  On-chip optical microresonators are essential building blocks in integrated
optics. The ability to arbitrarily engineer their resonant frequencies is
crucial for exploring novel physics in synthetic frequency dimensions and
practical applications like nonlinear optical parametric processes and
dispersion-engineered frequency comb generation. Photonic crystal ring (PhCR)
resonators are a versatile tool for such arbitrary frequency engineering, by
controllably creating mode splitting at selected resonances. To date, these
PhCRs have mostly been demonstrated in isotropic photonic materials, while such
engineering could be significantly more complicated in anisotropic platforms
that often offer more fruitful optical properties. Here, we realize the
spectral engineering of chip-scale optical microresonators in the anisotropic
lithium niobate (LN) crystal by a gradient design that precisely compensates
for variations in both refractive index and perturbation strength. We
experimentally demonstrate controllable frequency splitting at single and
multiple selected resonances in LN PhCR resonators with different sizes, while
maintaining high Q-factors up to 1 million. Moreover, we experimentally
construct a sharp boundary in the synthetic frequency dimension based on an
actively modulated x-cut LN gradient-PhCR, opening up new paths toward the
arbitrary control of electro-optic comb spectral shapes and exploration of
novel physics in the frequency degree of freedom.
