Geometrical Optics Ray Research Articles

Geometric optics-based design of laser beam shapers

When diffraction effects are not important, geometrical optics (ray tracing, conservation of energy within a bundle of rays, and the constant optical path length condition) can be used to design laser beam shapers by solving beam shaping equations or by optimizing a beam shaping merit function for the configurations, including aspheric ele- ments or spherical-surface gradient-index lenses, which are required to change the input irradiance and phase profile into a more useful form. Geometrical optics-based methods are presented for shaping both rota- tionally and rectangular symmetric laser beam profiles. Solutions of the beam shaping equations are presented for a two-plano-aspheric lens system for shaping a circular symmetric Gaussian beam into a top-hat output beam profile and a two-mirror system with no central obscuration for shaping an elliptical Gaussian input beam into a Fermi-Dirac output beam profile. © 2003 Society of Photo-Optical Instrumentation Engineers.

The Spectrum of the Damped Wave Operator for a Bounded Domain in R 2

The spectrum of the damped wave operator for a bounded domain in R 2 is shown to be related to the asymptotic average of the damping function by the geodesic flow. This allows the calculation of an asymptotic expression for the distribution of the imaginary parts of the eigenvalues for a radially symmetric geometry. Numerical simulations confirm the theoretical model. In addition, we are able to exhibit the beautiful structure of the spectrum and the close links between the eigenfunctions, the raysof geometrical optics, and the geometry of the damping region. The MATLAB code used in this paper is provided.

Application of Gaussian-ray basis functions for the rapid analysis of electromagnetic radiation from reflector antennas

A highly efficient Guassian beam (GB) method was recently developed to provide a relatively rapid analysis of large reflector antennas. This GB method is very efficient because it represents the feed radiation in terms of a set of very few rotationally symmetric GBs and provides an essentially closed form solution for the reflection and diffraction of each GB incidence on the reflector surface. In contrast, conventional approaches evaluate the radiated field via a numerical integration of the physical optics integral over the large reflector surface; such numerical techniques are time consuming for large reflectors. However, the fast GB method becomes less accurate if each GB illuminating spot on the reflecting surface is comparable to the overall reflector size, since its closed-form solution requires a small spot size. This limitation of the GB approach happens particularly in situations where the feed is located relatively far from the reflector, since the GBs are not able to remain narrow in the angular and spatial domains simultaneously, thereby creating too large a spot size on the reflector surface. The above limitation of the GBs is overcome here by introducing a Gaussian-ray basis function (GRBF), a hybridisation of a geometrical optics (GO) ray tube and a GB. Owing to its similarity with a GB in the amplitude variation transverse to the ray, its incorporation into any existing GB code is straightforward. Numerical results are presented to demonstrate the accuracy and robustness of the new GRBF approach.

Phase-contrast X-ray imaging with synchrotron radiation for materials science applications

Since Röntgen’s discovery of X-rays just over a century ago the vast majority of radiographs have been collected and interpreted on the basis of absorption contrast and geometrical (ray) optics. Recently the possibility of obtaining new and complementary information in X-ray images by utilizing phase-contrast effects has received considerable attention, both in the laboratory context and at synchrotron sources (where much of this activity is a consequence of the highly coherent X-ray beams which can be produced). Phase-contrast X-ray imaging is capable of providing improved information from weakly absorbing features in a sample, together with improved edge definition. Four different experimental arrangements for achieving phase contrast in the hard X-ray regime, for the purpose of non-destructive characterization of materials, will be described. Two of these, demonstrated at ESRF in France and AR in Japan, are based on parallel-beam geometry; the other two, demonstrated at PLS in Korea and APS in USA, are based on spherical-beam geometry. In each case quite different X-ray optical arrangements were used. Some image simulations will be employed to demonstrate salient features of hard X-ray phase-contrast imaging and examples of results from each of the experiments will be shown.

P‐41 Fresnel Diffraction Phenomena in Frontlights for Reflective LCDs

AbstractIn frontlights for reflective LCDs made from micro‐optically structured lightguides, unwanted shadows appear in certain viewing directions. It is shown that these shadows are wider than predicted from geometrical ray optics, and that they can not be eliminated by decreasing the micro‐optical structure size. These experimental observations can be explained accurately by taking Fresnel diffraction into account.

