Caustic Surface Research Articles

Optical Characterization of the Director Field in a Distorted Nematic Layer

We develop new optical methods using transmitted polarized light for the characterization of the out of plane component n z of the director field in a weakly distorted planar nematic layer. In extraordinary light, we relate the angles of aperture of the caustic surface to the local amplitudes of the n z distortion. In ordinary light, a new type of contrast due to the anisotropic light scattering is shown to give the map of n z 2 . We apply our methods to the study of the director distortions occuring in the thermal convection.

Bounds on fields from fast rotating sources, and others

General bounds are noted on the retarded wave field ψ ( r, t ) from a source distribution s ( r, t ) which is zero outside some volume. Specifically if s is bounded in magnitude then ψ is bounded in magnitude, and decays far away as r -1 (or faster). If the first derivatives of s are also bounded and continuous, the same applies to the gradient of the field. These are unremarkable properties, but being general, continue to apply in circumstances where they are less obvious. In particular they apply to a source distribution which is ‘rotating rigidly’ at such a high angular velocity that the outer parts are moving faster than the wave speed c. The bounds contradict some results derived and analysed by Ardavan (1984, 1989, 1991 a, b , 1994 a, b ) in terms, mostly, of an associated cusped caustic surface (bent Mach cone).

The whistler‐mode bow wave of an asteroid

The Galileo spacecraft has flown by two asteroids, Gaspra in 1990 and Ida in 1993. In both cases the magnetometer detected magnetic field perturbations that are believed to be produced by an interaction of the asteroid with the solar wind. Kivelson et al. have proposed that these perturbations are caused by whistler‐mode waves excited by the solar wind flow around the asteroid. This paper presents an analysis of whistler‐mode waves generated by the interaction of a small object with the solar wind. Three cases are considered, with the solar wind magnetic field (1) parallel, (2) perpendicular, and (3) at an arbitrary angle to the velocity vector. Using the Cerenkov condition and the quasi‐longitudinal approximation for the whistler mode, the angular limits of the wave pattern are determined. For the parallel case the waves are confined to the upstream region, and for the perpendicular case the waves are confined to two wedge‐shaped regions extending downstream from the magnetic field line through the object. For intermediate magnetic field orientations the accessibility region maintains the wedge‐shaped configuration, with the leading edge of the wedge roughly following the direction of the magnetic field. The outer boundary of the accessibility region is a caustic surface, along which large field amplitudes are expected, similar to the bow wave of a ship. In all cases the interaction involves a characteristic wavelength, λc = ƒcc²/(ƒp²V), that is comparable to the size of an asteroid.

Examination of the effect of the fibrous structure of a lens on the optical characteristics of the human eye: a computer-simulated model

We introduce a model of the human eye for which we take into consideration the laminated nature of lens fibers. The thickness of each lamina is 5.6 µm; thus the lens comprises 300 eccentric lenses of minute dimensions. The index gradient of the lens is such that the index of refraction increases exponentially from the lens core to its peripheral zone. A vector ray-tracing technique is employed to study the optical haracteristics of the system. Both paraxial and marginal rays are simulated, and the angles of incidence vary from 0° to ±20°. Special attention is given to the meridional caustic surfaces as well as the wave-front distortion of the refracted rays. A quasi-Newton optimization technique is employed to obtain the best parameters for the system. A computer modeling program, written in FORTRAN 77, is used to simulate a ray's refraction through the multisurfaces of the eye. The results show full agreement with previous data and that the cornea is responsible for eliminating possible spherical aberration of the system.

Refraction of the k function at spherical surfaces

The k function arises in obtaining the general solution of the eikonal equation for a homogeneous, isotropic medium and describes the shape and the structure of individual wave fronts in a wave-front train as well as the associated caustic surface. The effect that refraction by a spherical refracting surface has on this k function is examined. This general result is applied to the refraction of a train of spherical wave fronts arising from a finite object point and is illustrated by several special cases.

The k function in geometrical optics and its relationship to the archetypal wave front and the caustic surface

The k function is the arbitrary function that arises in the general solution of the eikonal equation, a nonlinear, first-order partial differential equation, and describes completely the geometrical properties of a wave-front train and the associated caustic in a homogeneous, isotropic optical medium. The archetypal wave front in such a train is that unique wave front whose optical path length from the ultimate object point is zero. As an example, the k function of a wave-front train resulting from a plane wave front incident upon a spherical refracting surface is calculated. From it the archetype and the caustic are obtained. The results agree exactly with ray-trace data.

Optimal corrections of the optical defects by minimizing the caustic merit function

Until recently corrections of the defects of the optical system in the lens design such as a triplet lens system have been based on classical conceptions of the image formation through refraction surfaces of the system, or based on the third-order Siedel's aberration coefficient. This paper continues emphasizing the use of the caustic merit function, that describes the spread of the caustic surfaces from an ideal image point. This caustic merit function is then used to optimize a triplet lens. Results show that optimizing the caustic merit function is still the fastest and an effective method in the optical design in the case of plane waves.

