Showing papers on "Paraxial approximation published in 1979"

Gaussian beam propagation beyond the paraxial approximation

Abstract: The propagation of a Gaussian beam in a homogeneous, isotropic, local, linear, and nonmagnetic dielectric medium is studied using the angular spectrum representation for the electric field. The electric field associated with the Gaussian beam inside the dielectric medium consists of the paraxial result and higher-order non-Gaussian correction terms. It is shown that the second-order correction term satisfies an equation consistent with the recent work of Lax, Louisell, and McKnight. Numerical results showing the corrections to the paraxial approximation are presented.

Variational approach to nonlinear self‐focusing of Gaussian laser beams

Abstract: The problem of nonlinear self‐focusing of Gaussian laser beams is reformulated in terms of a variational principle. By means of approximating Gaussian functions, expressions are obtained for the equilibrium radii and nonlinear frequency shifts of stationary self‐trapped laser beams. The nonsteady propagation is given an illuminating form in terms of a potential function description. The analysis confirms the recent results obtained by moment theory as opposed to those based on paraxial ray approximations.

Loss calculations in bent multimode optical waveguides

Abstract: We present a straightforward geometric optics method to calculate the power attenuation induced by a circular bend in the axis of a multimode optical waveguide. The method involves tracing of rays and use of generalized power transmission coefficients. Both slab and fibre waveguides (with either step-index or parabolic core profiles) have been considered; the source has been assumed to be Lambertian.

The self-focusing of laser beams in magnetoplasmas: the moment theory approach

Abstract: The self-focusing of electromagnetic waves in plasmas (both unmagnetised and magneto-active) has been studied by the moment theory approach. It is seen that the earlier method of studying the phenomenon by the paraxial ray approximation gives results very different from those in the present study. The divergence between the two theories is acute close to gyroresonance for a longitudinally propagating right-handed circularly polarised wave. The moment theory also shows that there is a critical power for self-focusing and no upper limit to the power that can be self-trapped. These conclusions are in qualitative agreement with those from the paraxial ray approach.

Theory of paraxial self‐focusing

Abstract: The phenomena of steady‐state self‐focusing and self‐trapping have been analyzed in the paraxial approximation through a formalism utilizing a generalization of the angular‐spectrum representation of electromagnetic beams. The method discards the a priori quasioptic approximation used in all previous theories; the effect of power loss from paraxial regions into beam aberrations during focusing and defocusing has also been incorporated. Results on self‐trapping suggest greater dominance of nonlinear effects over beam‐diffraction effects than predicted by earlier paraxial theories particularly in the region of nonlinear saturation. These results are in qualitative agreement with the moment theory of self‐trapping. The present theory indicates certain periodic focusing solutions much different from the earlier paraxial theory.

Optical apparatus for controlling the distribution of illumination

Abstract: In order to provide a laser beam having a relatively constant width (i.e. transverse spacing between points receiving a theshold irradiance), a converging lens 15 having substantial negative spherical aberration is used. In the far field, the beam comprises paraxial rays 18 which have diverged to provide the desired beam-width. In the near-field, the smaller, higher-irradiance beam formed by these rays is augmented by a `sheath` of marginal rays 19 which are not refracted towards the optical axis as strongly as the paraxial rays. The shape of the beam in the far field is determined primarily by the shape of the (laser) light source 10, while that of the near-field beam is controlled by a mask 14. In another arrangement, a lens 102 having substantial positive spherical aberration is used to create a `light source` having an accurately-controlled brightness distribution and very small size. Paraxial rays 109 are brought to a focus at a plane 108 defining the `position` of the `light source`, while the more strongly refracted marginal rays 110 cross the optical axis before reaching this plane, thus creating a halo round the bright central beam produced by the paraxial rays. The brightness distribution at the plane is controlled by adjusting the spherical aberration of the lens, and by masking.

Self-focusing of electromagnetic beams in semiconductors

Abstract: The self-focusing of an electromagnetic beam in parabolic and nonparabolic semiconductors has been investigated in the presence of a longitudinal magnetic field. The wave equation has been solved in paraxial ray and WKB approximations by expanding the dielectric tensor as epsilon = epsilon f(EE* mod r=0)+ gamma r2. No restriction is put on the amount of nonlinearity and the effect of the self-induced inhomogeneity is taken into account. For intense beams and strong magnetic fields, the nature of self-focusing is considerably different from that predicted by earlier analyses. The self-made waveguide propagation is discussed in detail.

Particle in a potential channel (waveguide-optical analogy)

Abstract: The motion of a particle in a potential channel (two-dimensional well without a potential gradient along the third coordinate) is characterized by a discrete set of allowed modes or travelling Ψ waves. A paraxial potential channel can transmit the initial distribution of the Ψ function to channel cross sections some distance from the entrance. The dispersion equation for a channel having transverse dimensions of the order of the Compton wavelength is the same as the free motion equation, the eigenvalue of the lower channel mode being identified with the rest energy of the particle.

