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## Parametrically driven Kerr cavity solitons - Nature Photonics

Abstract . Cavity solitons are optical pulses that propagate indefinitely in nonlinear resonators. They are attracting attention, both for their many potential applications and their connection to other fields of science. Cavity solitons differ from laser dissipative solitons in that they are coherently driven. So far the focus has been on driving Kerr solitons externally, at their carrier frequency, in which case a single stable localized solution exists for fixed parameters. Here we experimentally demonstrate Kerr cavity solitons driving at twice their carrier frequency, using an all-fibre optical parametric oscillator. In that configuration, called parametric driving, two backgroundless solitons of opposite phase may coexist. We harness this multiplicity to generate a string of random bits, thereby extending the pool of applications of Kerr cavity solitons to random number generators and Ising machines. Our results are in excellent agreement with a seminal amplitude equation, highlighting connections to hydrodynamic and mechanical systems, among others. You have full access to this article via your institution. Download PDF Download PDF Main . The spontaneous formation of patterns is encountered across many fields of science. Spatially extended nonlinear systems may be brought away from equilibrium, where spatiotemporal patterns emerge 1 . Examples include convection rolls in heated fluids 2 , vegetation patches in arid regions 3 and localized structures in vibrated layers of sand 4 . Such complex patterns can often be described by relatively simple reaction/diffusion equations that capture most of the nonlinear dynamics 1 . These so-called amplitude equations have been shown to be universal. Very different systems in terms of microscopical physical laws can, under some conditions, be governed by the same macroscopic equation, providing important connections between distinct fields of science. One such class of equations are the driven, damped, one-dimensional nonlinear Schr?dinger equations (NLSEs), which describe pattern formation in charge-density condensates, driven plasmas, surface waves and optical resonators among others (see refs. 5 , 6 , 7 and references therein). The conservative NLSE admits exact soliton solutions 8 and similar solitary waves can be found when dissipation and forcing are added. Two well-known configurations are external driving, also known as a.c.-driving in the context of plasma physics 5 , and parametric driving. In the former, the system is described by equation?(1), known as the LugiatoLefever equation in optics 9 , 10 . In the latter, the system is described by equation?(2), commonly called the parametrically driven, damped NLSE (or PDNLSE). The solitary waves of the PDNLSE have been observed in water tanks 11 and in pendulum lattices 12 , where the driving consists of periodically varying a parameter of the system at twice its oscillation frequency. In optics, they have been predicted to exist in amplifier chains 13 and resonators 14 , 15 , 16 , where the parametric driving term finds its origin in the nonlinear polarization. Such parametrically driven Kerr cavity solitons (PDCSs) constitute a subclass of optical dissipative solitons 17 along with temporal cavity solitons (CSs) 18 , 19 , 20 . CSs, which are solutions of equation?(1), have attracted a lot of attention over the past decade and have been shown to underpin the formation of ultra-coherent optical frequency combs in microresonators 21 , 22 . In that context, they are now commonly referred to as dissipative Kerr solitons (or DKSs) 23 . Both CSs and PDCSs are sech-shaped pulses phase locked to a driving laser, but there are important differences between the two (see Fig. 1 ). CSs are sustained by additive driving, which forms a homogeneous background next to the soliton. The soliton is phase locked to the background and a single attractor exists for fixed detuning and driving power 18 , 24 . PDCSs, on the other hand, are driven through phase-sensitive amplification, implemented, for instance, via four-wave mixing 14 , 16 or three-wave mixing 15 . The soliton lacks a homogeneous background and two stable, out-of-phase, solutions may coexist 14 , 15 , 25 , 26 . This multiplicity opens up several new avenues for soliton coalescence 27 , 28 as already demonstrated in hydrodynamics 11 . Moreover, stable optical pulses of opposite phases can be used to implement random number generators 29 and Ising machines 30 , 31 . Fig. 1: Illustration of the differences between externally driven and parametrically driven cavity solitons. a , CSs are solutions of the externally driven NLSE (equation?