Theory of coupled harmonics and its application to resonant and non-resonant electro-optic modulators

We present a unified theory for analyzing the frequency conversion between an RF voltage and an optical carrier in an electro-optic modulator. Our new approach results in the electro-optic coupled harmonic equation (EOCHE) that encompasses the electro-optic behavior of common types of modulators (lumped, traveling wave, and resonant) and effortlessly provides closed-form solutions for the small-signal frequency response and the large-signal intermodulation effects. In particular, the small-signal electro-optic frequency responses of lumped and traveling-wave Mach-Zehnder modulators, and resonant microring modulators are derived. We show that the concept of velocity-mismatch only applies to special cases when designing traveling-wave modulators. Therefore, to generalize the design of electro-optic modulators we propose a universal concept called the “effective RF phase” that governs how the optical power is exchanged between different RF harmonics of light. Using the EOCHE, we also propose a generalization of the widely used transfer matrix method for photonic circuits to fully capture the RF mixing effects in integrated photonic circuits, which is analogous to the steady-state AC analysis of electrical circuits. The simplicity and versatility of our approach make it a holistic tool for designing high-speed electro-optic modulators.