1. Introduction

Nonlinear and quantum optics in ring microresonators have been attracting a great deal of attention over the past decades [1]. In particular, the frequency comb generation and soliton photonics in microresonators are reaching unprecedented levels of practical relevance [2]. The most ubiquitous microresonator solitons remain the dissipative version of the Kerr solitons, which, thanks to the small losses of modern microresonators, remain well approximated by the fundamental bright solitons of the Nonlinear Schrödinger (NLS) equation [2–4].

The second-order, $\chi ^{(2)}$, nonlinearity has always been a viable alternative to the Kerr one. The $\chi ^{(2)}$ microresonators have the potential to extend comb generation to new wavelengths, elevate dispersion constraints, and reduce power requirements [5]. Using of the $\chi ^{(2)}$ response comes, however, with the caveats of the need to care about the phase and group velocity matching to take full advantage of it. One of the recent highlights in this area was the result by Bruch and colleagues [6] demonstrating the solitons due to parametric down-conversion in an integrated $\chi ^{(2)}$ microresonator. A sufficiently comprehensive list of references on the solitons in $\chi ^{(2)}$ resonators can be found in [7]. For further recent and historic theoretical contributions on the bright and dark solitons due to parametric-down conversion see, e.g., Refs. [8–17].

While Ref. [6] has used, or perhaps even coined, the term ’Pockels solitons’, it is also known that the impact of the $\chi ^{(2)}$ nonlinearity on the refractive index change depends on how well the index matching between the pump and either half- or second-harmonic is arranged. In the mismatched limit, the $\chi ^{(2)}$ susceptibility starts mimicking the Kerr effect, which could be called the cascaded-Kerr nonlinearity, see, e.g., [7]. Thus, the transition between the Pockels and the cascaded-Kerr solitons is a problem that requires further considerations. Our focus here is on the bright soliton pulses. It is known that, in the phase-mismatched cascaded-Kerr limit, the $\chi ^{(2)}$ microresonator model can be reduced to the parametric NLS equation [8,9], which has the sech-soliton solution [8,9,17–20]. One outstanding problem would be finding the solution for the bright parametric solitons in the practically important case of the phase-matched resonators.

Any intense intra-resonator waveform produces the nonlinear shift of the resonance frequencies. For the Kerr or Kerr-like solitons, this shift is naturally expected to be proportional to the power. This work aims to demonstrate that, in the phase-matched microresonator parametric-down conversion, there exists an explicit bright soliton solution producing the nonlinear resonance shift proportional to the field amplitude, i.e., like should be expected in the optical Pockels effect. The spatial profile of this soliton is given by the sech-squared function, unlike the sech-profile of the Kerr soliton. We note before proceeding, that the optical Pockels effect discussed below should be distinguished from the voltage controlled one, which is broadly used in Pockels cells, and as a frequency comb generation tool [21,22].

2. Model

Our model of the multi-mode half-harmonic generation in a microresonator follows the approach of Ref. [7]. The fundamental (pump) field, ’f’, is spectrally centred around the pump laser frequency, $\omega _p$, and the half-harmonic field, ’h’, is around $\omega _p/2$. The fields and detunings are defined as

Dispersion engineering is important for any type of modelocking, including the soliton modelocking. We set $D_{1\textrm {f}, 1\textrm {h}}/2\pi$ as the repetition-rate (free-spectral-ranges, FSR), and $D_{2\textrm {f}, 2\textrm {h}}$ as the dispersion parameters having units of Hz [7]. The FSR difference, $D_{1\textrm {f}}-D_{1\textrm {h}}$, scales inversely with $R$, while $D_{2\textrm {f}, 2\textrm {h}}\sim 1/R^{2}$ [23]. Therefore going from the mm to 10’s of micron radii leads to the relative reduction of the impact of the FSR difference on the dynamics, $|D_{1\textrm {f}}-D_{1\textrm {h}}|/|D_{2\textrm {f}}|\sim R$. The group-velocity matching can be implemented either across the zero-dispersion point for the modes of the same family or, for the same signs of $D_{2\textrm {f},2\textrm {h}}$, using the different mode families and the avoided mode-crossings, see, e.g., [24–27]. Applications of the quasi-phase matching [28,29] and optical fibres [30–32] offer further avenues for dispersion engineering in $\chi ^{(2)}$ resonators. In order to derive transparent analytical results for Pockels solitons, we consider the case of the FSR, i.e., group velocity, matching, $D_{1\textrm {f}}=D_{1\textrm {h}}=D_1$, for the same dispersion signs, i.e., $D_{2\textrm {f}}/D_{2\textrm {h}}>0$. The case of the opposite dispersion signs, $D_{2\textrm {f}}/D_{2\textrm {h}}<0$, is also proceedable analytically, but for the reasons of brevity we prefer to keep this work focused on the former.

