Use Of Geometrical Optics Research Articles

Scattering of radio frequency waves by cylindrical density filaments in tokamak plasmas

In tokamak fusion plasmas, coherent fluctuations in the form of blobs or filaments are routinely observed in the scrape-off layer. Radio frequency (RF) electromagnetic waves, excited by antenna structures placed near the wall of a tokamak, have to propagate through the scrape-off layer before reaching the core of the plasma. While the effect of fluctuations on the properties of RF waves has not been quantified experimentally, it is of interest to carry out a theoretical study to determine if fluctuations can affect the propagation characteristics of RF waves. Usually, the difference between the plasma density inside the filament and the background plasma density is sizable, the ratio of the density difference to the background density being of order one. Generally, this precludes the use of geometrical optics in determining the effect of fluctuations, since the relevant ratio has to be much less than one, typically, of the order of 10% or less. In this paper, a full-wave, analytical model is developed for the scattering of a RF plane wave by a cylindrical plasma filament. It is assumed that the plasma inside and outside the filament is cold and uniform and that the major axis of the filament is aligned along the toroidal magnetic field. The ratio of the density inside the filament to the density of the background plasma is not restricted. The theoretical framework applies to the scattering of any cold plasma wave. In order to satisfy the boundary conditions at the interface between the filament and the background plasma, the electromagnetic fields inside and outside the filament need to have the same k∥, the wave vector parallel to the ambient magnetic field, as the incident plane wave. Consequently, in contrast to the scattering of a RF wave by a spherical blob [Ram et al., Phys. Plasmas 20, 056110-1–056110-10 (2013)], the scattering by a field-aligned filament does not broaden the k∥ spectrum. However, the filament induces side-scattering leading to surface waves and can also couple some power to the cold plasma wave different from the incident wave. The changes induced by a filament in the propagation of electron cyclotron waves and lower hybrid waves are illustrated by numerical results displaying the properties of the Poynting vector. The Poynting flux in the wake of the filament, and directed towards the core of the plasma, develops a spatial structure due to diffraction and shadowing. Thus, the fluctuations affect the uniformity of power flow into the plasma.

Resonant inelastic scattering by use of geometrical optics.

We investigate the inelastic scattering on spherical particles that contain one concentric inclusion in the case of input and output resonances, using a geometrical optics method. The excitation of resonances is included in geometrical optics by use of the concept of tunneled rays. To get a quantitative description of optical tunneling on spherical surfaces, we derive appropriate Fresnel-type reflection and transmission coefficients for the tunneled rays. We calculate the inelastic scattering cross section in the case of input and output resonances and investigate the influence of the distribution of the active material in the particle as well as the influence of the inclusion on inelastic scattering.

Modelization of the Scattering of Electromagnetic Waves FROM the Ocean Surface

In this paper, two models for the solution of the electromagnetic bistatic scattering from sea surface are suggested. A rigorous formalism leading to weakly singular integral equations is presented, as well as the surface impedance approximation for low penetrable media and the beam simulation method to synthesize incident beams with arbitrary size. This rigorous integral method is used to test first order approximations, and it is shown that the Small Slope Approximation is very accurate in predicting the scattering crosssection from the high spatial frequencies of the sea surface. This result led us to suggest an improvement of the classical two-scale model, consisting in replacing the small perturbation theory by the small slope approximation. This change allows the cut-off spatial frequency to be shifted so that the use of geometrical Optics is restricted to the large scales.

Scattering of a Gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation: numerical results

We present numerical results concerning the properties of the electromagnetic field scattered by an infinite circular cylinder illuminated by a circular Gaussian beam. The cylinder is arbitrarily located and arbitrarily oriented with respect to the illuminating Gaussian beam. Numerical evaluations are provided within the framework of a rigorous electromagnetic theory, the generalized Lorenz-Mie theory, for infinite cylinders. This theory provides new insights that could not be obtained from older formulations, i.e., geometrical optics and plane-wave scattering. In particular, some emphasis is laid on the waveguiding effect and on the rainbow phenomenon whose fine structure is hardly predictable by use of geometrical optics.

Gross errors in upper-mantle discontinuity topography from underside reflection data

SUMMARY Currnet models of upper-mantle discontinuity topography derived from traveltime variations of long-period underside reflections (PdP or SdS waves) may contain gross errors due to the use of geometrical optics in relating observed traveltimes to depth variations of the discontinuity near the reflection point. The use of geometrical optics is not valid if variations in the depth of the discontinuities exist on a lateral scale smaller than the size of the Fresnel zone in the data. Geometrical optics does not take into account the large size and the complex structure of the traveltime surface of underside reflections and may introduce spurious structure when used in inversions. Examples of synthetic long-period PdP waveforms for reflections near subducting lithosphere show that scattering from expected small-scale depth variations of the discontinuities inside slabs causes complex waveform variations. For a 15–20s PdP wave these small-scale depth variations (with a scale-length of 4) are smaller than the Fresnel zone (roughly 20°) but larger than the PdP wavelength at 420 km depth (about 150–200 km). The synthetic waveforms are processed to obtain PdP traveltimes, which are in turn converted to apparent discontinuity depth at the reflection point using geometrical optics. The apparent discontinuity-depth variations are not only of the same order of magnitude as those obtained from real data but also exhibit artefacts that closely resemble features observed in current models. It is shown that large-scale (wavelengths of the order of 1000 km) downwarping of the 670-km discontinuity near subduction zones derived from underside reflection data can be explained as the long-wavelength manifestation of a single, small-scale (wavelengths of several hundreds of kilometres) deflection of the discontinuity inside and near subducting slabs.

