Beyond critical coupling: optimal design considerations for spontaneous four-wave mixing in microring resonators

TL;DR

The paper presents a comprehensive, perturbative, interaction-picture model for spontaneous four-wave mixing in microring resonators that unifies all-pass and add-drop geometries with both identical and distinct pump/biphoton couplings under CW and pulsed pumping. It yields explicit expressions for intracavity and extracavity fields, biphoton wavefunctions, and absolute generation rates, and it reveals time-frequency correlations along with the tradeoffs between coupling conditions and extraction efficiency. Key results identify conditions that maximize single-photon versus two-photon outputs, and show how distinct pump-biphoton coupling can enhance pair rates while affecting spectral purity, quantified by Schmidt numbers. The framework offers actionable design insights for integrated photon sources, linking device geometry and coupler coefficients to experimentally relevant performance metrics such as rate, heralding efficiency, and spectral factorability.

Abstract

We present a self-contained analytical model for biphoton generation in microring resonators. Encompassing both all-pass and add-drop geometries, identical and distinct pump and biphoton coupling coefficients, and continuous-wave and pulsed pumping, our interaction-picture-based approach reveals time-frequency biphoton correlations while also predicting absolute generation rates. Under continuous-wave excitation, we find critical coupling of both the pump and biphoton to maximize the rate of single photons extracted from the microring, whereas critical coupling of the pump but overcoupling of the biphoton maximize the two-photon rate. Under pulsed pumping, overcoupling of both pump and biphoton (to different degrees) maximizes photon extraction probabilities, albeit under a tradeoff with spectral factorability that we quantify via parameter scans over a range of coupler pairings. As a whole, our formalism should prove valuable for the practical design of integrated photon sources, merging a flexible and intuitive biphoton-centric depiction with quantitative predictions closely tied to experimental parameters.

Figures (3)

• Figure 1: Basic microring SFWM configurations considered. (a) All-pass geometry with identical pump-biphoton coupling rates. (b) Add-drop geometry with identical pump-biphoton coupling rates. (c) Add-drop geometry with distinct pump-biphoton coupling rates. Each case can be uniquely identified by the output mode $\hat{\mu}\in\{\hat{a},\hat{b}\}$ and triplet of coupling conditions defining the linewidth---namely, $\gamma=\gamma_a+\gamma_b+\gamma_c$ for the biphoton and $\tilde{\gamma}=\tilde{\gamma}_a+\tilde{\gamma}_b+\tilde{\gamma}_c$ for the pump.
• Figure 2: Calculated one- and two-photon rates for CW pumping of an exemplar AlGaAs microring. (a) All-pass geometry with identical pump-biphoton coupling rates ($\tilde{\gamma}_a=\gamma_a>0$, $\tilde{\gamma}_b=\gamma_b=0$). (b) Add-drop geometry with identical coupling rates ($\tilde{\gamma}_a=\gamma_a>0,\tilde{\gamma}_b=\gamma_b>0$). (c) Add-drop geometry with distinct rates ($\tilde{\gamma}_a>0,\gamma_b> 0,\gamma_a=\tilde{\gamma}_b=0$). Dotted lines highlight the optimal coupling conditions (normalized to $\gamma_c$) for each case as summarized in \ref{['tab:CWsummary']}.
• Figure 3: Calculated rates and Schmidt numbers for broadband pumping of an exemplar AlGaAs microring. (a) All-pass geometry, identical pump-biphoton coupling ($\tilde{\gamma}_a=\gamma_a>0$, $\tilde{\gamma}_b=\gamma_b=0$). (b) Add-drop geometry, identical coupling rates ($\tilde{\gamma}_a=\gamma_a>0,\tilde{\gamma}_b=\gamma_b>0$). (c) Add-drop geometry, distinct rates ($\tilde{\gamma}_a>0,\gamma_b> 0,\gamma_a=\tilde{\gamma}_b=0$). Cases (a,b) correspond to the same Schmidt number $K=1.091$ for all coupling combinations. (See text for parameter values.) Dotted lines show the rate-maximizing coupling values (normalized to $\gamma_c$) for each plot.