Nonlinear Characterization of Waveguide Index Profile: Application to Soft-Proton-Exchange in LiNbO$_3$

In integrated photonics, the precise knowledge of the waveguides refractive index profile is mandatory for the modeling of photonic chips and therefore implementing innovative circuits. Usual index profile determination relies on effective index measurement of propagating modes in planar waveguides coupled with numerical fitting tools. In this paper we propose an alternative technique based on the characterization of the second harmonic generation signature of a nonlinear waveguide. We include the characterization of high-order spatial modes showing their relevance to probe both vertical and lateral distributions. We finally provide an explicit profile ready-to-use for modeling soft-proton exchanged waveguides in lithium niobate and we test its prediction capability.