Crystals are ubiquitous in nature and are at the heart of material research, solid-state science, and quantum physics. Unfortunately, the controllability of solid-state crystals is limited by the complexity of many-body dynamics and the presence of defects. In contrast, synthetic crystal structures, realized by, e.g., optical lattices, have recently enabled the investigation of various physical processes in a controllable manner, and even the study of new phenomena. Past realizations of synthetic optical crystals were, however, limited in size and dimensionality. Here we theoretically propose and experimentally demonstrate optical frequency crystal of arbitrary dimensions, formed by hundreds of coupled spectral modes within an on-chip electro-optic frequency comb. We show a direct link between the measured optical transmission spectrum and the density of states of frequency crystals in one, two, three, and four dimensions, with no restrictions to further expanding the dimensionality. We demonstrate that the generation of classical electro-optic frequency comb can be modeled as a process described by random walks in a tight-binding model, and we have verified this by measuring the coherent distribution of optical steady states. We believe that our platform is a promising candidate for exploration of topological and quantum photonics in the frequency domain. (C) 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
