Geometrical Optics Limit Research Articles

Optical transfer function synthesis: a geometrical optics approach

It is shown that, within the limits of geometrical optics, it is possible to specify a pure-phase pupil function that will produce any desired optical transfer function (OTF) or point spread function (PSF) in an incoherent optical processing system. A general prescription for finding the pupil function is presented, and specific analytical results are given for simple power-law PSFs. As an example, the fabrication and testing of a logarithmic phase plate that produces an appropriate PSF for incoherent optical processing of transaxial tomography data are discussed.

Sensitivity of laboratory tests of gravitation beyond the geometrical optics limit

A report is given of a laboratory experiment that reached the sensitivity attained so far in extraterrestrial tests of gravitation beyond the limit of geometrical optics. The sensitivity achieved here in searching for a hypothetical relative change in the velocity of light ( δc/ c ≲ 2 × 10 -12) seems, however, already near to the presumable maximum for experiments carried out on the earth's surface with static fields.

An Empirical Formula for Broad-Band SAR Calculations of Prolate Spheroidal Models of Humans and Animals

An empirical relation for calculating approximate values of the average specific absorption rate (SAR) over a broad-frequency range for any prolate spheroidal model is derived for E-polarized incident plane waves. This formula provides a simple and inexpensive method for calculating the SAR for human and animal models, which otherwise requires complicated and expensive methods of calculation. The formula satisfies the f/sup 2/ SAR behavior at lower frequencies, the resonance characteristic at intermediate frequencies, the 1/f behavior past resonance, and the dependence on the dielectric constant at the geometrical optics limits. An expression for the resonance. frequency f/sub 0/ in terms of the dimensions of the model is also derived. The unknowm expansion coefficients were determined by curve-fitting all the data available in the second edition of the Radiofrequency Radiation Dosimetry Handbook. Numerical results obtained from the empirical relations are generally in good agreement with those calculated by other methods. Limitations of the formula and suggestions for its improvement are also discussed.

On the uniqueness question in the problem of phase retrieval from intensity measurements

The treatment of the problem of deducing the phase of an optical wavefield from image- and diffraction-plane intensity measurements recently presented by Robinson [ J. Opt. Soc. Am68, 87– 92 ( 1968)] is shown to be valid only in the limit of geometrical optics. A simple proof and example of the nonuniqueness of the phase retrieval problem in this limit is presented. It is argued that although the phase retrieval is not unique in the geometrical optics limit it does appear to possess a unique solution within the framework of the diffraction theory of image formation.

Elastic scattering of waves by a black hole

The properties of a Schwarzschild black hole as an elastic scatterer of waves are studied in detail. Phase shifts and elastic scattering cross sections are obtained for a wide range of energy and angular momentum from the exact analytical solutions of the wave equation by employing computational techniques developed by the author in an earlier paper. Features of the low-frequency elastic scattering are pointed out, and comparison with other approximations is made. The angular distribution of the scattered waves is obtained for different values of the frequency. As could be expected, it presents a peak ($\ensuremath{\propto}{\ensuremath{\theta}}^{\ensuremath{-}4}$) in the forward direction and a glory in the backward direction. For intermediate angles it shows a complicated oscillatory behavior as frequency increases that disappears only at the geometrical-optics limit. By using the elastic scattering parameters obtained here, we calculate the angular distribution of the absorbed waves. The differential absorption cross section is isotropic for low frequencies and gradually shows features of a diffraction pattern as frequency increases. It shows an absolute maximum in the forward direction which grows and narrows with the frequency. In the geometrical-optics limit there results a Dirac $\ensuremath{\delta}$ function.

Invariants and classification of gauge fields

A classification of gauge fields in terms of spinors and the rank of a matrix of Lorentz invariants is proposed. A peeling theorem for the Yang-Mills field in the asymptotic limit is shown to follow from an earlier work of Roskies. An analogous theorem is proved for all gauge fields in the geometric optics limit.

Absorption and emission spectra of a Schwarzschild black hole

The absorption spectrum of a Schwarzschild black hole is studied in detail. Accurate and useful computational methods based on the analytical resolution of the wave equation are developed. In this way phase shifts and absorption cross sections are obtained for a wide range of energy and angular momentum. Comparison with the explicit results valid for low and high frequencies is made. The total absorption cross section of the black hole is obtained as a function of the energy. It presents behavior characteristic of a diffraction pattern. The constant geometric-optics limit [$(\frac{27}{4})\ensuremath{\pi}r_{s}^{}{}_{}{}^{2}$] is approached in an oscillatory fashion. The physical interpretation of these results is given and a simple model which describes qualitatively the absorption of waves by the black hole is presented. From these absorption parameters, the Hawking emission rates are calculated and their properties discussed.