(1)), where u is the intracavity field envelope, T is a slow time, τ is a time reference travelling at the soliton group velocity and S is the intracavity driving field envelope. The forcing is provided through constructive interference with a driving laser at the soliton carrier frequency ω s . The coherent interference between the soliton and the driving laser results in the formation of dark pulses in the through port. In the frequency domain, the additive driving corresponds to providing energy to a single longitudinal mode (red dotted arrow). The other modes are sustained by parametric wave mixing (black dotted arrows) as the soliton propagates along the resonator. P out , output power; χ ( n ) , the n th-order susceptibility of the medium; ω , frequency. b , PDCSs are backgroundless sech-shaped solutions of the parametrically driven NLSE?(equation (2)), where μ is the driving field and u denotes the complex conjugate of u . They are sustained by the same parametric processes as CSs but in this case the driving itself is also parametric (red dotted arrows). The energy is provided directly from a driving laser at ω p to all the longitudinal modes composing the soliton through phase-sensitive amplification (PSA), which can be implemented in a χ (2) or χ (3) medium. Here we use degenerate parametric down conversion as the nonlinear driving process ( ω p ?=?2 ω s ). Full size image Despite recent interest in temporal patterns of optical parametric oscillators (OPOs) 32 , 33 and patterns excited through optical Faraday instabilities 34 , 35 , 36 , PDCSs have never been experimentally characterized. Soliton formation may have played a role in demonstrations of all-optical storage using phase-sensitive amplification 37 but no direct observation of solitons was reported. Here, using parametric down conversion as the driving process, we experimentally investigate PDCSs. We implement an all-fibre, singly resonant, degenerate OPO with both quadratic and cubic nonlinearities. We measure backgroundless sech-shaped optical waves at the signal frequency and show that solutions with different phases may coexist in the resonator. As a proof-of-principle experiment, we generate a short series of random numbers using PDCSs. Bifurcation analysis . We start by theoretically examining the dynamics of soliton formation in degenerate OPOs incorporating a Kerr section 15 . We use the dimensionless PDNLSE (equation?(2)). The derivation of the equation and its normalization are detailed in Supplementary Section 1 . In this equation, there are only two independent parameters: the detuning Δ and the driving strength μ . They determine the two-dimensional parameter space, plotted in Fig. 2a , where we show the different nonlinear attractors of the system. The degenerate OPO threshold is located at $$\mu =\sqrt{1+{{{\varDelta }}}^{2}}$$ and corresponds to a pitchfork bifurcation of the trivial state. For negative detunings, that bifurcation is supercritical and the trivial state is modulationally unstable above μ ?=?1 (ref. 32 ). The patterns emerging beyond this instability correspond to non-degenerate oscillation, which hence has a lower threshold than degenerate emission in that region. For positive detunings, the trivial solution is stable up to $$\mu =\sqrt{1+{{{\varDelta }}}^{2}}$$ and the pitchfork bifurcation is subcritical. An unstable homogeneous state emerges from the trivial solution and folds at the saddle-node bifurcation SN h located at μ ?=?1 (see Fig. 2b ). Beyond the fold, the upper branch is modulationally unstable, creating a region where a trivial solution and a modulated pattern coexist. In that region ( μ ?>?1), the PDNLSE admits exact solitary waves of the form $$u=\sqrt{2}\beta \,{\mathrm{sech}} (\beta \tau)\exp (i\phi )$$ where cos(2 ? )?=? μ ?1 and β 2 ?=? Δ ?+? μ ?sin(2 ? ) (refs. 7 , 15 , 25 , 38 ). There are two solitons of different amplitude and each can have one of two opposite phases. Both branches, defined as the soliton peak power, are shown in Fig. 2b as a function of the driving amplitude. They connect at the saddle-node bifurcation SN s ( μ ?=?1). The solutions corresponding to sin(2 ? )?