Holograms

Dear Dr. Duax

I agree with Profs. Spence and Cowley's letter, their historical remarks that "atomic" resolution was achieved previously in electron holography. However, I beg to differ with them that Tegze & Faigel's experiment is not a hologram.

Holograms are formed by the interference of a reference wave and an object wave. The intensity of the interference pattern depends on the cosine of the relative phase of the two waves. As the cosine of an angle is the same as the cosine of the negative angle, the object wave is not uniquely specified by the hologram. If the signs of all the "correct" relative phases are changed to their negative, one obtains another image that is called the dual image in holography. Of course, if some of the signs are changed and others are not, all sorts of "garbage" can be obtained.

The analogy with X-ray crystallography can be best described if part of the crystal structure is known, but another part is unknown. The X-rays scattered by the known part are analogous to the reference wave and those scattered by the unknown part are analogous to the object wave in holography. The measured intensity of a Bragg reflection contains an interference term, just like a hologram does. The crystallographic phase problem in this situation is analogous to the sign uncertainty described above. It can be shown that if the measured structure factors are combined with the phases of the known part in order to produce a "difference Fourier omit map" one obtains half the sum of the "correct" image and its dual image. In general, additional information has to be used in order to do better.

For example, the solution of a crystal structure by direct methods can be perceived as the use of positivity and atomicity in order to eliminate the dual image. Similarly, the use of multiple isomorphous replacement (MIR) and of anomalous dispersion (MAD) can be perceived as the simultaneous solution of several compatible holograms.

In the X-Ray holograms of Tegze & Faigel the reference wave is the characteristic X-ray emitted by an atom and the object wave is that part of the wave that is scattered by the surrounding atoms. The measured angular distribution of the X-rays from a particular atom is therefore analogous to a hologram. As the X-ray emission from each atom is completely incoherent with that of all other atoms, they obtain a simple sum of all the holograms. As long as the direction of observation avoids the directions of Bragg reflections, multiple scattering is negligible. Therefore in Tegze & Faigel's experiments there is no connection with Kossel lines. As a proof of the correctness of the theory and of their great experimental prowess they obtain the sum of the atomic structure and its enantiomorph using Gabor's holographic inversion algorithm. In that sense, what they have done is different from crystallography or from "ordinary diffraction". In my view, the difference comes from their knowledge of the point reference. I would like to comment on the ingenious work of Gog et al. They have shown that their experiment contains more information than Tegze & Faigel obtain. In principle, they can reconstruct the crystal structure ab initio (and eliminate the dual image). In Gog et al.'s experiments the incident beam is coherent, but the coherence is "lost" by the excitation of the atoms. Both experiments have to avoid the Bragg conditions (and Kossel lines) very carefully. Second, the difficulty with either methods of X-ray holography is the low contrast of the hologram. Therefore radiation resistant specimens are needed. Other, similar methods can be dreamt up that may be applicable to radiation sensitive crystals.