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.

I. Introduction

High-speed electro-optic modulators are the key enablers of the optical communication era [1]–[3]. Among different structures suitable for amplitude modulation, Mach-Zehnder interferometers (MZIs) in the form of lumped or traveling-wave [4]–[7] structures and the resonant microring [8]–[10] structures stand out as the widely adopted approaches in various material platforms such as silicon photonics or thin-film lithium niobate [11]–[15]. The two important performance metrics of a modulator equipped with an optical phase shifter are the phase shifter efficiency per volt and the small-signal electro-optic (EO) bandwidth (measured small-signal based on the electrical-to-optical-to-electrical conversion) [16]. Although the phase shifter efficiency is straightforward to calculate based on an electro-static simulation and coupling it to an optical eigen-mode simulation [17], the calculation of the EO bandwidth proves to be more difficult. This difficulty can be attributed to the nonlinear optical transfer function (i.e., optical intensity) of modulators [18] and the fact that RF harmonics of the electric field of light are mixed together once more when transformed into the optical intensity through a photodetector.