When holography meets coherent diffraction imaging

Modern imaging techniques at the molecular scale rely on utilizing novel coherent light sources like X-ray free electron lasers for the ultimate goal of visualizing such objects as individual biomolecules rather than crystals. Here, unlike in the case of crystals where structures can be solved by model building and phase refinement, the phase distribution of the wave scattered by an individual molecule must directly be recovered. There are two well-known solutions to the phase problem: holography and coherent diffraction imaging (CDI). Both techniques have their pros and cons. In holography, the reconstruction of the scattered complex-valued object wave is directly provided by a well-defined reference wave that must cover the entire detector area which often is an experimental challenge. CDI provides the highest possible, only wavelength limited, resolution, but the phase recovery is an iterative process which requires some pre-defined information about the object and whose outcome is not always uniquely-defined. Moreover, the diffraction patterns must be recorded under oversampling conditions, a pre-requisite to be able to solve the phase problem. Here, we report how holography and CDI can be merged into one superior technique: holographic coherent diffraction imaging (HCDI). An inline hologram can be recorded by employing a modified CDI experimental scheme. We demonstrate that the amplitude of the Fourier transform of an inline hologram is related to the complex-valued visibility, thus providing information on both, the amplitude and the phase of the scattered wave in the plane of the diffraction pattern. With the phase information available, the condition of oversampling the diffraction patterns can be relaxed, and the phase problem can be solved in a fast and unambiguous manner.Comment: 22 pages, 7 figure

Time-spliced X-ray Diffraction Imaging

Diffraction imaging of non-equilibrium dynamics at atomic resolution is becoming possible with X-ray free-electron lasers. However, there are unresolved problems with applying this method to objects that are confined in only one dimension. Here I show that one-dimensional coherent diffraction imaging is possible by splicing together images recovered from different delays in a time-resolved experiment. This is used to image the time and space evolution of antiferromagnetic order in a complex oxide heterostructure from measurements of a resonant soft X-ray diffraction peak. Mid-infrared excitation of the substrate is shown to lead to a magnetic front that propagates at a velocity exceeding the speed of sound, a critical observation for the understanding of driven phase transitions in complex condensed matter

Phaseless VLBI mapping of compact extragalactic radio sources

The problem of phaseless aperture synthesis is of current interest in phase-unstable VLBI with a small number of elements when either the use of closure phases is not possible (a two-element interferometer) or their quality and number are not enough for acceptable image reconstruction by standard adaptive calibration methods. Therefore, we discuss the problem of unique image reconstruction only from the spectrum magnitude of a source. We suggest an efficient method for phaseless VLBI mapping of compact extragalactic radio sources. This method is based on the reconstruction of the spectrum magnitude for a source on the entire UV plane from the measured visibility magnitude on a limited set of points and the reconstruction of the sought-for image of the source by Fienup's method from the spectrum magnitude reconstructed at the first stage. We present the results of our mapping of the extragalactic radio source 2200 +420 using astrometric and geodetic observations on a global VLBI array. Particular attention is given to studying the capabilities of a two-element interferometer in connection with the putting into operation of a Russian-made radio interferometer based on Quasar RT-32 radio telescopes.Comment: 21 pages, 6 figure

Computational Inverse Problems for Partial Differential Equations

The problem of determining unknown quantities in a PDE from measurements of (part of) the solution to this PDE arises in a wide range of applications in science, technology, medicine, and finance. The unknown quantity may e.g. be a coefficient, an initial or a boundary condition, a source term, or the shape of a boundary. The identification of such quantities is often computationally challenging and requires profound knowledge of the analytical properties of the underlying PDE as well as numerical techniques. The focus of this workshop was on applications in phase retrieval, imaging with waves in random media, and seismology of the Earth and the Sun, a further emphasis was put on stochastic aspects in the context of uncertainty quantification and parameter identification in stochastic differential equations. Many open problems and mathematical challenges in application fields were addressed, and intensive discussions provided an insight into the high potential of joining deep knowledge in numerical analysis, partial differential equations, and regularization, but also in mathematical statistics, homogenization, optimization, differential geometry, numerical linear algebra, and variational analysis to tackle these challenges