Diffraction Amplitudes Research Articles

Amplitude and phase clipping: Strehl ratio versus divergence

We consider the transverse characteristics of a Gaussian laser beam subject to a phase or amplitude clipping due to a pupil which is a π-plate or an opaque disc (stop). In particular, we consider the correlation between two features, the Strehl ratio and divergence angle, usually used for characterising the focusability of a diffracted beam. It is demonstrated that the Strehl ratio does not give systematically a global view, from a divergence point of view, on the transverse properties of a Gaussian beam suffering amplitude or phase diffraction. In addition, we consider the case of self-diffraction of a Gaussian beam upon a Gaussian phase aberration of same width, and it is found that the on-axis intensity describes correctly the whole diffracted beam cross-section, from a divergence point of view, only if the central phase shift is smaller than π. Another example showing that the focusability of a pure high-order Laguerre–Gauss TEM p0 beam, free from any clipping, cannot be correctly described by Strehl ratio is also considered.

Imaging of extreme-ultraviolet mask patterns using coherent extreme-ultraviolet scatterometry microscope based on coherent diffraction imaging

In extreme-ultraviolet (EUV) lithography, defect-free mask production is a critical issue for high-volume manufacturing. For mask inspection and metrology, we have developed a coherent EUV scatterometry microscope (CSM). It is a simple lensless system. An aerial image of the mask pattern is reconstructed with iterative calculation based on coherent diffraction imaging. Periodic patterns, aperiodic patterns, and phase structures were reconstructed well by the CSM. A defect in a line-and-space pattern was detected as a diffraction signal. The aerial image of the defect is also reconstructed. This paper demonstrates the capability of the CSM to observe complex diffraction amplitudes directly from the pattern and the defect.

Gain-phase grating based on spatial modulation of active Raman gain in cold atoms

In order to obtain an atomic grating which can diffract light into the high-order directions more efficiently, a gain-phase grating (GPG) based on the spatial modulation of active Raman gain is theoretically presented. This grating is induced by a pump field and a standing wave in ultracold atoms, and it not only diffracts a weak probe field propagating along a direction normal to the standing wave into the high-order directions, but also amplifies the amplitude of the zero-order diffraction. In contrast with electromagnetically induced grating or electromagnetically induced phase grating, the GPG has larger diffraction efficiencies in the high-order directions. Hence it is more suitable to be utilized as an all-optical router in optical networking and communication.

Two-dimensional shift-orthogonal random-interleaving phase-code multiplexing for holographic data storage

A novel phase encoding technology for phase-code multiplexing in holographic data storage (HDS) system called two-dimensional shift-orthogonal random-interleaving (2-D SORI) phase encoding is proposed. Compared with the traditional one-dimensional orthogonal phase-code multiplexing methods, the 2-D method is less sensitive to the variations of the diffraction amplitude and to the phase error of the phase mask. Phase masks for the 2-D SORI phase-code multiplexing can be generated by shifting an elaborately designed phase plate at a certain step, which can avoid the use of a high-cost phase spatial light modulator for the generation of multiple orthogonal phase masks. The cross-talk arising from the systematic phase defects of the static phase modulator is eliminated by the shift operation of the phase plate. Phase codes are interleaved under a predetermined random mapping rule to prevent unauthorized users from accessing the data and eliminate the Bragg degeneration cross-talk. A 2-D SORI phase plate with the size of 5.12 mm × 10.24 mm is designed and fabricated, from which 128 orthogonal phase patterns are generated. The feasibility of the 2-D SORI phase-code multiplexing method for HDS is experimentally demonstrated.

Breakdown of partial conservation of axial current in diffractive neutrino interactions

We test the hypothesis of partially conserved axial current (PCAC) in high energy diffractive neutrino production of pions. Since the pion pole contribution to the Adler relation (AR) is forbidden by conservation of the lepton current, the heavier states, like the a_1 pole, \rho-\pi-cut, etc., control the lifetime of the hadronic fluctuations of the neutrino. We evaluate the deviation from the AR in diffractive neutrino-production of pions on proton and nuclear targets. At high energies, when all the relevant time scales considerably exceed the size of the target, the AR explicitly breaks down on an absorptive target, such as a heavy nucleus. In this regime, close to the black disc limit, the off-diagonal diffractive amplitudes vanish, while the diagonal one, \pi->\pi, which enters the AR, maximizes and saturates the unitarity bound. At lower energies, in the regime of short lifetime of heavy hadronic fluctuations the AR is restored, i.e. it is not altered by the nuclear effects.

