Showing papers on "Diffraction published in 1968"

Parametric Interaction of Focused Gaussian Light Beams

Abstract: A theoretical study is presented on the optimization of second harmonic generation (SHG) and parametric generation (PG) by a laser beam in a uniaxial nonlinear crystal. Numerically computed curves show the dependence of the SHG power, and the reciprocal of the PG threshold power, on the parameter l/b, where l is the optical path length in the crystal and b is the confocal parameter (determined by the focal length of the focusing lens and the minimum radius of the laser beam, assumed to be in the TEM00 mode of an optical resonator). The calculations take full account of diffraction and double refraction. In the absence of double refraction, the optimum focusing condition is found to be l/b=2.84. For PG the optimization of the crystal length l is also discussed, and curves are given showing the dependence of the threshold on l for the case in which signal and idler have the same losses. It is shown that the computed functions are also relevant to the mixing of two Gaussian beams and to parametric amplificat...

Self-focusing and diffraction of light in a nonlinear medium

Abstract: CONTENTS 1. Introduction 609 2. Geometrical Optics of a Nonlinear Medium (Equations, Focal Points, Nonlinear Aberrations, Nonstationary Processes) 617 3. Wave Optics of a Nonlinear Medium (Diffraction Corrections to the Self-focusing Length, Formation of Proper Optical Waveguide, Nonstationary Processes) 624 4. Nonlinear Optical Effects in the Field of Self-focusing Beams (Stimulated Scattering in Liquids, Experimental Data, Self-focusing and Parametric Amplification) 629 5. Conclusion (Unsolved Problems, Diffraction in Nonlinear Interactions) 632 Cited Literature 634

X-ray Diffraction: New High-Speed Technique Based on X-ray Spectrography

Abstract: A method has been developed for obtaining lattice spacings of powder samples by x-ray diffraction in times potentially as short as 1 second. The sample is irradiated with polychromatic radiation from an x-ray tube, and the energy spectrum of x-rays scattered at a given angle is observed with a semiconductor radiation detector, coupled with a pulse-height analyzer.

The physics of vibrations and waves

Abstract: Introduction to First Edition. Introduction to Second Edition. Introduction to Third Edition. Introduction to Fourth Edition. Introduction to Fifth Edition. Introduction to Sixth Edition. 1 Simple Harmonic Motion. 2 Damped Simple Harmonic Motion. 3 The Forced Oscillator. 4 Coupled Oscillations. 5 Transverse Wave Motion. 6 Longitudinal Waves. 7 Waves on Transmission Lines. 8 Electromagnetic Waves. 9 Waves in More than One Dimension. 10 Fourier Methods. 11 Waves in Optical Systems. 12 Interference and Diffraction. 13 Wave Mechanics. 14 Non-linear Oscillations and Chaos. 15 Non-linear Waves, Shocks and Solitons. Appendix 1: Normal Modes, Phase Space and Statistical Physics. Appendix 2: Kirchhoff's Integral Theorem. Appendix 3: Non-Linear Schro..dinger Equation. Index.

Thermal and X‐Ray Diffraction Studies of the NaNbO3–KNbO3 System

Abstract: Thermal and x‐ray diffraction measurements on compositions from the system NaxK1−xNbO3 have revealed the presence of phase boundaries at approximately 825, 675, and 525 mole × NaNbO3 at 25°C, in addition to one very near the NaNbO3 end member The optimum piezoelectric properties in this system are apparently associated with compositions near the latter boundary Calorimetric determinations of the enthalpy and entropy changes associated with the ferroelectric→ferroelectric and ferroelectric→paraelectric transformations in the system clearly revealed the presence of the phase boundaries while measurements of the transformation temperatures alone did not readily reveal the boundaries Two different superlattice perovskite phases were observed in the system, one existing in the region ∼098≥x≥0825 and another in the region 0825≥x≥0675 No multiple‐cell structures were observed in the system at NaNbO3 concentrations below about 70 mole%

