Multidimensional synthetic frequency lattice in the dynamically modulated waveguides.

Here we propose an effective method to construct a higher-dimensional synthetic frequency lattice with an optical waveguide under dynamic modulation. By applying the traveling-wave modulation of refractive index modulation with two different frequencies that are not mutually commensurable, a two-dimensional frequency lattice could be formed. The Bloch oscillations (BOs) in the frequency lattice is demonstrated by introducing a wave vector mismatch of the modulation. We show that the BOs are reversible only as the amounts of wave vector mismatch in orthogonal directions are mutually commensurable. Finally, by employing an array of waveguides with each under traveling-wave modulation, a 3D frequency lattice is formed and its topological effect of one-way frequency conversion is revealed. The study offers a versatile platform for exploring higher-dimensional physics in concise optical systems and may find great application in optical frequency manipulations.