Diffraction Intensity Distribution Research Articles

Detecting the topological charge of vortex beams using an annular triangle aperture

The Fraunhofer diffraction pattern of a vortex beam by an annular triangle aperture is analyzed theoretically and experimentally. It is found that the pattern of the far-field diffraction intensity distribution exhibits a triangular lattice array, which becomes much clearer with the increase of the ratio of the inner to the outer side of the annular triangle aperture. The number of spot points of any external side of the triangular lattice array minus one is just equal to the topological charge value of the measured optical vortex. For the vortex beam with negative topological charge, the triangular diffraction pattern after the annular triangle aperture will be rotated by 180° in relation to the case of the positive topological charge. Based on the above properties, we propose a simple and feasible method to determine the magnitude and sign of the topological charge of an optical vortex beam.

Measuring the orbital angular momentum of elliptical vortex beams by using a slit hexagon aperture

The far-field diffraction pattern of an elliptical vortex beam by a slit hexagon aperture is investigated theoretically and experimentally. It is found that the number of the dark spots or stripes in the Fraunhofer diffraction intensity distribution is just equal to the topological charge value of the measured optical vortex, and that the centre of each dark spot or stripe is just a phase singularity point. Based on this property, it provides us a simple way to detect the orbital angular momentum (OAM) of an optical vortex beam.

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.

Diffraction of Lorentz-Gauss beam in uniaxial crystals: orthogonal to optical axis

Based on the diffraction theory of beams in uniaxial crystals, diffraction properties of a Lorentz-Gauss beam in uniaxial crystals orthogonal to the optical axis are derived in analytical forms. Diffraction fields, intensity distributions and effects of beam parameters are investigated by numerical examples, respectively. Results show that, upon propagation, initial field components and intensity distributions of Lorentz-Gauss beams would deteriorate due to effects of anisotropic media. When the Lorentz-Gauss beam diffracts into the far field, its intensity distribution would convert into a four-petal profile. Beam parameters w x and w y are shown to have a strong influence on intensity distributions. By selecting different values of them, profiles of Lorentz-Gauss beams would be different upon propagation.

Natural quasy-periodic binary structure with focusing property in near field diffraction pattern

A naturally-inspired phase-only diffractive optical element with a circular symmetry given by a quasi-periodic structure of the phyllotaxis type is presented in this paper. It is generated starting with the characteristic parametric equations which are optimal for the golden angle interval. For some ideal geometrical parameters, the diffracted intensity distribution in near-field has a central closed ring with almost zero intensity inside. Its radius and intensity values depend on the geometry or non-binary phase distribution superposed onto the phyllotaxis geometry. Along propagation axis, the transverse diffraction patterns from the binary-phase diffractive structure exhibit a self-focusing behavior and a rotational motion.

Modelling diffraction patterns from one‐dimensionally disordered structures of AIIBVI crystals by Monte Carlo Technique IV. The 6H structure with different kinds of stacking faults

AbstractTo investigate the one‐dimensionally disordered structures (ODDS) in the close packed (cp) crystals, the Monte Carlo computer simulation technique has been applied. Calculations of the diffraction intensity distributions along the 10.L reciprocal lattice row from the 6H(33) structure with the four different kinds of the stacking faults (SFs): growth, deformation, layer displacement and extrinsic fault are presented. In particular, using the simple frequency functions of the fault to fault distances, both random and non‐random distributions of the SFs are considered. Distinctive features of the diffraction patterns corresponding to the chosen examples of the transformations from the parent 6H(33) structure into another small‐period polytypes are discussed in detail. (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

X-ray diffraction effect from surface acoustic waves traveling on a deposited multilayer

We investigate the x-ray diffraction effects from surface acoustic waves (SAW) traveling along a multilayer. The diffraction intensity distribution depends on the incidence angle and the multilayer SAW (MLSAW) amplitude. Particularly, a small departure deviating from the Bragg incidence angle at a certain amplitude will produce a larger variation of the intensity distribution. This shows that the diffraction intensity from MLSAW has an extremely high sensitivity to the Bragg incidence angle, which is different from a SAW traveling along a solid surface without deposited layers. By carefully analyzing the relationship between the intensity distribution I and the incidence angle theta, the corresponding analytic expression of the intensity distribution is theoretically derived. Our theoretical prediction is in great agreement with the experimental results previously obtained. A theoretical model that can be applied to study the x-ray diffraction effect from MLSAW is developed. The extremely high sensitivity to the Bragg angle will help us in acousto-optic instrument research with MLSAW.

