Spatial coherence of fields from generalized sources in the Fresnel regime

Abstract

Analytic expressions of the spatial coherence of partially coherent fields propagating in the Fresnel regime in all but the simplest of scenarios are largely lacking, and calculation of the Fresnel transform typically entails tedious numerical integration. Here, we provide a closed-form approximation formula for the case of a generalized source obtained by modulating the field produced by a Gauss-Shell source model with a piecewise constant transmission function, which may be used to model the field's interaction with objects and apertures. The formula characterizes the coherence function in terms of the coherence of the Gauss-Schell beam propagated in free space and a multiplicative term capturing the interaction with the transmission function. This approximation holds in the regime where the intensity width of the beam is much larger than the coherence width under mild assumptions on the modulating transmission function. The formula derived for generalized sources lays the foundation for the study of the inverse problem of scene reconstruction from coherence measurements.

Passive sensing around the corner using spatial coherence

“…It is well known how coherence evolves in free-space propagation 17 . Thus, certain information about the source can always be extracted by measuring the coherence of the light at distant locations 18 . However, upon reflection from a scattering medium, it is expected that SCF is affected in a way that may complicate this reconstruction procedure.…”

On the inverse problem of source reconstruction from coherence measurements

“…This is demonstrated in Section V, where the experimental parameters violate the first inequality of (9a), yet the approximation is still dependable and allows for successful inversion. Further details on the physical meaning and the analysis of the result above can be found in [25].…”

“…We exploit the additional dimension in the coherence data to devise a global inversion method that applies local minimization to a family of residuals sharing a unique minimum, a task that would be difficult from intensity-only measurements. In our previous work [25], we studied the propagation of the spatial coherence of fields from generalized sources in the Fresnel regime. Such sources are modulations of the field produced by a Gauss-Schell source by a piecewise constant transmission function, thus modeling the field's interaction with objects and apertures.…”

“…Our focus here is on the inverse problem in which we seek to determine both the transmission function and the distance to the generalized source from the measurement plane from sampled coherence measurements. Leveraging the closed-form approximations obtained in [25] (which are explicit in the parameters of the transmission function) along with parametric modeling of the scene, we develop a gradient-descent-based approach to the inverse problem. The proposed algorithm yields accurate estimates of the parameters of the scene with low sampling complexity, i.e., only few measured samples of the coherence function suffice for the algorithm to converge to the actual parameters.…”

“…The paper is organized as follows. In Section II, we review the forward model of coherence propagation and the closed-form solution described in [25]. In Section III, we formulate the inverse problem and describe the gradientdescent-based algorithm.…”