Mode losses in unstable resonators with rounded edges

The waveguide approach to unstable resonators is employed for determining the eigenmode losses in unstable 2-D and 3-D resonators with perfectly reflecting round-edged mirrors. It is found that edge rounding in cylindrical mirror configurations leads to mode detachment at lower Fresnel numbers than for the sharp-edged case and that the loss behavior approaches more closely the geometric optical limit for infinite mirrors. Similar effects are observed in resonators with circular mirrors, where mode detachment does not occur at all when the mirror edges are sharp. It is also shown that edge rounding improves the separation between the low-loss detached mode and the next higher order modes.

Lasers with unstable resonators in the geometrical optics limit

A simple model is described for estimating the intensity distributions inside and emerging from lasers with high-Fresnel-number unstable resonators.

Sound propagation through a fluctuating stratified ocean: Theory and observation

We have derived expressions for the mean-square phase and intensity fluctuations and their spectra for cw sound propagating through a channeled fluctuating ocean. The ’’supereikonal’’ approximation reduces to the geometric optics (eikonal) limit for short acoustic wavelengths: λ≪2πL2H/R and λ≪L2V/(R tan2ϑ), where LH and LV are horizontal and vertical correlation lengths of the fluctuations, R is range, and tanϑ is the ray slope, replacing the traditional (and much more severe) Fresnel condition λ≪2πL2/R for a homogeneous isotropic ocean. The results can be expressed in closed form for an exponentially stratified ocean model and associated ’’canonical sound channel,’’ with superimposed fluctuations from an internal wave model spectrum based on oceanographic observations. The parameters are the stratification scale B, the inertial and buoyancy frequencies ωin and n (z), the scale j* of internal wave mode numbers, and the internal wave energy per unit area. The results are in reasonable agreement with numerical experiments based on the parabolic wave equation. For the ’’singlepath’’ 4-kHz transmission over Cobb Seamount the observed and computed rms fluctuations in phase are 1.6 and 2.5 cycles, respectively; in intensity these are 5.5 and 2.2 dB, respectively, with anomalous intensities measured at high frequencies (’’sporadic’’ multipathing?). For the multipath 406-Hz MIMI transmission, we obtain 4×10−3 and 5×10−3 sec−1, respectively, for the experimentally determined and the computed rms phase rates. Subject Classification: [43]30.20, [43]30.40; [43]20.15.

Transmission Characteristics of Specularly Reflecting Light Pipes Uniformly Irradiated by Obliquely Inclined Rays

The light-pipe efficiency or fraction of the incident energy transmitted from one end to the other is expressed as a function of the wall reflectivity rho and geometrical parameters. Light pipes with both circular and rectangular cross sections are considered. The analysis includes skew rays and is mathematically exact within the limits of geometrical optics and a constant specular wall reflectivity. The distribution along the light pipe of the average intensity incident upon the walls is also derived.

The statistical properties of echoes diffracted from rough surfaces

This work aims to provide the statistical theory behind recent wide-angle pulsed radar experiments to determine the topography and roughness of (a) polar glacier beds from measurements near the snow surface, and (b) the lunar surface from satellites in close orbit. W e analyse the statistical properties of the echo received back at the source when a quasimonochromatic pulse of radiation, isotropic in direction, is reflected from a rough surface which is a random perturbation of a horizontal plane. Kirchhoff diffraction theory is employed, so that multiple scattering and shadowing are neglected. But the r.m.s. surface height is unrestricted, and closed formulae are obtained which are valid through the transition from coherence (flat-mirror surface) to incoherence (surface of high relief, to which geometrical optics is applicable). For fixed source-receiver height the average echo wavefunction, and the average echo power, are calculated as functions of delay time, and the echo autocorrelation function is calculated as a function of the separation of two source-receiver points at fixed delay time. The duration of the echo is calculated, and the long-wave limit, the smooth-surface limit, the Fraunhofer limit, the continuous wave limit and the geometrical-optics limit are examined. A method is suggested for inferring the statistics of the surface from measurements of the average echo power. The theory of random noise is applied to the fluctuations about the calculated averages which occur as the source-receiver is moved horizontally. These fluctuations constitute 'spatial fading', and we calculate several measures of the spatial fading rate for the echo wavefunction and the time-smoothed echo power, as well as a measure of the degree of spatial periodicity of the fading. Finally, an estimate is made of the smallest detectable horizontal displacement of the source-receiver relative to the rough surface.

