The physical structure of materials is responsible for the different properties that these materials possess. Understanding structure-property relationships are one of the primary tasks of modern material science. The scattering of X-ray radiation from a sample region allows access to the electron density distribution, lattice spacing within crystals, materials composition, and other structural information.
Bragg coherent diffractive imaging (BCDI) is among the novel techniques where information from individual nanostructures is available to the experimenter. This shift of paradigm from studies of common structural signatures in ensembles of nanostructures to the imaging of individual goes together with the development of manufacturing techniques, where the quest for miniaturization has led far beyond sub-micron scales. Characterization of nanoscale materials requires X-rays with coherence larger than the volume of nanostructure under study. Coherent volume is defined by the spatial and temporal coherence lengths. Photon emission in radiation sources shows different degree of spontaneity, resulting in various degree of correlation between wavefronts at different coordinate points (spatial coherence) and at different moments in time (temporal coherence). To increase the degree of spatial coherence filters such as pinholes and slits are introduced in the beam while monochromators improve temporal coherence.
The advantage of X-ray techniques over alternative nanoscale imaging methods is reduced by the challenges in dedicated optics fabrication. This challenge forced the X-ray community to approach X-ray microscopes designs differently from optical analogies. Modern detectors of X-ray radiation are not capable of recording high-resolution wavefront information. The interferometric approach allows recording of phase but is currently limited by manufacturing precision of X-ray gratings. This inability to record the high-resolution phase is called the phase problem. Given both the phase and amplitude information are available, one can simply inverse Fourier transform the diffraction pattern and obtain real space image of the sample with high resolution, effectively performing computational image formation instead of optical. Importance of the phase information can be illustrated by swapping phases of two images in their Fourier transform (see Figure).
Importance of the phase information in the Fourier space. The swapping of phases in images’ Fourier domain results in swapping of spatial distribution of the signal in real space. This suggests that the phase plays an important part in encoding the underlying structure of images and is essential for their reconstruction. [D. Karpov & E. Fohtung, Journal of Applied Physics 125, 121101 (2019)]
To by pass the phase problem a number of algorithmic approaches emerged that attempt to iteratively reconstruct missing phase information. These approaches are largely based on initial observation of Sayre regarding the sampling requirements for direct structure determination. Series of algorithms for phase retrieval rely on the idea that if a finite support of the object in real space and partial information on the Fourier transform of the object in reciprocal space are available then the phase information can be retrieved. The groundwork was done by Fienup which resulted in the development of error-reduction and input-output algorithms. The oversampling requirements are discussed in detail in the work by Miao et al.
Birefringent Coherent Diffraction Imaging
The directional dependence of the index of refraction contains a wealth of information about anisotropic optical properties in semiconducting and transparent insulating materials. Here we develop a novel high-resolution lens-less technique that uses birefringence as a contrast mechanism to map the index of refraction and dielectric permittivity in optically anisotropic materials.
We have applied this approach successfully to a liquid crystal polymer film using polarized light from a helium-neon laser. This approach is scalable to imaging with diffraction-limited resolution, a prospect rapidly becoming a reality in view of emergent brilliant X-ray sources. Applications of this novel imaging technique are in disruptive technologies, including novel electronic devices, in which both charge and spin carry information as in multiferroic materials and photonic materials such as light modulators and optical storage.