Geometrical Optics Limit Research Articles

Diffraction, self-focusing, and the geometrical optics limit in laser produced plasmas

The effect of diffraction on the self-modulation of an intense laser beam in an initially uniform hydrogen plasma is investigated. A formalism is used in which the diffraction term in the paraxial wave equation can be arbitrarily reduced by the use of a weight factor ι. In the limit where ι approaches zero, it is shown that the paraxial wave equation correctly reduces to the geometrical optics limit and that the problem then becomes formally equivalent to solving the ray-tracing equations. When ι=1, the paraxial wave equation takes its usual form and diffraction is fully accounted for. This formalism is applied to the simulation of self-modulation of an intense laser beam in a hydrogen plasma, for which diffraction is shown to be significant.

Main optics equations for waveguides with a curved surface

An analysis is made of planar waveguides with a continuously curved surface, of the type used to form geodesic lenses in integrated optics. Two types of normal electromagnetic waves supported by these waveguides are considered. The Maxwell equations are used to derive a two-dimensional scalar wave equation describing the propagation, interference, and diffraction of these waves by a curved waveguide surface. The corresponding eikonal equation is formulated in the geometric-optics limit.

Statistical ray tracing in plasmas with random density fluctuations.

The propagation of light rays in a plasma is considered for small density fluctuations superimposed upon a given background density profile. The lowest-order angular spreading, focusing, and drift effects are calculated in the geometrical-optics limit using a statistical random-walk approach. This method is an economical semianalytic alternative to a purely numerical ray-trace approach and is of particular but not exclusive interest in describing the propagation of laser light in laser-produced plasmas. The model is applied to a simple fluctuation spectrum characterized by an rms amplitude and by a correlation length. Results are presented for the evolution of the intensity profiles of beams incident on a plane-parallel linear-profile plasma slab as a function of the angle of incidence. Comparisons of these results with numerical Monte Carlo ray-trace solutions show good agreement. Density fluctuations as small as a few percent of the critical density can, for example, produce significant angular broadening in the specularly reflected beam and reduce the sensitivity of the absorption fraction to the incidence angle, particularly near normal incidence.

Scattering of electromagnetic waves from a randomly perturbed quasiperiodic surface

The scattering of electromagnetic waves from a randomly perturbed quasiperiodic surface is studied for active remote sensing of plowed fields. The Kirchoff approximation is used. The narrow-band Gaussian random variation around the spatial frequency of the sinusoidal variation is used to introduce the quasiperiodicity. The physical optics integral is evaluated to obtain closed form solutions for coherent and incoherent bistatic scattering coefficients. In the geometrical optics limit, it is shown that the bistatic scattering coefficients are proportional to the probability of the occurrence of the slopes which will specularly reflect the incident wave to the observation direction. The theoretical results are illustrated for the various cases by plotting backscattering cross sections as a function of the angle of incidence. It is shown that there is a large difference between the cases where the incident wave vector is parallel or perpendicular to the row direction. When the incident wave vector is perpendicular to the row direction, the maximum value of the backscattering cross section does not necessary occur at normal incidence. The scattering coefficients can be interpreted as a convolution of the scattering patterns for the sinusoidal and the random rough surfaces. For the backscattering cross sections we observe the occurrence of peaks whose relative magnitudes and the locations are explained in terms of the scattering patterns for sinusoidal surfaces.

On: “Acoustic reciprocity—A paradox” by E. Eisner (GEOPHYSICS, v. 48, p. 1132–1134) or “Eisner’s paradox resolved”

Eisner has concocted a clever geometrical arrangement that appears to contradict the principle of acoustic reciprocity. In essence his example consists of a truncated opaque ellipsoidal reflector with the source and receiver situated at the foci as shown in Figure 1. In the limit of geometrical optics it is evident that only half the energy emitted by a source at 2 will contribute to the reflected arrival at 1 while a substantially larger fraction of the energy emitted by a source at 1 will appear at 2. Naively it seems as if the geometrical amplitude of the reflected arrival at 2 should exceed that at 1, in apparent violation of the reciprocity theorem.

Analytic solution for the two‐frequency mutual coherence function for spherical wave propagation

The quadratic approximation for the phase structure function is used to obtain the two‐frequency mutual coherence function Γ(Δρ, Δω) for spherical wave propagation through a finite slab with transmitter and receiver located in free space on opposite sides of the slab. General analytic solutions are derived for two cases. In the first case the random slab is represented by a one‐dimensional power spectrum of electron density fluctuations corresponding to propagation through elongated irregularities as would occur for an equatorial satellite link to a ground station. In the second case the random slab consists of isotropic ionization irregularities. Both cases taken together represent the extremes of the range of results to be expected for propagation through ionospheric fluctuations, solar wind irregularities, or ionization irregularities caused by barium cloud instabilities. For both cases the complex general analytic results are simplified by use of the thin phase screen approximation to obtain useful analytic expressions for Γ as well as the resulting impulse response function. It is shown that the impulse response to a transmitted power delta function reduces to an exponential form in the limit of strong diffraction and to a Gaussian form in the geometrical optics limit. The Gaussian form corresponds to pulse wander while the exponential form corresponds to diffractive spreading produced by multipath effects. The relationship between the generalized power spectrum and the impulse response function is given, and results are presented for the mean time delay and time delay jitter. The accuracy of the thin phase screen calculation of these quantities is investigated in detail.