>?0 are always unstable. These soliton branches are remininiscent of the ones describing CSs 39 . Conversely, when plotted as a function of the detuning, see Fig. 2c , both the homogeneous and soliton branches differ from those of CSs 40 . Unlike tilted resonances, the stable and saddle solutions do not connect, making the branches infinitely long. Along the main soliton branch, there are a couple of Hopf bifurcations. Between these bifurcations, the PDCSs are unstable and localized oscillatory behaviour as well as complex spatiotemporal dynamics can be found 7 . In what follows, we focus on the low detuning region where the formation of stable solitons is predicted. Fig. 2: Bifurcation structure of the PDNLSE. a , Phase diagram in the ( Δ , μ )-parameter space showing the main dynamical regions of the system. The bifurcation lines are the pitchfork bifurcation (PB ± ; green line), corresponding to the degenerate OPO threshold, and the Hopf bifurcation (HB; grey line). The black line at μ ?=?1 corresponds to the saddle-node bifurcation of both the non-trivial homogeneous state SN h and the soliton state SN s for Δ ?>?0, and to modulation instability (MI) for Δ ?diagram showing the soliton branches (red line) as well as the homogeneous states (black line) as a function of μ for Δ ?=?1.2. The solid lines correspond to stable states, the dashed lines correspond to homogeneously unstable states and the dotted line to modulationally unstable states. c , Bifurcation diagram as a function of Δ for μ ?=?1.37. The soliton develops breathing behaviour in-between the HBs (dash-dotted grey line). The colours and line descriptions are as in b . Full size image Experimental set-up . For our experimental investigation of the PDCS, we introduce an all-fibre degenerate OPO (see Fig. 3 ), with a signal oscillating around 1,550?nm. The cavity is composed of three main sections made of different fibres, each of a different length ( L n ): a periodically poled fibre (PPF; L 1 ?=?0.27?m) 41 , a standard single-mode fibre (SMF; L 2 ?=?21?m) and an erbium-doped fibre (EDF; L 3 ?=?0.52?m). The first two fibres provide, separately, the quadratic and cubic nonlinearities while the EDF is used to compensate the intracavity loss 42 . The 775?nm driving signal is generated by frequency doubling a highly coherent 1,550?nm laser, which can be phase- and amplitude-modulated, in a periodically poled lithium niobate (PPLN) crystal. It is sent into the cavity through a wavelength division multiplexer (WDM) and removed after the PPF. The total intracavity loss ( $$\Lambda\approx 40\%$$ ) is measured by removing the doped fibre $$\Lambda\approx 40\%$$ . The intracavity amplifier is pumped with 2?W at 1,480?nm, leading to a single-pass amplification g L 3 $$\approx35\%$$ where g is the fibre gain. The effective intracavity loss around 1,550 nm ( Λ e ?=? Λ ??? g L 3 ?=?5%) corresponds to a finesse of 122 ( Q -factor =?2.6?×?10 9 ). Note that this effective finesse is limited by gain dispersion in our experiment 42 . Increasing the amplification factor leads to lasing at shorter wavelengths (~1,548?nm), out of range of the highly coherent driving laser. Fig. 3: Experimental set-up. Temporal solitons ( $$P_{\mathrm{s}}=u^2$$ ) are excited in an all-fibre degenerate OPO. It includes a PPF ( L 1 ?=?27?cm), which provides the parametric gain, an SMF ( L 2 ?=?21?m) and a short EDF ( L 3 ?=?52?cm) pumped by a 1,480?nm laser ( $${P}_{{\mathrm{g}}}$$ ) for loss compensation. The cavity is driven by a 775?nm pump ( P p ), obtained by doubling the driving-laser frequency in a free-space PPLN crystal. Before its conversion, the driving laser is modulated and amplified through an EDFA. Two lenses are used to couple the light in (C L1 ) and out (C L2 ) of the fibre. Two phase plates ( λ /2 and λ /4) and a mode scrambler (MS) are used to limit polarization and modal losses. An output coupler (OC) is included for soliton analysis, using an optical spectrum analyser (OSA), a fast photodiode (PD) and an oscilloscope (OSC). The cavity is actively stabilized using a proportional–integral–derivative (PID) controller and a counter-propagating control signal ( P c ) frequency shifted from the driving laser. AM, amplitude modulator; PM, phase modulator; FS, frequency shifter; PC, polarization controller; BPF, bandpass filter; VOA, variable optical attenuator; DM, dichroic mirror. Full size image The oscillation threshold of the OPO, $$\mu =2\kappa \sqrt{{P}_{{\mathrm{p}}}}{L}_{1}/{{{\varLambda }}}_{{\mathrm{e}}}=1$$ , where κ is the effective second-order nonlinearity of the PPF, corresponds to a 775?nm driving power P p ?