Robust Pulse-Pumped Quadratic Soliton Assisted by Third-Order Nonlinearity

Abstract

The generation of a quadratic soliton in a pulse-pumped microresonator has attracted significant interest in recent years. The strong second-order nonlinearity and high peak power of pumps offer a straightforward way to increase efficiency. In this case, the influence of the third-order nonlinearity effect becomes significant and cannot be ignored. In this paper, we study the quadratic soliton in a degenerate optical parametric oscillator driven synchronously by the pulse pump with third-order nonlinearity. Our simulations verify that the robustness of quadratic soliton generation is enhanced when the system experiences a perturbation from pump power, cavity detuning, and pump pulse width. These results represent a new way of manipulating frequency comb in resonant microphotonic structures.

1. Introduction

Optical frequency combs based on microresonators represent a fast-growing research field that has developed in the last decade, as they can ultimately reduce the size, weight, and power consumption of conventional frequency comb systems [1,2,3]. They are routinely used in a wide range of scientific and technological applications, such as high-speed optical communication [4,5], time-frequency metrology [6,7,8], astronomical spectral calibration [9,10], and spectrometry [11,12,13]. In the past decade, Kerr-based frequency combs where a ${\chi }^{\left(3\right)}$ nonlinear resonator is coherently driven by a continuous-wave (cw) laser to excite temporal solitons has been extensively studied [14,15]. However, due to the relatively low strength of third-order nonlinearity, the generation of Kerr combs requires high pump power and the comb forms around the cw pump wavelength. In addition, the pump frequency must be close to the zero-dispersion point for the ideal phase-matching of effective four-wave mixing (FWM), which limits the wavelength range of the comb. These issues can be improved by operating with quadratic nonlinearity, which potentially offer lower pump power thresholds [16] and permit the direct generation of combs in spectral regions where the generation of conventional Kerr combs is difficult to achieve [17,18]. Another route to alleviate these issues is to use a pulse-pump driven arrangement [19,20,21,22,23,24,25,26]. The soliton can be generated by a periodic pulse pump instead of the continuous pump with higher pump-to-soliton conversion efficiency due to a significantly higher peak power [19,20,21,25,27,28]. Thus, soliton generation in the pulse-pumped quadratic nonlinear resonator is a viable route to realize highly efficient and widely tunable broadband frequency combs.

Recently, walk-off-induced temporal soliton in a degenerate optical parametric oscillator (DOPO) driven synchronously by the pulse pump with a temporal width of several picosecond based on pure quadratic nonlinearity has been demonstrated [26]. However, in general, quadratic nonlinear resonators also contain third-order nonlinearity. Compared with the second-order system, the participation of third-order nonlinearity can affect the comb-mode-locking behaviors by self and cross phase modulation (SPM and XPM), resulting in complex dynamics that are far from well-understood. In this paper, we numerically investigate the quadratic soliton generation in the DOPO driven synchronously by the pulse pump with the participation of the ${\chi }^{\left(3\right)}$ nonlinear effect. Compared to the pure ${\chi }^{\left(2\right)}$ system, the robustness of quadratic soliton generation is enhanced, benefiting from the ${\chi }^{\left(3\right)}$ nonlinear effect when the system experiences a perturbation from pump power, cavity detuning and pump pulse width. Our results provide a new way to control the quadratic soliton.

2. Theory and Simulation Results

Figure 1 shows a schematic of the DOPO system, which is driven synchronously by a pulse with a temporal width of several picoseconds. The nonlinear parametric interactions between pump at fundamental frequency and signal at half-harmonic frequency take place in a periodically poled lithium niobite (PPLN) waveguide. The cavity is composed of polarization-maintaining fibers and free-space to ensure that the pump repetition rate is approximately close to multiples of the cavity free spectral range. We consider slowly varying electric field envelopes A and B with their carrier frequencies ${\omega }_0$ and $2{\omega }_0$, respectively [29,30,31], which obey the coupled equations in the DOPO with both quadratic and cubic nonlinearity [32]:

$$\begin{array}{c}\hfill \frac{\partial A}{\partial z}=\left(-\frac{{\alpha }_s}{2}-i\frac{k_s^{{}^{″}}}{2}\frac{{\partial }^2}{\partial {\tau }^2}\right)A+i\kappa B{A}^{*}{e}^{-i\Delta kz}+i{\gamma }_1{\left|A\right|}^{2}A+i2{\gamma }_{12}{\left|B\right|}^{2}A,\end{array}$$

