Generation of photon pairs by spontaneous four-wave mixing in linearly uncoupled resonators

We present a detailed study of the generation of photon pairs by spontaneous four-wave mixing in a structure composed of two linearly uncoupled resonators, where energy can be transferred from one resonator to another only through a nonlinear interaction. Specifically, we consider the case of two racetrack-shaped resonators connected by a coupler designed to guarantee that the resonance comb of each resonator can be tuned independently and to allow the nonlinear interaction between modes that belong to different combs. We show that such a coupler can be realized in at least two ways: A directional coupler or a Mach-Zehnder interferometer. For these two scenarios, we derive analytic expressions for the pair-generation rate via single-pump spontaneous four-wave mixing and compare these results with that achievable in a single-ring resonator.

Figure 1

A sketch of the double-racetrack resonator structure.

Figure 2

(a) Transmission spectra for In to Through (I), Add to Drop (II), In to Drop (III), and Add to Through (IV). (b) Corresponding intensity enhancement. The dips in lines I and II in (a), which occur at the resonance frequencies, correspond to the intensity enhancement peaks of lines I and II shown in (b). We assumed $L_1=641\mu \mathrm{m}$, $L_2=432\mu \mathrm{m}$, ${\sigma }_1=0.933$, ${\sigma }_2=0.993$, $\xi =0.23{\mathrm{cm}}^{-1}$ (corresponding to $1\mathrm{dB}/\mathrm{cm}$), and a value of the coupling coefficient $X_{12}=X_{21}=-i0.00161$, the last value chosen to ensure that the resonators are nearly linearly uncoupled.

Figure 3

Sketch of the DC and schematic representation of the field distribution inside the channels.

Figure 4

Intensity distribution in the upper channel of the DC for two different values of the coupling coefficient: (a) ${\kappa }_{\text{DC}}=0.064\mu {\text{m}}^{-1}$ (perfect uncoupling) and (b) ${\kappa }_{\text{DC}}=0.068\mu {\text{m}}^{-1}$ (residual coupling).

Figure 5

Sketch of the Mach-Zehnder interferometer and a schematic representation of the overlap of the fields in the channels.

Figure 6

Effective straight-through coefficient ${\sigma }_{\text{MZI}}$ of the interferometer for different phases $\mathrm{\Delta }\varphi$ as a function of the coupling coefficients with the two PCs assumed to be identical.

Figure 7

Resonances involved in the SFWM process. The green arrow represents the coherent pump that generates the pair of signal and idlers photons (red and blue dots, respectively). The resonances of the upper resonator are plotted in gray, and those of the lower one are in violet. The interaction region, highlighted with the yellow dashed box in the structure's sketch, coincides with the coupling region.