Transforming to the rotating frame of reference, $\theta =\vartheta -D_1 t$, the equations for $\psi _{\textrm {f},\textrm {h}}$ are [7]

While the results derived below could be a guideline for a range of devices, our choice of parameters for the numerical estimates is geared towards the integrated resonators as used in Ref. [6]. Namely, we assume $\kappa _{\textrm {f},\textrm {h}}/2\pi =100$MHz, $D_{1\textrm {f},1\textrm {h}}/2\pi =300$GHz, $D_{2\textrm {f},2\textrm {h}}/2\pi =30$MHz, $\gamma _{\textrm {f},\textrm {h}}/2\pi =500$MHz/$\sqrt {\textrm {W}}$, and $\eta =1/2$. The linewidth here is two orders of magnitudes larger than in the high-Q bulk resonator samples [1], this explains relatively high pump power, ${{\cal W}}=80$mW, required to generate solitons in Ref. [6]. The sech-squared soliton derived below fits well into the above range of the parameter values, while the sech one requires even higher input powers ${{\cal W}}\sim 1$W. This is because the Pockels nonlinear response is achieved under the phase matching conditions, i.e., $|\varepsilon |/\kappa _{\textrm {h}}\sim 1$, while the sech-soliton is associated with the cascaded-Kerr nonlinear response triggered for $|\varepsilon |/\kappa _{\textrm {h}}\sim 10^{2}$. The large $|\varepsilon |$ lead to the inefficient conversion and push the soliton thresholds up.

3. Sech-squared soliton: Pockels regime

To find the Pockels soliton we first neglect the loss terms by assuming $|\delta |\gg \kappa _{\textrm {h}}$, $|2\delta -\varepsilon |\gg \kappa _{\textrm {f}}$. Practically, it suffices to take $|\delta |/\kappa _{\textrm {h}}\sim 10$ and $|\varepsilon |/\kappa _{\textrm {h}}\lesssim 1$. Then, Eq. (3) become

The soliton solution is sought in the form

We substitute Eq. (6) in Eq. (5), and separate the real and imaginary parts. The first outcomes of this procedure are the explicit expressions for the soliton background

The further result of the substitution is a pair of the differential equations for $\psi$,

$\psi$ in Eq. (9) must simultaneously solve Eqs. (8b) and (8a), which requires acknowledging the following conditions

We first look into how the peak power of the sech-squared soliton scales with the detuning by taking the important practical case of the exact phase matching, $\varepsilon =0$. Fixing dispersions to comply with Eq. (10), $D_{2\textrm {f}}/D_{2\textrm {h}}=2$, is a soft assumption to make for the sake of dealing with a transparent analytical solution. For $\delta /\kappa _{\textrm {h}} =10$, the peak soliton power in the fundamental and half-harmonic fields are

The total power and the spectrum of the half-harmonic component of the sech-squared Pockels soliton are

 

Fig. 1. The normalized power spectra for the Pockels (sech-squared) and cascaded-Kerr (sech) soliton solutions vs the normalised mode number, $m=\mu /\mu _0$. The full line is $\textrm {log}\left (m^{2}/\sinh ^{2}m\right )$, see Eq. (13), and the dashed one is $\textrm {log}\left (\textrm {sech}^{2}m\right )$, see Eq. (25).

