Integrated optical wave analyzer using the discrete fractional Fourier transform

In this work, we detail a proposal for optical signals to be represented and analyzed in phase-space. Our proposal aims to integrate a series of operations in waveguide realization, as a compact all-together platform that takes an initial wavefield and returns a two-dimensional representation of the information. We show, step by step, that the quantum harmonic oscillator can be considered as a propagator of initial fields, and when a discretized version is implemented, the fractional order Fourier transform emerges. This last is crucial, since the Wigner-Radon theorem is used to establish a path between the propagated wavefield and the phase-space representation. We show by example that this integration offers a direct and efficient method for characterizing optical signals by reconstructing their Wigner phase-space in the scope of integrated optics.