1. Introduction

Nowadays, dissipative Kerr soliton frequency combs [1,2] have attracted great interest in both nonlinear optical physics and applications, such as integrated optical atomic clocks [3], coherent optical communication [4,5], low noise microwave and terahertz wave signal generations [6–8], and spectroscopy [9,10]. Usually, when the Kerr soliton combs are generated in amorphous materials such as silica and silicon nitride, they will experience a self-frequency shift [11–14] due to the interactions between the Raman nonlinearity and the solitons. This Raman soliton self-frequency shift (SSFS) has an impact on the dynamics [15,16], efficiency [12], bandwidth [17], as well as on the quantum diffusion [18] of the generated solitons. On the other hand, such SSFS, which is important for some practical applications, can be utilized in dual-comb technologies [19,20] based on a single microresonator and low-noise microwave generation with a quiet point [21].

Recently, to model the SSFS in Kerr soliton frequency combs [11,12,15], the Raman response of the materials is usually simplified as a Lorentzian profile [22]. However, the real Raman response in amorphous material exhibits complicated dynamics due to the excess density of vibrational states at Terahertz frequencies [23,24].

To accurately model the Raman response of the amorphous material such as silica, the Boson peak (BP) [24–26], which refers to the significant excess of vibrational modes in the THz range, should be considered. The BP is a characteristic feature of many glassy and amorphous materials, and is believed to play a crucial role in understanding the vibrational states of these materials. Although it has been used to model conventional temporal solitons propagating in optical fibers [27], the influence of the BP on the dissipative Kerr solitons has not been investigated thus far.

In this work, we theoretically investigate the SSFS in silica resonators with a more accurate Raman response function [27]. We surprisingly find that the BP will lead to even more SSFS than that induced by the Lorentzian part of the Raman response. We also derive a revised Raman shock time and point out that the Raman shock time is related to the soliton pulse width even for a relatively long pulse duration. Moreover, we demonstrate that the interference between the continuous-wave (cw) background and the soliton will affect the Raman SSFS.

2. Theoretical analysis

We begin with the solitonary solution of the generalized mean-field Lugiato-Lefever equation (LLE) [28–30] by taking into account the delayed Raman term [14,17]. Such modified equation that is capable of describing the temporal evolution of the envelope of the optical field circulating in a resonator takes the following form in a co-rotating frame:

Taking into account the cw background light [29] and treating the Raman term as a perturbation [12], the solitonary solution of single soliton for Eq. (1) can be expressed in the following mathematical form:

To analyze the soliton evolution in the system, we employ the method of moments [31] and calculate the frequency shift of the single soliton state as

Note that $E = \frac{1}{{2\pi }}\mathop \smallint \nolimits_{\frac{{ - \pi }}{{{D_1}}}}^{\frac{\pi }{{{D_1}}}} {|A |^2}d\tau $ is the average energy of the soliton state inside the cavity. By first taking the time derivative of ${\mu _c}$ in Eq. (3), and then assuming the boundary conditions $A\left( {\tau ={\pm} \frac{\pi }{{{D_1}}}} \right) = {A_0}{e^{i{\varphi _0}}}$, and finally performing the integral by parts, one will eventually reach the following dynamical equation of ${\mu _c}$ with the help of Eq. (1):

Next, under the condition of the dynamical equilibrium by setting $\frac{{\partial {\mu _c}}}{{\partial t}} = 0$, we obtain

From Eq. (5), one can compute the SSFS denoted by ${\mathrm{\Omega }_{Raman}}$, as long as the Raman fraction ${f_R}$ and the response function ${h_R}(\tau )$ are available.

Following the previous studies [12,14,22] on the soliton pulses (duration > 100 fs) propagating in the cavities, one can simplify the Raman term to the first order

Under the deep-red-detuned pumping we can adopt a simple sech-shape envelop of the optical field to compute ${\mathrm{\Omega }_{Raman}}$ [29]:

By plugging Eq. (6) into Eq. (5) and utilizing Eq. (8), we finally arrive at [12]

To theoretically estimate the value of ${\mathrm{\Omega }_{Raman}}$ in a silica resonator, we will work with two different forms of the Raman response function ${h_R}(\tau )$. By the first one we model the Raman-gain spectrum of the fused silica with a single Lorentzian profile [22]:

Due to the existence of the BP, we here introduce another improved response function [27], which can explain the appearance of the hump at the frequencies ∼2.5 THz of the actual Raman-gain spectrum [see Fig. 1(b)].

 

Fig. 1. (a) Illustration for the Raman-induced self-frequency shift of the Kerr-soliton frequency comb. (b) Raman-gain spectrum of the fused silica.