ASYMPTOTIC THEORY OF LIGHT SCATTERING FROM INHOMOGENEOUS PARTICLES

ABSTRACT The various contributions to backscattering from inhomogeneous particles are analyzed rigorously. These contributions are extracted from the exact solution by employing the Watson transformation. For wavelengths much shorter than the diameter of the inhomogeneous particles (λ < < a), the dominant contributions included in this paper are the rainbow and glory, and the front and rear axial rays. Partial surface waves, creeping waves and whispering gallery modes are small contributors in the geometrical optics limit and will not be treated in this paper.

Efficiency of nonimaging concentrators in the physical-optics model

The consequences of Liouville’s theorem that are known to be valid in the geometrical-optics limit are shown to hold when wavelength-dependent effects, for example, diffraction and scattering, are taken into account.

Neutrinos in the Kerr and Robertson-Walker geometries

The perturbative behaviour of neutrinos is examined in the framework of the Hertz potential formalism in two important space-times, namely the Kerr and Robertson-Walker space-times. In particular, the angular functions of the neutrino field are studied in detail on the background of the Kerr geometry. It is found that the properties of the solution depend crucially on the value of the separation constant Q defined in the text. It is found that the Robertson-Walker model (k=+1) harbours a repulsive effective potential for neutrinos. The behaviour of the neutrinos in the cases of the spherically collapsing dust interior and the k=-1 Robertson-Walker model can be deduced from the k=+1 case. Also exact solutions in term of known functions are obtained for the neutrino perturbations for the k=0 case. In the geometric optics limit ( omega M>>1) all the results agree with the classical ones obtained in the null geodesic formalism.

On coupled-wave theory of two-beam self-diffraction

Coupled-wave equations are derived, which describe, in the geometrical-optics limit, the interaction between two beams in a medium having intensity dependent permittivity. These equations are interpreted physically and a solution is presented for the special case of non-uniform plane waves. New results on energy exchange and beam profile distortion are obtained and are contrasted with known results for static grating.

Thermodynamic limitations of the concentration of electromagnetic radiation

The temperature formally assigned to spectral electromagnetic radiance Lν by solving Planck’s equation for T permits the application of the Second Law of Thermodynamics to passive concentrating systems. (1) Concentrators operating within the limits of geometrical optics, (2) concentrators changing the frequency of the radiation but conserving the total radiant flux, and (3) systems in which the frequency is changed and part of the absorbed power transferred to the surroundings as heat are discussed. The attainable concentration ratios are given. Particularly for systems of category (3), perspectives are encouraging for further development and application. Such systems resemble, in certain respects, heat pumps. High concentration ratios are allowed by thermodynamics. In this category concentrators that use Stokes fluorescence are discussed.

The free oscillations of an anelastic aspherical earth

Summary. Some century-old results, due to Rayleigh and Routh, have been adapted to investigate the normal mode eigenfrequencies and eigenfunctions of an earth with laterally variable anelasticity and to determine the transient response of such an earth to earthquakes. Using degenerate perturbation theory, the eigenfrequencies are found to first order and the associated eigenfunctions to zeroth order in the small deviations of the Earth away from a spherical perfectly elastic reference earth model. Both the eigenfrequencies and the eigenfunctions are complex and, in addition, the latter are not mutually orthogonal, reflecting the non-Hermitian character of the normal mode eigenvalue problem. The effect of laterally heterogeneous attenuation on the shape of an unresolved multiplet spectrum has been investigated in the surface-wave geometrical-optics limit. Singlet cancellation leads in that limit to the appearance of a single resonance peak whose decay rate or apparent Q-' depends only on the average attenuation structure underlying the source-receiver great-circle path.

Light Scattering by Hexagonal Ice Crystals

Abstract A complete and traceable geometric ray-tracing solution for finite hexagonal columns and plates arbitrarily oriented in space has been developed by means of analytic geometry. In addition, an analytic expression for the cross-sectional area for arbitrarily oriented hexagons also has been derived based on which Fraunhofer diffraction and extinction and scattering cross sections in the limit of geometric optics can be computed exactly. The program involving geometrical reflection and refraction and Fraunhofer diffraction was used to compute the scattered intensities corresponding to two components of polarization for randomly oriented columns and plates in a horizontal plane and three-dimensional space. The scattered intensities were subsequently normalized to yield the nondimensional phase function commonly used in radiative transfer analyses. Numerical computations have been performed to study the effects of size, shape, orientation, and absorption on the scattering phase function and linear pola...

