OPTIMIZATION OF K-SPACE SAMPLING IN ATOMIC IMAGING BY ELECTRON-EMISSION HOLOGRAPHY

Two limiting-case algorithms have previously been proposed for hologra phically imaging atoms near surfaces using photoelectron diffraction d ata and other diffraction data associated with electron emission:(i) a phased sum of Fourier transforms of scanned-angle data taken at sever al energies from Barton, (ii) and a phased sum of Fourier transforms o f scanned-energy data taken along several directions due to Tong et al . We first point out that both methods are equivalent three-dimensiona l transforms in the wave Vector k of the emitted electron, differing o nly in the way they sample k-space. A continuum of different sampling densities in the direction and magnitude of k exists in such holograph y, spanning the two limits previously discussed. An additional variant on these methods involves using only a small cone of data in k-space for each transform. Using model diffraction calculations for localized electron emission (e.g., core photoelectron emission) from Cu(001) cl usters, we have explored the full range of k-space sampling possible, and find that optimum image quality is expected for choices intermedia te between the extreme limits of scanned-angle or scanned-energy. Gene ral rules for optimizing image quality for a given data-set range are also discussed, and used to evaluate the sampling choices made in some prior experimental studies.