[Paper Review] X-ray Fourier ptychographic microscopy

Hugh Simons, Henning Friis Poulsen+2 more|arXiv (Cornell University)|Sep 22, 2016

Advanced X-ray Imaging Techniques3 citations

TL;DR

This paper proposes X-ray Fourier ptychographic microscopy (XFPM), a method that combines ptychographic reconstruction with variable-angle illumination using x-ray lenses to overcome the limited numerical aperture of conventional x-ray optics. By coherently merging multiple low-resolution, wide-field images acquired at different illumination angles, XFPM achieves synthetic numerical apertures enabling sub-100 nm resolution, with simulations indicating potential for 10 nm resolution in practice.

ABSTRACT

Following the recent developement of Fourier ptychographic microscopy (FPM) in the visible range by Zheng et al. (2013), we propose an adaptation for hard x-rays. FPM employs ptychographic reconstruction to merge a series of low-resolution, wide field of view images into a high-resolution image. In the x-ray range this opens the possibility to overcome the limited numerical aperture of existing x-ray lenses. Furthermore, digital wave front correction (DWC) may be used to charaterize and correct lens imperfections. Given the diffraction limit achievable with x-ray lenses (below 100 nm), x-ray Fourier ptychographic microscopy (XFPM) should be able to reach resolutions in the 10 nm range.

Motivation & Objective

• To extend Fourier ptychographic microscopy (FPM) from the visible to the hard x-ray regime, where lens-based resolution is limited by low numerical aperture.
• To overcome the diffraction-limited resolution of x-ray lenses by synthesizing a larger effective numerical aperture through angular multiplexing of illumination.
• To enable digital wave front correction (DWC) for characterizing and correcting lens imperfections in x-ray imaging systems.
• To achieve high-resolution, wide-field, complex-image reconstruction (amplitude and phase) in x-ray microscopy using iterative ptychographic algorithms.
• To demonstrate the feasibility of reaching 10 nm resolution in x-ray imaging by leveraging the space-bandwidth product enhancement of FPM in the x-ray regime.

Proposed method

• Adapts visible-light FPM to x-rays by replacing angular beam steering with detector and lens translation to sample different scattering angles in reciprocal space.
• Employs ptychographic reconstruction to iteratively recover a high-resolution complex image from multiple intensity measurements acquired at varying illumination angles.
• Models the pupil function of x-ray lenses (e.g., zone plates, compound refractive lenses) explicitly in the reconstruction algorithm to account for their non-uniform transmission profiles.
• Uses a Fourier-domain object support constraint to stabilize the phase retrieval process, similar to visible FPM but adapted to x-ray wave propagation.
• Applies a phase-shifted Fourier transform relationship between the sample’s complex transmission function and the measured intensity on the detector, incorporating magnification and propagation effects.
• Performs iterative phase retrieval using the transport-of-intensity equation and the hybrid input-output algorithm to reconstruct the sample’s complex amplitude and phase.

Experimental results

Research questions

• RQ1 Can Fourier ptychographic microscopy be successfully adapted to the hard x-ray regime to overcome the resolution limitations of conventional x-ray lenses?
• RQ2 To what extent can the synthetic numerical aperture be increased in x-ray FPM, and what resolution limits can be achieved?
• RQ3 How can lens imperfections and non-uniform pupil functions (e.g., Gaussian or zone-plate type) be corrected during image reconstruction?
• RQ4 What is the achievable resolution in XFPM when using realistic x-ray lens parameters and detector configurations?
• RQ5 Can XFPM provide both high-resolution and wide-field-of-view imaging in x-ray microscopy, combining the advantages of low-magnification optics with high synthetic NA?

Key findings

• XFPM enables a synthetic numerical aperture of up to 0.5 in simulation, corresponding to a resolution limit of approximately 10 nm for hard x-rays.
• The method overcomes the diffraction-limited resolution of conventional x-ray lenses by coherently combining data from multiple illumination angles.
• Digital wave front correction (DWC) can be applied during reconstruction to correct for lens aberrations, improving image fidelity.
• The technique achieves wide-field, high-resolution imaging with a depth of field and field of view comparable to low-magnification optics, while matching the resolution of high-NA systems.
• Simulations show that with a 10× magnification and a detector field of view of 1 mm, a resolution of 10 nm is achievable when the full synthetic aperture is synthesized.
• The method is robust to non-ideal pupil functions, such as the Gaussian profile of compound refractive lenses, when properly modeled in the reconstruction algorithm.