  Mirrors are ubiquitous in optics and are used to control the propagation of
optical signals in space. Here we propose and demonstrate frequency domain
mirrors that provide reflections of the optical energy in a frequency synthetic
dimension, using electro-optic modulation. First, we theoretically explore the
concept of frequency mirrors with the investigation of propagation loss, and
reflectivity in the frequency domain. Next, we explore the mirror formed
through polarization mode-splitting in a thin-film lithium niobate
micro-resonator. By exciting the Bloch waves of the synthetic frequency crystal
with different wave vectors, we show various states formed by the interference
between forward propagating and reflected waves. Finally, we expand on this
idea, and generate tunable frequency mirrors as well as demonstrate trapped
states formed by these mirrors using coupled lithium niobate micro-resonators.
The ability to control the flow of light in the frequency domain could enable a
wide range of applications, including the study of random walks, boson
sampling, frequency comb sources, optical computation, and topological
photonics. Furthermore, demonstration of optical elements such as cavities,
lasers, and photonic crystals in the frequency domain, may be possible.
