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Showing papers on "Diffraction published in 1971"


Journal ArticleDOI
TL;DR: In this paper, it was shown that the field of a Gaussian beam can be represented by a function G(P) =eikr/r, where r is the distance from the observation point P to a fixed point having a complex location.
Abstract: It is observed that the function G(P)=eikr/r, where r is the distance from the observation point P to a fixed point having a complex location, represents the field of a Gaussian beam. This can be used to justify, without further computation, the application of formulas of ordinary optics to the transformation of beams through optical systems. It can also be used to solve very simply some problems of diffraction and scattering of Gaussian beams.

557 citations


Book
01 Jan 1971
TL;DR: In this article, the authors present a list of symbols and their applications in general spherical coordinates, including the Wave Equation in Cylindrical Coordinates and its application in sound propagation and propagation in ideal channels and tubes.
Abstract: Preface The Symbols Historical Introduction I. Equations and Units II. Complex Notation and Symbolic Methods III. Analytic Functions: Their Integration and the Delta Function IV. Fourier Analysis V. Advanced Fourier Analysis VI. The Laplace Transform VII. Intergral Transforms and the Fourier Bessel Series VIII. Correlation Analysis IX. Wiener's Generalized Harmonic Analysis V. Transmission Factor, Filters, and Transients ("Kupfmuller’s Theory") XI. Probability Theory, Statistics, and Noise XII. Signals and Signal Processing XIII. Sound XIV. The One-Dimensional Wave Equation and Its Solutions XV. Reflection and Transmission of Plane Waves at Normal Incidence XVI. Plane Waves in Three Dimensional XVII. Sound Propagation in Ideal Channels and Tubes XVIII. Spherical Waves, Sources, and Multipoles XIX. Solution of the Wave Equation in General Spherical Coordinates XX. Problems of Practical Interest in General Spherical Coordinates XXI. The Wave Equation in Cylindrical Coordinates and Its Applications XXII. The Wave Equation in Spheroidal Coordinates and Its Solutions XXIII. The Helmholtz Huygens Integral XXIV. Huygens Principle and the Rubinowicz—Kirchhoff Theory of Diffraction XXV. The Sommerfeld Theory of Diffraction XXVI. Sound Radiation of Arrays and Membranes XXVII. The Green's Functions of the Helmholtz Equation and Their Applications XXVIII. Self and Mutual Radiation of Impedance Tables References Subject Index List of Symbols

411 citations


Journal ArticleDOI
TL;DR: In this paper, the omega phase is shown experimentally to be formed by cooling certain metastable β-Ti alloys to cryogenic temperatures, and the gradual progression from diffuse streaking in the diffraction patterns of the β phase to the sharp ω reflections is interpreted in terms of a displacive type reaction.

313 citations


Journal ArticleDOI
TL;DR: The Huygens-Fresnel principle is extended to a medium that exhibits a spatial (but not temporal) variation in index of refraction, and the formula used is used to calculate the mean intensity distribution for a plane wave diffracting from a circular aperture and to give approximate expressions for the beam spreading at various ranges.
Abstract: The first part of this paper is devoted to extending the Huygens-Fresnel principle to a medium that exhibits a spatial (but not temporal) variation in index of refraction. Utilizing a reciprocity theorem for a monochromatic disturbance in a weakly inhomogeneous medium, it is shown that the secondary wavefront will be determined by the envelope of spherical wavelets from the primary wavefront, as in the vacuum problem, but that each wavelet is now determined by the propagation of a spherical wave in the refractive medium. In the second part, the above development is applied to the case in which the index of refraction is a random variable; a further application of the reciprocity theorem results in a formula for the mean intensity distribution from a finite aperture in terms of the complex disturbance in the aperture and the modulation transfer function (MTF) for a spherical wave in the medium. The results are applicable for an arbitrary complex disturbance in the transmitting aperture in both the Fresnel and Fraunhofer regions of the aperture. Using a Kolmogorov spectrum for the index of refraction fluctuations and a second-order expression for the MTF, the formula is used to calculate the mean intensity distribution for a plane wave diffracting from a circular aperture and to give approximate expressions for the beam spreading at various ranges.

