Hybridized-band parametric oscillations in coupled Kerr microresonators

TL;DR

The paper develops a supermode-based framework to describe Kerr-driven interband interactions in a system of three coupled silicon-nitride microring resonators, revealing how a dispersive band structure enables multiple phase-matching pathways for hybridized optical parametric oscillations (OPOs). Through theory and experiment, it identifies two dominant OPO pathways in symmetric |ooo| structures (diagonal Type-II-like OPO1 and horizontal Type-I-like OPO2) and provides closed-form expressions for the azimuthal orders of the primary sidebands, demonstrating excellent agreement with observed spectra. It further shows that an asymmetric |oOo| design can suppress competing channels via avoided-mode crossing (AMX), enabling a competition-free intra-FSR OPO at 7 GHz spacing with high spectral purity, a feature advantageous for photonic quantum information tasks. Overall, the work establishes design principles for engineering nonlinear dynamics in coupled-resonator platforms and highlights implications for coherent photonic computing and quantum information processing.

Abstract

Coupled resonators form band-like optical states that support rich nonlinearities beyond what is possible in single resonators. In these systems, four-wave mixing mediates interband coupling, displaying multimode dynamics that span both spatial and spectral degrees of freedom. In this study, we propose a framework describing the onset and control of hybridized optical parametric oscillation in three coupled silicon nitride microring resonators. In a symmetric configuration, we observe the emergence of diverse phase-matching pathways defined by the dispersive band structure. We develop an analytical model that captures the parametric gain of these interband processes and derive closed-form expressions for the dominant gain maxima; the analytical framework itself readily extends to more complex coupled networks. We further report an asymmetric design that co-engineers mode overlap and dispersion to operate on a compact 7-GHz spacing, free from mode competition. Our findings establish design principles for engineering nonlinear dynamics in coupled-resonator platforms, with implications for coherent photonic computing and quantum information processing.

Figures (5)

• Figure 1: (a) Schematic of a microring resonator driven by a monochromatic pump (red arrow). (b) Representative integrated dispersion of a single resonator operating in the normal-dispersion regime. (c) Generation of hybridized optical parametric oscillations (OPOs) in a symmetric three-ring resonator under monochromatic pumping (red arrow). (d) Dispersive band structure of the $|$ooo$|$ design, showing the S (red), C (green), and AS (blue) supermode bands. The associated spatial profiles are also shown: dark shading indicates a relative $\pi$ phase shift, while blank rings denote non-resonant excitation. (c) Hybridized OPO operation in an asymmetric three-ring resonator formed by larger and smaller rings, in which geometric asymmetry suppresses competing comb-generation channels. (d) Dispersive band structure of the $|$oOo$|$ design, highlighting an avoided mode crossing between the dispersions of the smaller and larger rings (black dashed and solid lines). The transition of the supermode eigenvectors from the strongly coupled (colored) to the uncoupled regime (black solid) is also shown.
• Figure 2: Phase-matching topologies of hybridized OPOs. Phase matching in coupled microresonators can be classified according to the geometry of the interaction within the dispersive band structure. (a) Horizontal phase matching, corresponding to a Type-I-like OPO, where signal and idler are generated on the same supermode branch (AS–AS) at finite azimuthal order through an effective anomalous dispersion induced by band curvature. (b) Vertical phase matching, corresponding to a Type-II-like OPO at $\Delta\mu = 0$, where signal and idler occupy different branches (S and AS) and frequency matching is enabled solely by interband splitting. (c) Diagonal phase matching, also Type-II-like, where signal and idler are generated on different branches at finite and opposite azimuthal orders, requiring simultaneous interband and intraband dispersion compensation. While panels (b) and (c) share the same modal degeneracy class, they represent distinct dynamical regimes with different gain maxima and competition behavior. In (a-c), the color coding indicates the frequency hierarchy of the interaction
• Figure 3: Hybridized OPOs in the $|$ooo$|$ coupled resonator system. (a) Left: Transmission map as a function of frequency offset ($\Delta\omega/2\pi$) and relative mode number $\mu$. Each FSR supports a hybridized supermode triplet, with splitting increasing at longer wavelengths due to stronger coupling. Additional splittings are visible at some $\mu$ due to CW–CCW backscattering. Right: Transmission spectra for selected $\mu$ values. Dashed lines represent the theoretical transmission. (b) Schematic of the two observed multi-FSR OPOs. Pumping the C supermode excites distinct oscillations: OPO 1 (solid) generates signal and idler in the S and AS supermodes, while OPO 2 (dashed) produces both within AS supermodes. (c) Integrated dispersion of the supermodes, referenced to the pumped C mode at $\mu = 0$. Colored markers represent experimental data, while solid curves show polynomial fits. Modes participating in OPO 1 and OPO 2 are highlighted by solid and dashed black lines, corresponding to diagonal and horizontal phase-matching, respectively. (d) Calculated parametric gain spectra. For OPO 1 (solid), gain peaks for the S and AS modes stem near $\mu = \pm 4$. For OPO 2 (dashed), the AS–AS gain peaks arise at $\mu = \pm 21$. (e) Optical spectrum showing simultaneous excitation of OPO 1 and OPO 2. Inset: zoom at $\mu = 0$, revealing an intra-FSR OPO enabled by vertical phase-matching.
• Figure 4: OPO 1 (a) and OPO 2 (b) under individual excitation. (i) Broadband optical spectrum, with insets showing the corresponding phase-matching pathways. (ii–iii) Zoom into the signal and idler frequencies at $\mu = \pm 5$ for (a) and $\mu = \pm 21$ for (b), with incremental adjustments of pump power and detuning to track their evolution. In (a, ii–iii), the excited supermodes are identified. In (b, ii–iii), additional lines are observed.
• Figure 5: Intra-FSR OPO in the $|$oOo$|$ coupled resonator system. (a) Left: Transmission map as a function of frequency offset ($\Delta\omega/2\pi$) and relative mode number $\mu$. As the wavelength changes, the different dispersion between the larger and smaller rings results in an AMX, modifying the coupling from hybridized triplets at $\mu = 0$ to uncoupled singlets at distant $\mu$. Right: Transmission spectra for selected $\mu$ values. Dashed lines represent the theoretical transmission. (b) Schematic of the intra-FSR OPO that takes place within a single triplet. Pumping the C supermode results in signal and idler excitation at the AS and S modes. (c) Mode overlap for the OPO process while pumping the C supermode at $\mu=0$. Purple lines assume the signal-idler excitation at AS and S supermodes (solid for $|$oOo$|$ and dashed for $|$ooo$|$). Green is the self-overlap of the C supermode. (d) Integrated dispersion of the supermodes, referenced to the pumped C mode at $\mu = 0$. Colored markers represent experimental data, while solid curves show polynomial fits. Modes participating in the intra-FSR OPO at $\mu = 0$ are highlighted, corresponding to vertical phase-matching. (e) Calculated parametric gain for the parametric oscillator, which peaks at $\mu = 0$. (f) Optical spectrum of the competition-free intra-FSR OPO. Inset: zoom at $\mu = 0$, showing the signal and idler sidebands.