Download Full Size | PDF

4. Limits of existence of the sech-squared solitons

Accepting a small level of complication of the algebra as a necessity lets us to trace the effects of the so far neglected loss terms. This needs to be limited by the half-harmonic loss, $\kappa _{\textrm {h}}$, which, however, reveals the net effect well, and allows to remain within the comfortably transparent analytical considerations. Equation (5) is now replaced with

Equations (6), (14), (15) lead to the refreshed system for $\psi$

The soliton solution of Eq. (16b) is then,

 

Fig. 2. The normalised total power of the half-harmonic soliton vs the normalized detuning, $\delta /\kappa _{\textrm {h}}$. (a) is the sech-squared soliton case. The red line shows ${{\cal P}}/{{\cal P}}_\textrm {Pock}=|\delta /\kappa _{\textrm {h}}|^{3/2}$, ${{\cal P}}_\textrm {Pock}\approx 0.2$W, $\varepsilon /2\pi =0$. The blue lines show the effect of the losses, see Section 4. The plus and minus branches correspond to the respective signs in Eq. (18). (b) is the sech soliton case. The red line shows ${{\cal P}}/{{\cal P}}_\textrm {cKerr}=|\delta /\kappa _{\textrm {h}}|^{1/2}$, ${{\cal P}}_\textrm {cKerr}\approx 6$W, $\varepsilon /2\pi =-10$GHz. The plus and minus branches correspond to the respective signs in the in-line $\bar \delta$ equation in the first sentence of Section 6.

Download Full Size | PDF

5. Sech soliton: cascaded-Kerr regime

The parametric sech-solitons in resonators [8,17] are like the ones previously found in a few other physical contexts, see, e.g., Refs. [17–20]. They can be derived via the reduction of Eq. (3) to the parametric NLS equation. The reduction starts from the assumption that the resonator is pumped well away from the resonance, $|\varepsilon |\gg |\delta |$, while the half-harmonic is generated close to the resonance. This practically inefficient phase-mismatched arrangement allows to approximately resolve Eq. (3b) with

The total power and spectrum for the cascaded-Kerr solitons are

6. Pump power threshold: Sech-squared vs sech solitons

Retaining the $\kappa _{\textrm {h}}$-loss term in Eq. (21), leads to the substitution $\delta \to \bar \delta =\delta \pm \sqrt {\kappa _{\textrm {f}}^{2}\gamma _{\textrm {h}}^{2}{{\cal H}}^{2}/\kappa _{\textrm {h}}^{2}\varepsilon ^{2}-1}$ in Eq. (22). Plots of the soliton power, that account (blue) and disregard (red) $\kappa _{\textrm {h}}$, are shown in Fig. 2(b). Because of the dominance of the index-matching parameters $\varepsilon$ in this regime, the square-root is independent from $\delta$. The existence of the cascaded-Kerr soliton requires ${{\cal H}}>{{\cal H}}_\textrm {cKerr}=|\varepsilon |\kappa _{\textrm {h}}/\gamma _{\textrm {h}}\kappa _{\textrm {f}}$, i.e., the laser power is

If $\varepsilon =0$, or $|\varepsilon |\ll |\delta |$, then the Pockels soliton given by Eqs. (17), (18) exists for the much smaller threshold powers determined by the detuning,

7. Summary

We have elaborated the explicit sech-squared Pockels solitons describing the soliton regime of the phase and group velocity matched parametric down-conversion in microresonators with either normal or anomalous dispersion. We have compared these solitons with the cascaded-Kerr sech-solitons existing away from the phase-matching condition and revealed the differences in (i) the values of the threshold powers, see Eqs. (26), (27), (ii) scaling of the soliton power with the detuning parameter, see Fig. 2, and (iii) how the Pockels-soliton spectra deviate from the triangular spectra of the sech-solitons, see Fig. 1 and, cf., Eqs. (13) and (25). The sech and sech-squared shapes of the half-harmonic field are embedded, as the two limit cases, inside the wider and numerically accessible family of the bright soliton solutions, see, e.g., [6,8,9,14–16].