Ewald Sphere Construction Research Articles

Fast generation of spectrally shaped disorder.

Media with correlated disorder display unexpected transport properties, but it is still a challenge to design structures with desired spectral features at scale. In this work, we introduce an optimal formulation of this inverse problem by means of the nonuniform fast Fourier transform, thus arriving at an algorithm capable of generating systems with arbitrary spectral properties, with a computational cost that scales O(NlogN) with system size. The method is extended to accommodate arbitrary real-space interactions, such as short-range repulsion, to simultaneously control short- and long-range correlations. We thus generate the largest-ever stealthy hyperuniform configurations in 2d (N=10^{9}) and 3d (N>10^{7}) and demonstrate the flexibility of the approach by generating structures with designed spectral features at scale. By an Ewald sphere construction we link the spectral and optical properties at the single-scattering level and show that stealthy hyperuniform structures generically display transmission gaps, providing a concrete example of fine-tuning of a physical property. We also show that large 3d power-law hyperuniformity in particle packings leads to single-scattering properties nearly identical to those of simple hard spheres. Finally, we demonstrate generalizations of the approach to impose features in either continuous or discrete real space, using constraints in either continuous or discrete reciprocal space. In particular, enforcing large spectral power at peaks with the right symmetry leads to the nondeterministic generation of quasicrystalline structures in 2d and 3d. This technique should become an essential tool to embed, and understand the role of, long-range correlations in disordered metamaterials.

EwaldSphere: an interactive approach to teaching the Ewald sphere construction

EwaldSphere is a Microsoft Windows computer program that superimposes the Ewald sphere construction onto a small-molecule single-crystal X-ray diffractometer. The main objective of the software is to facilitate teaching of the Ewald sphere construction by depicting our classical description of the X-ray diffraction process as a three-dimensional model that can be explored interactively. Several features of the program are also useful for introducing students to the operation of a diffractometer. EwaldSphere creates a virtual reciprocal lattice based on user-defined unit-cell parameters. The Ewald sphere construction is then rendered visible, and the user can explore the effects of changing various diffractometer parameters (e.g. X-ray wavelength and intensity, goniometer angles, and detector distance) on the resulting diffraction pattern as captured by a virtual area detector. Additional digital resources are provided, including a simple but comprehensive program manual, a PowerPoint presentation that introduces the essential concepts, and an Excel file to facilitate calculation of lattice dhk spacings (required for the presentation). The program and accompanying resources are provided free of charge, and there are no restrictions on their use.

Response to Fraser & Wark's comments on A new theory for X-ray diffraction.

The criticisms of my theory, as given by Fraser & Wark [(2018), Acta Cryst. A74, 447-456], are built on a misunderstanding of the concept and the methodology I have used. The assumption they have made rules out my description from which they conclude that my theory is proved to be wrong. They assume that I have misunderstood the diffraction associated with the shape of a crystal and my calculation is only relevant to a parallelepiped and even that I have got wrong. It only appears wrong to Fraser & Wark because the effect I predict has nothing to do with the crystal shape. The effect though can be measured as well as the crystal shape effects. This response describes my reasoning behind the theory, how it can be related to the Ewald sphere construction, and the build-up of the full diffraction pattern from all the scatterers in a stack of planes. It is the latter point that makes the Fraser & Wark analysis incomplete. The description given in this article describes my approach much more precisely with reference to the Ewald sphere construction. Several experiments are described that directly measure the predictions of the new theory, which are explained with reference to the Ewald sphere description. In its simplest terms the new theory can be considered as giving a thickness to the Ewald sphere surface, whereas in the conventional theory it has no thickness. Any thickness immediately informs us that the scattering from a peak at the Bragg angle does not have to be in the Bragg condition to be observed. I believe the conventional theory is a very good approximation, but as soon as it is tested with careful experiments it is shown to be incomplete. The new theory puts forward the idea that there is persistent intensity at the Bragg scattering angle outside the Bragg condition. This intensity is weak (∼10-5) but can be observed in careful laboratory experiments, despite being on the limit of observation, yet it has a profound impact on how we should interpret diffraction patterns.