Refractive-index distribution for a prescribed ray path

Abstract: A theory is presented of the refractive-index distribution which produces a prescribed two-dimensional ray path. When two inhomogeneous media give the same paraxial ray equation with respect to a prescribed path, they are referred to as the paraxially equivalent media. Various types of the equivalent media pertaining to a circular ray path are discussed, and numerical examples of the paraxial ray paths are given. Furthermore, a mechanical model of a ray is used to clarify the physical meaning of the condition imposed on the index profile to produce a prescribed ray path.

Graphical ray tracing for conic surfaces

Abstract: Alejandro Cornejo-Rodriguez and Alberto Cordero Davila Instituto Nacional de Astrofisica, Optica y Electronica, A. P. 216, Puebla, Pue, Mexico. Received 6 June 1979. 0003-6935/79/183075-02$00.50/0. © 1979 Optical Society of America. The known method of graphical ray tracing is an efficient method used by lens designers to check rapidly diameters, thicknesses, and the trajectory of particular rays through an optical surface or system. The situation is straightforward for spherical surfaces, but for conic surfaces, the method requires the knowledge of the direction of the normals, and only for the parabolic surface can this be done easily. If instead of the direction of the normals we can determine the locus of some reference points corresponding to the different zones of the surface, we can trace the normal from such reference points to the points on the surface. In a recent paper, both the longitudinal aberration of the normals and the coordinates of the centers of curvature of the different zones of a conic surface (caustic coordinates) were determined with reference to the paraxial center of curvature. Since, to our knowedge, these expressions had not been used in the past for graphical ray tracing, we are proposing their use as follows. From Fig. 1 and using the formula for the sagitta Z of a conic of revolution given by

General theory of the magnetic einzel lens based on the axial field distribution model H(z)=H0C∣ z/a ∣m-1/(1+∣ z/a ∣ 2m)

Abstract: Investigations are described which show that the magnetic einzel lens given by the axial field distribution H(z)=H0C mod z/a mod m-1/(1+ mod z/a mod 2m), where m is the parameter depending on the double-gap pole-piece geometry and the magnetic flux concentration, is the simple magnetic einzel lens whose solutions of paraxial ray paths can be obtained in the form of hypergeometric functions. Based on these solutions, the important focal quantities and third-order aberrations are expressed analytically in terms of the maximum width amax of the field and the lens strength k, mostly depending on the ampere-turns for various magnetic einzel lenses specified by m. The important parameters m and amax and their dependence upon double-gap pole-piece geometry are derived approximately in accordance with the potential theory for a cylindrical magnetic einzel lens. The optimum lenses with the minimum focal length and aberrations, and satisfying the anastigmatic conditions, are described and discussed in terms of practical design.

Scattering of low‐energy electrons on micron‐size pinholes

Abstract: In exploring forces between charged particles and surfaces, we have sent a beam of low‐energy (0.5–40 eV) electrons through two (laser drilled) micron‐size pinholes in thin films of gold spaced 10 cm apart. Image forces associated with the solid around the second pinhole cause the beam to diverge into an angular distribution which is scanned with a movable slit and Channeltron detector. Distributions have widths which depend inversely on electron energy as expected from transit time considerations. The paraxial portion of the distribution appears to diverge as though from a weak negative electron lens. The impulse approximation for the radial component of the electron motion is used to derive a radial potential averaged along a line parallel to the pinhole axis. This potential approximately equals the expected value e/4x at x=10 A from the pinhole edge but deviates from this function at greater distances. This deviation may be due to geometric effects, to the impulse collision model employed, or to geometric irregularities at the edge. The results are useful in estimating efficiencies of filters for charged aerosol collection.

The Optical Level System

Abstract: A zoom projection lens is described which enables the user of a Super-8 projector to adjust the level of the projected image. The paraxial characteristics are discussed with the aid of the Delano and aperture diagram. The aberrational behaviour is given in terms of the second and third order aberration coefficients.

The magnetic vector potential in a gapped tunnel

Abstract: The problem of computing a matrix of magnetic vector potentials in a gapped tunnel is studied. First, a method is given for deriving the parameters of a set of ideal current loops that will serve as sources for the field of a PPM-focused TWT, then the magnetic vector potentials are derived from the loop parameters. The axial and radial fields at any point are then obtained by nine-point interpolation and differencing of the potential matrix. The method is free from any paraxial approximations.

Filamentation of a Gaussian laser beam in a plasma

Abstract: Investigates the growth of filamentation instability arising from the interaction of an ion-acoustic wave with a Gaussian laser beam using the two fluid model of plasma and the paraxial approximation. The pump wave (laser beam) on account of its non-uniform character modifies the carrier density and thereby gets self-focused provided its initial power exceeds the critical value. This process further modifies the growth rate of filamentation instability.