Lensless imaging with high-harmonic sources
Authors:   , de Beurs Anne Christiaan Cornelis 
Published: October 2023

Coherent diffractive imaging (CDI) is a family of computational imaging techniques that uses iterative reconstruction algorithms to decipher the information encoded in one or more interference patterns to reconstruct an image of an object located in another propagation plane.
The lensless nature of these techniques makes them well-suited for imaging with coherent extreme ultraviolet (EUV) or x-ray illumination as refractive optics are limited at these wavelengths.
In particular, this work investigates the use of CDI techniques in combination with high-harmonic generation. High-harmonic generation~(HHG) sources can generate EUV illumination beams with a high degree of spatial coherence in a compact tabletop setup.
In this work we use Fourier-Transform spectroscopy~(FTS) to separate sets of nearly monochromatic diffraction patterns from a broadband HHG diffraction pattern. These monochromatic diffraction patterns can used to reconstruct spectrally resolved images through reconstruction methods that are similar to those applied in conventional CDI.
In Chapter 4 we describe how we use a common path interferometer and a noncollinear chirped pulse amplifier system to generate phase locked 25 fs pulse pairs with a central wavelength of approximately 800 nm and a combined pulse energy of 10 mJ.
These infrared driving laser pulses are focused at slightly separated locations in a noble gas jet to upconvert them into a pair of almost identical high-harmonic pulses.
In FTS-based imaging experiments, we illuminate a sample with the HHG pulse pairs and record the far-field diffraction pattern as a function of pulse-to-pulse time delay. The spatial separation of our two harmonic beams results in spatial interference between two laterally sheared copies of the diffraction pattern.
As a consequence of the geometry, the spectrally separated diffraction patterns obtained in these measurements are similar, but not identical to the standard CDI case.
In this work, we demonstrated an algorithm, called diffractive shear interferometry (DSI), to reconstruct images from such diffraction patterns. Using this algorithm, the information present in these diffraction patterns is used to reconstruct complex images of the sample. The reverse problem is either constrained by combining an diffraction pattern with a finite object support prior in Chapter 5 or with other diffraction patterns with a different relative orientation between the shear and the object.
One of the advantages of coherent diffractive imaging techniques is that it they reconstruct the full complex electric field at the sample. In reflection mode, such phase difference can be easily attributed to height differences of the reflecting surface.
However, most research in diffractive imaging has focused on transmission mode imaging.
At the EUV wavelengths generated by HHG sources normal incidence reflection coefficients are vanishingly small. However towards grazing incidence the reflection coefficients approach one.
Such a geometry does come at a cost of added experimental and computational complexity. While far-field diffraction between colinear planes can be described by a straight forward Fourier transform of the electric field, for the propagation between non-collinear planes, an additional non-linear coordinate transformation is required. This coordinate transformation depends on the tilt angle of the fields and becomes very sensitive to the exact tilt-angle towards grazing incidence.
While CDI itself requires accurate knowledge of the wave propagation, a technique known as ptychography offers more flexibility, as it is often possible to solve for more variables than just the object field.
In Chapter 7 we use that property to demonstrate an auto-calibration algorithm that can iteratively calibrate the tilt-angle during a ptychographic reconstruction.
Using this approach we were able to refine the tilt angle close to the correct value even when the initial estimates were off by more than 5 degrees, greatly improving flexibility in reflection-mode lensless imaging.