A universal framework for nonlinear frequency combs under electro-optic modulation

Nonlinear frequency combs, including electro-optic and Kerr combs, have become central platforms for chip-scale frequency synthesis. Recent breakthroughs in strong-coupling electro-optic modulation further expanded their accessible nonlinear dynamics, unlocking new phenomena and functionalities, but the underlying foundation remains largely unexplored. Here we establish a universal theoretical and experimental framework for nonlinear combs under arbitrary electro-optic modulation by introducing a general evolution equation (GEE) that transcends the mean-field Lugiato-Lefever equation. The GEE reduces to a discrete-time Integration Hamiltonian that provides a frequency-domain formalism unifying strong-coupling electro-optic modulation with photonic synthetic dimensions. Together with a band-wave correspondence linking modulation waveforms to synthetic band structures, the formalism enables programmable spectral control. We further show compatibility between Kerr nonlinearity and strong-coupling electro-optic modulation, highlighting their cooperative dynamics. Our work provides a foundational model for strong-coupling electro-optics in nonlinear combs, opening a route toward chip-integrated, microwave-programmable comb sources for metrology, spectroscopy, and emerging photonic technologies.