Crystallographic Heterodyne Phase Detection Technique for Highly-Sensitive Lattice-Distortion Measurements

A main topic of inverse problems in crystallography appears to have been phase recovery of diffracted fields in diffraction crystallography1,2. In this paper we deal with a problem of detecting yet another type of phase in a structure image (or a lattice fringe image) obtained by the direct observation of a crystal using a high-resolution electron microscope. The phase in our problem represents spatial distortion of lattices in a crystal rather than the phase of X-ray or electron wave fields. We note that a quasi periodic structure of atoms observed in a TEM (transmission electron microscopy) image or in a STM (scanning tunnel microscopy) image bears a close similarity to an optical interferometric fringe pattern having spatial carrier frequencies, where lattice distortion or atom displacement may be regarded as a spatial fringe shift. The interpretation of the distorted lattice image as an interferogram permits us the use of spatial heterodyne technique for highly sensitive detection of the lattice distortion, where a phase change by 2π corresponds to the displacement of an atom by a lattice constant. Based on this interpretation, we propose a crystallographic heterodyne technique for precisely determining the positions of dislocated atoms using the Fourier fringe analysis technique originally developed for optical heterodyne interferometry3,4.