Photon-pair generation in lossy coupled-resonator optical waveguides via spontaneous four-wave mixing

We theoretically model pair generation and evolution via the nonlinear process of spontaneous four-wave mixing in a coupled-resonator optical waveguide in a photonic crystal slab. Using the adjoint master equation for a system of lossy coupled cavities, we calculate a symmetrized second-order coherence function to determine pair detection probability. We find that the scattering loss can have as large an effect on pair generation as waveguide dispersion. In particular, the wave-vector dependence of the loss can shift the frequency of the maximum detection probability and therefore cannot, in general, be ignored or treated via a simple overall loss factor.

Figure 1

Schematic of the nonlinear pair generation experiment via SFWM in a CROW device. It shows the top view of a CROW structure in a rectangular PCS that is made by filling every other hole along a straight line. The pump pulse enters from the left and passes through by tunneling from one cavity to another. In this slow propagation scheme, the nonlinearity is enhanced, and therefore, the probability of pair generation is increased.

Figure 2

The dispersion and quality factor are plotted in (a) and (b), while the corresponding group velocity and the effective loss propagation coefficient ${\alpha }_P\equiv c/\left(v_{g}Q\right)$ are plotted in (c) and (d). The three vertical dashed lines represent the wave vectors of the idler, pump, and signal photons, from left to right, for which the results plotted in Fig. 5 are obtained.

Figure 3

Spatial dispersion of the pump pulse due to propagating a full CROW length. Three different pulse widths are considered as labeled on each graph, ranging from $\mathrm{\Delta }=0.1\pi /D$ to $\mathrm{\Delta }=0.02\pi /D$. On the left, (a)–(c) show the pulse shape before entering the CROW, and on the right, (d)–(f) show the pulse shape at the exit time. In all of the above cases we set $k_p=0.6\pi /D$.

Figure 4

The second-order coherence function $G_s^{\left(2\right)}$ as a function of the signal and idler Bloch vectors. In (a), the system is pumped at $k_p=0.6\pi /D$, while in (b) the system is pumped at $k_p=0.3\pi /D$. Both plots are normalized such that the maximum detection probability is of equal brightness.

Figure 5

The detection probability is plotted as a function of the pump Bloch vector for the resonant condition, ${\omega }_s+{\omega }_i=2{\omega }_p$, for two different cases: in (a) and (c) $\mathrm{\Delta }{f}_{sp}=7\mathrm{\Delta }{f}_P$ and in (b) and (d) $\mathrm{\Delta }{f}_{sp}=25\mathrm{\Delta }{f}_P$. Also, (a) and (b) show the case when the loss has been turned off, and (c) and (d) show the case when the true loss characteristic is used. Note that (b) and (d) are plotted only up to $k=0.6\pi /D$ (roughly), beyond which frequency matching between the signal, idler, and pump is not possible. All values here are normalized to the maximum detection probability in (a) such that a fair comparison between different plots can be made.

Figure 6

The detection probability ratio $R$ due to CROW losses and the phase mismatch $\mathrm{\Delta }k$ plotted as a function of $\mathrm{\Delta }{f}_{sp}$. The inset shows the value of $k_p$ that maximizes $G_{\mathrm{Lossless}}^{\left(2\right)}$.