Survival probability of large rapidity gaps in the QCD and N=4 SYM motivated model

In this paper we present a self-consistent theoretical approach for the calculation of the Survival Probability for central dijet production. These calculations are performed in a model of high energy soft interactions based on two ingredients: (i) compatibility with the results of N=4 SYM, which at the moment is the only theory that is able to deal with a large coupling constant; and (ii) the required matching with high energy QCD. Assuming, in accordance with these prerequisites, that the soft Pomeron intercept is rather large and the slope of the Pomeron trajectory is equal to zero, we derive analytical formulae that sum both enhanced and semi-enhanced diagrams for elastic and diffractive amplitudes. Using parameters obtained from a fit to the available experimental data, we calculate the Survival Probability for exclusive central dijet production at energies accessible at the LHC. The results presented here, which include the contribution of semi-enhanced and net diagrams, are considerably larger than our previous estimates.

Mie scattering in the time domain Part II The role of diffraction

The p=0 term of the Mie-Debye scattering amplitude contains the effects of external reflection and diffraction. We computed the reflected intensity in the time domain as a function of the scattering angle and delay time for a short electromagnetic pulse incident on a spherical particle and compared it to the predicted behavior in the forward-focusing region, the specular reflection region, and the glory region. We examined the physical consequences of three different approaches to the exact diffraction amplitude, and determined the signature of diffraction in the time domain. The external reflection surface wave amplitude gradually replaces the diffraction amplitude in the angular transition region between forward-focusing and the region of specular reflection. The details of this replacement were studied in the time domain.

Improved molecular replacement by density- and energy-guided protein structure optimization

Molecular replacement procedures, which search for placements of a starting model within the crystallographic unit cell that best account for the measured diffraction amplitudes, followed by automatic chain tracing methods, have allowed the rapid solution of large numbers of protein crystal structures. Despite extensive work, molecular replacement or the subsequent rebuilding usually fail with more divergent starting models based on remote homologues with less than 30% sequence identity. Here we show that this limitation can be substantially reduced by combining algorithms for protein structure modelling with those developed for crystallographic structure determination. An approach integrating Rosetta structure modelling with Autobuild chain tracing yielded high-resolution structures for 8 of 13 X-ray diffraction data sets that could not be solved in the laboratories of expert crystallographers and that remained unsolved after application of an extensive array of alternative approaches. We estimate that the new method should allow rapid structure determination without experimental phase information for over half the cases where current methods fail, given diffraction data sets of better than 3.2 Å resolution, four or fewer copies in the asymmetric unit, and the availability of structures of homologous proteins with >20% sequence identity.

Influence of Surface Roughness on Evaluation of Stress Gradients in Coatings

Roughness influence on the residual stress gradient evaluation in the case of a grazing incidence X-ray diffraction setup is considered. In this geometry the surface roughness changes essentially the X-ray wave fields of the transmitted and diffracted beams inside the coatings and subsurface regions of bulk samples, and thus influences the refractive properties of the investigated sample area. In turn, the change in the refraction index enforces the re-scale of the informational depth and, consequently, the evaluated stress depth profile. The diffracted amplitude from the crystalline grain located beneath the surface is calculated. The surface roughness is shown to contribute into reconstruction of the real stress gradient profile of the coating.

N=4 SYM and QCD motivated approach to soft interactions at high energies

In this paper we construct a model that satisfies the theoretical requisites of high energy soft interactions, based on two ingredients:(i) the results of N=4 SYM, which at present is a unique theory that allows one to deal with a large coupling constant; and (ii) the requirement of matching with high energy QCD. In accordance with these ideas, we assume that the soft Pomeron intercept is rather large, and the slope of the Pomeron trajectory is equal to zero. We derive analytical formulae that sum both enhanced and semi-enhanced diagrams for elastic and diffractive amplitudes. We fit the available experimental data, and predict the valuefor cross sections at the energies accessible at the LHC. The main corrections to the model are studied and evaluated.

Self-imaging by a volume grating

The self-image phenomenon by a volume grating is proposed and theoretically analyzed. A theoretical model based on a path integral formulation to describe wave propagation through the grating inhomogeneous medium is applied. A modified version of the scalar diffraction theory Fresnel propagator is obtained which allows calculating the diffracted field amplitude by the grating. The proposed model is applied to amplitude and/or phase volume gratings. Remarkable features appear, in particular at the fractional Talbot distance 0.125 z T. In this case, if an in-phase real and imaginary grating modulation is considered a self-image intensity profile is observed for determined values of the absorptive and refractive parameters. On the other hand, a spatial comb intensity profile for a near half period shift between the real and imaginary grating modulations is found.

