects of SRS and SS, and set the pump wavelength to 1.7 μm i.e., the minimum anomalous dispersion wavelength. A single CS with octave-spanning Kerr frequency comb is generated when increasing Δ from 3 (phase detuning 0.0105) to 30 at X = 90 (1.9 W pump power), shown in Fig. 1.

## 3. SRS and SS induced breather soliton

When pump power increases, the CS could develop Hopf bifurcations and breather instability [14, 21, 24]. Breather soliton can also be understood as a result from phase-mismatching in cascaded for-wave-mixing [15]. Here, when increasing X to 240, the CS becomes unstable, shown in Fig. 2(a), with the soliton peak power changing randomly. SRS and SS can change the power distribution of the comb spectrum, thus the phase-mismatching condition and soliton stability. In addition, gain from SRS could counterbalance the cavity loss, causing strong perturbations on the CS, and induce soliton breathing [21]. If we add the SRS term to our model, while still leaving SS out, the random fluctuation of the spectrum in Fig. 2(a) will change to a quasi-periodic pattern in Fig. 2(b). However, breather soliton is not excited yet, as the peak power doesn’t change periodically. If we add SS instead of SRS, the random fluctuation of the spectrum will be stabilized (Fig. 2(c)). When we add both SRS and SS, breather soliton is excited, and its spectrum breathes deeply with the 20-dB comb bandwidth changing from 35 to 116 THz in a breathing period of 0.5 ns, as seen in Fig. 2(d). To describe the breathing strength quantitatively, we define the breathing contrast of the peak power as *(P _{max}-P_{min})/(P_{max} + P_{min})*, where

*P*and

_{max}*P*are the maximum and minimum peak power, respectively. The breathing contrast in Fig. 2(d) is as high as 0.73. From Fig. 2, it is obvious that SRS and SS significantly affect the soliton stability for an octave-spanning comb. SRS tends to trigger soliton breathing, which can be greatly enhanced by the presence of SS, although SS alone suppresses the soliton breathing in this case. SRS and SS makes it easier to excite breather solitons, transforming random fluctuation into periodic oscillation.

_{min}To show this impact is not limited to a specific pump condition, we vary the pump power and detuning to test the soliton’s stability. Breather soliton is found to exist in a wider parameter space with SRS and SS considered. We summarize the breathing of the peak power and 20-dB bandwidth (Δω) in Fig. 3. *P _{max}* increases with the pump power, whereas

*P*decreases. Thus, the breathing contrast increases from 0.40 to 0.75, when the pump strength X increases from 140 to 260. The 20-dB bandwidth (Δω) changes in a similar manner with peak power. Further increasing the pump power will cause the breathing to become chaotic [14]. On the other hand, the breathing contrast decreases from 0.80 to 0.09 when detuning Δ increases from 20 to 34. With a larger detuning, breathing ceases, and the CS is stabilized again. Figure 3 shows that the breathing is stronger at a large pump power and small detuning. Such a general trend is similar to the situation where SRS and SS are not considered [14].

_{min}In addition, the breathing contrast is enhanced with SRS and SS included for pump conditions, under which breather soliton can be excited regardless of considering SRS and SS or not. For instance, at X = 160, Δ = 20, comparing Figs. 4(a)-4(b), we see that the CS breathes slower and stronger, due to SRS and SS. Such an enhancement of the breathing dynamics can also be found in Fig. 4(c), which shows the breathing of the 20-dB bandwidth is nearly two times stronger with SRS and SS considered. This can be understood as SRS (Raman gain on red-side) and SS (larger Kerr nonlinear coefficient on blue-side) lead to higher gain at the wings of the spectrum, pushing the spectrum away from the center.

## 4. Degradation of the coherence

It has been shown in mode-locked fiber lasers pulse dynamics affects the comb linewidth [25] and SRS can degrade the coherence of the pulse train [23]. Therefore, it is important to examine how SRS and SS affect the coherence property of breather solitons. We evaluate the first order coherence property as [20]:

_{1}-t

_{2}stands for the time delay between the pulses.

In Figs. 5(a)-5(b), we show the coherence property of the generated combs in Figs. 4(a)-4(b) respectively. For the breather soliton without SRS and SS, the coherence changes periodically, similar to [20]. When SRS and SS are included, the coherence property still breathes periodically, but is significantly deteriorated in general, shown in Fig. 5(b).

To show the deteriorated coherence due to breathing dynamics more clearly, we pad 3001 successive pulses into a pulse train and take the Fast Fourier Transform (FFT) of the pulse train. The generated combs in these two cases are shown in Figs. 5(c)-5(d). There are ripples in the spectrum in Fig. 5(c), because the output pulse train is truncated in a limited time window, and thus the spectrum is a convolution of the comb spectrum with a sinc function. If we add enough zero to the padded pulse train and extend the time window, we can remove the ripples, obtaining an envelope of the black line in Fig. 5(c). The comb with SRS and SS considered exhibits a stronger dispersive wave emission at the high frequency side. Also, the central peak of the soliton spectrum is shifted away from the pump to the lower frequency side [17], while staying at the pump wavelength 1.7 μm for the case without SRS and SS. In Fig. 5(d), a spike can be observed near 150 THz, due to the good coherence of that narrow band (Fig. 5(b)).

By padding pulses, we have a finer spectral resolution. Details of the comb line around 169.96 THz are shown in the insets of Figs. 5(c)-5(d) respectively. Without SRS and SS, the comb line has multiple discrete sub-teeth, since the spectral breathing can be regarded as a periodic modulation on the spectrum. In contrast, with SRS and SS included, more sub-teeth appear inside the comb line, as it breathes slower (see Fig. 4(b)). The sub-teeth arising from the breathing modulation of the spectrum have comparable intensity to the main comb line and can be detrimental to many frequency comb applications (e.g., frequency metrology).

## 5. Breather soliton at different pump wavelengths

It is desirable to prove the impact of SRS and SS on breather solitons is generic to a wide parameter space. The complicated expressions of SRS and SS hinder us from proving it analytically. Here, we scan the pump wavelength, which changes the dispersion and nonlinearity profile (see Fig. 1), to examine the universality of our observation. Scanning the pump wavelength also keeps the overall dispersion low, allowing the generation of octave-spanning comb. We see similar results, when scanning across the whole anomalous dispersion regime. Here, we choose 1.55 and 1.3 μm (the dotted and dashed lines in Fig. 1) as two examples to prove the universality of the impact of SRS and SS on soliton dynamics. For 1.55 μm, at X = 160, Δ = 20, a breather soliton is excited regardless of considering SRS and SS or not. Nevertheless, the CS breathes slower and deeper with SRS and SS taken into account (Figs. 6(a)-6(b)), similar to the results in Fig. 4. The breathing contrast for the peak power increases from 0.71 to 0.85, with SRS and SS included. For 1.3 μm, at X = 160, Δ = 34, the CS with octave-spanning spectrum is stable without SRS and SS, shown in Fig. 6(c). However, with SRS and SS considered, a breather soliton is excited. From Fig. 6, the impact of SRS and SS on breather soliton dynamics is generic to various microresonator parameter combinations.

## 6. Conclusion

In conclusion, SRS and SS make breather soliton exist in a wider parameter space and enhance soliton’s breathing dynamics in octave-spanning comb generation. Such a conclusion remains applicable when parameters of a microresonator change. The coherence of breather soliton will be deteriorated due to SRS and SS, which implies exploiting breathing dynamics to scale the comb bandwidth may not be feasible for applications requiring a stabilized comb.

## Acknowledgment

C. Bao and C. Yang acknowledge support from Natural Science Foundation of China (61177046, 61377039).

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