Theoretical and Experimental Analysis of Optical Frequency Combs Synchronized to a Microwave Reference Achieving 10⁻¹⁹-Level Additional Stability

This paper presents a combined theoretical and experimental method for noise suppression in the repetition frequency (<i>f_r</i>) locking of erbium-doped fiber optical frequency combs (OFCs). This study proposed a novel mathematical model to bridge the noise relationship of <i>f_r</i> between the free-running and locked modes, and analyzed this relationship from two perspectives: the additional phase noise and the frequency stability. In addition, to integrate theoretical modeling with experimental validation, this study designed <i>f_r</i> locking strategy that uses a phase-locked loop (PLL) with PFD + PIID (a phase frequency detector and a proportional, first-order integer, second-order integer, first-order differential controller). Under synchronization of the <i>f_r</i> with a microwave reference (REF), this study achieved OFC additional frequency stabilities of 2.81 × 10⁻¹⁵@1 s and 8.08 × 10⁻¹⁹@10,000 s at 200 MHz fundamental frequency locking and 4.25 × 10⁻¹⁶@1 s and 1.91 × 10⁻¹⁹@10,000 s at 1200 MHz harmonic locking. The simulated and experimental results are in good agreement, confirming the consistency of the theoretical model and experiment. This work provides a reliable theoretical model that can be used to predict stability for OFC locking and significantly improves the additional frequency stability of OFCs.