Ab Initio Electron Diffraction Structure Analysis of Zeolites – Direct Methods Determination of NaY

Abstract The crystal structure of NaY was re-determined in space group Fd-3m from 87 unique electron diffraction amplitudes obtained at 1250 kV by A. Carlsson, et al. (Chem. Eur. J. 5 (1999) 244), where a = 24.7 Å. The determination, based on maximum entropy and likelihood (computer program MICE), leads to a clearly resolved structure, in contrast to the use of crystallographic phases derived from the Fourier transforms of experimental high-resolution images cited in the above paper. Potential maps from a trial solution contain density sites that correspond closely to the known T-site locations of this FAU framework. Although only framework T-site positions and not linkage oxygens are observed in the map, they can be linked with theoretical oxygens. Subsequent distance least squares (DLS) refinement of the linked model converges to a complete framework structure very close to the one found in earlier X-ray diffraction studies. Contrasting to the earlier study, where no counterion positions were identified, two possible Na positions were clearly visible in the direct phasing map One of these corresponds to a site postulated in previous studies. The other is near a site proposed in a separate electron microscopic/electron diffraction study of Na,FeY. Least squares refinement verifies the former position while eliminating the latter. On the other hand, a separate direct analysis of a dataset from the Y zeolite containing both Na and Fe (Carlsson et al., loc. cit.) was not successful, owing to the smaller number of recorded amplitudes.

Fraunhofer diffraction of a Laguerre–Gaussian laser beam by fork-shaped grating

In this article we present a theoretical study for Fraunhofer diffraction of a Laguerre–Gaussian laser beam with zeroth radial mode number and azimuthal mode number l by a diffractive grating with embedded fork-shaped dislocations of integer order p. Analytical expressions describing the diffracted wave field amplitude and intensity distributions in the Fourier plane are deduced and analyzed. They are also followed by the vortex radii expressions.

Back Cover (Phys. Status Solidi A 12/2010)

Direct methods in crystallography derive diffraction phases directly from experimentally measured diffraction amplitudes. This is why they have played the central role in diffraction analysis of solving crystal structures. In 1987, Haifu Fan suggested that direct methods should turn to exploring new applications outside their traditional field. For this purpose, there are four ways to go: (i) from single crystals to powder samples, (ii) from X-ray crystallography to electron microscopy, (iii) from periodic structures to incommensurate structures, and (iv) from small molecules to macromolecular structures. In his Review Article on pp. 2621–2638, Haifu Fan focuses on works belonging to the last three categories done at the Institute of Physics in Beijing. The cover image shows partial electron-density maps of the protein TT0570 at different resolutions with the final structure model superimposed. Phases corresponding to each set of truncated data were derived separately and improved by density modification.

Images of paraffin monolayer crystals with perfect contrast: Minimization of beam-induced specimen motion

Quantitative analysis of electron microscope images of organic and biological two-dimensional crystals has previously shown that the absolute contrast reached only a fraction of that expected theoretically from the electron diffraction amplitudes. The accepted explanation for this is that irradiation of the specimen causes beam-induced charging or movement, which in turn causes blurring of the image due to image or specimen movement. In this paper, we used three different approaches to try to overcome this image-blurring problem in monolayer crystals of paraffin. Our first approach was to use an extreme form of spotscan imaging, in which a single image was assembled on film by the successive illumination of up to 50,000 spots, each of a diameter of around 7 nm. The second approach was to use the Medipix II detector with its zero-noise readout to assemble a time-sliced series of images of the same area in which each frame from a movie with up to 400 frames had an exposure of only 500 electrons. In the third approach, we simply used a much thicker carbon support film to increase the physical strength and conductivity of the support. Surprisingly, the first two methods involving dose fractionation in space or time produced only partial improvements in contrast whereas the third approach produced many virtually perfect images, where the absolute contrast predicted from the electron diffraction amplitudes was observed in the images. We conclude that it is possible to obtain consistently almost perfect images of beam-sensitive specimens if they are attached to an appropriately strong and conductive support; however great care is needed in practice and the problem remains of how to best image ice-embedded biological structures in the absence of a strong, conductive support film.