Modified electron whistler dispersion law

A modified electron whistler dispersion law is derived in the cold-plasma approximation for analytical treatment and simplified numerical calculations of wave propagation in a wide range of ratios ωc/ωp of electron gyro- to plasma frequencies if the wave frequency is much less than ωp. The net contribution of ions to the wave dispersion law is expressed through the value of the lower-hybrid resonance frequency ωlhr only. This approximate dispersion law is valid in a wide frequency domain, that is, from the range of ωlhr until the domain where the contribution of ions can be neglected. A comparison of geometrical-optics ray trajectories calculated by the use of modified and total cold-plasma electron whistler dispersion laws is presented for the case of the Earth's plasma environment. Computer simulations of dynamical spectra of whistler waves excited by lightning discharges and registered in remote regions of the Earth's plasmasphere reveal good numerical stability of the developed ray-tracing code.

Wave packets in smoothly inhomogeneous dispersive media

In the quasi-optical aberration-free approximation, a parabolic equation for the envelope of an electromagnetic wave packet propagating along a geometric optical ray in a smoothly inhomogeneous isotropic medium with time dispersion is obtained. The corresponding Green’s function is found whose parameters are determined by integrating a system of ordinary differential equations. The effects of the combined influence of the refraction, diffraction, and dispersion on the evolution of the packet are analyzed; in particular, the effects that cause precession of the envelope waves about the binormal to the propagation trajectory are considered.

Excitation, propagation, and damping of electron Bernstein waves in tokamaks

The conventional ordinary O-mode and the extraordinary X-mode in the electron cyclotron range of frequencies are not suitable for core heating in high-β spherical tokamak plasmas, like the National Spherical Torus Experiment [M. Ono, S. Kaye, M. Peng et al., in Proceedings of the 17th International Atomic Energy Agency Fusion Energy Conference (International Atomic Energy Agency, Vienna, 1999), Vol. 3, p. 1135], as they are weakly damped at high harmonics of the electron cyclotron frequency. However, electron Bernstein waves (EBW) can be effective for heating and driving currents in spherical tokamak plasmas. Power can be coupled to EBWs via mode conversion of either the X-mode or the O-mode. The two mode conversions are optimized in different regions of the parameter space spanned by the parallel wavelength and wave frequency. The conditions for optimized mode conversion to EBWs are evaluated analytically and numerically using a cold plasma model and an approximate kinetic model. From geometric optics ray tracing it is found that the EBWs damp strongly near the Doppler-broadened resonance at harmonics of the electron cyclotron frequency.

Comparative analysis of radioholographic methods of processing radio occultation data

Two methods for the determination of refraction angle profiles from radio occultation measurements in multipath areas are analyzed and compared: (1) the radio‐optic method based on the analysis of the local spatial spectra of the measured wave field and (2) the back propagation of the received wave field to a single‐ray region. The basic limitations of the radio‐optic method are (1) the restriction of the resolution of refraction angle profiles due to the uncertainty relation of refraction angle and impact parameter and (2) diffractive effects in subcaustic areas, where the spatial spectra cannot be interpreted in terms of geometric optical rays. The basic limitation of the back propagation method is related to ray and caustic structures, which may not contain single‐ray areas. It is shown that strong refraction reduces the uncertainties of refraction angle and impact parameter. On the other hand, strong refraction or superrefraction is responsible for complicated caustic structures that cannot be resolved by the back propagation technique. The two methods are thus complementary to each other and can be combined for processing lower tropospheric occultation data. This analysis is corroborated by numerical simulations based on global fields of atmospheric variables from analyses of the European Centre for Medium Range Weather Forecast.

Use of Gaussian ray basis functions in ray tracing methods for applications to high frequency wave propagation problems

Traditional ray tracing approaches require a large number of geometrical optical (GO) ray tubes, and can become inefficient in many practical applications. The pulse-type basis functions of GO ray tubes cannot provide a good representation in the field expansion if the field to be expanded is not locally planar or the size of the GO ray tubes is too large. A previously developed Gaussian ray basis function (GRBF) has a Gaussian-type amplitude distribution and is suitable to replace a GO ray tube in the ray tracing approaches because the Gaussian-type distribution tends to yield a more efficient field expansion. The GRBFs are incorporated into existing computer programs of generalised ray expansion (GRE). Numerical results validate its accuracy and efficiency in comparison with GO-based GRE for the prediction of EM scattering from open cavities.