=?5.4?W. We drive the cavity with short flat-top pulses 43 , synchronized to the cavity free spectral range around 1,550?nm (9.2?MHz), to minimize the average driving power and to ensure that the average intracavity power remains well below the saturation power of the intracavity amplifier (600?mW). The phase detuning of the intracavity degenerate signal ( δ 0 ?=? Λ e Δ ) is controlled by tuning the frequency of the 1,550?nm driving laser. Light propagation in this synchronously driven, singly resonant OPO is, under some conditions (see Supplementary Section 1 ), described by equation?(2). Characterization of the parametrically driven Kerr cavity soliton . In a first experiment, we use 650-ps-long, 10-W-peak driving pulses (corresponding to μ ?=?1.37) and scan the laser frequency (–230?kHz?ms ?1 ). Our results are shown in Fig. 4 . The signal resonance, measured around 1,550?nm, is reminiscent of that observed in externally pumped Kerr resonators 20 . The signal average power gradually increases until it reaches the bistable region where it suddenly drops, indicating the formation of localized structures. The small plateau emerging at that point corresponds to the soliton branch shown in Fig. 2c . In the context of externally driven Kerr resonators, it is often called the soliton step as pulses tend to merge one by one, leading to a stair-shaped transmission curve 20 . Additional higher-resolution measurements of the nonlinear transmission of the cavity, including multi-soliton steps, are shown in Supplementary Fig. 2 . We readily note an important difference between our experimental scans and the analytical branch shown in Fig. 2c . The soliton step in our experiments has a finite extension while the theoretical branch grows indefinitely with increasing Δ . First, we stress that frequency scans are inherently dynamic such that the measured output power is not necessarily representative of steady-state solutions at the corresponding detuning. Second, higher order effects, not included in equation?(2), such as parametric gain saturation, limit the soliton existence range. In our experiment, the soliton collapse is due to the 5?nm, flat-top intracavity filter we use to prevent lasing at shorter wavelengths 42 . As the detuning is ramped up, so is the soliton’s spectral width, such that the filter eventually prevents stable soliton propagation. Fig. 4: Characterization of the PDCS. a , Forward scan (black line) through a resonance for P p ?=?10?W. The dot highlights the stabilization setpoint ( Δ ?=?1.2). The blue line corresponds to the output power when the cavity is actively stabilized around that level. b , Oscilloscope recording–taken several seconds after the excitation process–showing a stable, resolution-limited pulse exiting the cavity. c , Experimental (blue line) and theoretical (red line) autocorrelation traces. The inset shows the theoretical profile of the corresponding background-free soliton. d , Experimental (blue line) and theoretical (red line) spectra at the cavity output. The narrow peak corresponds to back-reflections of the control signal. Full size image Next, we use a control signal to stabilize the system in the soliton region (see Methods ). The average output power when the detuning is set to Δ ?=?1.2 (corresponding to a phase detuning δ 0 ?=?0.03) is shown in Fig. 4a . A high-resolution (80?ps) recording of the corresponding cavity output is shown in Fig. 4b . A resolution-limited pulse can be seen exiting the cavity every rountrip time. Further temporal (Fig. 4c ) and spectral (Fig. 4d ) characterizations confirm that a short (3.6?ps) pulse is circulating in the cavity. The agreement with the analytic soliton solution of the PDNLSE is excellent. The experimental spectral background corresponds to the intracavity-amplified spontaneous emission 42 . These measurements confirm that our novel system is governed by the PDNLSE in that region and constitute an experimental observation of its well-known soliton in optics. Random bit generation . PDCSs are phase locked to a driving laser, as are externally driven CSs that attract a lot of attention because of their inherent stability. The additional advantage of the PDCS is its multiplicity. Owing to the $${{\mathbb{Z}}}_{2}$$ -symmetry of the PDNLSE, two attractors, which have the same amplitude but are out of phase, may coexist in the cavity, adding a degree of freedom to Kerr resonators. In particular, it opens the possibility of using Kerr solitons in applications that require two different attractors, such as random bit generators 29 and Ising machines 30 . To confirm this potential, we design a proof-of-principle experiment of random number generation. The