Electromagnetic scattering from a radially moving spherical discontinuity

The scattering of a spherical electromagnetic wave from a radially moving spherical discontinuity in index of refraction is analyzed. The Doppler shifts are found to depend on the radius as well as the velocity of the discontinuity. As an example the scattering from a tenuous evaporating dielectric sphere is found. The plane-wave scattering problem is formulated and it is shown that the scattered frequencies have a satisfying physical interpretation in the geometrical optics limit.

Geometrical Optics in Thin Film Light Guides

We consider thin film light guides consisting of a transparent film of high refractive index deposited on a substrate of lower index. The propagation of light in such a two-dimensional transmission medium can be described within the limits of geometrical optics by an effective index of refraction N. Its value depends on the film thickness. Therefore, a light beam in the thin film guide is refracted or totally reflected at a step of film thickness. We discuss these phenomena (Snell's law) and demonstrate them experimentally, using ZnS films on glass as guides. As applications, we show a thin film prism and thin film lenses for guided light beams. By properly choosing the film thicknesses at both sides of the step, one can obtain an unusually large positive or negative wavelength dispersion of the refraction or, if desired, achromatic refraction.

Propagation of the Fourth-Order Coherence Function in a Random Medium (a Nonperturbative Formulation)*

In a previous paper, we considered a perturbation solution for propagation of the fourth-order coherence function in a random medium. In this paper, it is shown how under certain conditions this solution may be extended to treat the propagation problem when the field fluctuations need not be small. A differential equation governing the fourth-order coherence function is derived. A solution is obtained in the geometric-optics limit when the four points of the coherence function are distinct.

Shadowing of Randomly Rough Surfaces

Measurements have been made of the reduction in illuminated area of a randomly rough surface due to shadowing in the geometrical optics limit. These measurements were made using extremely high-contrast photographs of a randomly rough model surface having known statistics. The illumination was quasicollimated incoherent light. The resulting photographs were measured by two different methods in order to determine the ratio of total area to shadowed area. Data are given for several surfaces having different mean-square slopes. The resulting shadowing curves, given as a function of grazing angle, are compared with a theory developed by Wagner. The agreement was found to be generally good, and the theory for area reduction due to shadowing appears to be sufficiently accurate to warrant its incorporation in the theory of scattering from randomly rough surfaces.

Zum Übergang von der Wellenoptik zur geometrischen Optik in der allgemeinen Relativitätstheorie

The transition from the (covariantly generalized) MAXWELL equations to the geometrical optics limit is discussed in the context of general relativity, by adapting the classical series expansion method to the case of curved space time. An arbitrarily moving ideal medium is also taken into account, and a close formal similarity between wave propagation in a moving medium in flat space time and in an empty, gravitationally curved space-time is established by means of a normal hyperbolic optical metric.

Optical Resonators in the Unstable Region

The spectrum of characteristic oscillation frequencies and modal losses for spherical mirror opticalresonators is derived in an approximation equivalent to geometrical optics. This is a significant approximation for unstable or high loss configurations when the Fresnel number is large. The analysis is based on the scalar integral equations of Fox and Li and agrees with the known result for the dominant mode losses in the geometrical optics limit.

Propagation of an electromagnetic pulse from a nuclear burst

The general study of nuclear detection clearly identifies the theoretical problem of predicting the effect of the propagation medium on the form or shape of the low-frequency electromagnetic pulse of nuclear origin propagated to great distance in the terrestrial waveguide. This is accomplished with a theory of propagation into which an arbitrary electron density profile of the ionosphere can be introduced using a quite general cold magnetoplasma theory. In a previous paper by the author, the propagation of the ground wave electromagnetic pulse was discussed in detail and the engineering significance of the propagation theory, to nuclear detection systems was considered quantitatively. At distances from the source greater than approximately 100 km, the ionosphere reflections become an important consideration. The theory of propagation for ionospheric waves is introduced into the analysis in this paper. A transient waveform reconstructed theoretically at great distance from the source can be analyzed in the time domain with the aid of a geometric series expansion of the solution in the space domain. Thus, different times on the propagated pulse can be identified with particular reflection regions of the ground and ionosphere along the propagation path. In the geometric-optical limit, the individual ionospheric waves correspond approximately to rays traveling to and fro between the ionosphere and the ground, and the composite pulse at great distance can be considered to be a superposition of a multiplicity of pulses, each pulse delayed in time by the delay time of the ray.

The diffraction theory of large-aperture spherical reflector antennas

The field along the axis of a spherical reflector is determined from the geometry of the system rather than from each term of the aberration taken separately. The procedure shows how the field distribution changes from the case of small aberration, where there is a well-defined focus, to the geometric optics limit. A spherical reflector becomes an efficient antenna when a set of feed elements is located along the axis to reduce the effects of spherical aberration. The number and position of these elements is dictated by the size and curvature of the reflector and the allowable distortion of the wavefront.