Depth-resolved holographic reconstructions by three-dimensional deconvolution: erratum.

Correction of Eq. (20) in our published article [Opt. Express18, 22527 (2010)10.1364/OE.18.022527].

Three-Dimensional Structure from Single Two-Dimensional Diffraction Intensity Measurement.

Conventional three-dimensional (3D) imaging methods require multiple measurements of the sample in different orientation or scanning. When the sample is probed with coherent waves, a single two-dimensional (2D) intensity measurement is sufficient as it contains all the information of the 3D sample distribution. We show a method that allows reconstruction of 3D sample distribution from a single 2D intensity measurement, at the z resolution exceeding the classical limit. The method can be practical for radiation-sensitive materials, or where the experimental setup allows only one intensity measurement.

Three-dimensional volumetric deconvolution in coherent optics and holography.

Methods of three-dimensional deconvolution (3DD) or volumetric deconvolution of optical complex-valued wavefronts diffracted by 3D samples with the 3D point spread function are presented. Particularly, the quantitative correctness of the recovered 3D sample distributions is addressed. Samples consisting of point-like objects can be retrieved from their 3D diffracted wavefronts with non-iterative (Wiener filter) 3DD. Continuous extended samples, including complex-valued (phase) samples, can be retrieved with iterative (Gold and Richardson-Lucy) 3DD algorithms. It is shown that quantitatively correct 3D sample distribution can be recovered only with iterative 3DD, and with the optimal protocols provided. It is demonstrated that 3DD can improve the lateral resolution to the resolution limit, and the axial resolution can be at least four times better than the resolution limit. The presented 3DD methods of complex-valued optical fields can be applied for 3D optical imaging and holography.

Convergent beam electron holography for analysis of van der Waals heterostructures.

The van der Waals heterostructures, which explore the synergetic properties of 2D materials when assembled into 3D stacks, have already brought to life a number of exciting phenomena and electronic devices. Still, the interaction between the layers in such assembly, possible surface reconstruction, and intrinsic and extrinsic defects are very difficult to characterize by any method, because of the single-atomic nature of the crystals involved. Here we present a convergent beam electron holographic technique which allows imaging of the stacking order in such heterostructures. Based on the interference of electron waves scattered on different crystals in the stack, this approach allows one to reconstruct the relative rotation, stretching, and out-of-plane corrugation of the layers with atomic precision. Being holographic in nature, our approach allows extraction of quantitative information about the 3D structure of the typical defects from a single image covering thousands of square nanometers. Furthermore, qualitative information about the defects in the stack can be extracted from the convergent diffraction patterns even without reconstruction, simply by comparing the patterns in different diffraction spots. We expect that convergent beam electron holography will be widely used to study the properties of van der Waals heterostructures.

Iterative phase retrieval for digital holography: tutorial: publisher's note.

This publisher's note corrects the paper type and title of J. Opt. Soc. Am. A36, D31 (2019)JOAOD60740-323210.1364/JOSAA.36.000D31.

Imaging proteins at the single-molecule level.

Imaging single proteins has been a long-standing ambition for advancing various fields in natural science, as for instance structural biology, biophysics, and molecular nanotechnology. In particular, revealing the distinct conformations of an individual protein is of utmost importance. Here, we show the imaging of individual proteins and protein complexes by low-energy electron holography. Samples of individual proteins and protein complexes on ultraclean freestanding graphene were prepared by soft-landing electrospray ion beam deposition, which allows chemical- and conformational-specific selection and gentle deposition. Low-energy electrons do not induce radiation damage, which enables acquiring subnanometer resolution images of individual proteins (cytochrome C and BSA) as well as of protein complexes (hemoglobin), which are not the result of an averaging process.

Iterative phase retrieval for digital holography: tutorial.

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 a complex-valued wavefront using 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.

Lateral and axial resolution criteria in incoherent and coherent optics and holography, near- and far-field regimes.