concept is simple. When a soliton is spontaneously excited, it has a 50% chance of locking to the pump with one of the two possible phase relations. By exciting multiple solitons, and extracting the phase, we can generate a random sequence of bits. For this demonstration, we phase modulate the pump beam so as to excite a series of equally spaced single solitons. The physics behind soliton attraction to phase maxima is similar to that of CSs 44 and is detailed in Supplementary Section 3 . A low modulation frequency (4.6?GHz) is chosen to be able to resolve individual solitons on the oscilloscope. We extract a portion of the 1,550?nm driving laser before its frequency doubling and use it as a local oscillator for coherent detection (see Fig. 5a ). We excite two solitons in the cavity and send both the reference and the combined beams to a fast photodetector. The results are shown in Fig. 5b,c . As expected, the reference, corresponding to the intensity, displays identical traces separated by 220?ps. After interfering with the local oscillator, however, two different amplitudes are measured. These measurements confirm that solitons of different phases are excited in the cavity. In a second series of experiments, we expand the pulse width to host four solitons and perform three distinct resonance scans. Our results are shown in Fig. 5d–f . By assigning a binary value to each soliton, our results correspond to a series of four-bit random numbers, highlighting the potential of PDCSs for applications. Moreover, our measurements confirm that the solitons are phase locked, as only two distinct amplitudes are measured across the different scans. Fig. 5: Random bit generation. a , Experimental set-up for coherent detection. CW, continuous wave. b , Direct detection of two PDCSs. c , Coherent detection of two PDCSs, highlighting the two different phases. d – f , Sequences of four random bits generated through PDCS formation. Full size image Discussion . In summary, we investigated Kerr soliton formation in singly resonant OPOs. We built a system that is well described by the seminal parametric NLSE when driven with a frequency close to twice that of a longitudinal mode. We theoretically showed that a couple of stable solitons exist in a broad region of experimental parameters. Our measurements confirm the existence of a backgroundless, sech-shaped and phase-locked optical pulse in that region. Its temporal and spectral profiles are in excellent agreement with the soliton solution of the PDNLSE. The same profile corresponds to the well-known non-propagating hydrodynamic soliton 11 , 25 . Here, the soliton propagates along the resonator and forms an ultra-stable pulse train at the output. The phase locking ensures minimal jitter and the output spectrum consists of an ultra-coherent frequency comb. Importantly, the driving laser is at twice the comb central frequency. It can be easily filtered out, which is important for applications 45 , or it may be harnessed for self-referencing 46 . Moreover, we showed that applications of PDCSs go beyond frequency comb generation. The two different phases can be leveraged for random number generation, as demonstrated above, or physical Ising machines. The Ising machine has already been implemented using a synchronously pumped degenerate OPO 30 , but the number of individual spins is limited by the repetition rate of the pump laser. Our results show that a grid of individual spins, as dense as the input phase modulation, can be generated in a long fibre cavity. Because the number of potential connections scales as N 2 , a 40?GHz phase modulation would lead to a three orders of magnitude increase in the number of spin–spin couplings as compared to the state of the art 30 . Methods . Linear stability analysis . The temporal linear stability of the steady-state solutions, shown in Fig. 2 , has been computed by solving the eigenvalue problem Lψ ?=? σψ , obtained from the linearization of equation?(2) around a given steady state, where L is the linear operator evaluated at such a state, and σ and ψ are, respectively, the eigenvalues and eigenfunctions of L . This problem can be easily solved analytically for the homogeneous u h state as shown in Supplementary Section 2 . For the soliton state stability, we have adopted a numerical approach. We compute the eigenvalues of the Jacobian matrix obtained from L after spatial discretization in an N ?=?1,024 points grid. Experimental set-up . The all-fibre OPO is made of a section ( L 1 ?=?27?cm) of PPF, a section ( L 2 ?