Diffraction effects due to modulation of orthorhombic crystals by aperiodically arranged twin boundaries

A number of the orthorhombic crystals formed due to diffusionless transformations, of which Ni2MnGa crystal is an example, show diffraction patterns suggesting a modulation along the <110> directions. Nanotwined substructures with aperiodically arranged twin boundaries (TBs) on the (110) planes are found in such crystals. In the work, peculiarities of the diffraction intensity distributions for orthorhombic crystals with the aperiodic TBs arrangement are analyzed. It is shown that the intensity distributions for the twin-modulated structures formed in given orthorhombic crystals differ from those resulted from a wave-modulation of the crystals.

Polarization dependence of the quasi-Talbot effect of the high-density grating

Diffractions by the one-dimensional high-density grating in the near field with TM and TE polarization illuminations are studied, and the diffraction intensity distributions are calculated with the finite-difference time-domain technique. The calculation results show that the diffractions of the high-density grating with different polarization illuminations are different. The quasi-Talbot image of the grating depends on the polarization of the incident wave, and the existence condition of the quasi-Talbot image of the grating in the near field also changes with the polarization of the incident wave. We present explanations based on the vector distribution of the energy flow density. These studies on the polarization dependence of the quasi-Talbot imaging of the high-density grating are helpful for the application of the grating to near-field photolithography.

Non-iterative imaging of inhomogeneous cold atom clouds using phase retrieval from a single diffraction measurement

We demonstrate a new imaging technique for cold atom clouds based on phase retrieval from a single diffraction measurement. Most single-shot diffractive imaging methods for cold atoms assume a monomorphic object to extract the column density. The method described here allows quantitative imaging of an inhomogeneous cloud, enabling recovery of either the atomic density or the refractive index, provided the other is known. Using ideas borrowed from density functional theory, we calculate the approximate paraxial diffracted intensity derivative from the measured diffracted intensity distribution and use it to solve the Transport of Intensity Equation (TIE) for the phase of the wave at the detector plane. Back-propagation to the object plane yields the object exit surface wave and then provides a quantitative measurement of either the atomic column density or refractive index. Images of homogeneous clouds showed good quantitative agreement with conventional techniques. An inhomogeneous cloud was created using a cascade electromagnetically induced transparency scheme and images of both phase and amplitude parts of refractive index across the cloud were separately retrieved, showing good agreement with theoretical results.

Diffraction theory of nanotwin superlattices with low symmetry phase: Adaptive diffraction of imperfect nanotwin superlattices

The diffraction behaviors of imperfect nanotwin superlattices were considered and the effects of random variations in nanotwin layer thicknesses and variant volume fractions on the diffraction peak intensity profiles investigated. Imperfect nanotwin superlattices exhibit a pronounced adaptive diffraction phenomenon when the Brillouin zone-dependent condition, , is met, where T is twin-related bilayer thickness, γ is twinning shear strain magnitude, s is twinning shear direction unit vector, and K is the reciprocal lattice vector of the constituent crystal denoting the fundamental reflection spots and Brillouin zones in reciprocal space. Nanotwin layer thickness variations produce little effect on the diffraction intensity distributions of adaptive Bragg reflection peaks, whereas variant volume fraction variations significantly reduce the height and broaden the width of adaptive peaks, but only affect their integrated intensities slightly and do not change their positions. In spite of imperfections, an extraordinary peak still appears at position k along twin peak splitting vector ΔK, as determined by a lever rule according to the average twin variant volume fraction : . Imperfect nanotwin superlattices exhibit a certain degree of diffuse reflection around the adaptive peaks with strong Lorentzian shape characteristics; adaptive peaks exhibit anisotropic broadening, i.e. peak is broader in the direction perpendicular to twin plane, and sharper parallel to the twin plane. The width of the adaptive peak does not provide direct information of nanodomain size, but the Brillouin zone-dependent diffraction behavior can be used to determine the nanodomain sizes.