Photonic glass for high contrast structural color

Non-iridescent structural colors based on disordered arrangement of monodisperse spherical particles, also called photonic glass, show low color saturation due to gradual transition in the reflectivity spectrum. No significant improvement is usually expected from particles optimization, as Mie resonances are broad for small dielectric particles with moderate refractive index. Moreover, the short range order of a photonic glass alone is also insufficient to cause sharp spectral features. We show here, that the combination of a well-chosen particle geometry with the short range order of a photonic glass has strong synergetic effects. Using a first-order approximation and an Ewald sphere construction the reflectivity of such structures can be related to the Fourier transform of the permittivity distribution. The Fourier transform required for a highly saturated color can be achieved by tailoring the substructure of the motif. We show that this can be obtained by choosing core-shell particles with a non-monotonous refractive index distribution from the center of the particle through the shell and into the background material. The first-order theoretical predictions are confirmed by numerical simulations.

Ewald sphere construction for structural colors.

Disordered structures producing a non-iridescent color impression have been shown to feature a spherically shaped Fourier transform of their refractive-index distribution. We determine the direction and efficiency of scattering from thin films made from such structures with the help of the Ewald sphere construction which follows from first-order scattering approximation. This way we present a simple geometrical argument why these structures are well suited for creating short wavelength colors like blue but are hindered from producing long wavelength colors like red. We also numerically synthesize a model structure dedicated to produce a sharp spherical shell in reciprocal space. The reflectivity of this structure as predicted by the first-order approximation is compared to direct electromagnetic simulations. The results indicate the Ewald sphere construction to constitute a simple geometrical tool that can be used to describe and to explain important spectral and directional features of the reflectivity. It is shown that total internal reflection in the film in combination with directed scattering can be used to obtain long wavelength structural colors.

The Ewald sphere construction for radiation, scattering, and diffraction

In most electrodynamics textbooks, the directional gain of an antenna is calculated using analytical integration, and the resulting expression is plotted as an afterthought. From a student's perspective, the analysis may be difficult, mysterious, or unrevealing. In this paper, we show that the Ewald sphere construction, a powerful tool for predicting crystallographic diffraction patterns, can also be used to help students gain direct geometrical insight into antenna radiation patterns. The radiation pattern from a sinusoidally varying current distribution can be obtained intuitively by sketching the reciprocal-space current density and examining how it behaves on an “Ewald” sphere centered at the origin. Furthermore, the nodes of the radiation pattern can be determined quantitatively by locating the intersections of the Ewald sphere with the nodes of the reciprocal-space current density. We illustrate this procedure with several examples, in the context of quantum mechanics, acoustics (sound), and electrodynamics (light). We provide an alternative formulation using the reciprocal-space polarization and magnetization, which treats loop antennas and coil antennas as easily as linear antennas. We make the connection to the original Ewald construction for scattering. We also show how the Ewald construction applies to diffraction through a planar aperture, within the Kirchhoff approximation.

Visible and near UV light-induced scattering of LiNbO3:Fe crystals and material characterization

We present an experimental study of reconstructing parasitic gratings that produce simultaneous wide-angle polarization-anisotropic and -isotropic light induced scattering. The gratings were recorded in lithium niobate crystals doped with iron (LiNbO3:Fe) using an ordinary-polarized pump beam in the visible and the near UV spectral ranges. The reconstruction was performed at different readout angles, wavelengths and linear polarization states. The main features of reconstructing the parasitic gratings at different readout conditions are qualitatively explained by a simple phenomenological model based on the Ewald sphere construction. The obtained results and the scattering pattern were applied to determine the birefringence, its sign and further to estimate the relative contribution of diffusion and bulk photovoltaic effect to the photorefractive effect, as well as the ratios of product of the Pockels and the photovoltaic tensors components that contribute to the readout of the gratings at different readout polarizations.

Full field measurements and identification in Solid Mechanics Depth-resolved Phase Imaging

Traditional full-field interferometric techniques (speckle, moiré, holography etc) encode the surface deformation state of the object under test in the form of 2-D phase images. Over the past 10 years, a family of related techniques (Wavelength Scanning Interferometry, Phase Contrast Spectral Optical Coherence Tomography (OCT), Tilt Scanning Interferometry and Hyperspectral Interferometry) has emerged that allows one to measure the volume deformation state within weakly-scattering objects. The techniques can be thought of as combining the phase-sensing capabilities of Phase Shifting Interferometry and the depth-sensing capabilities of OCT. This paper provides an overview of the techniques, and describes a theoretical framework based on the Ewald sphere construction that allows key parameters such as depth resolution and displacement sensitivity to be calculated straightforwardly for any given optical geometry and wavelength scan range. Finally, the related issue of robust phase unwrapping of noisy 3-D wrapped phase volumes is also described.

