Terms Of Geometrical Optics Research Articles

Transformation of Gaussian beams by spherical and cylindrical lenses: the scalar diffraction approach

The transformation of a Gaussian beam by a lens has often been described in terms of geometrical optics by the formal application of the ABCD-matrix law of the lens to the complex beam parameter. In this paper we present as a wave optical method the scalar diffraction approach to the same problem. It is shown that the fresnel approximation yields results which are analytically identical to those obtained by the ABCD-law, if truncation effects of the lens can be neglected.

Change of blaze wavelength as a function of position on the surface of a concave grating

Conventional ruling of a concave diffraction grating with blaze angle constant relative to the optical axis produces a grating whose blaze wavelength changes continuously with displacement across the face of the ruled area. A simple explanation of the effect is given in terms of geometrical optics, and quantitative expressions are developed that relate the expected blaze-wavelength shift to the grating constant and aperture ratio. Results are compared with experimental efficiency measurements, and the magnitude of the effect in a typical grating is discussed.

On the refractive properties of media with poles or zeros in the index of refraction

The refractive properties of inhomogeneous fibers are examined with emphasis being placed on the limiting situation where the index of refraction possesses poles or zeros. If, e.g., the index of refraction is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n(\xi) = \xi^{m}, \xi = r/a</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</tex> is the radius of the fiber and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> an arbitrary constant, it is found that energy integrability is satisfied if <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m &gt; -1</tex> . When <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m \leq -1</tex> energy infinities occur. The ray behavior of such media is examined in terms of geometrical optics, and corrections to geometrical optics are obtained by an asymptotic analysis of the exact solution. For <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m &gt; 0</tex> , the lens is of the diverging type, and when the angle of incidence is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\alpha = 0</tex> , geometrical optics predicts that rays "reflect" at various angles from the origin (depending on the value of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> ). When <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m &lt; 0</tex> , rays "wrap" around the origin several times with a zero radius of curvature before they leave the lens ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\alpha = 0</tex> ). For <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1 &lt; m &lt; -\frac{1}{2}</tex> , it is found that when <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\alpha= [(2m + 1)/2m] \pi</tex> caustics occur ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\alpha = 0, \pi/2</tex> excluded). Pictorial diagrams show the behavior of these caustics and the correction coefficients to geometrical optics are obtained.

Convolution scattering model for ferroelectric ceramics and other display media

A theoretical model has been developed which is based upon the approximation that the multiple scattering from ferroelectric ceramics and other display media can be described in terms of geometrical optics. The model has been used to describe the scattering from PLZT, liquid crystals, and display screens and has agreed well with exprimental data. The scattering characteristics of several current display materials have been compared with regard to their use in wide-angle direct-view displays. The model has predicted the material parameters for new and more effective display materials. Measurements of a simulated new medium have shown close agreement with the predicted scattering characteristics.

Diffraction by a conducting wedge in the near field

The purpose of this paper is to derive a near-field asymptotic solution for the diffraction of waves by a perfectly conducting wedge. Thus, starting from the series solution for cylindrical-wave incidence and using an integral expression for products of the Bessel functions involved, the total field is represented as a geometrical-optics term plus a diffraction integral. Using an integral form for the field of a spherical wave, the solution for the cylindrical-wave excitation is extended to the spherical-wave case. The method of steepest descent is used to obtain an asymptotic expression for the diffraction integral in terms of Fresnel integrals. The accuracy of our expressions is established by comparison with previous results as well as the exact solutions.

Reflectographic method for inspection of the internal surface of smooth cylinders

A simple technique is described for the inspection of the interior of smooth or polished cylinders. The technique may be described purely in terms of geometric optics and either a laser or a white-light source can be used. The point source of light is arranged to be coaxial with the cylinder and at a short distance from one end of it. The light emerging from the source is reflected by the walls of the cylinder in a controlled fashion, and is collected on a photographic film or viewing screen on the far side of the cylinder from the source. Minor deviations from parallelism of the cylinder walls are thereby greatly magnified and may easily be measured using the formulae given in the text. This method can be used by unskilled soldiers to take reflectographs of gun barrels for inspection, and for following the working life of a particular barrel. The technique can be further refined by replacing the screen by a cylindrical one.

