Digital in-line holography

H.]. Kreuzer and R.A. Pawlitzek" 'Department ofPhysics, Dalhousie University, Halifax, Canada "IBM Zurich Research Laboratory, Switzerland out by larger objects, classical diffraction becomes more important. When it eventually dominates the image,we are in the regime of classical wave optics. A hologram is just a complicated interference pattern from which little information about the shape and structure of the object can be obtained by simple inspection. What object, as an example, produced the hologram in panel (a) of Fig.2? Holographywas therefore conceived by Gabor as a two-step process: first, a hologram must be recorded, and second, reconstruction must yield an "image" of the object, i.e. the intensity of the scattered wavefront at the object. The reason why the second step is possible is the fact that the holographic information is linear in the scatteredwave so that an inversion ofthe scattering process is feasible. This, Gabor envisaged, could be achieved by recording the hologram photographically on a transparent plate and then "looking" through the hologram from the back. As Gabor put it: "One must expect that looking through such a properlyprocessed diagram one will see behind it the original object, as if it were in place:' Gabor tested this idea with visible light but the quality of the reconstruction was not overwhelming. More importantly, the application in electron microscopy needed modification for two reasons: (1) A photographic plate is not transparent to electrons, and (2) using the same wavelength for the reconstruction would not yield any magnification. Therefore Gabor suggested to use visible light for the reconstruction after scaling up all dimensions in the ratio of light waves to electron waves, that is, by about a factor 100,000. Although attempts at electron holography were made early on, their success was very limited, essentially because the condenser lenses for electrons at the time were so poor that no focus of the size ofthe electron wavelength ofmuch less than angstroms could be achieved. A breakthrough came in the late 1980's when Fink and collaborators showed that an ultrathin metal tip with one or a few atoms at its apex serves as a "point" source with a virtual source size ofatomic dimensions for a coherent beam ofelectrons with energies in the 10-200eV range. A new kind of microscopy with point sources, now called Low EnergyElectron Point Source (LEEPS) microscopy, evolved from combining the technological tools of scanning probe microscopywith new ideas in projection microscopy.We refer the reader to a recent review article with further details and references.[] In the optical realm holography took off in the early 1970's after the availability oflasers.Although holography with spherical waves, as originally proposed by Gabor and now called in-line holography, is the simplest realization ofthe holographic method, because it works without lenses, its applications had been limited until recently due to the fact that reconstruction of the object image with another wave is not practical. To avoid this problem various schemes of off-line holography have been developed in which a laser beam is split to provide an undisturbed reference wave while the other beam is focussed onto the object. There are too manyvariations on this scheme to allow an adequate overview here and the reader is referred to two books [4, 5] in which the principles and the practical implementations are described in detail. The remainder of this article will deal exclusively with our work on digital in-line holography with numerical reconstruction. l In digital in-line holography-DIH-the hologram is recorded by a detector array, such as a CCD camera for photons, and transferred to a computer in which the reconstruction is done numerically. The role of reconstruction is to obtain the threedimensional structure of the object from the two-dimensional hologram on the screen, or, in physical terms, to reconstruct the