The science of holography was born with Dennis Gabor’s monumental idea of coherent imaging as a two-step process, namely, recording followed by reconstruction. Gabor’s original formulation of holography treated the problem of reconstructing both the amplitude and the phase of a coherent monochromatic optical disturbance from a single Fresnel diffraction pattern of an object of interest. Unfortunately this original formulation, now known as “inline holography,” suffered from the famous “twin-image problem” in which the complex field to be reconstructed was polluted with a spatially overlaid “twin” image. This twin-image problem was later overcome by the method of “off-axis holography” due to Leith and Upatnieks.

Inline holography encodes information regarding a complex optical disturbance, via the pattern of fringes created when the unscattered beam (“reference wave”) interferes with the beam scattered by the object (“object wave”). Off-axis holography encodes phase information by interfering a separate reference beam with the beam scattered by the object. Fourier transform holography—the origins of which are associated with names such as Baez, Stroke, Winthrop, and Worthington—is a special case of off-axis holography in which the reference wave and the object wave emanate from the two respective pinholes of a Young-type interferometer. The first (larger) pinhole contains the sample and provides the object wave, while the second (smaller) pinhole contains no sample and provides the reference wave. The Fourier hologram is the far-field diffraction pattern obtained when this modified Young-two-pinhole arrangement is coherently illuminated. Importantly, the complex disturbance at the exit surface of the object-containing pinhole, can be reconstructed by simply taking the inverse Fourier transform of the Fourier hologram.

A drawback of the above scenario for Fourier holography is that the object is typically rigidly attached to the modified Young-two-pinhole mask described above. Very recently Stickler et al. and Tieg et al. have overcome this limitation, in the context of Fourier transform holography using soft X-rays, by separating the sample from the Young-type mask (see Refs. 10 and 11 of Guehrs et al.). This allows a sample to be scanned, thereby building up a larger field-of-view image via a sequence of images obtained through the “keyhole” of the object-containing pinhole. However, the previously mentioned mask–sample separation introduces an inevitable gap between the object and the mask, leading to a defocus-induced error in reconstructions obtained using the standard method of processing Fourier holograms (i.e., using a single inverse Fourier transform).

As shown in the present work by Guehrs et al., the effects of the mask–sample gap can be accounted for by suitably modifying this standard procedure for analysis of Fourier transform holograms. In an elegant and beautifully simple result, one need only multiply the Fourier hologram by the free space transfer function corresponding to the gap distance, prior to taking the inverse Fourier transform. The efficacy of this procedure, which results in “refocusing” the reconstruction of the complex wavefield at the exit surface of the sample, is clearly and convincingly demonstrated through a sequence of soft-X-ray experiments employing a diatom as an object, undertaken at the BESSY II UE52-SGM undulator in Germany.

This extension to the standard method of Fourier holography has ramifications wider than the present context of soft X-rays, insofar as it is also applicable to a variety of other coherent radiation and matter wavefields. In addition to its fundamental contribution to the science of Fourier holography, the present work by Guehrs et al. also has important wider ramifications for related hot areas of contemporary work in coherent X-ray optics, such as coherent diffractive imaging and ptychographical phase retrieval. The spotlighted work also touches upon the concept of “virtual optics,” in which the computer forms a meaningful extension to a coherent imaging system, with both hardware and software optics processing optical information at the complex-field level.

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