While holography truly constitutes an ingenious concept, ever since its invention by Gabor it has been troubled by the so-called twin-image problem limiting the information that can be obtained from a holographic record. For symmetry reasons there are always two images appearing in the reconstruction of a hologram and the unwanted out of focus twin-image obscures the object. Here we show a universal method of reconstructing a hologram completely free of twin-image disturbances while no assumptions about absorbing or phase shifting properties of the object need to be imposed. Thus, truthful amplitude and phase distributions are retrieved.

Here we present practical methods for simulation and reconstruction of in-line digital holograms recorded with plane and spherical waves. The algorithms described here are applicable to holographic imaging of an object exhibiting absorption as well as phase-shifting properties. Optimal parameters, related to distances, sampling rate, and other factors for successful simulation and reconstruction of holograms are evaluated and criteria for the achievable resolution are worked out. Moreover, we show that the numerical procedures for the reconstruction of holograms recorded with plane and spherical waves are identical under certain conditions. Experimental examples of holograms and their reconstructions are also discussed.

We demonstrate the application of graphene as a support for imaging individual biological molecules in transmission electron microscope (TEM). A simple procedure to produce free-standing graphene membranes has been designed. Such membranes are extremely robust and can support practically any submicrometer object. Tobacco mosaic virus has been deposited on graphene samples and observed in a TEM. High contrast has been achieved even though no staining has been applied.

Terahertz waves provide a better contrast in imaging soft biomedical tissues than X-rays and unlike X-rays, they cause no ionisation damage, making them a good option for biomedical imaging. Terahertz absorption imaging has conventionally been used for cancer diagnosis. However, the absorption properties of a cancerous sample are influenced by two opposing factors: an increase in absorption due to a higher degree of hydration and a decrease in absorption due to structural changes. It is therefore difficult to diagnose cancer from an absorption image. Phase imaging can thus be critical for diagnostics. We demonstrate imaging of the absorption and phase-shift distributions of 3.2 mm × 2.3 mm × 30-μm-thick human hepatocellular carcinoma tissue by continuous-wave terahertz digital in-line holography. The acquisition time of a few seconds for a single in-line hologram is much shorter than that of other terahertz diagnostic techniques and future detectors will allow acquisition of meaningful holograms without sample dehydration. The resolution of the reconstructions was enhanced by sub-pixel shifting and extrapolation. Another advantage of this technique is its relaxed minimal sample size limitation. The fibrosis indicated in the phase distribution demonstrates the potential of terahertz holographic imaging to obtain a more objective, early diagnosis of cancer.

This paper provides a tutorial of iterative phase retrieval algorithms based on the Gerchberg–Saxton (GS) algorithm applied in digital holography. In addition, a novel GS-based algorithm that allows reconstruction of 3D samples is demonstrated. The GS-based algorithms recover complex-valued wavefront-by-wavefront back-and-forth propagation between two planes with constraints superimposed in these two planes. Iterative phase retrieval allows quantitatively correct and twin-image-free reconstructions of object amplitude and phase distributions from its in-line hologram. The present work derives the quantitative criteria on how many holograms are required to reconstruct a complex-valued object distribution, be it a 2D or 3D sample. It is shown that for a sample that can be approximated as a 2D sample, a single-shot in-line hologram is sufficient to reconstruct the absorption and phase distributions of the sample. Previously, the GS-based algorithms have been successfully employed to reconstruct samples that are limited to a 2D plane. However, realistic physical objects always have some finite thickness and therefore are 3D rather than 2D objects. This study demonstrates that 3D samples, including 3D phase objects, can be reconstructed from two or more holograms. It is shown that in principle, two holograms are sufficient to recover the entire wavefront diffracted by a 3D sample distribution. In this method, the reconstruction is performed by applying iterative phase retrieval between the planes where intensity was measured. The recovered complex-valued wavefront is then propagated back to the sample planes, thus reconstructing the 3D distribution of the sample. This method can be applied for 3D samples such as 3D distribution of particles, thick biological samples, and other 3D phase objects. Examples of reconstructions of 3D objects, including phase objects, are provided. Resolution enhancement obtained by iterative extrapolation of holograms is also discussed.

Significance: We report a method to image and reveal structural details of proteins on a truly single-molecule level. Low-energy electron holography is used to image individual proteins electrospray deposited on freestanding graphene. In contrast to the current state of the art in structural biology, we do away with the need for averaging over many molecules. This is crucial because proteins are flexible objects that can assume distinct conformations often associated with different functions. Proteins are also the targets of almost all the currently known and available drugs. The design of new and more effective drugs relies on the knowledge of the targeted proteins structure in all its biologically significant conformations at the best possible resolution.