282 citations


07 Sep 1971
TL;DR: In this article, the diffraction of plane electromagnetic waves by ideally conducting bodies, the surface of which has discontinuities, is investigated, and the linear dimensions of the bodies are assumed to be large in comparison with the wavelength.
Abstract: : The diffraction of plane electromagnetic waves by ideally conducting bodies, the surface of which have discontinuities, is investigated in the report. The linear dimensions of the bodies are assumed to be large in comparison with the wavelength. The method developed takes into account the perturbation of the field in the vicinity of the surface discontinuity and allows one to substantially refine the approximations of geometric and physical optics. Expressions are found for the fringing field in the distant zone. A numerical calculation is performed of the scattering characteristics, and a comparison is made with the results of rigorous theory and with experiments.

242 citations


Journal ArticleDOI
TL;DR: The problems in the analysis of surface structures by means of electron diffraction, particularly at low energy, are reviewed in this paper, where the basic scattering and diffraction phenomena occurring at a solid surface is described.

198 citations



Journal ArticleDOI
TL;DR: In this paper, the diffraction of a plane dilatational wave of arbitrary profile by a finite line crack in an infinite elastic medium is analyzed by the generalized Wiener-Hopf technique.

141 citations



Journal ArticleDOI
TL;DR: In this paper, a method to separate the diffraction pattern into its various components so that each may be interpreted independently is presented, and assumptions are made about the nature of the atomic displacements δ, except that they are sufficiently small.
Abstract: X-ray diffuse scattering from disordered alloys is the primary direct source of information concerning the Warren parameters which describe the existing state of order. However, the short range order diffuse intensity is obscured by other contributions to the diffraction pattern resulting from static atomic displacements from the average lattice, and from effects related to thermal motion. Described here is a method to separate the diffraction pattern into its various components so that each may be interpreted independently. No assumptions are made about the nature of the atomic displacements δ, except that they are sufficiently small, and that the diffraction vector k/2π is sufficiently small, that terms beyond the term quadratic in k . δ may be neglected in the series expansion of exp [ik . δ].

137 citations


Journal ArticleDOI
TL;DR: In this paper, the reflection and transmission of a plane wave incident on a thin perfectly conducting screen perforated periodically with circular holes are considered, where the spacings between holes may be comparable to or less than the wavelength.
Abstract: The reflection and transmission of a plane wave incident on a thin perfectly conducting screen perforated periodically with circular holes are considered. The spacings between holes may be comparable to or less than the wavelength. The metal screen is assumed to be either backed by a flat dielectric sheet, or sandwiched between two or more dielectric sheets that are used to match the susceptsnce of the screen. The techniques of transmission line analysis and the method of moments are employed to obtain a solution for this general class of boundary-value problems. Numerical results for several cases of interest in both X and Ku bands were obtained and shown to be in good agreement with experimental measurements.

Journal ArticleDOI
TL;DR: This work uses the techniques of optical diffraction and optical filtering to study electron micrographs of myofibrils and of paracrystals of my ofibrillar proteins, and compares the optical Diffraction patterns with the X-ray diffraction patterns obtained from living muscle.
Abstract: We have used the techniques of optical diffraction and optical filtering to study electron micrographs of myofibrils and of paracrystals of myofibrillar proteins. The optical diffraction patterns provide information about periodic structure in the micrographs, and sometimes may reveal periodicities not apparent to the eye. We compare the optical diffraction patterns with the X-ray diffraction patterns obtained from living muscle, and this comparison can assist our interpretation of both the X-ray diffraction patterns and the electron micrographs. The optical diffractometer we have used is essentially similar to those described by Taylor & Lipson (1964), and by Klug & DeRosier (1966). The apparatus incorporates several refinements to facilitate operation. The recombining lens has a focal length, f , of about 1 m, and is placed so that the recombined image is formed at 2 f and has the same size as the subject. The diffraction subjects are not usually the electron micrographs themselves but copies on film. The film is of more uniform optical thickness than the glass electron micrograph, and is less fragile. Moreover, a set of films of varying contrast can be made from one micrograph.

