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From Classical Four-Wave Mixing to Parametric Fluorescence in Silicon micro-ring resonators

Four-wave mixing can be stimulated or occur spontaneously. The first process is intrinsically much stronger, and well understood through classical nonlinear optics. The latter, also known as parametric fluorescence, can be explained only in the framework of a quantum theory of light. We experimentally demonstrate that, in a micro-ring resonator, there exists a simple relation between the efficiencies of these two processes, which is independent of the nonlinearity and size of the ring. In particular we show that the average power generated by parametric fluorescence can be immediately estimated from a classical FWM experiment. These results suggest that classical nonlinear characterization of a photonic integrated structure can provide accurate information on its nonlinear quantum properties.

Mirco-ring resonators have been investigated for more than a decade, with applications ranging from signal processing to optical sensing [1,2]. These resonators are extremely appealing for on-chip integrated optics and already exploited in commercial devices, as they can be realized either in silicon [3], silicon nitride [4], or hydex [5,6] following CMOS compatible processes.
The large field intensity that can be obtained by constructive interference within the ring has inspired several theoretical proposals and experiments in the field of nonlinear optics, including nonlinear bistability [3], parametric frequency conversion [5,7,8], parametric fluorescence [9], and frequency comb generation [4,6]. In this regard, four-wave mixing (FWM) is probably the most investigated process. This is a third-order nonlinear effect that can be viewed as the elastic scattering of two photons of a beam (pump), which results in the generation of two new photons at different frequencies (idler and signal) [10]. This nonlinear effect can occur with (stimulated FWM) or without (spontaneous FWM) an input signal beam. Spontaneous FWM, also known as parametric fluorescence, is particularly interesting, as it can be exploited for the generation of correlated photonpairs [11], used to construct qubits in quantum information and quantum computation.
For a given device, there is obviously a connection between the efficiencies of spontaneous and classical FWM, as the two processes originate from the same material nonlinearlity. Thus, it is somehow expected that systems designed to enhance stimulated FWM can be utilized for the generation of correlated photon-pairs. Yet, it would be extremely helpful to know of the efficiency of spontaneous FWM in a device from nonlinear characterization  via stimulated FWM [12]. The stimulated process is indeed intrinsically much more efficient, and thus it can be observed much more easily.
In this work we study stimulated and spontaneous FWM in micro-ring resonators, and we investigate their efficiencies with respect to the characteristic parameters of the ring: the radius R and the resonance quality factor Q. We consider single-channel side-coupled ring resonators. In this system, for any triplet of resonances equally separated in energy, assuming ω s ≃ ω p ≃ ω i , and considering a similar Q and group velocity v g , the generated powers integrated over the idler resonance are for the stimulated and spontaneous processes, respectively [13]. These expressions can be calculated follow-ing [14], assuming the critical coupling condition for the CW pump of power P p , tuned to the central resonance, and considering, in the stimulated case, a CW signal of power P s tuned to the lowest energy resonance. Here γ is the nonlinear parameter of the ring waveguide [11]. It is interesting that, given the same input pump power P p the ratio between spontaneous and stimulated powers is independent of the ring size, but it depends uniquely on the resonance quality factor, the signal power in the stimulated experiment, and a characteristic powerhω 2 p (e.g. athω p = 0.8eV , we havehω 2 p ≃ 160µW ). This makes it possible, given Q, to determine the number of pairs generated in the quantum process solely by means of the corresponding stimulated FWM experiment, once the value of Q is known.
The devices were fabricated on a silicon-on-insulator (SOI) wafer via e-beam lithography and inductively coupled plasma etching. Rings with radii of 5, 10, 20, and 30 µm were fabricated side-coupled to a single bus waveguide: the waveguides and rings have a 220×500 nm 2 rectangular cross section. Spot-size converters are used for efficient coupling [15]. The silicon chip was finally coated with a protective PMMA layer. The samples were characterized using a swept laser: a transmission curve for R=5µm is shown in Fig. 1 (a). The spectrum shows well distinct resonances (Q ∼ 7900) with the transmission falling to less than 1%, a sign that the ring is in critical coupling with the access waveguide. In the other samples the Q factors are Q ∼ 8400 for R=10µm, Q ∼ 12000 for R=20µm and Q∼15000 for R=30µm [16].
