1. Introduction

Generation of a regular sequence of narrow spectral lines (optical frequency comb) in micro-ring resonators is an active research area aiming to develop micro-comb sources for precision spectroscopy, optical to RF conversion and other applications, see, e.g., [1–5]. Microcombs can either complement or represent a viable alternative to the commercially available fiber laser based combs [6]. Directions of research are focused not only on improving and tailoring microcombs based on Kerr nonlinearity in semiconductor and crystalline resonators, but also on exploring novel non-ring geometries and non-Kerr materials, see, e.g., [7–12]. For example, so called bottle resonators open an opportunity to create compact comb sources having free spectral range (line separation) as small as in the fiber lasers and thereby proving microcombs with repetition rates in the MHz range [7, 8, 10]. Development of microresonators using materials with quadratic nonlinearities allows to reduce nonlinear thresholds and to introduce tunable frequency coverage [11, 12]. Modelocking of hundreds and thousands of resonator modes, each corresponding to a comb line, is associated with the pulse train generation in time domain. These pulses can often be understood as dissipative solitons, see, e.g., [1, 4, 5, 13, 14] and references therein.

Photonic devices have dominated the microcomb and more generally the preceding modelocking research, see, e.g., [15, 16], though few papers have explored these concepts in the contexts of the matter-wave modelocking [17–19] and of the half-light half-matter polariton modelocking [20–22]. Considering exciton-polaritons, which are the mixed states between the photonic modes of a microresonator and quantum-well excitons, there was a recent proposal to generate combs using polaritonic Josephson junction, where strongly anharmonic regimes of oscillations result in a narrow few-line comb spectrum [21]. An advantage of using polaritons is their strong nonlinearity, which exceeds values of the photonic nonlinearities in semiconductors by several orders of magnitude. The most common microcomb geometry of a ring resonator coupled to a bus waveguide [1–5] has not been so far explored in the polariton comb context and it is studied in this communication. Recent demonstration of optical to RF conversion with Kerr microcombs in the practical setting of the ultrafast information processing [23], makes it timely to develop the concept of polaritonic microcombs. The low threshold generation of the relatively narrow band combs with the THz-GHz repetition rates for all-optical processing and wave shaping can be thought as a practical goal of research into polaritonic frequency combs. Noting that the exciton-polariton nonlinearity is intrinsically narrow band, one can forecast that spectroscopy related applications are unlikely. Research into polaritonic microcombs also fits into the context of a wider push towards development of all-polariton on-chip devices [24, 25].

2. Mathematical model

Here we study microcavity wires, where polaritons are confined vertically through Bragg mirrors and laterally through the total internal reflection [26–28]. An alternative geometry for polariton applications is semiconductor waveguides guiding exclusively through the total internal-reflection and with quantum wells embedded inside the substrate [29], which requires using a mathematical model different from the one used below. Despite being more difficult to fabricate, microcavity wires have an advantage of providing significant group velocity reduction and employing well-established resonant or nonresonant pumping schemes for loss compensation. Microcavity wires also provide a flexible platform for realisation of couplers and inter connectors. For our analysis we assume a ring resonator etched through the upper Bragg mirror of the planar microcavity with embedded quantum wells, Fig. 1(a). The light is coupled in and out of the ring by means of the evanscent fields of the waveguide and resonator. To describe polariton dynamics in a microring we use a coupled system of equations for the photon and exciton amplitudes and neglect the spin and polarization degrees of freedom [30–32]:

 

Fig. 1 (a) Sketch of the microring resonator etched in the upper Bragg mirror of a planar microresonator with embedded quantum wells. (b) Dispersion (energy-momentum diagramme) of the linear polaritons, see Eq. (3). (c) Linear response of the ring resonator vs pump frequency for the fixed pump momentum m_p = 10. (d) Nonlinear response of the ring resonator in the vicinity of the resonances with m = 23, 24 for H = 0.01, 0.02 and 0.035 (blue). The angular width of the pump ϑ₀ = 2π/100. Red double arrows in (d) indicate bistability of the steady-state solution. Other parameters: d = 0.003, γ_c = γ_e = 0.002, and (Ω_c − Ω_e)/Ω_R = −1.

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For our numerical analysis it is convenient to normalise Eqs. (1) further. Introducing dimensionless t = TΩ_R and substituting A, ψ with $A{e}^{-i{\mathrm{\Omega }}_{P}T}$, $\psi e^{-i{\mathrm{\Omega }}_{p}T}$ we find

3. Linear and nonlinear resonances

First, we discuss linear properties of our microring model. We are looking for the solution of Eqs. (2) in the form A = a_me^imθ−iωt, ψ = b_me^imθ−iωt and assume that the losses, pump amplitude and frequencies are all nil and the nonlinear term is disregarded. Then the two-branch dispersion of linear polaritons is:

As a next step we accounted for the excitonic nonlinearity in our numerical modelling, which results in the density dependent shifts of the resonance frequencies and leads to the tilts of the resonance curves towards higher frequencies. This is a well known blue shift effect of the lower branch resonance, which in the ring geometry happens for a discrete set of the resonance frequencies. Fig. 1(d) shows transformation of the m = 23 and m = 24 resonances as we increase the pump intensity. The cavity response becomes bistable for the large enough pump meaning that the strongly tilted resonance peaks overlap with their own low amplitude tails, which are only weakly perturbed by nonlinearity [10, 13].

