Uniqueness results for the phase retrieval problem of fractional Fourier transforms of variable order

TL;DR

This paper establishes conditions under which the phase retrieval problem for fractional Fourier transforms of variable order has unique solutions, with implications for optics and quantum physics.

Contribution

It provides new uniqueness results for phase retrieval using fractional Fourier transforms, including cases with minimal transform sets and specific function classes.

Findings

Uniqueness up to a phase factor when modulus of fractional Fourier transforms match.

Single fractional Fourier transform can suffice for uniqueness in certain cases.

Results apply to functions with compact support, pulse trains, Hermite functions, and Gaussian combinations.

Abstract

In this paper, we investigate the uniqueness of the phase retrieval problem for the fractional Fourier transform (FrFT) of variable order. This problem occurs naturally in optics and quantum physics. More precisely, we show that if $u$ and $v$ are such that fractional Fourier transforms of order $\alpha$ have same modulus $|F_\alpha u|=|F_\alpha v|$ for some set $\tau$ of $\alpha$'s, then $v$ is equal to $u$ up to a constant phase factor. The set $\tau$ depends on some extra assumptions either on $u$ or on both $u$ and $v$. Cases considered here are $u$, $v$ of compact support, pulse trains, Hermite functions or linear combinations of translates and dilates of Gaussians. In this last case, the set $\tau$ may even be reduced to a single point (i.e. one fractional Fourier transform may suffice for uniqueness in the problem).