For the nonlinear experiment a tunable laser (Santec TSL-510) set at the pump and, for the stimulated FWM experiments another set at the signal frequencies, were injected into the bus waveguide of the rings with a tapered lensed fiber. The emitted light was spectrally filtered and sent to a CCD detector. Fig. 2 shows the results of FWM experiments on the R=5µm ring resonator, the sample with the smallest footprint, using the resonances marked in Fig 1. The power coupled into the ring and that generated within the ring are estimated by measuring the losses of each component of the experimental setup and consequently rescaling the output power from the laser or the power measured by the CCD (which was calibrated before the experiment). Couplipg losses to/from the sample are obtained as half its total insertion loss (measured to be about 7 dB). We evaluate a total error of about 10% on the estimated powers. Fig. 2 (a) shows an example of spontaneously generated signal and idler beams. The widths of the peaks are identical to those measured in transmission experiments, indicating that there is little degradation due to free carrier absorption. The integrated intensities for the signal and idler beams for spontaneous FWM, as well as the integrated intensity of the idler beam for stimulated FWM at fixed signal input power (P s =200µ W), are shown in Fig. 2  (b) as a function of P p . In all cases the intensities follow the expected quadratic dependence of (1) and (2). Notice that the saturation observed for P p > 2 mW is due to the thermo-optic effect induced by two-photon absorption, which produces a small redshift of the ring resonances [3,17]. We have also verified that in classic FWM the idler intensity scales linearly with P s . We performed a best fit of the stimulated FWM data (black dashed curve in Fig. 2 (b)) using γ as the only fit parameter (we take v g = c/n ef f where n ef f =2.47 is estimated by numerical simulation and is a typical mode effective index for this kind of waveguides [8]), obtaining γ=190 W −1 m −1 , a value consistent with those already reported in [8]. We then used this value to compute the expected spontaneous emission rate using eq. (2). The result is shown as a short dashed blue line in Fig. 2(b) and is in very good agreement with the experiments.
FWM experiments were carried out on rings with radii of 5, 10, 20 and 30 µm to verify the scaling of (1), (2), and (3) with R . In all cases, we chose rings near the critical coupling condition, and we worked with the same separation in energy between the signal, pump, and idler resonances: this means we have chosen the first neighbors in the case of the R=5µm ring (as in Fig 1(b)), second neighbors for R=10µm, fourth neighbors for R=20µm and sixth neighbors for R=30µm. In Fig. 3 (a) we show stimulated and spontaneous idler powers as a function of R: the data were averaged on a set of measures taken varying P p and P s . Notice that, for all data of Fig. 3, we kept P p < 1 mW to avoid the thermo-optic effect. As expected from (1) and (2), both of the generated powers scale as R −2 . Finally, in Fig 3 (b) we show the ratio between the measured generated idler powers in the spontaneous and in the classical processes as a function of Q · P s . The data show an excellent agreement with (3) and confirm that the ratio is independent of the ring radius. More importantly, these results prove that data from stimulated FWM can be used to precisely predict spontaneous FWM generation rates. For clarity, in Fig. 3(b) we show only data for P p =0.12 mW, but we obtained the same results for all the investigated pump powers varying between 60 µW and 1 mW.
In conclusion we have reported classical and spontaneous Four-wave mixing in silicon micro-ring resonators. We have verified that the efficiency of the spontaneous process is related to that of the classical process by a simple analytical formula. This result is of great practical interest, since it is general and not limited to ring resonators [14]: given any photonic structure, the application of this method allows the derivation of the number of photon pairs generated in the spontaneous process through a simple measure of the classical process, greatly simplifying the characterization of any structure designed for the generation of quantum photonic states. This work was supported by MIUR funding through the FIRB "Futuro in Ricerca" project RBFR08XMVY and from the foundation Alma Mater Ticinensis.