Journal ArticleDOI
TL;DR: In this article, sets of integral equations are obtained that describe the X-ray diffraction in defective crystals and a simple description of defects is suggested both for weakly and strongly distorted regions.
Abstract: Sets of integral equations are obtained that describe the X-ray diffraction in defective crystals. A simple description of defects is suggested both for weakly and strongly distorted regions. In the case of ideal crystals, the solution for the wave fields with arbitrary incident beam distribution is given for crystals of arbitrary thickness. For distorted crystals, the integral equations give the universal method of treatment of both weakly and strongly distorted regions. The problem of image determination from strongly distorted regions is reduced to the solution of a simple one-dimensional integral equation. The first iteration approximation of initial integral equations is shown to give results similar to those of a Fourier analysis method with the defect being treated as a small perturbation.

Journal ArticleDOI
TL;DR: In this article, it was found that the intensity of a reflection from a polyoxymethylene single erystal decays exponentially after a short period, of reproducible duration, when the form of decay is variable.
Abstract: Irradiation in the electron microscope causes many changes in crystalline polymers, but the most easily measurable of these is the change in the diffraction pattern from single crystals. It has been found that the intensity of a {1010} reflection from a polyoxymethylene single erystal decays exponentially after a short period, of reproducible duration, when the form of decay is variable. The intensity of all electrons emerging from the column except those assosciated with the directly transmitted beam behaves in the same way. The period of variable decay corresponds to the period when diffraction contrast remains visible on the dark-field image. The decay of diffraction from polyethylene is more complex because radiation damage produces long-range disorder in the lattice, and this is accompanied by motion of the crystal, even when it is firmly supported on a carbon film. Measurement of these decay curves have provided the basis for comparison of radiation damage rates at different beam voltages a...

Journal ArticleDOI
TL;DR: In this paper, a theoretical model is developed that predicts the steady-state self-induced thermal distortion of a laser beam in an absorbing medium flowing transverse to the direction of propagation.
Abstract: A theoretical model is developed that predicts the steady-state self-induced thermal distortion of a laser beam in an absorbing medium flowing transverse to the direction of propagation A general perturbation solution is obtained that is appropriate when the thermal distortion effects are small A computer iteration of this solution is performed to obtain results that are applicable over an increased range of distortion conditions The effects of diffraction are neglected, and therefore the solution is limited to the near-field range of the laser beam or to a range where a caustic or infinite intensity is predicted as a result of the self-focusing associated with the thermal distortion effects of the flow The thermal distortion of an initially Gaussian collimated laser beam causes a shift of the beam into the direction of the flow and, at the outset of the thermal effects, a decrease in the peak intensity As the thermal distortion becomes more severe, the self-induced effects cause an increase in the peak intensity, ie, a focusing to occur Experimental results are presented for the thermal distortion of a CW CO 2 laser beam propagating in CS 2 that moves transverse to the propagation direction The data verify the predicted beam deflection and the defocusing as well as focusing effects of the self-induced thermal distortion


Journal ArticleDOI
TL;DR: In this article, the angular spectrum of straight-crested surface waves offers a valid means of solving surface-wave diffraction problems for homogeneous anisotropic crystals, and it has been demonstrated that beam steering is inherent in the diffraction process itself.
Abstract: : It has been shown that agreement can be obtained between experimental and calculated diffraction patterns over a frequency range extending from 107-905 MHz for propagation directions having very high phase-velocity anisotropy. One may therefore conclude that the angular spectrum of straight-crested surface waves offers a valid means of solving surface-wave diffraction problems for homogeneous anisotropic crystals. It has also been demonstrated that beam steering is inherent in the diffraction process itself and that calculations for diffraction losses may be undertaken for delay lines that are oriented either along pure-mode axes or with various types of misorientation, the only major requirement being that the surface-wave phase velocities be known for that particular crystal cut. (Author)

Journal ArticleDOI
TL;DR: In this article, the interference of the low-energy electron multiply scattered amplitudes can be eliminated by averaging data taken with different diffraction geometry but at fixed momentum transfer, which essentially yields a measurement of the kinematic intensities, which are most directly useful for surface crystallography.
Abstract: It is demonstrated that the interference of the low-energy-electron multiply scattered amplitudes can be in large part eliminated by averaging data taken with different diffraction geometry but at fixed momentum transfer. This essentially yields a measurement of the kinematic intensities, which are most directly useful for surface crystallography.

