Parametric solitons in optical resonators

Kerr cavity solitons (CSs) are pulses that propagate unperturbed in an optical resonator. They were first observed in fiber resonators [1] and subsequently in microresonators [2], where they are often called dissipative Kerr solitons (DKSs). So far, the focus has been on CSs driven at their natural oscillation frequency, i.e. with a driving laser at the carrier frequency of the soliton. However, CSs can also be parametrically driven by a laser at twice their carrier frequency. This parametric Kerr cavity soliton (PCS) is solution of the parametrically driven nonlinear Schrödinger equation (PDNLSE) [3]-[5], which describes a singly-resonant optical parametric oscillator:\begin{equation*}{t_R}\frac{{\partial A}}{{\partial T}} = \left( { - \frac{{{\Lambda _e}}}{2} - i\frac{{{\beta _2}L}}{2}\frac{{{\partial ^2}}}{{\partial {\tau ^2}}} - i{\delta _0}} \right)A + \kappa {B_{in}}{L_1}{A^{\ast}} + i\gamma {L_2}|A{|^2}A\tag{1}\end{equation*}where A is the intracavity electric field envelope, Λ
e
is the effective loss, L is the total length of the resonator and $\underline {{\beta _2}}$ its average group velocity dispersion. T=nt
R
, where n is an integer, is a slow time and τ is a time reference traveling at the group velocity. δ
0
the phase detuning from the closest cavity resonance. κ and ɣ are the second-and third-order nonlinear parameters and B
in
is the driving amplitude. PCSs differ from CSs in their lack of homogeneous background, but also in their multiplicity as two different solitons of the same amplitude but opposite phase, may exist for the same set of parameters. To demonstrate the existence of this new soliton as well as its multiplicity, we built the experimental set-up depicted in Fig. 1a. It is a resonator made of three different fibers. A periodically poled fiber (PPF, L
1
= 30cm) and a single-mode fibre (SMF, L
2
= 21m) provide the second- and third-order nonlinearity, respectively. In addition, a short piece of erbium doped fiber (EDF, 50cm) is used for loss compensation [6]. Using 650-ps flat top pulses at 775 nm (P
p
=|B
in
|
²
=10W), we manage to excite a single PCS. Detailed temporal and spectral analysis (not shown), show the presence of a 5W, 3.6 ps long, background-less sech-shaped pulse circulating inside the cavity for δ0=0.03, in perfect agreement with Eq. (1). To demonstrate their multiplicity, we perform a coherent detection. We extend the pump pulses duration to 1 ns to generate several PCSs and imprint a phase modulation to resolve them individually on the oscilloscope [5]. Once PCSs are excited, we send half of the cavity output power $P_s^{out}$ to a fast photodiode for direct measurement (Fig. 1b) and make the other half beating with a local oscillator (Fig. 1c). The direct measurement clearly shows that the two solitons have the same amplitude. On the other hand, the result of the interference confirms that they have two different phases. Since there are only two distinct phases [4] and the probability of having one or the other is the same, PCSs can be used to generate random bits. As a proof of principle, we generate four PCSs and associate the result of the interference to a binary value. This leads to the generation of random numbers (Figs. 1d,e). Optical parametric oscillator have already been used for random bits generation [7] and Ising machines [8]. Yet, complex set-ups had to be used to force the bits localization. Our demonstration therefore opens up new applications for Kerr cavity solitons.