When holography meets coherent diffraction imaging

Abstract

The phase problem is inherent to crystallographic, astronomical and optical imaging where only the intensity of the scattered signal is detected and the phase information is lost and must somehow be recovered to reconstruct the object's structure. 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. We demonstrate the reconstruction of various diffraction patterns of objects recorded with visible light as well as with low-energy electrons. Although we have demonstrated our HCDI method using laser light and low-energy electrons, it can also be applied to any other coherent radiation such as X-rays or high-energy electrons. ABSTRACTThe phase problem is inherent to crystallographic, astronomical and optical imaging where only the intensity of the scattered signal is detected and the phase information is lost and must somehow be recovered to reconstruct the object's structure. 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 mus...

Coherent microscopy at resolution beyond diffraction limit using post-experimental data extrapolation

“…4(c). The intrinsic resolution of the recorded diffraction pattern, 7 according to the Abbe criterion, amounts to 1.8 lm, being in good agreement with the quality of the reconstruction, shown in Fig. 4(c).…”

“…Here an iterative routine is applied, which includes the following steps: holography, the integral transformation is given by the Fresnel-Huygens principle and must be computed. 6,7 (iii) In the object plane, the following constraints are applied to the reconstructed complex-valued object distribution o(x o ,y o ). Since the object exhibits a finite size, the distribution o(x o ,y o ) is multiplied with a loose mask and the values outside the mask are set to zero.…”

Low-energy electron holographic imaging of individual tobacco mosaic virions

“…The spatial resolution attained in a hologram can be estimated using the Abbe criterion 43,44 and by measuring the largest angle under which interference fringes are observable. 39,40,45 This is illustrated in Fig. 4(b), where an intensity profile along the blue line in the hologram is displayed.…”

“…The numerical reconstruction from the hologram is essentially achieved by back propagation to the object plane, which corresponds to evaluating the Fresnel-Kirchhoff integral transformation. [35][36][37][38][39][40] In low-energy electron holography, a lens-less technique not suffering from lens aberrations, the resolution limit is given by the deBroglie wavelengths (k) and by the numerical aperture (NA) of the detector system. With k being as small as 0.7 Å and NA ¼ 0.54, Angstrom and even atomic resolution shall eventually be possible.…”

Graphene Unit Cell Imaging by Holographic Coherent Diffraction

“…Since the phase of the scattered wave is not known a priori, we also record a hologram of the very same sample and, hence, gain coarse information about the phase distribution. The combination of this initial phase distribution with the single high-resolution diffraction pattern [9] allows us to iteratively reconstruct the entire 210 nm diameter graphene sheet displaying about 660.000 unit cells at once. A schematic of the experimental setup with its two modes of operation, coherent diffraction and holography, is displayed in Fig.…”