This work presents an overview of spatial resolution criteria in classical optics, digital optics, and holography. Although the classical Abbe and Rayleigh resolution criteria have been thoroughly discussed in the literature, there are a few issues that still need to be addressed, e.g., the axial resolution criteria for coherent and incoherent radiation (which is a crucial parameter in 3D imaging), the resolution criteria in the Fresnel regime, and the lateral and the axial resolution criteria in digital optics and holography. This work discusses these issues and provides a simple guide on which resolution criteria should be applied for a particular imaging scheme: coherent/incoherent, far- and near-field, lateral and axial resolution. Different resolution criteria such as two-points resolution and the resolution obtained from the image spectrum (diffraction pattern) are compared and demonstrated with simulated examples. It is shown that, for coherent light, the classical Abbe and Rayleigh resolution criteria do not provide accurate estimation of the lateral and axial resolution. The lateral and axial resolution criteria based on the evaluation of the spectrum of the diffracted wave provide more precise estimation of the resolution for coherent and incoherent light. It is also shown that the resolution criteria derived in approximation of the far-field can be applied for the near-field (Fresnel) imaging regime.

Metal Adsorption and Nucleation on Free-Standing Graphene by Low-Energy Electron Point Source Microscopy.

The interaction of metals with carbon materials (and specifically with graphene) is of importance for various technological applications. In particular, the intercalation of alkali metals is believed to provide a means for tuning the electronic properties of graphene for device applications. While the macroscopic effects of such intercalation events can readily be studied, following the related processes at an atomic scale in detail and under well-defined experimental conditions constitutes a challenge. Here, we investigate in situ the adsorption and nucleation of the alkali metals K, Cs, and Li on free-standing graphene by means of low-energy electron point source microscopy. We find that alkali metals readily intercalate between the layers of bilayer graphene. In fact, the equilibrium distribution of K and Cs favors a much-higher particle density between the layers than on the single-layer graphene. We obtain a quantitative value for the difference of the free energy of the binding between these two domains. Our study is completed with a control experiment introducing Pd as a representative of the nonalkali metals. Now, we observe cluster formation in equal measure on both single-layer and bilayer graphene; however, there was no intercalation. Our investigations thus constitute the first in situ study of metal-atom sorption of different specificity on free-standing graphene.

Three-dimensional double helical DNA structure directly revealed from its X-ray fiber diffraction pattern by iterative phase retrieval.

Coherent diffraction imaging (CDI) allows the retrieval of an isolated object's structure, such as a macromolecule, from its diffraction pattern. CDI requires the fulfillment of two conditions: the imaging radiation must be coherent and the object must be isolated. We discuss that it is possible to directly retrieve the molecular structure from its diffraction pattern, which was acquired neither with coherent radiation nor from an individual molecule. This is provided that the molecule exhibits periodicity in one direction, as in the case of fiber diffraction. We demonstrate that, when we apply iterative phase retrieval methods to a fiber diffraction pattern, the repeating unit; that is, the molecule structure, can directly be reconstructed without any prior modeling. For example, we recover the the DNA double helix's structure in three-dimensions from its two-dimensional X-ray fiber diffraction pattern, Photograph (Photo) 51, which was acquired in Raymond Gosling and Rosalind Franklin's famous experiment at a resolution of 3.4 Å.

Iterative phase retrieval in coherent diffractive imaging: practical issues.

In this work, issues in phase retrieval in the coherent diffractive imaging (CDI) technique, from discussion on parameters for setting up a CDI experiment to evaluation of the goodness of the final reconstruction, are discussed. The distribution of objects under study by CDI often cannot be cross-validated by another imaging technique. It is therefore important to make sure that the developed CDI procedure delivers an artifact-free object reconstruction. Critical issues that can lead to artifacts are presented and recipes on how to avoid them are provided.

Filming the formation and fluctuation of skyrmion domains by cryo-Lorentz transmission electron microscopy.