4. Mode-locked pulses and associated frequency combs

To this end the problem is, if for the set of resonances bounded to the spectrally narrow window order of 5Thz with the rapidly narrowing FSR at both ends of the interval, one can achieve generation of the pulse train consisting from the dissipative quasi-solitons in a way similar to what happens in the weakly dispersing photonic microresonators with effectively unbounded transmission window, see, e.g., [2, 3].

First we choose the pump frequency to be above the inflection point of the dispersion relation in Fig. 1(b), in order to excite polaritons with negative effective mass, i.e. with the momenta |m | > m_in = 16. This would fulfil the soliton existence condition in a planar cavity [31, 32]where such dissipative polariton solitons can form robust trains of moving wave-packets [32]. Then we prepare the system in the low-intensity state, which we are expecting to serve as a stable background for the pulse train. In order to excite the pulse train we use the seed pulse of the duration of several Rabi oscillations, ≈ $15\cdot {\mathrm{\Omega }}_R^{-1}$ or about 2.5ps, which excites a stable pulse travelling around the ring and coupling out every round-trip through the bus waveguide, see Figs. 2, 3. The pulse shape is well defined with the spatial extend being 7µm, corresponding to the angular spread ≃ 0.2π and temporal duration ≃ 3.6ps. However, the pulse intensity profile is strongly modulated, see. Fig. 3(a). Physical origin of this modulation is most likely related to two factors. First, and most obvious, is that spectral content of the pulse is far from being quasi-continuous, since the ring is relatively short and therefore FSR is large, which, in combination with the narrowness of the energy bandwidth of the lower polariton branch, allows only for less than 10 modes to play a significant role in the pulse formation, see Fig. 3(d). Also, for the shorter rings a polariton pulse looses comparatively less energy before it reaches the pump spot again, which brings down the pump power levels required to sustain a pulse train. Second, is that the pulse spectrum spreads equally on both sides of the inflexion point, while the repulsive exciton nonlinearity is able to support the soliton related pulse shaping only in the range of momenta associated with the negative mass, |m| > m_in. We have also observed that if the seed pulse amplitude is too large, then the spatio-temporal dynamics inside the cavity becomes irregular and appears to represent a complex superposition of several counter-propagating pulses, see Figs. 2(c) and (d). Fig. 3(d) shows the spectral content of the stable pulse train representing a frequency comb of 9 modes. The comb lines are up shifted with respect to linear resonances indicated in grey, as is expected due to repulsive polariton interaction. The comb FSR is ≃ 29GHz, which also matches the pulse repetition rate. Note, that we used a relatively narrow pump spot, therefore its spectrum has a noticeable tail extending into the range of negative momenta, which creates a relatively fragile counterpropagating pulse, see Fig. 2.

 

Fig. 2 (a) Initial spatio-temporal evolution of the light intensity resulting from the local excitation of the low intensity polariton state (H = 0.035 and Ω_p − Ω_e = −0.74Ω_R) by a seed pulse with the amplitude 0.1 and duration $15\cdot {\mathrm{\Omega }}_R^{-1}$. (b) Long-range propagation of the single-peak soliton-pulse. (c) Initial dynamics resulting from the excitation with the seed pulse having the amplitude 0.2 and resulting two-peak soliton state (d). Other parameters as in the Fig. 1.

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Fig. 3 (a) A snap-shot of the soliton profile in the ring resonator. Local temporal dynamics of the light intesity over the short (b) and long (c) time interval. (d) shows the corresponding nonlinear comb spectum (full blue lines) and the linear polariton spectrum (grey dashed lines). H = 0.035, Ω_p − Ω_e = −0.74Ω_R and other parameters as in the Fig. 1.

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Summary

We demonstrated that a microring resonator with the radius of few tens of microns etched in the planar polaritonic microcavity can be used to generated a train of pulses with repetition rate of 10’s of GHz and durations of 2 – 4ps. This sequence of pulses corresponds to the frequency comb of around 10 spectral lines. Our results demonstrate feasibility of using microcavity polaritons to generate GHz combs to be used in future polariton based optical information processing schemes. Relative smallness of the polariton comb devices is related to the fact that the polariton group velocity is about 10 to 100 times less than the group velocity of light, which implies that the effective length of our device is 1–2 orders of magnitude larger than the physical one.