Inverse Diffraction and a New Reciprocity Theorem

Abstract: A solution is obtained to the inverse diffraction problem for a monochromatic scalar wave field propagated into the half space z>0. It is shown how to determine the field distribution throughout the region 0≤z 0. The solution takes a particularly simple form when the spatial-frequency spectrum of the distribution in the plane z=z1 (or in any other plane z=const>0) is bandlimited to a circle whose radius is equal to the wavenumber of the field. In this case, the solution to the inverse diffraction problem may be expressed in a form strictly similar to that for the direct-propagation problem (exterior boundary-value problem), given by Rayleigh’s diffraction formula of the first kind. A comparison of these two solutions leads to the formulation of a new reciprocity theorem, valid for a wide class of wave fields.

Approximations of the dynamical theory of diffraction contrast

Abstract: The approximations of diffraction contrast theory, in particular the two-beam approximation and the column approximation, are critically examined. Contrast calculations for dislocations, spherically symmetric coherency strains and magnetic domain walls are described. In most cases the effect of the column approximation is small. This is because θD, the half angle of divergence of the energy flow in the crystal is usually very much less than θB, the Bragg angle. The results are discussed in terms of a perturbation theory of scattering which may prove useful for the determination of the structure of small, unknown defects.

Spin Waves in 3d Metals

Abstract: A review is given of recent measurements of spin‐wave dispersion relations of the 3d metals by the Brookhaven neutron diffraction group using the diffraction technique and triple‐axis spectrometry. The parameters D and β in the relation ħω = Dq2(1 − βq2) have been determined at 295°K for Fe, Co, Ni, and some of their alloys. These values are compared with those obtained by thin‐film resonance and small‐angle scattering.The most extensive measurements were carried out on Fe using a triple‐axis spectrometer. The dispersion relation was measured along the three principal symmetry directions for wavevectors up to q/qmax = 0.4. The stiffness constant D as well as the linewidth of selected spin waves were studied for the temperature range between 77°K and the Curie temperature, 1042°K. Well‐defined magnons were observed up to a reduced temperature T/TC = 0.995, but not above TC.

Ray Theory of Reflection from the Open End of a Waveguide

Abstract: Ray methods have been widely used to calculate high frequency scattering by objects in an unbounded medium, but they have not been used much for waveguide problems. To show that they can be used for waveguides, we treat by ray methods reflection from an open-ended parallel plane waveguide propagating several modes. The incident mode is decomposed into two plane waves whose scattering by the edges at the termination produces the reflected field. The singly diffracted cylindrical wave originating at each. edge, which is known from the asymptotic theory of diffraction by a single wedge, is represented by means of diffracted rays. Then the stem of the fields on the multiply reflected rays is converted into modal form. This yields formulas for the reflection coefficients in the various modes due to single diffraction. In addition double and multiple diffraction are also taken into account, yielding improved formulas for the reflection coefficients. The results of extensive numerical calculations for $TE$ and $...

Diffraction of light by opals

Abstract: Gem opals were examined in an optical diffractometer and found to give several types of diffraction pattern which are interpreted by analogy with conventional three-dimensional X-ray theory to give the structure of the opal. The results show that the spherical silica particles in opals are arranged hexagonally in layers which are usually stacked randomly. In some specimens there are parallel domains of ordered packing, commonly in a f.c.c. sequence and sometimes in a h.c.p. sequence. Although the silica particles cannot be resolved by optical microscopy, bands parallel to the layers and fringes across the bands can be seen in images with the diffracted light.