Role of nanostructure on the optical waveguiding properties of epitaxial LiNbO3 films

High-quality LiNbO3 (LN) epitaxial films have been grown on (0 0 1)Al2O3 substrates by pulsed laser deposition. The deposition conditions have been optimized using experimental designs to promote the growth of highly (0 0 1)-oriented, phase pure and light-guiding LN films. The structural and nanostructural properties have been investigated by x-ray diffraction reciprocal space mapping and the guiding properties have been investigated by m-line spectroscopy and measurements of the light propagation losses. In particular it is shown that the film composition, the state of strain, the film thickness, the film roughness, the lateral extension of the crystallites building up the film as well as the mosaicity can be determined by a careful examination of the x-ray reciprocal space maps associated with simulation of the diffracted intensity distribution in reciprocal space. The guiding properties have been correlated with the nanostructural properties of the films: whereas light guiding has been clearly observed in single-crystal-like films, the existence of a mosaic structure, made of nanocrystalline domains, is shown to be detrimental to the guiding properties.

Microbeam X-ray diffraction from twisted lamellar crystals: theory and computer simulation

The diffraction peak position, width and intensity distribution are calculated for the case of a helicoidally twisted crystalline lamella, both analytically and numerically. It is shown that the diffraction peak broadening depends on the orientation of the corresponding reciprocal-space vector with respect to the helicoid axis and the normal to the lamellar basal plane. The equatorial peaks, which are close to the normal direction to the lamellar basal plane, are characterized by the highest azimuthal width. By contrast, the reflections positioned close to the lamellar surface have the smallest azimuthal width. For non-equatorial peaks in the proximity of the twisting axis the intensity has an unusual asymmetric shape. The shape of the microbeam, as well as its position and direction with respect to the lamella, influences the shape of the diffraction peaks in reciprocal space and their appearance in two-dimensional diffractograms. The proposed approach can be useful, for example, for the interpretation of microbeam diffractograms of banded polymer spherulites.

Diffraction from quasi-one-dimensional crystals

General expressions of diffraction intensity distribution of quasi-one-dimensional crystals are evaluated within kinematical approximation. In order to gain the generality, diffraction intensity has been derived for each of the fifteen conformation classes. Characteristic features of the diffraction patterns are discussed and it is shown how the symmetry can be fully determined from the diffraction intensity distribution. General results are tabulated and their application is illustrated on the (10,10) molybdenum disulfide nanotube.

Diffraction features due to ordered distribution of twin boundaries in orthorhombic Ni–Mn–Ga crystals

The diffraction intensity distributions for orthorhombic crystals containing the ordered twin boundaries of the system (110), 〈1{\overline 1}0〉 were calculated in the kinematic approach using the Monte Carlo computer simulation technique. A simple model of short-range order in the arrangement of the twin boundaries was assumed to generate the nano-twinned structure. Analysis of the intensity distributions showed that such a twin boundary distribution results in the formation of extra peaks along the reciprocal lattice rods that are perpendicular to the close-packed planes. These extra peak intensities and positions are dependent on the most probable distances between the twin boundaries and on the degree of short-range order.