High-frequency scattering from a conducting cylinder with an inhomogeneous plasma sheath

The high-frequency backscattering characteristics of an infinite conducting cylinder enveloped in a radially inhomogeneous plasma sheath are studied in detail. When the permittivity of the sheath has a general power-law dependence, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\epsilon(\rho) = \epsilon(\rho_{0}) (\rho/\rho_{0})^{p}</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\rho_{0}</tex> is the outer radius of the sheath and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</tex> is any real number <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\neq -2</tex> , the wave function in the plasma is expressible in terms of Bessel functions of a fractional order. The slowly convergent series form of the backscattering coefficient is first recast into an integral by means of the Watson transformation, which is then asymptotically evaluated by the method of stationary phase. The mathematical result is conveniently interpreted in terms of geometrical optics by identifying the contributions due to the central ray and the trapped "interrupted" and "uninterrupted" rays. In contrast to the rather unpredictable and violent backscattering coefficient variations with frequency when a conducting cylinder is clad in a homogeneous plasma sleeve, the change in the coefficient for the same cylinder enveloped in a plasma sheath with a power-law radial inhomogeneity is much more smooth and, in most cases, the approximate locations of the maximums and minimums can be predicted. Numerical results showing the dependence of the backscattering coefficient on the type of power-law inhomogeneity, sheath thickness, and permittivity level are presented in graphical form.

Refraction: Neurophysiological and Psychological Viewpoints.

This brief monograph presents the author's views on refraction which are based on an extensive and long experience. He strives to inject some new life into what he believes is an unpopular subject. The central idea is that cerebral visual mechanisms play a fundamentally important role in refraction and that it is a fallacy to evaluate visual acuity solely in terms of geometrical optics. The heart of the book is a presentation of the author's own investigative work and statistical evaluation of clinical patients with astigmatism. He presents evidence to show that central mechanisms are important in the ability of patients to adapt and adjust (accommodate) to astigmatism. He emphasizes that subjective tests are primarily tests of neural function. The idea that astigmatic accommodation has both a peripheral and central component is interesting and worth discussing. It is refreshing to have an experienced ophthalmologist set down his ideas on a

A Review of Optical Data-processing Techniques

Various optical data-processing techniques are reviewed in this article. Both coherent and incoherent optical systems are considered. Coherent systems are described in terms of spatial filtering concepts and two principal methods for implementing a general linear integral operator are discussed: one using space-domain reference function masks and one using frequency-domain spatial filters. Incoherent systems are described in terms of geometrical optics.

Wave-Optical Aspects of Lorentz Microscopy

The customary defocused and Foucault modes of Lorentz microscopy of magnetic films are usually described in terms of geometric optics. However, Wohlleben has shown that geometric optics has a restricted range of validity; a more fundamental approach is provided by wave optics. The defocused and Foucault modes may be discussed in terms of wave optics, and for the defocused mode, it can be shown explicitly that the geometric theory is simply the first approximation to the wave-optics theory. Consideration of wave optics also leads to the proposal of two additional modes of Lorentz microscopy: Zernike phase-contrast and interference microscopy; these modes cannot be described on the basis of geometric optics. The most fundamental problems in magnetic films which are amenable to study by Lorentz microscopy are investigations of the fine structures of domain walls and magnetization ripple. These problems are discussed in terms of wave optics for all four modes of Lorentz microscopy; in particular, the intensity distribution of the zero-width divergent domain wall is explicitly calculated for each mode. For practical experiments, the importance of coherence, i.e., of the illumination source size, is emphasized, and the experimental aid of holography is suggested. Since the Wohlleben limit is valid for all four modes, however, there is no resolution advantage inherent in any one mode. The choice of modes for solution of the domain wall and ripple problems therefore depends upon experimental convenience. It is concluded that the defocused mode seems most promising for practical application; Fresnel diffraction is preferred for the domain-wall problem, while Fraunhofer diffraction using low-angle electron-diffraction techniques is fruitful for the ripple problem.