=?21?m) of standard telecommunication single-mode silica fibre (SMF-28) and a section ( L 3 ?=?52?cm) of EDF. The PPF has a second-order nonlinear parameter of κ ?=?0.04?W ?1/2 ?m ?1 and a phase-matching wavelength of 1,548.8?nm at room temperature. This wavelength is increased up to 1,549.72?nm to be in the tuning range of the driving laser by placing the fibre in a stabilized oven at 36?°C. Two WDMs are used to combine the 775?nm pump with the intracavity signal, and to reject the remaining pump power at the fibre output. Two different polarization controllers are used: one to align the pump polarization with the phase-matched eigenmode of the PPF, and the other to align the signal polarization with one of the two eigenmodes of the cavity. The EDF (Liekki ER16-8/125) provides the optical gain. Two WDMs are inserted in the cavity to combine the 1,480?nm pump with the intracavity signal and to reject the unabsorbed power at the EDF output. The EDF length is first empirically set so that the gain is slightly larger than the intrinsic cavity loss. We then use a variable optical attenuator to increase the loss and ensure that the cavity is below the lasing threshold. An optical bandpass filter (5?nm at 0.5?dB, centred on 1,550?nm) hinders laser emission at shorter wavelengths. The cavity contains a 99/1 coupler used both to inject the control signal into the cavity and to extract part of the intracavity power. The total intracavity loss, excluding the doped fibre, is 40%. The driving continuous-wave laser is a Koheras Adjustik E15 with a sub-100?Hz linewidth. Its wavelength is set to 1,549.72?nm, on the edge of the tuning range (1?nm) to coincide with the PPF phase-matching wavelength. The laser output is first modulated using a Mach–Zehnder amplitude modulator (bandwidth: 12?GHz, extinction ratio: 30?dB), driven by a pattern generator connected to a radiofrequency clock. The pulsed beam is then amplified with an erbium-doped fibre amplifier (EDFA) and converted to its second harmonic through a 4-cm-long PPLN crystal ( κ ?=?2.5?W ?1/2 ?m ?1 ; MSHG1550-0.5-40, Covesion) in a free-space section. Using dichroic mirrors, the unconverted field is attenuated by 125?dB such that only the second harmonic is injected into the fibre. To minimize both polarization and modal losses at the first WDM, a half- and quarter-wave plate (free-space) and a mode scrambler are used. The cavity resonances are measured by scanning the frequency of the driving laser and recording the average power at the output coupler, $${P}_{{\mathrm{s}}}^{{\mathrm{out}}}$$ , using a 200?kHz photodiode (see, for example, Fig. 4a ). To stabilize the cavity at a fixed detuning, we use a frequency-shifted counter-propagating control signal 47 . It is generated by extracting a portion of the driving-laser power through a 95/5 coupler and sending it to a tunable frequency shifter (110?±?5?MHz). Using a circulator and a polarization controller, the counter-propagative control signal, with a power of $${P}_{{\mathrm{c}}}$$ , is sent to the cavity on the orthogonal polarization eigenmode to avoid seeding the OPO. The cavity detuning is stabilized through feedback on the driving-laser wavelength to maintain a constant control signal output power $${P}_{{\mathrm{c}}}^{{\mathrm{out}}}$$ . The feedback signal is generated by a proportional–integral–derivative controller (DigiLock 110, Toptica), driven by a photodiode. In our experiments, we fix the stabilization setpoint ( $${P}_{{\mathrm{c}}}^{{\mathrm{out}}}/{P}_{{\mathrm{c}}}=0.9$$ ) and modify the detuning of the OPO signal by shifting the control signal frequency. To retrieve the detuning of the OPO signal, we map it to the frequency of the control signal by injecting the 1,550?nm laser in the cavity through the output coupler and fitting the cavity transmission in the linear regime. Part of the intracavity power is extracted at the output coupler to characterize the solitons. The spectrum of the PDCS is recorded using an optical spectrum analyser (0.1?nm resolution bandwidth). Time measurements are carried out with a fast photodiode (45?GHz bandwidth) and an oscilloscope (12?GHz bandwidth, 20?gigasample per second). The intensity autocorrelation trace is directly acquired at the cavity output. For this measurement, a commercial EDFA is used to increase the average output power to ~70?mW. Periodically poled fibre fabrication . The PPF is a 125?μm outside diameter cladding fibre with a germania-doped glass core of 4?μm diameter and a numerical aperture NA?=?0.17. Two 27-μm-diameter channels run adjacent to the fibre core at a distance of 13.6?