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The form of the Raman response thus becomes a linear combination

One can simply apply the Raman response function Eq. (13), instead of Eq. (11), to compute ${\tau _R}\; $(denoted as ${\tau _{R,2}}$) with Eq. (7), and it is found to be ∼1.57 fs [the blue dash-dotted line in Fig. 2(a)]. After a further examination, however, we find that the result is still far away from the realistic one because the approximation in Eq. (6) has implicitly overlooked the BP response. To clarify this issue, we first notice the integral,

 

Fig. 2. (a) Raman shock time ${\tau _R}$ as a function of the pulse width ${\tau _s}$: black solid line, the Raman shock time ${\tau _{R,1}}$ acquired by Raman response function ${h_{R,1}}(\tau )$; red solid curve, the corrected Raman shock time $\tau _{R,2}^\mathrm{^{\prime}}$ obtained by modified Raman response function ${h_{R,2}}(\tau )$. The blue and green dash-dotted lines represent the contributions of the Lorentzian response ${\tau _{R,2}}$ and the BP response $\frac{{15}}{{16}}{\tau _{R,BP}}$ to $\tau _{R,2}^\mathrm{^{\prime}}$, respectively. Brown dashed curve, the Raman shock time $\tau _{R,2}^{\mathrm{^{\prime\prime}}}$ directly calculated by Eq. (5) with the function ${h_{R,2}}(\tau )$. Violet dashed curve, the evaluated contribution of the Lorentzian response ${\tau _{R,L}}$ to $\tau _{R,2}^{\mathrm{^{\prime\prime}}}$. (b) Corresponding computed values of contributions to the Raman soliton self-frequency shift ${\mathrm{\Omega }_{Raman}}$ plotted versus $1/\tau _s^4$, where ${\tau _s}$ ranges from 105 to 300 fs. In the plots, the parameters of the free spectral range, dispersion, and damping rate are set to ${D_1} = 2\mathrm{\pi } \times 22\;\textrm{GHz}$, ${D_2} = 2\mathrm{\pi } \times 4\;\textrm{kHz}$, and $\gamma = 2\mathrm{\pi } \times 4\;\textrm{MHz}$, respectively.

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Then, under the approximation of the first-order Taylor-series expansion, the contribution of the BP response to the Raman shock time approaches to zero and is thus entirely neglected. To account for the influence of the BP response to the SSFS, we go back to the precise form of the frequency shift, Eq. (5), while adopting the response function in Eq. (13), and then obtain

In Eq. (17), the Lorentzian-profile part of the Raman-gain spectrum takes the same form as Eq. (12) but with the factor ${\tau _{R,2}}$, while the BP contribution

Putting together the above results involving the BP contribution, we eventually find

Equation (22) shows that the time constant $\tau _{R,2}^\mathrm{^{\prime}}$ is related to the pulse width ${\tau _s}$. In Fig. 2(a), we plot $\tau _{R,2}^\mathrm{^{\prime}}$ (red solid curve) as a function of the pulse width, which clearly reveals that the Raman shock time obtained by the complete response function [Eq. (13)] is generally more than twice longer than the time obtained by Eq. (6) for certain pulse durations. In our calculations, we also compute ${\tau _{R,2}}$ [the contribution of Lorentzian response in linear approximation to $\tau _{R,2}^\mathrm{^{\prime}}$, the blue dash-dotted line of Fig. 2(a)] and $\frac{{15}}{{16}}{\tau _{R,BP}}$ (the contribution of BP response, the green dash-dotted line), and find that the contribution of the BP response in $\tau _{R,2}^\mathrm{^{\prime}}$ even surpasses that of the Lorentzian response for certain pulse durations. This is due to the fact that as the pulse duration increases, the low-frequency Raman gain gradually contributes more to the Raman shock time. When the pulse duration increases to a certain extent (70-260 fs), the BP response [resonance at ∼2.5 THz in Raman spectral, see Fig. 1(b)] dominates in the low-frequency domain. Meanwhile, when the pulse continues to increase (greater than 260 fs), the Lorentzian response contribution in the lower frequency domain (less than 1 THz) dominates again. On the other hand, when the pulse duration is especially short (less than 100 fs), we need to reexamine the contribution of Lorentzian response (denoted as ${\tau _{R,L}}$ as the violet dashed curve) to Raman shock time through Eq. (5), due to the invalidation of the approximation in Eq. (6). From the calculation [the violet dashed curve in Fig. 2(a)], one may find that the Lorentzian response will dominate when the pulse duration is less than 70 fs. Intriguingly, we notice that ${\tau _{R,L}}$ will reach a maximum around 23 fs, which can be attributed to the resonant frequency ∼12.5 THz of Lorentzian response. In addition, the brown dashed curve in Fig. 2(a) shows the Raman shock time $\tau _{R,2}^{\mathrm{^{\prime\prime}}}$ directly computed from Eq. (5), which indicates that Eq. (22) [the red solid curve in Fig. 2(a)] is a well approach to the Raman shock time given by modified Raman response function when the pulse duration ${\tau _s}\textrm{} > \textrm{}100\;\textrm{fs}$.