A computer simulation of XPD degradation due to multipath propagation

A computer program has been constructed to study degradation inXpd(cross-polarization discrimination) due to multipath propagation. The geometric optics limit is used. The program will accept any analytic index of refraction including those with discontinuities in n or ∇n. Antenna effects are included and all reflected, refracted, direct and ground reflected rays can be considered. Using a realistic index of refraction and realistic discontinuity, we have found that, after the antenna pattern, the major influence onXpdis interference between the ground-reflected wave and the direct wave. There is also significant interference due to the waves reflected from the refractive index discontinuity which may arrive without ground reflection or may be reflected once or twice from the ground.

Optical Function of the Convexiclivate Fovea with Particular Regard to Notosudid Deep-sea Teleosts

The optical function of the deep symmetrical convexiclivate purecone fovea of deep-sea notosudids has been examined within the limits of geometrical optics, assuming, inter alia, that the full lens aperture is available for formation of foveal images, that the spherical aberration of the lens is negligible, and that the refractive index of the retina is slightly higher than that of the vitreous. Refraction at the vitreo-retinal boundary in the deep part of the fovea distorts a decentred image of a point source into an elongated spot, almost a double spot, while light from an extended source (residual daylight) will produce uneven illumination within the fovea, with a ring-shaped area of least illumination concentric with the foveal centre. It is suggested that the main optical function of the notosudid fovea is to break the camouflage of mesopelagic animals using small photophores for countershading. The theory implies the angular distribution of the photophore light to differ from that of the residual da...

Geometric Optics of Lower Hybrid Waves in Nonuniform Plasmas

The propagation of lower hybrid waves in the presence of strong density and field gradient-produced refractive effects is studied in the geometric optics limit. Attention is focused on the electrostatic wave and the modification of the parallel wavenumber k‖ due to gradients. It is found that in cases where the gradients occur along the direction of the magnetic field the resulting decrease of k‖ gives rise to a cusp or reflection of the ray at the resonance layer. An asymptotic series for the field amplitudes is developed which is valid in the short wavelength limit everywhere except near the cusp point. It is found that the cusp represents a singularity where the geometric optics assumption is invalid; however, the inclusion of thermal effects removes the singularity by linear mode conversion (LMC). The analysis is applied to toroidal geometry and is extended to include distributed sources.

Flux density and caustics for sound multiply reflected or refracted in variable index media

A formula is given for the flux density along a ray path when sound or light is propagated in variable index media. The sound may be reflected or refracted by an arbitrary interface between media of different indices of refraction. Results are applicable in the geometrical optics limit, that is, the wavelength must be small relative to other dimensions. The only requirement is that the equation for the geodesics in the media be solvable so that an explicit expression is available for the ray path. Results are applicable to multiple reflection of sound from underwater objects and in assessing the performance of variable index lenses. Equations are also given for the associated caustics.

On the detection of a stochastic background of gravitational radiation by the Doppler tracking of spacecraft

Consideration is given to the possibility of detection of an isotropic background gravitational radiation of a stochastic nature by the method of Doppler tracking of spacecraft. Attention is given in the geometrical optics limit, to the general formula for the frequency shift of an electromagnetic signal in the gravitational radiation field, and it is shown to be gauge independent. The propagation of a free electromagnetic wave in a gravitational radiation field is examined with the conclusion that no resonance phenomena can be expected. Finally, the 'Doppler noise' due to a stochastic background is evaluated, and it is shown to depend on the total energy density of the background and a parameter that is a characteristic of the radiation spectrum and the detection system used.

Diffraction of electromagnetic waves in Schwarzschild's space-time

In order to study the gravitational lens effect in detail this paper investigates the electromagnetic radiation generated by an electric dipole oscillating with high frequency in the Schwarzschild metric outside the horizon. Expressions for the Newman-Penrose tetrad components characterizing the radiation field are derived in a suitable approximation. The Poynting vector is discussed in the region behind the deflecting mass (black hole, neutron star) in the fully coherent case for large distances of source and observer from the centre of the lens. Intensity and gain in the focal region coincide with the corresponding values applying to scalar radiation. An observer sees double images of the source outside the focal region. The results (positions of the images, relative intensities) are compared with those occurring in the case of an incident plane wave (recently treated by Herlt and Stephani) and with the properties of the images in the (geometrical optics) limit of incoherent radiation.

Optical extraction efficiency in lasers with high Fresnel number confocal unstable resonators

Using the formulation of Moore and McCarthy for the equation describing the dominant mode in a positive branch confocal unstable resonator in the geometrical optics limit and a simple two-level kinetics model, we analyze the dependence of the optical extraction efficiency (eta) on resonator magnification (M), length (L), small signal gain (go), and (nonsaturable) background absorption (alpha). The model has cylindrical symmetry and spatially uniform small signal gain, absorption, and index of refraction. For fixed gamma= g(0)/alpha and M, a value of g(0)L that maximizes eta is found. For different gamma, the maximum obtainable value of al is found to be independent of M and to depend upon gamma in a simple way.