Effects of source size and wavefront propagation on the energy resolution of a bent-crystal polychromator

Polychromators, or elliptically bent diffracting crystals that focus a broad-bandwidth X-ray beam onto a sample, have become a common device at synchrotron beamlines specializing in X-ray absorption spectroscopy (XAS) because they allow a full absorption spectrum to be collected in one shot. Such a device is being planned for the XAS beamline I20 of the Diamond Light Source. A bent silicon crystal diffracting 7 keV X-rays with the (1 1 1) reflection is taken as a model for the simulations of this report. Instrumental resolution is determined by the demagnification of the source, the spread of the diffracted beam during propagation and the pixel size of the position-sensitive detector placed behind the sample. The first is calculated by geometrical optics. The second is calculated by a full wave-optical treatment, which includes Takagi–Taupin integration to find the diffracted amplitude at the crystal's surface and Huygens–Fresnel propagation of the diffracted wave to the sample or detector. This sets the polychromator's intrinsic energy resolution. The pixel size of the detector is then added to find the total instrumental resolution at various sample–detector distances.

Sub-15 nm beam confinement by two crossed x-ray waveguides

We have combined two high transmission planar x-ray waveguides glued onto each other in a crossed geometry to form an effective quasi-point source. From measurements of the far-field diffraction pattern, the phase and amplitude of the near-field distribution is retrieved using the error-reduction algorithm. In agreement with finite difference field simulations (forward calculation), the reconstructed exit wave intensity distribution (inverse calculation) exhibits a full width at half maximum (FWHM) below 15 nm in both dimensions. Finally, holographic imaging is successfully demonstrated for the crossed waveguide device by translation of a lithographic test structure through the waveguide beam.

Coherent diffraction surface imaging in reflection geometry

We present a reflection based coherent diffraction imaging method which can be used to reconstruct a non periodic surface image from a diffraction amplitude measured in reflection geometry. Using a He-Ne laser, we demonstrated that a surface image can be reconstructed solely from the reflected intensity from a surface without relying on any prior knowledge of the sample object or the object support. The reconstructed phase image of the exit wave is particularly interesting since it can be used to obtain quantitative information of the surface depth profile or the phase change during the reflection process. We believe that this work will broaden the application areas of coherent diffraction imaging techniques using light sources with limited penetration depth.

Direct methods and refinement in electron and X-ray crystallography – diketopiperazine revisited

AbstractTexture diffraction data in pioneering Russian electron crystallographic studies inspired the invention of modern precession methods for collecting single crystal data. (Although oblique texture patterns contain overlapped reflections, the similarity to precession methods is a result of averaging intensities over an angular distribution of crystallites within the large area sampled by the incident electron beam.) Using texture diffraction amplitudes originally measured by Vainshtein, the crystal structure of diketopiperazine has been re-determined using automated direct methods (SIR97) and refined by block diagonal least squares. This determination was carried out in parallel with an equivalent direct methods study of a larger X-ray data set from the same material published by Degeilh and Marsh, followed by least squares refinement, reproducing the original results. After both refinements, the electron crystallographic bonding parameters for heavy atoms are found to be similar to derived X-ray crystallographic parameters. On the other hand, C–H and N–H distances are more accurately determined by electron crystallography than by X-ray crystallography since the light atom positions are more easily detected in ensuing maps. Surprisingly the least squares refinement against the electron diffraction data did not require a restraint on the magnitude of atomic incremental movement; moreover the atomic temperature factors could be refined, producing results that were more reasonable than expected from the very low overall B value found from a Wilson plot.

Nonperturbative and spin effects in the central exclusive production of the tensorχc(2+)meson

We discuss central exclusive production (CEP) of the tensor $\chi_c(2^{+})$ meson in proton-(anti)proton collisions at Tevatron, RHIC and LHC energies. The amplitude for the process is derived within the $k_t$-factorisation approach. Differential and total cross sections are calculated for several unintegrated gluon distributions (UGDFs). We compare exclusive production of all charmonium states $\chi_c(0^+),\,\chi_c(1^+)$ and $\chi_c(2^+)$. Equally good description of the recent Tevatron data is achieved both with Martin-Ryskin phenomenological UGDF and UGDF based on unified BFKL-DGLAP approach. Unlike for Higgs production, the main contribution to the diffractive amplitude of heavy quarkonia comes from nonperturbative region of gluon transverse momenta $Q_{\perp}<1\,\GeV$. At $y \approx$ 0, depending on UGDF we predict the contribution of $\chi_c(1^+,2^+)$ to the $J/\Psi + \gamma$ channel to be comparable or larger than that of the $\chi_c(0^+)$ one. This is partially due to a significant contribution from lower polarization states $\lambda=0$ for $\chi_c(1^+)$ and $\lambda=0,\,\pm 1$ for $\chi_c(2^+)$ meson. Corresponding theoretical uncertainties are discussed.