Fig. 3: Reflection induced interference in the frequency synthetic crystal using polarization coupling.

a On the x-cut thin-film lithium niobate, TE mode that propagates along the y- and z-direction has different indexes \({n}_{o,{TE}}\) and \({n}_{e,{TE}}\), while the TM mode has index of \({n}_{o,{TM}}\) for all directions. Dispersion engineering can be used to make the indices of TE and TM modes degenerate, leading to strong coupling \(\mu\) between TE and TM resonances. b The frequency mirrors induced by mode-splitting only happens when the TE and TM modes have both index and frequency degeneracy. Difference in FSR of TE and TM modes, guarantees that they frequency overlaps at some frequencies, leading to formation of periodic frequency mirrors. c The group index \({n}_{o,{TM}}\) can be designed to be between the \({n}_{o,{TE}}\) and \({n}_{e,{TE}}\) via dispersion engineering. Then, TE mode circulating inside a x-cut lithium niobate resonator, it experiences different averaged indices (ranging from \({n}_{o,{TE}}\) to \({n}_{e,{TE}}\)) at different bending points of the resonator. As a result, index degeneracy between TE and TM modes can be achieved over a broad wavelength range (850 nm to 1450 nm for \(w=1.4{{\upmu }}{{{{{\rm{m}}}}}}\), \(h=350\) nm, \(t=250\) nm). d Measured transmission spectrum of TE modes on the x-cut dispersion-engineered lithium niobate device. The mode-splitting breaks the translation symmetry of the crystal, leading to frequency mirrors. Arb. units: arbitrary units. e Experimental verification of the reflection due to frequency mirror using polarization-crossing. Optical energy propagates along the frequency dimension when there is no mirror. Applying frequency mirror leads to interference states and varying the Bloch wavevectors can adjust the shape of the state. Due to the discrete nature of the crystal, our output signal measures the oscillation with a discrete sampling in frequency domain with a sampling period equal to the lattice constant. As a result, for \(k=0.5\pi /a\), the energy distribution on each lattice point shows constructive/destructive interference at every other lattice points. The patterns for \(k=0.345\pi /a\) and \(k=0.21\pi /a\) correspond to different oscillation period compared to the case of \(k=0.5\pi /a\), leading to destructive interference at every three/four lattice points.