Sound radiation from a cylindrical duct. Part 1. Ray structure of the duct modes and of the external field

This paper determines the ray structure of a spinning acoustic mode propagating inside a semi-infinite circular cylindrical duct, and thereby determines the ray structure of the field radiated from the end of the duct. Inside the duct, but outside of a caustic cylindrical surface, the rays are piecewise linear helices; on striking the rim of the end-face of the duct, these rays produce ‘Keller cones’ of diffracted rays. The cones determine the structure of the radiated field: for example, no rays penetrate two cone-shaped far-field quiet zones centred on the duct axis; two rays pass through each point in a forward loud zone; and one ray passes through each point in a rearward loud zone. The two rays through each point in the forward loud zone interfere to produce an oscillatory directivity pattern. One quarter of the rays on each cone point back inside the duct and produce the reflected field. Thus the rim of the end-face of the duct acts as a ‘ring source’, in which the radiated and reflected fields have their origin. Every propagating duct mode determines a polar angle and an azimuthal angle; these are taken as parameters specifying the mode and are used to calculate the positions and angles of all the rays. The mathematical method on which the paper is based is Debye's approximation for the Bessel function which appears in the expression for the duct modes; the approximation shows also that the duct contains a region of smooth helical rays on which the field consists of inhomogeneous waves: this region is the inner cylinder, lying inside the annulus of piecewise linear helical rays. The results of the paper are very promising for the application of Keller's geometrical theory of diffraction to detailed calculations of the sound radiated from aeroengine ducts. An alternative description of the field, using Cargill's meridional rays, is summarized.

Simultaneous observations of VLF ground transmitter signals on the DE 1 and COSMOS 1809 satellites: Detection of a magnetospheric caustic and a duct

Khabarovsk transmitter signals (15.0 kHz, 48°N, 135°E) were observed on the high‐altitude (∼15000 km) Dynamic Explorer 1 (DE 1) and the low‐altitude (∼960) km COSMOS 1809 satellites during a 9‐day period in August 1989. On 7 out of 9 days the linear wave receiver (LWR) on the DE 1 satellite detected signals from the Khabarovsk transmitter. In addition, the DE 1 satellite also detected signals from the Alpha transmitter (11.9‐15.6 kHz) in Russia and an Omega transmitter (10.2‐13.6 kHz) in Australia, as well as natural VLF emissions such as hiss, chorus, whistlers, and wideband impulsive signals. On two days, August 23 and 27, 1989, observations of the Khabarovsk transmitter signals were simultaneously carried out at high altitude on the DE 1 satellite and at low altitude on the COSMOS 1809 satellite. Analysis of data from these 2 days has led to several new results on the propagation of whistler mode signals in the Earth’s magnetosphere. New evidence was found of previously reported propagation phenomena, such as (1) confinement of transmitter signals in the conjugate hemisphere at ionospheric heights (∼1000 km), (2) observation of direct multipath propagation on both DE 1 and COSMOS 1809, (3) detection of ionospheric irregularities of ≤ 100 km scale size with a few percent enhancement in electron density, believed to be responsible for the observed multipath propagation. We report the first detection of an exterior caustic surface near L ∼ 3.5 for VLF ground transmitter signals injected into the magnetosphere; the location of the caustic surface depended on the signal frequency, and the electric and magnetic fields decreased by several hundred decibels per L shell in the dark (shadow) side of the caustic. We also report the first direct detection of a magnetospheric duct at L = 2.94 which was believed to be responsible for the ducted propagation of Khabarovsk signals observed on the COSMOS 1809 satellite; the measured duct parameters were: ΔL ∼ 0.06 and ΔNe, ∼ 10 ‐ 13%. The duct width at the equator was ∼367 km. Our study also indicates that duct end points can extend down to at least ∼1000 km. The peak electric and magnetic fields of ducted Khabarovsk transmitter signals at ∼1000 km were 520 µV/m and 36 pT respectively. Estimated field strengths of these signals inside the duct at the geomagnetic equator were 57 µV/m and 12 pT for electric and magnetic field respectively. The results of two‐dimensional ray tracing simulations were consistent with the observations of the nonducted whistler‐mode propagation of Khabarovsk (15 kHz) and Alpha (11.9 kHz) signals from the transmitter location to the DE 1 and COSMOS 1809 satellites. Our results have direct implications for the question of accessibility of waves injected from the ground to various regions of the ionosphere and the magnetosphere. In situ measurements of electric and magnetic fields of Khabarovsk transmitter signals inside a duct may well prove to be the critical measurements needed to differentiate between the small signal and large signal theories of wave particle interactions.