Development of LEED Apparatus Using Nano-Tips Fabricated by Field-Assisted Etching

We have developed a low-energy electron diffraction (LEED) apparatus using field emission (FE) from scanning tunneling microscopy (STM) tips. Tips fabricated by field-assisted H2O etching enabled us to observe LEED patterns of few layer graphene grown on SiC(0001) reproducibly. The obtained hexagonal patterns showed a bias dependency and those obtained at sample biases of 93 V, 56 V, and 28 V were interpreted as patterns derived from the graphene, the SiC(0001) substrate, and a (6 √ 3 × 6 √ 3)R30◦ buffer layer, respectively. The analysis of Ewald sphere construction indicated that the opening angles of FE from the tip could be deduced directly from the LEED patterns, and their experimentally obtained value was ∼13◦. From the opening angle of FE and the dataset of tip-to-sample distances, the irradiated areas were estimated at 350 nmφ when the sample bias was 100 V. [DOI: 10.1380/ejssnt.2012.292]

Determination of Correlation Functions of Scattering Potentials of Stochastic Media from Scattering Experiments

The classic "Ewald-sphere construction" for determining the structure of crystalline objects from x-ray and neutron diffraction experiments is generalized to determine the correlation functions of scattering potentials of stationary random media from scattering experiments.

Holographic scattering in SiO_2 nanoparticle-dispersed photopolymer films

We describe an experimental study of holographic (coherent) scattering due to parasitic noise gratings recorded in SiO2 nanoparticle-dispersed photopolymer films. Dependences of film thickness and nanoparticle concentration on holographic scattering losses are evaluated. It is shown that the geometric feature of the holographic scattering pattern in the two-beam recording setup can be explained by the Ewald sphere construction. It is found that holographic scattering becomes noticeable when a film with nanoparticle concentrations higher than 10 vol.% is thicker than 100 microm. The significance of holographic scattering in the characterization of a volume grating recorded in a thick (>100 microm) nanoparticle-dispersed photopolymer film is also discussed.

Holographic scattering in photopolymer-dispersed liquid crystals

Strong polarization-conserving holographic scattering was observed in a photopolymer-dispersed liquid crystal film fabricated from the UV curable mixture of commercially available constituents. During the photopolymerization process a bright corona of diffracted light evolves around the pump beam. The intensity of the rotationally symmetric light distribution increases upon exposure. By rotating the sample, two characteristic diffraction rings appear which can be explained by the Ewald sphere construction. Our results demonstrate that the associated parasitic holograms are very pronounced. Hence, their presence must be accounted for whenever preparing and utilizing holographic polymer-dispersed liquid crystals in any application.

Heteroepitaxial diamond nucleation and growth on silicon by microwave plasma-enhanced chemical vapor deposition synthesis

Heteroepitaxial diamond films were successfully nucleated and deposited on 1-inch diameter Si(001) substrates by microwave plasma-enhanced chemical vapor deposition (MPECVD). The precursor gases for the synthesis were methane and hydrogen. Before the application of a negative d.c. bias to the substrate, an in-situ carburization pre-treatment on the silicon was found to be an indispensable step towards the heteroepitaxial diamond on the silicon. Morphologies of the films were characterized by scanning electron microscopy (SEM). Interface observations based on the cross-sectional HRTEM directly reveal the heteroepitaxial diamond nucleation phenomena in detail. No interlayers of silicon carbide and/or amorphous carbon phases were observed. Tilt and azimuthal misorientation angles between the heteroepitaxial diamond crystals and the substrate were determined by combining the Ewald sphere construction in the reciprocal lattice space and the selected area diffraction (SAD) patterns taken across the interface.

21] Data collection strategy

Publisher Summary This chapter emphasizes the importance of determining the outer resolution limit to define the parameters to be used in data collection. The chapter describes the behavior of the reciprocal lattice during rotation photography, the effect of crystal mosaicity, the myth of the blind region, and the ways in which crystal symmetry can help. To design a data-collection protocol so that a data set is complete is not straightforward. Geometric considerations must be taken into account that depend on a crystal's orientation, its cell dimensions, and its symmetry. The geometric principles of the rotation method and individual cases of different symmetries of a crystal are discussed in the chapter. As the crystal is rotated during exposure of X-rays, the diffraction geometry may be explained by the Ewald-sphere construction, which, in turn, illustrates Bragg's law in three dimensions. The Ewald sphere with the radius 1/λ represents the radiation and therefore is stationary, whereas the crystal is represented by the reciprocal lattice with the origin at the point where the primary beam leaves the sphere. The crystal is rotated around one axis, usually perpendicular to the beam.