Journal ArticleDOI
TL;DR: The first observation of the diffraction of atoms from a clean metal surface was made in this paper, when the azimuthal angle of the incident beam relative to the surface lattice corresponds to the [110] direction of the lattice, the zeroth and first order diffraction peaks are sharp and well resolved and their positions agree satisfactorily with the predictions of simple diffraction theory.

Journal ArticleDOI
TL;DR: In this paper, an analytical expression for the irradiance pattern, in both the far field and near field, associated with the field propagating from an aperture with a truncated-gaussian field distribution was obtained by evaluating the diffraction integral subject to the Fresnel approximation.
Abstract: In this paper we obtain an analytical expression for the irradiance pattern, in both the far field and near field, associated with the field propagating from an aperture with a truncated-gaussian field distribution. This is accomplished by evaluating the diffraction integral subject to the Fresnel approximation. Numerical results are presented, which indicate how the patterns vary as the width of the truncated-gaussian field distribution changes.

Journal ArticleDOI
TL;DR: One-dimensional focusing and diffraction techniques for thin-film optical waveguides are examined in terms of their effectiveness and fabrication requirements in this paper, which is derived from potential application of optical planar guided waves to analog signal processing systems.
Abstract: One-dimensional focusing and diffraction techniques for thin-film optical waveguides are examined in terms of their effectiveness and fabrication requirements. Impetus for the study is derived from potential application of optical planar guided waves to analog signal-processing systems. Methods of altering modal-phase velocities to achieve ray deflection are investigated and include the use of substrate and film-embedded refractive-index discontinuities, multilayer films, thickness-profiled layers, and absorptive-material deposition on the guiding films. Thin-film lenses having f numbers as low as 2 to 3 may be fabricated by vacuum deposition of high-index materials, such as CeO2. Experimental results obtained with glass-sputtered films illustrating basic operation of thin-film lenses, gratings, and prisms are presented.

Journal ArticleDOI
TL;DR: In this paper, a diffraction minimum for 6Li at 500 MeV and values of q sup 2, the momentum transfer squared, up to 13 f(m sup (-2)).

Journal ArticleDOI
TL;DR: In this paper, a simple design formula is developed from a two-dimensional diffraction theory in order to characterize Gaussian beam diffraction in terms of three readily determined anisotropic propagation coefficients.
Abstract: The diffraction of any distribution can be discussed in terms of its constituent Gaussian modes. Simple design formulas are developed from a two‐dimensional diffraction theory in order to characterize Gaussian beam diffraction in terms of three readily determined anisotropic propagation coefficients. The relationship between these coefficients and known anisotropic propagation phenomena is explored. Beam steering, diffraction, and focusing effects are discussed in some detail. Attention is drawn to two particular effects: autocollimation and ``negative'' phase front curvature. Situations in which both occur in practice are discussed. The former should play a significant role in acoustic matched filter design. A cylindrical transducer is shown, within limits, to be an optimum structure for obtaining a single acoustic beam convergence, even in the presence of beam scattering. The parabolic diffraction theory is applied to the design of an experimental anisotropic focusing system.

Journal ArticleDOI
TL;DR: X-ray diffraction patterns from sperm-tail microtubules indicate that subunits with a 40 to 50 A packing diameter form filaments, alternately half-staggered, parallel to the tubule axis, which suggest a 12- or a 13-stranded structure fits best with the X-ray diagram.

Journal ArticleDOI
TL;DR: In this paper, a simplified wave equation involving the lattice potential averaged along the direction of the incident beam was derived by means of several distinct stages of approximation, and the way in which wave propagation in a lattice becomes classical at high energies was analyzed.
Abstract: By means of several distinct stages of approximation, the way in which wave propagation in a lattice becomes classical at high energies is analysed. First, the principle that deflection angles (whether caused by 'quantum' or 'classical' processes) are small at high energies is used to derive a simplified wave equation involving the lattice potential averaged along the direction of the incident beam. Next, the many-beam solution of this equation for the case of systematic reflections is presented in a form which emphasises the spatial variation of the potential, rather than its Fourier components. Third, approximate analytical expressions for the Boch eigenvalues and eigenfunctions, and for the amplitudes of the diffracted beams, are derived by means of the WKB method; this leads to easily calculable expressions for the number of diffracted beams expected in a given situation, as well as for the number of Bloch waves contributing to these beams.

