Understanding electronic structure at nanometer resolution is crucial to understanding physics such as phase separation and emergent behavior in correlated electron materials. Nondestructive probes which have the ability to see beyond surfaces on nanometer length and sub-picosecond time scales can greatly enhance our understanding of these systems and will impact development of future technologies, such as magnetic storage. Polarized x-rays are an appealing choice of probe due to their penetrating power, elemental and magnetic specificity, and high spatial resolution. The resolution of traditional x-ray microscopy is limited by the nanometer precision required to fabricate x-ray optics. In this thesis, a novel approach to lensless imaging of an extended magnetic nanostructure is presented. We demonstrate this approach by imaging ferrimagnetic "maze" domains in a Gd/Fe multilayer with perpendicular anisotropy. A series of dichroic coherent diffraction patterns, ptychographically recorded, are numerically inverted using non-convex and non-linear optimization theory, and we follow the magnetic domain configuration evolution through part of its magnetization hysteresis loop by applying an external magnetic field. Unlike holographic methods, it does not require a reference wave or precision optics, and so is a far simpler experiment. In addition, it enables the imaging of samples with arbitrarily large spatial dimensions, at a spatial resolution limited solely by the coherent x-ray flux and wavelength. It can readily be extended to other non-magnetic systems that exhibit circular or linear dichroism. This approach is scalable to imaging with diffraction-limited resolution, a prospect rapidly becoming a reality in view of the new generation of phenomenally brilliant x-ray sources.
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