To gain an intuitive comparison of the amount of the Raman SSFS with (the red solid curve) and without (the black one) a consideration of the BP response, we examine Eqs. (12) and (21) using the different pulse widths and respectively plot the corresponding results in Fig. 2(b). As one can see, the SSFS in the presence of the BP is significantly larger than the one in the absence of the BP. The contribution from the BP response (the green dash-dotted line) to the SSFS is also drawn in Fig. 2(b) and, intriguingly, one can see that this contribution is generally larger than that of the Lorentzian response (the blue dash-dotted line) to the SSFS. In Fig. 2, we further compare the analytical values with the numerical ones (the brown dashed curve) by performing the integral of Eq. (5) with the use of Eq. (13). Despite a little discrepancy, we can conclude that Eq. (21) gives a good approximation for the Raman SSFS taking into account the generalized response function in Eq. (13).

3. Numerical simulations

To show the exact SSFS under the effect of BP response, we perform the numerical simulations based on Eqs. (1) and (13) and with the parameters of real silica microresonators. For this purpose we introduce the mode-coupled equations [29] to simulate Eq. (1) and further employ the fast-Fourier-transform method [34] to efficiently deal with the Kerr term. Meanwhile, we process the Raman term as $i{f_R}g{F^{ - 1}}\{{F[{{h_R}(\tau )} ]\times F[{A{{({t,\tau } )}^2}} ]} \}A({t,\tau } )$, where the symbols F[…] and F^-1[…] represent the Fourier transform and inverse Fourier transform, respectively. In the numerical simulations, we set the wavelength of the pump laser to be 1550 nm and the intrinsic and external decay rates to be ${\gamma _0} = 2\mathrm{\pi } \times 2\;\textrm{MHz}$ and $\kappa = 2\mathrm{\pi } \times 2\;\textrm{MHz}$, respectively. Then, we take the refractive and Kerr nonlinear indexes of the silica to be ${n_0} = 1.45$ and ${n_2} = 2.2 \times {10^{ - 20}}\textrm{}\;{\textrm{m}^2}/\textrm{W}$, respectively, as well as setting the FSR to be ${D_1} = 2\mathrm{\pi } \times 22\;\textrm{GHz}$. Here, we consider two different resonators with the group velocity dispersions of ${D_{2,1}} = 2\mathrm{\pi } \times 4\;\textrm{kHz}$ (resonator #1) and ${D_{2,2}} = 2\mathrm{\pi } \times 17\;\textrm{kHz}$ (resonator #2), respectively. In addition, the effective volumes of the optical modes are estimated as ${V_{eff}} \approx 5.65 \times {10^{ - 4}}\;\textrm{}m{m^3}$ from the 3-mm-diameter microdisks, which give the coupling coefficients of the Kerr nonlinearity at $g = 6.75 \times {10^{15}}\; \textrm{Hz}/\textrm{J}$ [29]. These parameters are obtained based on previous works [11,12]. Finally, to trigger the Kerr solitons, we seed the system with the Gaussian pulses [30].

The soliton spectrum of resonator #1 obtained from numerical simulation is displayed in Fig. 3(a). There, we set the pump-cavity detuning to $\Delta \omega = - 2\pi \times 7.5\;\textrm{MHz}$, to have the corresponding Raman self-frequency shift ∼−97.81 GHz. In Figs. 3(b) and 3(d), we depict the numerically acquired Raman SSFSs (the red dots) against the fitted pulse width in the two silica resonators. We can find that the SSFSs grow with $1/\tau _s^4$, in good agreement with the tendency given by Eq. (21). In addition, one can see that the SSFSs of resonator #1 are much larger than that of resonator #2. This is due to the smaller group velocity dispersion of resonator #1 than that of resonator #2, resulting in the narrower pulse widths of the solitons generated in resonator #1. Correspondingly, the calculated Raman shock times are ∼3.1 fs for ∼120 fs of pulse width [see Fig. 3(c)] will decrease to ∼2.4 fs for ∼300 fs pulse width [see Fig. 3(e)], and it is in agreement with the previously measured results [11,21,33]. Furthermore, based on numerical simulations, we can calculate that the BP contributes approximately 1.7 fs and 1.1 fs to the Raman shock time in Fig. 3(c) and Fig. 3(e), respectively. The ratios of the BP’s contribution to Raman shock time are consistent with that in Fig. 2.