Journal ArticleDOI
TL;DR: In this article, the reflection of a monochromatic plane electromagnetic wave by an electrically perfectly conducting grating is investigated, and the vectorial electromagnetic problem is reduced to two separate scalar problems: those corresponding to E- and H-polarization respectively.
Abstract: The reflection of a monochromatic plane electromagnetic wave by an electrically perfectly conducting grating is investigated. The vectorial electromagnetic problem is reduced to two separate scalar problems: those corresponding to E- and H-polarization respectively. A Green's function formulation of the problem is employed. For both cases an integral equation of the second kind for the remaining unknown function on the surface of the grating is derived. A numerical solution of this integral equation is obtained with the aid of either a (discrete) Fourier transform or a cubic spline approximation. Some numerical results of both the echellette grating and the sinusoidal grating are presented.

Patent
Lo D1, Norman D1, Torok E1
01 Sep 1971
TL;DR: In this paper, a magneto-optic light deflection system that utilizes the stripe domains in a magnetic film as a diffraction grating is described, where the angle of deflection of the light from the plane of the film is varied in two dimensions by varying the separation and orientation of the stripe domain.
Abstract: A magneto-optic light deflection system that utilizes the stripe domains in a magnetic film as a diffraction grating. The angle of deflection of the light from the plane of the film is varied in two dimensions by varying the separation and orientation of the stripe domains. The wall separation is varied by varying the intensity of a DC field in the plane of the film parallel to the stripe domains, or by varying the intensity of a DC field normal to the plane of the film. The orientation of the stripe domains is varied by varying the direction of the DC field in the film plane. Hysteresis is overcome by an AC tickle field perpendicular to the stripe domains.

Journal ArticleDOI
TL;DR: In this article, two wave fields, one of which is usually neglected, are considered as the solution of the dispersion equation, and the boundary conditions of the waves are considered for the gradient of amplitude normal to the surface besides the amplitude.
Abstract: X-ray diffraction phenomenon, for which the glancing angle of the incident beam is so small that the effect of specular reflection can not be neglected, is studied theoretically. Two wave fields, one of which is usually neglected, are considered as the solution of the dispersion equation, and the boundary conditions of the waves are considered for the gradient of amplitude normal to the surface besides the amplitude. Numerical calculations are made for (220) reflection of CuKα1 from a germanium single crystal. The diffracted beam is stronger in intensity and broader in half-value width than that where specular reflection is not considered.

Journal ArticleDOI
TL;DR: In this article, the angular patterns that result from the scattering of solar radiation by hexagonal ice prisms are computed under the assumption that the sizes of the prisms were many times the wavelengths of the incident radiation.
Abstract: The angular patterns that result from the scattering of solar radiation by hexagonal ice prisms were computed under the assumption that the sizes of the prisms were many times the wavelengths of the incident radiation. Using the results of geometric ray optics, a sufficiently large number of equally spaced, parallel rays were mathematically traced through the prisms (including internal reflections) in order to determine the contribution to the scattering pattern that was due to refraction and reflection. The contribution due to forward diffraction was approximated by the Kirchhoff formula. Results for a distribution of prisms in random orientation show the expected large diffraction peak in the intensity in the forward direction as well as a prominent relative maximum at an angle corresponding to the well-known halo phenomenon occasionally seen around the sun or moon. In comparison, the scattering pattern for a sphere whose size was such as to scatter out of an incident beam the same total amount of radiation as scattered out by the prisms, is also shown. The major difference in the two patterns is the absence of the halo in the case of the sphere. It can be readily shown that the energy of the radiation scattered in the angular region of the halo is comparable in magnitude to that in the forward peak, thereby showing the importance of the halo.

Journal ArticleDOI
TL;DR: In this article, the moments are directed along the c-axis of KFeF 4 and they show a two-dimensional antiferromagnetic order and below T N = 137°K the three-dimensional magnetic structure.