(1)

$$\begin{array}{c}\hfill \frac{\partial B}{\partial z}=\left[-\frac{{\alpha }_p}{2}-\Delta k^{{}^{\prime }}\frac{\partial }{\partial \tau }-i\frac{k_p^{{}^{″}}}{2}\frac{{\partial }^2}{\partial {\tau }^2}\right]B+i\kappa A^2{e}^{i\Delta kz}+i{\gamma }_2{\left|B\right|}^{2}B+i2{\gamma }_{21}{\left|A\right|}^{2}B.\end{array}$$

(2)

The dynamics of the signal outside the waveguide are obtained as follows (the pump is non-resonant, no feedback contribution) [26]:

$$\begin{array}{c}\hfill A_{m+1}\left(0,\tau \right)={\Gamma }^{-1}\left[\frac{1}{\sqrt{G_0}}{e}^{-i\phi }\Gamma \left[A_m\left(L,\tau \right)\right]\right],\end{array}$$

(3)

where A and B are the signal and pump field envelope, respectively, z is the longitudinal coordinate, ${\alpha }_{s,p}$ is the propagation losses, $\Delta k$ is the wave–vector mismatch, $\Delta k^{{}^{\prime }}$ is the group velocity mismatch, $k_{s,p}^{{}^{″}}$ is the group-velocity dispersion (GVD) coefficient, $\tau$ is a fast time variable in a reference frame moving at the group velocity of the signal field frequency ${\omega }_0$, L is the length of the PPLN waveguide, m is an integer representing the mth roundtrip. $(1-G_0^{-1})$ is the roundtrip cavity loss (including the out-coupling loss and propagation loss). The coupling phase shift is $\phi =\Delta \phi +l\omega {\lambda }_s/2c+{\phi }_2{\omega }^2/2$, where $\Delta \phi =\pi l$ is the cavity detuning, with parameter l changed in integers results in the cavity being detuned by ${\lambda }_s/2$, ${\phi }_2$ represents the cavity GVD contribution. $\Gamma$ and ${\Gamma }^{-1}$ represents= Fourier transform and inverse Fourier transform. ${\gamma }_1$ and ${\gamma }_2$ are the nonlinear coefficients of self-phase modulation, ${\gamma }_{12}$ and ${\gamma }_{21}$ are the cross-phase modulation coefficients and ${\gamma }_1={\gamma }_{12}=2\pi n_3/{\lambda }_s{A}_{eff}$, ${\gamma }_2={\gamma }_{21}=2\pi n_3/{\lambda }_p{A}_{eff}$, where $n_3$ is the nonlinear index, ${\lambda }_{s,p}$ is the wavelength of signal and pump field, $A_{eff}$ is the effective mode area. The normalized second-order nonlinear coupling coefficient is $\kappa =\sqrt{2}{\omega }_0{d}_{eff}/(A_{eff}\sqrt{c^3{n}_s^2{n}_p{ϵ}_0})$, where $d_{eff}$ is the effective second-order nonlinear coefficient, c is the speed of light, ${ϵ}_0$ is the vacuum permittivity and $n_{s,p}$ are the linear refractive indices. High-order dispersion and nonlinearity are neglected for simplicity and we assume there is a flat-top pump profile.