Quasi-Talbot effect of the high-density grating in near field

On the basis of the finite-difference time-domain technique, the diffraction of the high-density grating in the near field is developed, and the gray-scale pattern of the diffraction intensity distribution of a one-dimensional grating is presented. A detailed analysis shows that the near-field diffraction of the grating is the result of the diffraction of a single slit, the interference of two evanescent waves from neighboring slits, and the interference of the homogeneous waves from the slits. Through many numerical calculations, the condition for obtaining the quasi-Talbot imaging of the grating in the near field is explored, i.e., the period of the grating d is larger than the incident wavelength lambda but smaller than 4lambda. The influence of the opening ratio of the grating on the quasi-Talbot imaging of the grating in the near field is also discussed. This study of the near-field diffraction of the high-density grating may be helpful for understanding the diffraction characteristics of subwavelength structures, and the quasi-Talbot imaging of the high-density grating will contribute to the application of the grating in near-field photolithography.

Distribution of Diffraction Intensity in Diffraction Orders for Thin Anisotropic Diffraction Gratings

The Raman-Nath diffraction in thin anisotropic slanted holographic gratings is studied. Also, dielectric and absorption modulation with common grating vector and of arbitrary relative phase shift is considered. Solutions for the wave amplitudes, diffraction efficiencies are given in transmission geometries for the case of dielectric and absorption modulations. The effect of slanted fringes of diffraction gratings is discussed and for comparison is presented the dependences for diffraction gratings in unslanted cases too. This theoretical method can be extended to a study of the mechanism of distribution and control of diffraction intensities within diffraction orders in materials such as organic crystals, ordered polymers, liquid crystalline and polymer-dispersed liquid crystalline cells.

Crystallography without crystals. I. The common-line method for assembling a three-dimensional diffraction volume from single-particle scattering

It is demonstrated that a common-line method can assemble a three-dimensional oversampled diffracted intensity distribution suitable for high-resolution structure solution from a set of measured two-dimensional diffraction patterns, as proposed in experiments with an X-ray free-electron laser (XFEL) [Neutze et al. (2000). Nature (London), 406, 752-757]. Even for a flat Ewald sphere, it is shown how the ambiguities due to Friedel's law may be overcome. The method breaks down for photon counts below about 10 per detector pixel, almost three orders of magnitude higher than expected for scattering by a 500 kDa protein with an XFEL beam focused to a 0.1 microm diameter spot. Even if 10(3) orientationally similar diffraction patterns could be identified and added to reach the requisite photon count per pixel, the need for about 10(6) orientational classes for high-resolution structure determination suggests that about 10(9) diffraction patterns must be recorded. Assuming pulse and readout rates of approximately 100 Hz, such measurements would require approximately 10(7) s, i.e. several months of continuous beam time.

Crystal truncation rod X-ray scattering: exact dynamical calculation

A new method is presented for a calculation of the reciprocal-space distribution of X-ray diffracted intensity along a crystal truncation rod. In contrast to usual kinematical or dynamical approaches, the method is correct both in the reciprocal-lattice points and between them. In the method, the crystal is divided into a sequence of very thin slabs parallel to the surface; in contrast to the well known Darwin dynamical theory, the electron density in the slabs is constant along the surface normal. The diffracted intensity is calculated by a matrix formalism based on the Fresnel reflection and transmission coefficients. The method is applicable for any polarization of the primary beam and also in a non-coplanar scattering geometry.

Diffraction of a one-dimensional phase grating in the deep Fresnel field

We analyze theoretically the diffraction of phase gratings in the deep Fresnel field on the basis of the theory of scalar diffraction and Green's theorem and present the general formula for the diffraction intensity of a one-dimensional sinusoidal phase grating. The numerical calculations show that in the deep Fresnel region the diffraction distribution can be described by designating three characteristic regions that are influenced by the parameters of the grating. The microlensing effect of the interface of the phase grating provides the corresponding explanation. Moreover, according to the viewpoint that the diffraction intensity distribution is the result of the interference of the diffraction orders of the grating, we find that the diffraction patterns, depending on the carved depth of the phase grating, are determined by the contributing diffraction orders, their relative power, and the quasi-Talbot effect of the phase grating, which results from the second meeting of the diffraction orders carrying most of the power of the total field, as in the case of the amplitude grating.