A method to overcome the limitations of G.O. in axis-symmetric dielectric lens shaping

The design of a cylindrically symmetric dielectric lens antenna is presented for the case of a very squeezed fan-beam. One makes use of geometrical optics (G.O.) to establish the profile of the lens in a first step. A half-integrated solution of Green's formula for bodies with cylindrical symmetry and circularly polarized surface fields is then derived to calculate the radiation pattern. Since the use of G.O. is restricted to some conditions which were observed unsatisfied in our study, a two-steps procedure to reach an acceptable lens shape is then presented. The first step is based on weighting the source beam of the antenna; and the second one is based on simple geometrical interpolation. Prototypes provided experimental results that are consistent with the predicted radiation characteristics and interesting features of such antennas are pointed out.

Depth resolution in stereoscopic systems

In television stereoscopic systems the magnification and the effective eye separation can be varied at will. This study shows that, under carefully chosen experimental conditions, depth resolution can be closely predicted by the use of geometrical optics and can be enhanced by increases in magnification and effective eye separation. The limited spatial resolution of conventional television or other photoelectronic systems may, however, prevent high levels of stereoacuity, comparable with naked-eye viewing, from being achieved in a wide-field system. In more complex environments the inevitable conflict of perceptual cues might be expected to reduce depth resolution further.

Interpretation of out-of-focus star images in terms of wave-front curvature

The wave-front-curvature-sensing method used by F. Roddier for the University of Hawaii’s curvature-adaptive-optics system is derived from the interpretation in 1979 of the out-of-focus images obtained by use of the Multiple Mirror Telescope. I summarize those original Multiple Mirror Telescope observations and the simple interpretation given in terms of wave-front curvature by use of geometrical optics at that time.

Mathematical modeling of absorbent highway noise barriers

Analytical modeling methods of complex noise barriers are presented in this paper. The models are developed by use of the geometrical theory of diffraction which allows for modeling of simple and complex shaped noise barriers. These models involve diffraction coefficients that include the influence of frequency, grazing angle, surface impedance and the source and receiver positions. The diffraction coefficients for complex shaped barriers are constructed from those for simple shapes by use of geometrical optics and physical acoustics. The influence of the ground on barrier diffraction is included through complex reflection coefficients for point sources off impedance covered planes. For highway noise barrier applications, the barrier noise reduction efficiency is calculated for a line of incoherent point sources. Scale model tests were conducted on barriers of various shapes. These include barriers shaped as thin walls, wedges, trapezoids with vertical or sloped walls and double thin walls. These barriers were tested with hard or absorbent surfaces erected over hard or absorbent ground.

Computer generated holograms: cylindrical, conical, and helical waves

In computer generation of holograms the wave propagation is numerically simulated. Usually this propagation is approximated by the discrete Fourier transform or by geometrical optics. Use of these methods restricts the kind of object which can be recorded synthetically. We demonstrate that the reconstruction of holographic optical elements and 3-D images can be composed from line segments by superimposing several analytic distributions in the hologram plane. To implement the algorithm, the wave equation is considered in cylindrical coordinates. A solution is given which describes propagating waves denoted cylindrical, conical, and helical waves. Since the complex amplitude is known as an analytic expression throughout space, calculating the transmission of holograms on curved surfaces is possible. In the reconstruction step the waves converge to line foci of arbitrary position and orientation even parallel to the optical axis. The length is controlled by the use of geometrical optics. The feasibility is demonstrated by an optical reconstruction of a 3-D object.

The design of correcting secondary reflectors

A computer program called DERIVE is described for the design of the reflecting surfaces of two-mirror radio telescopes. It calculates the shape of a secondary reflector to any given primary by use of geometrical optics, yielding zero path length errors for all paraxial rays. The only condition to be fulfilled is (for Cassegrain types) that incoming rays, after reflection at the primary, do not cross each other before reaching the secondary. Equations for this condition are given. This method can be used to improve existing telescopes at short wavelengths by correcting for the surface deviations of the primary reflector with a specially shaped secondary. The results of 17 calculated examples are described, for various types of deviations

A ray theory for wave propagation in a non-uniform medium

The study of wave propagation by geometrical optics is applied to a consideration of propagation through a non-uniform, statistically homogenous medium. Following the trajectory of a phase point, two effects are examined; the retardation of the phase point by the variable local wave speed along its trajectory and the further retardation and dispersion resulting from its meandering. Mean and mean-squared displacements are obtained to describe the retardation of the wave front, and the dispersion of the phase point from the incident direction.The theory has application as a correction to the use of geometrical optics wherever the latter can be employed. In particular, it is shown by an estimate of the magnitude of the pertinent parameters that an application may well be found in the study of the tsunami (a long ‘shallow’ ocean wave).