We solve the coupled-wave equations, Equations (1) and (2), by using the split-step Fourier method and fourth-order Runge-Kutta method with the following physical parameters: $L=40$ mm, ${\alpha }_s$ = 30 dB/m, ${\alpha }_p$ = 30 dB/m, $k_s^{{}^{″}}$ = 112 fs${}^2$/mm, $k_p^{{}^{″}}$ = 406 fs${}^2$/mm, $\kappa$ = 17 W${}^{-1/2}$ m${}^{-1}$, $\Delta k^{{}^{\prime }}$= 329 ps/m, $G_0$ = 7.9, ${\phi }_2$ = −21.6 fs${}^2$/mm, and the width of pump pulse $t_p$ = 14 ps. Our simulations, with noise, are iterated for two thousand roundtrips, until a stable soliton is formed, which is similar to the method commonly used for the excitation of cavity soliton in Kerr resonators [33]. For comparison, we first consider the case of pure ${\chi }^{\left(2\right)}$ nonlinearity by setting ${\gamma }_1={\gamma }_2={\gamma }_{12}={\gamma }_{21}=0$ with the quasi-phase-matched condition $\Delta k=0$. As the pump energy continues to enter the cavity, the OPO oscillation threshold occurs when the parametric gain overcomes the loss, and the quadratic soliton is formed. Figure 2a–c show the dynamics of the soliton generation under pure ${\chi }^{\left(2\right)}$ nonlinear interaction for the different peak powers of the pump. Figure 2a shows the evolution of the intracavity power of signal field A and there is a stable state when the pump peak power is 14 W and 15 W. The temporal envelopes and frequency spectra of the last round of the intracavity signal field A are shown in Figure 2b and c, respectively. For the stable state, the corresponding temporal envelopes are single soliton and the spectra are smooth sech${}^2$ profiles. The inner cavity power becomes unstable when the pump power increases to 16 W. The quadratic soliton with a pulse pump is formed under the condition of timing balance, which is determined by the collective interplay between linear cavity detuning, gain saturation and gain clipping. Higher pump power will induce increases temporal width of the gain-clipping region, which breaks this timing balance and results in an unstable state.

Then, we add third-order nonlinearity ${\chi }^{\left(3\right)}$ by setting ${\gamma }_1={\gamma }_{12}$ = 0.66 W${}^{-1}$m${}^{-1}$, ${\gamma }_2={\gamma }_{21}$ = 1.32 W${}^{-1}$m${}^{-1}$ to simulate the waveform inner cavity evolutionary processes using the same parameters mentioned. Figure 2d–f show the evolution of the inner cavity power and the corresponding time and frequency domain profiles, respectively. The phase modulation from ${\chi }^{\left(3\right)}$ nonlinearity influences the ${\chi }^{\left(2\right)}$ nonlinear conversion efficiency, which leads to a higher OPO threshold. The quadratic soliton is also formed for a higher pump peak power of 16 W because the gain saturation and clippings are alleviated and the timing balance is maintained. It is obvious that ${\chi }^{\left(3\right)}$ nonlinearity can make the quadratic soliton more stable and enhance the robustness when it experience the pump power fluctuation, although the ${\chi }^{\left(3\right)}$ effect reduces the OPO conversion efficiency.

Cavity detuning is an important factor affecting the generation of quadratic soliton, because it causes a timing mismatch from exact synchrony with the pump repetition rate. The drift induced by large detuning is counterbalanced by the combined effect of gain saturation and gain-clipping. Figure 3a shows the evolution of the intracavity power of the signal field A with pure ${\chi }^{\left(2\right)}$ nonlinear interaction for different detuning parameter l. The larger cavity detuning is, the more unstable intracavity energy is. For $l=10$, the spectrum profile has two-peaks and the relative phases between comb lines are not a constant, which means the mode-locking state is not reached, as shown in Figure 3c. By setting ${\gamma }_1={\gamma }_{12}$ = 0.66 W${}^{-1}$m${}^{-1}$, ${\gamma }_2={\gamma }_{21}$ = 1.32 W${}^{-1}$m${}^{-1}$, the intracavity power becomes stable under the combined action of the ${\chi }^{\left(2\right)}$ and ${\chi }^{\left(3\right)}$ nonlinearity, even the cavity detuning l increases to 10 (Figure 3b). The spectrum also maintains a smooth sech${}^2$ profile and the relative phases are constant, as shown in Figure 3d. This phenomenon shows that the ${\chi }^{\left(3\right)}$ effect can not only modulate the relative phase of OPO comb lines, but also alleviate the drift caused by cavity detuning, which will benefit the timing balance situation. In practice, cavity detuning is easily interfered with by various factors; the ${\chi }^{\left(3\right)}$ effect will help the system to maintain the stable pulse output in the case of large cavity detuning, which reduces the difficulty of experimental operation.