Comparison of recording densities in three-dimensional optical storage systems: multilayered bit recording versus angularly multiplexed holographic recording

We compare the geometrical limits of the recording density that can be attained in three-dimensional optical memory systems that employ the multilayered bit-recording method and the angularly multiplexed holographic recording method. Both recording methods have the potential to overcome the recording density limitations in current optical storage systems. Using the Ewald sphere construction, we analyze the lateral and longitudinal bandwidths for each recording method. The respective recording densities of the two methods are also derived directly in the space domain and compared with each other. With the bit-recording method we found that the memory density increases inversely with the f-number of the recording lens. On the other hand, the density that can be achieved using holographic recording first increases with the f-number values, attains a maximum, and then decreases at larger f-number values. This implies that the bit-recording method yields larger memory densities when lenses of smaller f-numbers are used in the optical system.

XRayView: a teaching aid for X-ray crystallography

A software package, XRayView, has been developed that uses interactive computer graphics to introduce basic concepts of x-ray diffraction by crystals, including the reciprocal lattice, the Ewald sphere construction, Laue cones, the wavelength dependence of the reciprocal lattice, primitive and centered lattices and systematic extinctions, rotation photography. Laue photography, space group determination and Laue group symmetry, and the alignment of crystals by examination of reciprocal space. XRayView is designed with "user-friendliness" in mind, using pull-down menus to control the program. Many of the experiences of using real x-ray diffraction equipment to examine crystalline diffraction can be simulated. Exercises are available on-line to guide the users through many typical x-ray diffraction experiments.

Scatter rings in bismuth silicate illuminated by a single beam

We have observed cones of diffracted light by illuminating a crystal of bismuth silicate with a single beam of light while applying an electric field across the crystal. The scatter rings are a result of the preferential amplification of noise, which we explain here using the Ewald sphere construction.

Image restoration of thick biological specimens using a through focus series as applied to electron tomography

The main focus of our laboratory is the study of higher order chromatin structure using three dimensional electron microscope tomography. Three dimensional tomography involves the reconstruction of an object by combining multiple projection views of the object at different tilt angles. It is thus imperative to obtain an accurate representation of the projected object mass density to reconstruct the object correctly in three dimensions. Due to the effects of electron-specimen interactions and microscope lens aberrations, image intensities are not always related to the projected mass density in a simple fashion. We are using a variety of techniques to interpret collected images. In previous work, we have analyzed image formation for thick (0.3-0.7um) biological specimens by electron energy loss spectroscopy and imaging, as well as Ewald sphere construction analysis. In this work, we have modified existing techniques to restore images with only three focus levels using the exit surface wave reconstruction as a reference for comparison.

Interpretation of low-voltage field-emission projection interferograms

Remarkable transmission electron interference patterns have been reported from very thin crystals at energies Eo< 400 volts using a field emission tip, distance z1 ≈ 200 nm from a semi-transparent crystal sample, which acts as the grounded anode. As predicted, “atomic resolution” Fourier images ar observed on a screen distance z2 from the sample, with magnification z2/z1 ≈ 106at interior sample regions, confused by Fresnel fringes at edges. The same geometry is used to observe lattice images without scanning in coherent CBED patterns with overlapping orders. The interpretation of these patterns must be based on the theory of transmission LEED (TLEED), including multiple scattering. Figure 1 shows the Ewald sphere construction for 250 volt electrons along [110] gold. Image resolutio is limited to the inner reflections by the small sphere (large wavelength). TLEED computations using the Bloch-wave method of Collela are compared with the single scattering approximation in figure 2 Convergence tests show that 58 forward and backscattered beams are sufficient (Backscattered beams hop along the surface under the repulsive influence of the tip field).

OPTICALLY INDUCED LIGHT DIFFRACTION IN LiNbO3: Fe

In This paper, Diffraction cones are studied in LiNbO3:Fe. These diffracted cones are shown to result from the degenerate four wave mixing among the incident reflected and scattered beams induced by imperfections in the crystals. The angles of the cones are determined by phase matching condition △k = 0, which can be achieved by Ewald sphere construction from diffraction theory, while diffraction cones induced by degenerate four wave mixing of time-resolution are also discussed.