Magnetic skyrmions are promising candidates as information carriers in logic or storage devices thanks to their robustness, guaranteed by the topological protection, and their nanometric size. Currently, little is known about the influence of parameters such as disorder, defects, or external stimuli on the long-range spatial distribution and temporal evolution of the skyrmion lattice. Here, using a large (7.3 × 7.3 µm(2)) single-crystal nanoslice (150 nm thick) of Cu2OSeO3, we image up to 70,000 skyrmions by means of cryo-Lorentz transmission electron microscopy as a function of the applied magnetic field. The emergence of the skyrmion lattice from the helimagnetic phase is monitored, revealing the existence of a glassy skyrmion phase at the phase transition field, where patches of an octagonally distorted skyrmion lattice are also discovered. In the skyrmion phase, dislocations are shown to cause the emergence and switching between domains with different lattice orientations, and the temporal fluctuation of these domains is filmed. These results demonstrate the importance of direct-space and real-time imaging of skyrmion domains for addressing both their long-range topology and stability.

Resolution enhancement in in-line holography by numerical compensation of vibrations.

Mechanical vibrations of components of the optical system is one of the sources of blurring of interference pattern in coherent imaging systems. The problem is especially important in holography where the resolution of the reconstructed objects depends on the effective size of the hologram, which is on the extent of the interference pattern, and on the contrast of the interference fringes. We discuss the mathematical relation between the vibrations, the hologram contrast and the reconstructed object. We show how vibrations can be post-filtered out from the hologram or from the reconstructed object assuming a Gaussian distribution of the vibrations. We also provide a numerical example of compensation for directional motion blur. We demonstrate our approach for light optical and electron holograms, acquired with both, plane- as well as spherical-waves. As a result of such hologram deblurring, the resolution of the reconstructed objects is enhanced by almost a factor of 2. We believe that our approach opens up a new venue of post-experimental resolution enhancement in in-line holography by adapting the rich database/catalogue of motion deblurring algorithms developed for photography and image restoration applications.

Order/Disorder Dynamics in a Dodecanethiol-Capped Gold Nanoparticles Supracrystal by Small-Angle Ultrafast Electron Diffraction.

The design and the characterization of functionalized gold nanoparticles supracrystals require atomically resolved information on both the metallic core and the external organic ligand shell. At present, there is no known approach to characterize simultaneously the static local order of the ligands and of the nanoparticles, nor their dynamical evolution. In this work, we apply femtosecond small-angle electron diffraction combined with angular cross-correlation analysis, to retrieve the local arrangement from nanometer to interatomic scales in glassy aggregates. With this technique we study a two-dimensional distribution of functionalized gold nanoparticles deposited on amorphous carbon. We show that the dodecanethiol ligand chains, coating the gold cores, order in a preferential orientation on the nanoparticle surface and throughout the supracrystal. Furthermore, we retrieve the dynamics of the supracrystal upon excitation with light and show that the positional disorder is induced by light pulses, while its overall homogeneity is surprisingly found to transiently increase. This new technique will enable the systematic investigation of the static and dynamical structural properties of nanoassembled materials containing light elements, relevant for several applications including signal processing and biology.

Direct Observation of Individual Charges and Their Dynamics on Graphene by Low-Energy Electron Holography.

Visualizing individual charges confined to molecules and observing their dynamics with high spatial resolution is a challenge for advancing various fields in science, ranging from mesoscopic physics to electron transfer events in biological molecules. We show here that the high sensitivity of low-energy electrons to local electric fields can be employed to directly visualize individual charged adsorbates and to study their behavior in a quantitative way. This makes electron holography a unique probing tool for directly visualizing charge distributions with a sensitivity of a fraction of an elementary charge. Moreover, spatial resolution in the nanometer range and fast data acquisition inherent to lens-less low-energy electron holography allows for direct visual inspection of charge transfer processes.

Creating Airy beams employing a transmissive spatial light modulator.