In addition, the generation of the quadratic soliton is sensitive to the pump pulse width due to the walk-off effect. If the pump pulse width is smaller, part of the crystal is not utilized and only provides dispersion. For the larger pulse width, multiple solitons will be generated. As shown in Figure 4a,b, a single soliton and a smooth sech${}^2$ spectrum can be obtained for a pump pulse-width of 12 ps. The generated soliton broadens and the corresponding spectrum has sidebands with increased pump pulse width. However, by setting ${\gamma }_1={\gamma }_{12}$ = 0.66 W${}^{-1}$m${}^{-1}$, ${\gamma }_2={\gamma }_{21}$ = 1.32 W${}^{-1}$m${}^{-1}$, the single soliton and smooth sech${}^2$ spectrum can be generated, even when the pump puls -width increases to 14 ps, as shown in Figure 4c,d. This indicates that the ${\chi }^{\left(3\right)}$ effect alleviates the sensitivity to the pump pulse width of quadratic soliton, which reduces the requirement of experimental conditions.

To illustrate the role played by the value of ${\chi }^{\left(3\right)}$ nonlinearity, we investigate the intracavity dynamics with varied ${\chi }^{\left(3\right)}$. Figure 5 shows the temporal and spectral profiles with ${\chi }^{\left(3\right)}$ coefficient ${\gamma }_1$ increases from 0 to 0.5 W${}^{-1}$m${}^{-1}$ for the pump pulse-width of 16 ps. As shown in Figure 5a, the output pulse changes from a two peaks profile to single soliton as the ${\chi }^{\left(3\right)}$ increases. The corresponding frequency spectra (Figure 5b) exhibit variations from the two peaks profile to smooth the sech${}^2$ profile. It is obvious that ${\chi }^{\left(3\right)}$ nonlinearity helps to realize the mode-locking state compare to the pure ${\chi }^{\left(2\right)}$ case.

3. Discussion and Summary

In this paper, we theoretically study quadratic soliton generation in DOPO, driven synchronously by the pulse pump with participation of the ${\chi }^{\left(3\right)}$ nonlinear effect.For applications of the frequency comb, it is key to realize and maintain the mode locking between comb lines, namely single soliton generation. In our work, the strong second-order nonlinearity and high peak power of the pump offer a straightforward way to increase the efficiency of comb generation.In this case, the influence of the third-order nonlinearity effect becomes significant and cannot be ignored. In general, third-order nonlinearity is considered as a disadvantage factor to the quadratic soliton. We show for the first time that third-order nonlinearity can enhance the robustness of quadratic soliton generation when its synchronization experiences perturbations from pump energy, cavity detuning and pump pulse width.

Third-order nonlinearity can help in maintaining the stable single quadratic soliton, although it will affect the efficiency through phase matching. In practice, stability is essential for the comb system when it comes to application. The presence of third order nonlinearity can enhance the stability of this system and relax the fabrication requirements, paving the way to compact system integration. The microcomb will be used as a seed source and its output power can be increased by using the optical amplifier. We expect that our results will be useful for understanding the dynamics of nonlinear processes in pulse-pumped quadratic resonators and in the application of on-chip quadratic solitons integrated with the nanophotonic platform [34,35]. In addition, combined ${\chi }^{\left(2\right)}$ and ${\chi }^{\left(3\right)}$ effects provide a unique opportunity for quadratic soliton generation at the mid-infrared spectral range that may lead to the significant enhancement of the precision measurements and optical signal processing.