Modulating incoherence for phase recovery with single-pixel intensity correlation

Recovery of missing phase information from the intensity correlation is crucial in various applications ranging from astronomy to biology. Interferometry and phase retrieval algorithms are common methods for phase recovery. Here, we present a non-interferometric and iteration-free approach for recovering the phase of the Fourier spectrum. This is implemented in an in-line configuration with a structured illumination and a single-pixel estimation of the intensity correlations. Instead of many temporal measurements, single exposure speckle is used to acquire the complex Fourier spectrum from the intensity correlation. Mapping of the complex Fourier spectrum is demonstrated by discretely varying the frequency of the structured illumination and then extracting the intensity correlations for a set of three phase-shifted structured illuminations at each frequency. Furthermore, the recovered Fourier spectrum is used to image the two-dimensional amplitude object through scatterer. Experimental implementation of the proposed technique is demonstrated by determining the complex Fourier spectrum of different objects, and results are compared with theory and simulations.