A class of boundary value problems

A simple method is presented for the solution of the partial differential equation of diffusion type with constant values of the solution or its normal derivative prescribed on the surfaces of a wedge of arbitrary angle. It is shown that this solution may be transformed in such a manner that it yields the solution to a similar class of boundary value problems for the time-harmonic wave (i.e. Helmholtz’s) equation. Direct solutions are also obtained for the problem of the diffraction of acoustic or electromagnetic plane waves by a perfectly absorbing or a perfectly reflecting wedge. In one solution the difraction problem is solved by modifying the solution of a similar boundary value problem for the diffusion equation. In the second formulation the solution of the diffraction problem is obtained by expressing the total solution as a sum of a diffracted field and geometrical optics terms. The diffracted field is then obtained by imposing the conditions of continuity across the shadow lines of geometrical optics. The solution for the diffraction of an arbitrary plane pulse by a wedge is also obtained; the solution being valid even if the boundary conditions are of the impedance type.

Focusing Procedures for Electrostatic Accelerators

Ion optics for Van de Graaff accelerators are reviewed in elementary terms of geometric optics with particular reference to the Oak Ridge 3-Mv accelerator. This machine produces well-focused beams of over 100 μa and pulsed peak currents of several hundred μa. The beam diameter measured at a distance of 2.7 tube lengths from the accelerator tube exit decreases from 0.26 in. at 0.38 Mv to 0.10 in. at 2.47 Mv. The beam divergence at 1.8 tube lengths from the exit is nearly constant, ≃2.8×10−3 rad. The system is analyzed in terms of five separate lenses: (1) einzel lens, (2) gap lens, (3) accelerator tube, (4) 90-degree analyzing magnet, and (5) strong-focusing lens. It is shown that the crossovers formed in the gap lens and near the exit of the analyzing magnet are of fundamental importance. Brief reference is made to the small energy spread, ≃0.025%, for the beams obtained from this accelerator in spite of the large control slit spacing demanded by a large crossover.

A Design for an Absolute Light-Scattering Photometer

An optical design for the receiver of a light-scattering photometer is described. The design lends itself to a simple analysis, in terms of geometrical optics, which shows that absolute measurements of the Rayleigh factor are easily made.Modifications of the receiver of a Brice-Phoenix photometer in use in this laboratory are described and analyzed.A preliminary value of 46×10−6 cm−1 at 436 mμ and 23° C is given for the Rayleigh factor for benzene.

The calibration of disc recordings by light-pattern measurements

The paper sets out an analysis, in terms of geometrical optics, of the Buchmann-Meyer, or light, pattern which is formed by reflection from the modulated grooves of a recorded tone-disc. The pattern is considered as lying in the focal plane of the circular groove walls and not on the disc surface, as is usually assumed. A new apparatus for measuring the width of the patterns is described and the errors in disc-surface and focal-plane measurements found when using it are compared. The latter results are shown to possess the higher degree of consistency and their correctness is confirmed by an examination of the pattern formation in terms of physical optics. The origin of two sets of interference fringes which are visible in the patterns is also described. The decrease of luminous intensity which occurs at the edge of the patterns is then computed and it is shown that the new method of measurement described should give a higher degree of accuracy than hitherto available, even under some practical conditions when the edge of the pattern is not sharply defined.

The calibration of disc recordings by light-pattern measurements

The paper sets out an analysis, in terms of geometrical optics, of the Buchmann-Meyer, or light, pattern which is formed by reflection from the modulated grooves of a recorded tone-disc. The pattern is considered as lying in the focal plane of the circular groove walls and not on the disc surface, as is usually assumed. A new apparatus for measuring the width of the patterns is described and the errors in disc-surface and focal-plane measurements found when using it are compared. The latter results are shown to possess the higher degree of consistency and their correctness is confirmed by an examination of the pattern formation in terms of physical optics. The origin of two sets of interference fringes which are visible in the patterns is also described. The decrease of luminous intensity which occurs at the edge of the patterns is then computed and it is shown that the new method of measurement described should give a higher degree of accuracy than hitherto available, even under some practical conditions when the edge of the pattern is not sharply defined.