Geometric-optical Rays, Poynting's Vector, and Field Momenta

Get PDF Email Share Share with Facebook Tweet This Post on reddit Share with LinkedIn Add to CiteULike Add to Mendeley Add to BibSonomy Get Citation Copy Citation Text MASUD MANSURIPUR, "Geometric-optical Rays, Poynting's Vector, and Field Momenta," Optics & Photonics News 10(3), 53-56 (1999) Export Citation BibTex Endnote (RIS) HTML Plain Text Citation alert Save article

Laser Guidance and Trapping of Mesoscale Particles in Hollow-Core Optical Fibers

Mesoscale particles are guided and trapped in micron-sized hollow optical fibers using a $\frac{1}{2}\ensuremath{-}\mathrm{W}$ near-infrared diode laser. The optical scattering force calculated using geometrical ray optics is $3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}11}\mathrm{N}$ for a $7\ensuremath{-}\ensuremath{\mu}\mathrm{m}$ polystyrene sphere, and agrees well with velocity measurements of optically guided spheres in water. We have levitated a variety of particles using laser-fiber guidance, and have mixed micron-sized glycerin droplets in two-beam laser-fiber traps.

Optical realization of a circular billiard

We briefly review the classical mechanics of a point particle in a circular billiard system, classifying the possible trajectories. We then show how this system can be approximately realized experimentally, utilizing an analogy with geometrical ray optics, using a laser and cylindrical reflecting surface.

Beam tracing in inhomogeneous anisotropic plasmas

An asymptotic method for solving the wave equation in the short-wavelength limit is presented. This method, called beam tracing, takes into account the wave properties, i.e., diffraction and interference. It reduces the full wave equation to a set of ordinary differential equations. In this respect, it differs from all other asymptotic techniques describing diffraction which end up with much more complicated partial differential equations. The resulting system of beam tracing equations is expressed in terms of the same Hamiltonian function as in geometric optics (ray tracing) and, similar to the ray tracing, allows powerful numerical solving algorithms. Thus the beam tracing combines the simplicity of ray tracing with a description of the wave phenomena, which are not included in the ray tracing. The beam tracing technique provides an efficient tool for calculation of wave fields in all problems where the short-wave approximation is applicable such as rf heating, current drive and plasma diagnostics with microwave beams.

SABOR: description of the methods applied for a fast analysis of horn and reflector antennas

The program presented in this paper is intended to provide a valuable aid for teaching antennas to electrical engineers, and to provide fast and accurate pre-designs for professionals. This paper covers the theory and numerical techniques used in SABOR. In summary, this program computes the radiated field of an aperture antenna (horn or reflector), using a common engine based on the Gauss-Legendre quadrature method for evaluating the radiation integrals. For horn, the aperture fields are the usual dominant modes of the feed waveguide, with a quadratic phase correction. For reflectors, the aperture fields are computed using geometrical optics ray tracing from the feed horn. Also, equivalent-reflector concepts are applied for dual-reflector antennas. The paper includes some examples to demonstrate the most important features of the program.

Theory of convective saturation of Langmuir waves during ionospheric modification of a barium cloud

Abstract In recent experiments (Djuth, F. T., Sulzer, M. P., Elder, J. H. and Groves, K. M. (1995) Journal of Geophysical Research, 100, 17,347), a parametric decay instability was excited by an ordinarywave HF pump during an ionospheric chemical release from a rocket over Arecibo, PR, which created an artificial ‘barium ionosphere,’ with peak plasma frequency above the pump frequency, and a density gradient with a (short) 5 km scale length. Simultaneous incoherent scattering measurements revealed a strong initial asymmetry in the amplitudes of almost vertically upgoing versus downgoing measured plasma waves. We can account for this asymmetry in terms of linear convective saturation of parametrically unstable plasma waves propagating over a range of altitudes along geometric optics ray paths. Qualitative features of the frequency spectrum of the measured downgoing wave are in agreement with this model, although the theoretically predicted spectrum is narrower than observed. The observed altitude localization of the enhanced spectrum to a few range cells is consistent with the theory.

