Direct single-shot phase retrieval from the diffraction pattern of separated objects.

Ben Leshem,Oren Raz,Jianwei Miao,Rui Xu,Nirit Dudovich,Dan Oron,Yehonatan Dallal,Boaz Nadler

doi:10.1038/ncomms10820

Abstract

The non-crystallographic phase problem arises in numerous scientific and technological fields. An important application is coherent diffractive imaging. Recent advances in X-ray free-electron lasers allow capturing of the diffraction pattern from a single nanoparticle before it disintegrates, in so-called ‘diffraction before destruction' experiments. Presently, the phase is reconstructed by iterative algorithms, imposing a non-convex computational challenge, or by Fourier holography, requiring a well-characterized reference field. Here we present a convex scheme for single-shot phase retrieval for two (or more) sufficiently separated objects, demonstrated in two dimensions. In our approach, the objects serve as unknown references to one another, reducing the phase problem to a solvable set of linear equations. We establish our method numerically and experimentally in the optical domain and demonstrate a proof-of-principle single-shot coherent diffractive imaging using X-ray free-electron lasers pulses. Our scheme alleviates several limitations of current methods, offering a new pathway towards direct reconstruction of complex objects.

Highlights

The non-crystallographic phase problem arises in numerous scientific and technological fields
We show that when the single measured diffraction pattern is obtained from two sufficiently separated objects, the phase problem can be reduced to a set of linear equations that can be efficiently solved using standard numerical algebra tools
Our method consists of three main steps: (i) The sum of the objects autocorrelations, as well as their cross-correlation, are reconstructed from the Fourier transform of the measured diffraction pattern. (ii) The individual objects autocorrelations are reconstructed from their sum and the cross-correlation. (iii) Using the two intensities and the interference cross term as in refs 27–29, double-blind Fourier holography (DBFH) is applied to recover the phase by solving a set of linear equations

Summary

Introduction

The non-crystallographic phase problem arises in numerous scientific and technological fields. The Fourier phase has to be retrieved from this diffraction pattern alone. This method enables phase retrieval from a single diffraction pattern via a convex approach.

Results

Conclusion