Mapping Nonlinear Mode Interactions in Coupled Kerr Resonators

TL;DR

The paper addresses how spatial overlaps between nonlinear Kerr modes in coupled resonators influence cross-phase modulation. It introduces a pump–probe method in a three-ring silicon nitride system to extract relative mode overlaps between pump and probe supermodes by leveraging the distinct timescales of Kerr (fast) and thermal (slow) XPM, quantified via the overlap proxy $\eta_{p,b}=\Gamma_{p,b}/\Gamma_{p,p}$ and the overlap integral $\Gamma_{p,b}=\frac{1}{4}\int_V \epsilon^2 |\vec{\Upsilon}_b^* \cdot \vec{\Upsilon}_p|^2 dV$. The experimental results for all pump–probe combinations agree with predictions from the coupled-mode eigenvectors, with minor discrepancies for the AS mode explained by residual detuning. The methodology generalizes to larger networks and other nonlinear processes, providing a practical tool for modeling and optimizing nonlinear phenomena in complex multimode resonator arrays, with potential impacts on microcomb generation, wavelength conversion, and quantum photonics.

Abstract

We present a method for resolving spatial mode overlaps in coupled microresonators based on Kerr and thermal cross-phase modulation. Through a pump-probe setup, we measure experimental overlap in a three-ring resonator with good agreement with analytical theory. Our technique can be generalized for describing nonlinear interactions in more complex multi- and coupled-mode systems.

Figures (3)

• Figure 1: (a) Chain of three coupled microresonators. (b) Uncoupled (bare-ring) schematic representation. Top: degenerate resonance frequencies of the uncoupled microrings. Bottom: uncoupled optical profiles showing the electric field amplitude in each resonator. (c) Frequency-split supermodes under strong coupling: symmetric (S, red), central (C, green), and anti-symmetric (AS, blue). Top: optical resonances corresponding to the hybridized supermodes. Bottom: supermode field profiles projected onto the bare-ring basis (unnormalized). “0” signifies non-resonant excitation, and negative amplitudes with dark shading indicate a relative $\pi$ phase difference.
• Figure 2: (a) Schematic of the mode overlap experiment. An intensity-modulated pump ($\lambda_\mathrm{p} \approx 1546$ nm), resonant with a supermode, induces cross-phase modulation on a probe ($\lambda_\mathrm{b} \approx 1542$ nm) via Kerr and thermo-optic effects. This leads to a modulation of the probe’s resonance frequency and, consequently, its transmission. (b) Experimental setup for the overlap characterization. EOM electro-optic modulator, PC polarization controller, BS 50:50 beam splitter, DUT device under test, OF optical filter, PD photodetector, FPD fast photodetector, VNA vector network analyzer. (c) Spectrum of the pumped and probed TE-polarized supermode triplets, centered at $\lambda_p$ and $\lambda_b$, respectively. S, C, and AS resonances are indicated in red, green, and blue. Measured loaded Q-factors are approximately $3.5 \times 10^5$ for the S and AS modes, and $2 \times 10^5$ for the C mode. (d) Probe AC transmission versus modulation frequency, with the pump fixed at the S supermode and the probe set into S (red), C (green), and AS (blue). Shaded regions highlight dominant Kerr (yellow) and thermal (red) contributions. Dashed lines are fits from the theoretical model.
• Figure 3: Thermal response of the S, C, and AS supermodes as a function of the modulation frequency. Inset: Temperature profile of the modulated absorbed power for the S/AS (i.) and C (ii.) supermodes, simulated in COMSOL Multiphysics with the Heat Transfer in Solids module.