Reciprocity in electron diffraction and microscopy

Abstract: The Reciprocity Theorem of scattering theory is shown to hold generally for electrons scattered elastically in an absorbing medium, and also for inelastically scattered electrons, to a certain approximation. Two scattering symmetry conditions are defined by applying the theorem to crystals having mirror symmetry parallel to the crystal surface, and to centrosymmetric crystals. These are general forms of conditions demonstrated by Fukuhara (J. Phys. Soc. Japan (1966), 21, 2645). A number of symmetry effects observable in electron microscope images of crystals result from one or other of these conditions. The variation of intensity of diffracted beams as a function of angular deviation from the Bragg condition is considered in detail. The symmetry of bright-field images of defects lying at equal distances from the crystal surfaces was first explained by Howie & Whelan (Proc. Roy. Soc. (1961), A 263, 217); such symmetry properties can be conveniently classified and explained by the reciprocity theorem. A method of obtaining high-resolution dark-field images, due to Cowley, is based on the use of the reciprocity theorem. Examples of its application for both Bragg and diffusely scattered electrons are given.

The diffraction of low-energy electrons by crystals

Abstract: A formulation of the problem of low-energy electron diffraction from crystal surfaces is given using multiple scattering theory. Only elastic processes are considered. The resultant formulae are straightforward to compute and can be applied to model surfaces of arbitrary complexity, provided they are regular.

Observations of Thin Crystals of TiNi in Premartensitic States

Abstract: Thin films of TiNi have been examined in the temperature range 0°–200°C, using transmission electron microscopy and selected‐area diffraction. The CsCl diffraction patterns are accompanied by strong localized diffuse scattering, which becomes more pronounced as the temperature is reduced, approaching that of a phase transformation. A series of experiments designed to determine the directions of maximum diffusely scattered intensity suggest that these are 〈210〉 and 〈321〉, and that scattering is largely confined to reciprocal lattice directions subtending small angles with the incident electron beam. Since the phase transformation is thought to be of the second kind, and diffusionless, the results are interpreted in terms of large‐amplitude, short‐wavelength atomic displacements, reflecting incipient mechanical instability. Several diffraction patterns and dark‐field photomicrographs are presented to indicate the scale of the domain structure of the transformation product.

Algebraic Formulation of Diffraction Applied to Self Imaging

Abstract: One of Rayleigh’s diffraction integrals is generalized as to the nature of the functions on which it operates. These functions comprise the well-known Hilbert space ℒ2 of square-integrable functions. The algebraic properties are apparent in the existence of a left inverse for the diffraction operator, and in other places where the ordering of a sequence of operations is important. The conditions for self imaging are completely characterized by a certain part of the mathematical spectrum of the diffraction operator. The contribution of the evanescent waves to this spectrum is clearly shown. Both the infinite- and finite-aperture cases of diffraction are treated. In the infinite-aperture case we can employ a vector theory in which we obtain a complete characterization of self imaging. This self imaging is of an approximate nature. For the finite-aperture case we employ a scalar theory and obtain some partial results. However, in contrast to the infinite aperture we prove, for the case of an arbitrary finite aperture in a black screen, that there exist fields which image themselves exactly through the diffraction process. These fields are the eigenfunctions of a diffraction operator which corresponds to the sequential operations of inserting a screen into the incident field, discarding the evanescent-wave contribution, and diffracting through a distance z. The operator which corresponds to just the first two processes of inserting the screen and discarding the evanescent waves is of the type which generates the so-called generalized prolate spheroidal functions. These functions correspond to fields which have the interesting optical property of giving the same diffraction pattern whether the screen is applied or not (i.e., to within a constant positive factor). A conjecture is put forth concerning the evanescent-wave contribution to measurements that have been taken near the aperture. If the conjecture proves correct, it may be possible to show a consistency between the first Rayleigh diffraction integral and near-aperture measurements.