We present a detailed study of two novel methods for shaping the light optical wavefront by employing a transmissive spatial light modulator (SLM). Conventionally, optical Airy beams are created by employing SLMs in the so-called all-phase mode. In the first method, a numerically simulated lens phase distribution is loaded directly onto the SLM, together with the cubic phase distribution. An Airy beam is generated at the focal plane of the numerical lens. We provide for the first time, to the best of our knowledge, quantitative properties of the formed Airy beam. We derive the formula for deflection of the intensity maximum of the so-formed Airy beam, which is different from the quadratic deflection typical of Airy beams. We cross-validate the derived formula by both simulations and experiment. The second method is based on the fact that a system consisting of a transmissive SLM sandwiched between two polarizers can create a transmission function with negative values. This observation alone has the potential for various other wavefront modulations where the transmission function requires negative values. As an example for this method, we demonstrate that a wavefront can be modulated by passing the SLM system with transmission function with negative values by loading an Airy function distribution directly onto the SLM. Since the Airy function is a real-valued function but also with negative values, an Airy beam can be generated by direct transfer of the Airy function distribution onto such an SLM system. In this way, an Airy beam is generated immediately behind the SLM. As both new methods do not employ a physical lens, the two setups are more compact than conventional setups for creating Airy beams. We compare the performance of the two novel methods and the properties of the created Airy beams.

Practical algorithms for simulation and reconstruction of digital in-line holograms.

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.

Holographic time-resolved particle tracking by means of three-dimensional volumetric deconvolution.

Holographic particle image velocimetry allows tracking particle trajectories in time and space by means of holography. However, the drawback of the technique is that in the three-dimensional particle distribution reconstructed from a hologram, the individual particles can hardly be resolved due to the superimposed out-of-focus signal from neighboring particles. We demonstrate here a three-dimensional volumetric deconvolution applied to the reconstructed wavefront which results in resolving all particles simultaneously in three-dimensions. Moreover, we apply the three-dimensional volumetric deconvolution to reconstructions of a time-dependent sequence of holograms of an ensemble of polystyrene spheres moving in water. From each hologram we simultaneously resolve all particles in the ensemble in three dimensions and from the sequence of holograms we obtain the time-resolved trajectories of individual polystyrene spheres.

Terahertz in-line digital holography of dragonfly hindwing: amplitude and phase reconstruction at enhanced resolution by extrapolation.

We report here on terahertz (THz) digital holography on a biological specimen. A continuous-wave (CW) THz in-line holographic setup was built based on a 2.52 THz CO(2) pumped THz laser and a pyroelectric array detector. We introduced novel statistical method of obtaining true intensity values for the pyroelectric array detector's pixels. Absorption and phase-shifting images of a dragonfly's hindwing were reconstructed simultaneously from single in-line hologram. Furthermore, we applied phase retrieval routines to eliminate twin image and enhanced the resolution of the reconstructions by hologram extrapolation beyond the detector area. The finest observed features are 35 µm width cross veins.

Double-slit holography-a single-shot lensless imaging technique.

In this study, we propose a new method for single-shot, high-resolution lensless imaging called double-slit holography. This technique combines the properties of in-line and off-axis holography in one single-shot measurement using the simplest double-slit device: a plate with two apertures. In double-slit holography, a plane wave illuminates the two apertures giving rise to two spherical waves. While diffraction of one spherical wave from a sample positioned behind the first aperture (the object aperture) provides the object wave, the other spherical wave diffracted from the second (reference) aperture provides the reference wave. The resulting interference pattern in the far-field (hologram) combines the properties of an in-line (or Gabor-type) hologram and an off-axis hologram due to the added reference wave from the second aperture. Both the object and reference waves have the same intensity, which ensures high contrast of the hologram. Due to the off-axis scheme, the amplitude and phase distributions of the sample can be directly reconstructed from the hologram, and the twin image can be easily separated. Due to the object wave being the same as in-line holography with a spherical wave, imaging at different magnifications is similarly done by simply adjusting the aperture-to-sample distance. The resolution of the reconstructed object is given by the numerical aperture of the optical setup and the diameter of the reference aperture. It is shown both by theory and simulations that the resolution of the reconstructed object depends on the diameter of the reference wave aperture but does not depend on the diameter of the object aperture. Light optical proof-of-concept experiments are provided. The proposed method can be particularly practical for X-rays, where optical elements such as beam splitters are not available and conventional off-axis holography schemes cannot be realised.