 

Fig. 3. (a) The soliton spectrum obtained from numerical simulation in resonator 1#, and the insert is the corresponding temporal waveform. (b) and (d), soliton self-frequency shift ${\mathrm{\Omega }_{Raman}}$ plotted versus $1/\tau _s^4$. The black solid and dashed lines are the analytical results of the Raman SSFSs calculated from the model including [Eq. (21)] and not including [Eq. (12)] the BP, respectively. The red dots indicate the numerically simulated results. Moreover, the blue stars are the values with the BP and cw background light (BL) obtained from Eq. (5), while the green squares represent the theoretical values acquired with Eq. (23). (c) and (e), the obtained Raman shock time against the pulse width. The red dots are the values acquired from the simulations (with $1/\tau _s^4$ fitting), and the black curve depicts the analytical values of $\tau _{R,2}^\mathrm{^{\prime}}$, in addition to the blue stars as the theoretical results considering background fields. (b) and (c) are the results of resonator #1, and (d) and (e) are about resonator #2. The width of the soliton pulse varies with changes in the pump-cavity detuning Δω (from -2π×5.2 MHz to -2π×7.5 MHz), and the corresponding pump power (80 mW-130 mW) also changes accordingly.

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The analytical predictions in Figs. 3(b) and 3(d) (the black solid lines) do not exactly fit the values of the numerically simulated SSFSs (the red dots). Correspondingly, the analytical Raman shock times [the black curves in Figs. 3(c) and 3(e)] are also deviating from the results obtained by $1/\tau _s^4$ fitting of the simulated SSFSs [the red dots in Figs. 3(c) and 3(e)]. A main reason is that the condition $B \gg {A_0}$ is not usually well satisfied under the actual parameters so that Eq. (10) becomes invalid. Therefore, we need to improve Eq. (21) to

 

Fig. 4. (a) An example of the soliton temporal profile in resonator #1. The red (black) curve presents the intracavity power of the soliton field with (without) consideration of its interference with the cw background. (b) For different pulse widths ${\tau _s}$ in resonator #1, we fit the intracavity peak powers of the solitons without (${P_{sol}}$) and with ($P_{sol}^\mathrm{^{\prime}}$) considering the interference with the cw background, as well as the powers of corresponding background fields (${P_0}$) inside the resonator.

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To better understand the significance and importance of our work, we here provide an example to compare our simulations with the previous experimental results [11]. According to Ref. [11], the Raman shock time and Raman SSFS of a soliton pulse with a pulse width of 150 fs are ∼2.4 fs and 0.37 THz in the experiment, respectively. Based on Eqs. (1) and (13) (the numerical calculations are based on the parameters derived from Ref. [11]), we generate a soliton with the pulse width of 150 fs and find its corresponding Raman shock time and Raman SSFS are ∼2.8 fs and 0.43 THz, respectively, which match well with the experimental results in Ref. [11]. Furthermore, these simulated results are also consistent with the theoretical ones considering the background field, which are computed with Eq. (5). However, as for the model without including the effect of the BP, its correspondingly simulated Raman shock time and Raman SSFS are respectively ∼1.3 fs and 0.20 THz, which considerably deviate from the actual experimental data in Ref. [11]. In addition, we conclude that, in Ref. [11], the BP contributes approximately 1.4 fs and 0.21 THz (58% of the total Raman response) to the Raman shock time and Raman SSFS, respectively, when the soliton has a pulse width of 150 fs. All these results validate the accuracy of our model for the prediction of the Raman SSFS in silica microresonators.

4. Conclusion

In conclusion, we have examined the SSFS of the dissipative Kerr soliton in the presence of the BP in the Raman-gain spectrum of a silica resonator. It is surprisingly found that the contribution from the BP response to the SSFS is non-negligible and even larger than the Lorentzian response part of the Raman effect for certain pulse durations. We also point out that the cw background field will attenuate the peak power of the soliton through an interference, leading to a decrease of the SSFS. One prospect of our study is the modification of the theoretical limitations on the span or efficiency of the dissipative Kerr soliton combs. Although our model is based on silica resonators, the theory is expected to be relevant to other amorphous material platforms such as high-index doped silica [35] or Tantalum oxide [36,37].