X-Ray Stress Topography of Thin Films on Germanium and Silicon

Abstract: X‐ray stress topography of thin films deposited on single‐crystal substrates is established. This method permits a nondestructive rapid analysis of the stress environment of film‐crystal interfaces. It is based on dynamical diffraction phenomena that occur at such interfaces. This technique is principally useful for the measurement of the sign of the stress in film and/or substrate. It is used to detect stress buildups and stress reversals connected with processing procedures, in particular, with those of the planar semiconductor device technology. It is also an elegant way to measure the adhesion properties of thin films deposited on single‐crystal slices as large as 2 in. in diameter. Dynamical diffraction phenomena of film‐crystal interfaces are discussed in terms of the Penning and Polder theory and shown to be caused through tie‐point migration. Topographical x‐ray contrast of film‐crystal interfaces is interpreted in terms of Bragg plane curvature. Relations between lattice curvature and film stress are given.

Diffraction Theory of Scattering by Hydrogen Atoms

Abstract: Diffraction theory is applied to elastic and inelastic scattering of charged particles by hydrogen atoms. At small momentum transfers the calculated intensities for elastically scattered electrons are significantly greater than those calculated by the first Born approximation.

Modulated structures in the system TiO2-SnO2

Abstract: Tetragonal crystalline solutions in the system TiO2–SnO2 were annealed at 850°C and 1000°C to study the mechanism of separation within the exsolution dome. X-ray diffraction results indicated a unidirectional composition variation along [001]. The formation of the modulated structure was verified by electron microscopy, which showed alternately dark and light lamellae composed of SnO2-rich and TiO2-rich plates perpendicular to [001]. The sum of the thicknesses of the dark and light lamellae (‘wavelength') decreased in the early stages of the separation and then increased at longer annealing times, the thickness varying between 180 and 400 A. It is proposed that the precipitation proceeds as follows: crystalline solution→intermediate modulated phase → equilibrium phases. The first step shows all the characteritics of the spinodal type of mechanism. It is shown theoretically that a one-directional fluctuation of composition parallel to [001] can be expected in the system TiO2-SnO2.

Single crystals of transfer RNA: an x-ray diffraction study.

Abstract: Large single crystals of formylmethionyl transfer RNA have been prepared. An x-ray diffraction study shows that the material crystallizes in a hexagonal lattice with a equal to 170 angstroms, c equal to 234 angstroms. The diffraction pattern extends to spacings just under 20 angstroms at present. The crystals are heavily hydrated, containing 88 percent water.

X‐ray Pendellösung fringes in Darwin reflection

Abstract: Oscillations have been observed in the tails of the Darwin curve from thin specimens of silicon. Dynamical theory predicts such oscillations (Pendellosung) for plane wave incident conditions in which two wave points on the same branch of the dispersion surface interfere and produce beating in the diffracted intensity. The usual Pendellosung is observed in transmission in which beating occurs between wavefields on different branches of the dispersion surface related by spherical rather than plane wave conditions. The present experiment uses an asymmetrically cut first crystal to increase the effective width of the incident wave and therefore approach the plane wave condition. Quantitative fringe measurements versus thickness are in fair agreement with theory. The intensity and contrast of the fringes are in poor agreement with theory. Several of the more important factors which reduce the contrast are discussed. With a relatively simple assumption, the observed and expected intensity dislocations can be brought into good agreement.

Etude théorique de la propagation des rayons X dans un cristal parfait ou légèrement déformé

Abstract: Takagi's theory is used to calculate the propagation of X-rays in perfect and nearly perfect crystals. In the general case, the equations have to be solved on a computer. The principle of the calculation is given. It has been applied to the case where an incident plane wave is collimated by a slit. The separation of wavefields is observed, each presenting a fine structure shown to be due to the diffraction by the slit. The same calculation is extended to a crystal submitted to a thermal gradient.The propagation and the intensities of wavefields are in good agreement with the predictions of Penning and Polder.

Mössbauer Diffraction. I. Quantum Theory of Gamma-Ray and X-Ray Optics

Abstract: X rays and gamma rays interaction with crystals analyzed by quantum electrodynamics Green function methods, emphasizing Mossbauer gamma ray diffraction optics