μm and 7.2?μm, respectively, from the core’s edges. The fibre is first thermally poled in single-anode configuration at 265?°C with an electric potential of +8?kV applied to the embedded electrode, for two hours 48 . The second-order nonlinearity created via thermal poling is then erased periodically by means of a continuous-wave argon-ion laser frequency doubled to 244?nm, equipped with an acousto-optic modulator used to modulate the laser output. The laser is focused to a circular spot, 20?μm in diameter, while the poled fibre is clamped onto a linear stage using two fibre rotator clamps. The laser is modulated using the acousto-optic modulator while translating the fibre core through the spot to achieve a grating of the desired duty cycle and period. The latter was chosen to be 55?μm to have quasi-phase matching at a wavelength of around 1,550?nm. Coherent detection measurement . To demonstrate the existence of PDCSs with opposite phases, the cavity is synchronously pumped with 1?ns or 1.9?ns flat-top pulses. On these driving pulses, we also imprint a 4.6?GHz phase modulation using a phase modulator. As with CSs 44 , 49 , PDCSs are attracted by phase-modulation maxima (see Supplementary Section 3 ). When scanning the resonance, we generate up to four PDCSs, separated by 220?ps. Using a 90/10 coupler, most of the cavity output power $${P}_{{\mathrm{s}}}^{{\mathrm{out}}}$$ is sent to a 10?GHz photodiode (that is, the reference beam in Fig. 5a ). The remaining power is combined with part of the driving-laser power, obtained by bypassing the frequency shifter, through another 90/10 coupler. The result of the interference is sent to a 45?GHz photodiode for coherent detection. Data availability . The data that support the findings of this study are available from the corresponding author on reasonable request. References . 1. Cross, M. C. & Hohenberg, P. C. 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F.D.L. acknowledges the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement no. 842676. F.L. and P.P.-R. acknowledge the support of the Fonds de la Recherche Scientifique (FNRS). Author information . Affiliations . Service OPERA-Photonique, Université libre de Bruxelles (ULB), Brussels, Belgium Nicolas Englebert,?Francesco De Lucia,?Pedro Parra-Rivas,?Carlos Mas Arabí,?Simon-Pierre Gorza?&?Fran?ois Leo Optoelectronics Research Centre, University of Southampton, Southampton, UK Francesco De Lucia?&?Pier-John Sazio Authors Nicolas Englebert View author publications You can also search for this author in PubMed ? Google Scholar Francesco De Lucia View author publications You can also search for this author in PubMed ? Google Scholar Pedro Parra-Rivas View author publications You can also search for this author in PubMed ? Google Scholar Carlos Mas Arabí View author publications You can also search for this author in PubMed ? Google Scholar Pier-John Sazio View author publications You can also search for this author in PubMed ? Google Scholar Simon-Pierre Gorza View author publications You can also search for this author in PubMed ? Google Scholar Fran?ois Leo View author publications You can also search for this author in PubMed ? Google Scholar Contributions . N.E. designed and performed the experiments, supervised by S.-P.G. Both F.D.L. and P.-J.S. manufactured the periodically poled fibre. N.E. derived and simulated the mean-field model. P.P.-R. and C.M.A. performed the bifurcation and linear stability analysis of the mean-field model. F.L. supervised the overall project and wrote the manuscript. All authors discussed the results and contributed to the final manuscript. Corresponding author . Correspondence to Nicolas Englebert. Ethics declarations . Competing interests . N.E., S.-P.G. and F.L. have filed patent applications on the active resonator design and its use for frequency conversion (European patent office, application number EP20188731.2). The remaining authors declare no competing interests. Additional information . Peer review information Nature Photonics thanks the anonymous reviewers for their contribution to the peer review of this work. Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Supplementary information . Supplementary Information . Supplementary Sections 1–3 and Figs. 1 and 2. Rights and permissions . Reprints and Permissions About this article . Cite this article . Englebert, N., De Lucia, F., Parra-Rivas, P. et al. Parametrically driven Kerr cavity solitons. Nat. Photon. 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