New calculations of the acoustic weighting function for ocean internal-wave tomography

In acoustic tomography observables like travel time can be expressed in terms of integrals along geometrical optics ray paths where the integrand is called the weighting function. In internal-wave tomography, the travel-time variance is an important acoustic observable and in this case a well-known weighting function has been derived which has the form F(zray,θray)=〈μ2(z)〉Lp(θ,z)/c(z)2 where 〈μ2(z)〉 is the fractional-sound-speed variance, Lp(θ,z) is the effective internal-wave correlation length along a ray-path tangent with angle θ, and c(z) is sound speed. Direct numerical solution for the travel-time variance weighting function shows that acoustic signals are most sensitive to internal waves near the ray upper apex but not as strongly as suggested by F. Examples comparing the direct calculation of the weighting function with F, suggest that the greatest sensitivity to internal waves exists within hundreds rather than tens of meters vertically of the ray upper apex. The inaccuracy of the weighting function, F, can be attributed to the assumptions of small ray curvature and the depth dependence of 〈μ2(z)〉. Comparisons between observations and predictions of travel-time variance for basin-scale acoustic transmissions based on the new weighting function show excellent agreement for internal waves at half the Garrett–Munk spectral level.

A hybrid GO-Gaussian ray beam

In modern geometrical optics (GO) the field solutions are described as bundles of rays in which the field associated with each ray is independent of the fields of the other rays. But this description is not valid in caustic regions where neighboring rays intersect. The Gaussian beam provides an approximate global description of a modal-type field solution, which is valid even in caustic regions where GO is invalid. In this article a hybrid GO–Gaussian ray beam is presented. It behaves as a GO ray bundle in one cross section along the propagation direction and as a Gaussian beam in the other. We also present the equations for the reflection of such a beam from an arbitrary curved reflector. © 1997 John Wiley & Sons, Inc. Microwave Opt Technol Lett 15: 278–282, 1997.

Wavefront folding, chaos, and diffraction for sound propagation through ocean internal waves

This work presents an analysis of the effects of ocean internal waves on long-range acoustic pulse propagation from the geometrical-optics point of view. The chaotic behavior of rays and the microfolding of timefronts are investigated. The extent of the region of the timefront in which strongly chaotic rays appear, and the strength of the rays’ sensitivity to initial conditions, are found to depend on the average (range-independent) sound-speed profile, on the range from the source to the receiver, and on the internal-wave spectral model, but not on the specific realization of the internal waves. For a particular experiment (SLICE89), it is concluded that the observed depth diffusion of energy in the late-arriving portion of the timefront is a result of refraction (of geometrical-optics rays), not diffraction. It is found that internal-wave effects cause an upper turning point of a ray to be spread to the extent of 10 km horizontally and 100 m vertically, which affects the resolution of ocean-acoustic tomography. The validity and usefulness of ray-based, semi-classical (WKB) waveforms to represent received arrivals are evaluated by comparing with waveforms generated with multifrequency, parabolic-equation simulations. Center frequencies of 250 Hz and 1000 Hz with 100-Hz bandwidth are used. The semi-classical waveforms reproduce the correct arrival time and temporal shape of almost all arrivals, even those that are made of dozens of microrays induced by the internal waves. The overall intensities of the 1000-Hz arrivals are reasonably accurate, while the 250-Hz intensities show differences of order 5 dB.

Asymptotic analysis of whispering‐gallery mode radiation from the aperture of cylindrical concave conducting boundary

AbstractAn asymptotic analysis is discussed for the radiating field when a whispering gallery (WG) mode is radiated from the edge aperture plane of a cylindrical concave conducting boundary and is propagated as a beam. the integral representing the radiation field derived by the method similar to the one used in the problem of diffraction of a plane wave from a semiinfinite conducting plane is evaluated by means of the saddle point method taking into account the existence of the poles and the residue theorem.An asymptotic representation for the beam wave is a combination of the geometric optical ray and the edge diffracted ray. the phenomenon of the radiation of a geometric optical ray from the edge aperture and the phenomenon of diffraction of the modal ray at the terminating edge are clarified. the asymptotic solutions applicable in the shadow boundary, reflection boundary, and the nearby transition region caused by the incidence of the modal ray onto the edge are compared with those obtained by the Fresnel‐Kirchhoff formula so that the effectiveness of the asymptotic expression is determined. Also, it is demonstrated numerically that the radiation field expression based on the diffraction coefficients by the GTD cannot provide good results in the transition region. Further, propagation phenomena of beam fields are clarified from numerical results and the geometrical picture obtained from the ray tracing technique.