Sound radiation and caustic formation from a point source in a wall shear layer

The propagation of acoustic waves from a high-frequency point source in a shear layer flowing over an infinite rigid plate is considered. Asymptotic expansions of the solution are obtained as k = omegaD/c0 --> infinity, using a previously developed method. Here, omega is the circular frequency of the point source, D is the thickness of the shear layer, and c0 is the ambient sound speed. Stationary and moving sources are analyzed. An infinite sequence of caustic surfaces are created downstream of the source and adjacent to the wall, by the refraction of rays from the source and their subsequent reflection from the wall. The acoustic fields on and off the caustics are obtained. The vorticity of the acoustic field is large on a caustic, but it decays rapidly away from the ''center'' of the caustic surface. That is, the energy of the acoustic field on a caustic is localized. In addition, the caustic surfaces are ''swept back'' by the moving source. The striking similarity between the moving caustic surfaces and the experimentally determined coherent fluid structures, which originate in the laminar sublayer of a turbulent boundary layer and are localized regions of intense vorticity, suggests that the previously proposed mechanism for the formation of these localized coherent fluid structures might be valid. The bursting of these structures and the violent ejection of the fluid away from the sublayer could be one of the mechanisms producing turbulence in the boundary layer.

The relationship between the second order moment width and the caustic surface radius of laser beams

It is found that the width of an arbitrary cylindrical symmetric laser beam, defined as twice the mean square deviation of its transversal intensity profile, and the radius of its caustic surface coincide within the parabolic approximation of the phase. Some examples are given.

Strain measurements inside solids subjected to lithotripter pulses

To study the direct stress wave effects on the fragmentation of kidney stones or gallstones during lithotripsy, it is important to know the evolution of the stress state inside concretions subjected to lithotripter pulses. Knowledge of these stress fields can be used, along with measurements of the mechanical properties of the stone, to predict locations and modes of failure. A technique of implanting monocrystalline silicon strain gauges within plaster concretions to obtain information about these internal stress fields was developed. Concretions of simple geometries, i.e., disks and spheres, were chosen for this initial study so that the reflections from the boundaries can be more easily identified and compared with theoretical models. Experimental results were obtained for both spherically diverging and focused shock wave sources. Theoretical models based on geometrical acoustics were developed for a spherical wave front incident on a disk and on a sphere. Predictions from these models match well with the experimental results. In addition, the caustic surfaces for a spherically diverging wave incident on a sphere are present. [Work supported by NIH Grant No. DK39796.]

Wave scattering from a rough ground in an inhomogeneous medium.

An exact analytical expression of the Green’s function is obtained for the case of a time-harmonic linesource embedded in a medium with the linear sound-speed profile. This is the first time an analytical two-dimensional Green’s function has been found for wave propagation in an inhomogeneous medium. Substituting the exact analytical expression of the Green’s function into the Helmholtz–Kirchhoff integral equation and using the method of moments, the scattered fields from a ground with barriers or troughs in such a medium are computed. The effects of scattering in the region beyond the caustic surface are also studied. [Work supported by U.S. Army Construction Res. Lab.]

Peculiar velocities of field spiral galaxies near and beyond the Great Attractor

We measure peculiar velocities for a sample of 48 late-type spiral galaxies which are located in the general vicinity of the Great Attractor (GA). Relative distances are derived using the I-band Tully-Fisher relation. We confirm the existence of positive velocity residuals, with amplitude 500-2000 km s^-1^ in the Hydra-Centaurus region, for galaxies located at kinematic distances of 20-30 Mpc. Beyond 45 Mpc, we have a very weak signal of negative velocity residuals whose amplitude is very sensitive to the form of the Malmquist correction. However, positive velocity residuals can be detected in galaxies with distances as large as 80 Mpc. We test the origin of these motions by the use of a nonlinear spherical infall model which predicts the position of the caustic surface as a function of distance from the GA. We also use all available velocities to search for the caustics in redshift space. We explicitly incorporate distance errors into the models and adopt a correction which is appropriate for a highly clustered distribution. Comparison of the data with the model yields the following results: (1) we cannot identify a symmetric infall pattern centered on the GA (2) & bulk flow " infall pattern, centered on a kinematic distance of 4350 km s^-1^ provides the best means of correcting the observed velocity so as to produce a Tully- Fisher relation with the least amount of scatter, and (3) galaxies in the vicinity of OCA 3574 appear to participate in a small-scale infall pattern. The overall behavior of the data shows that we have not yet obtained a sample over a sufficiently large volume to isolate the true source of the observed peculiar velocities. The possibility of large peculiar motions on the scale of 10,000 km s^-1^ remains viable. However, the morphology in redshift space of the Centaurus Cluster, coupled with the large observed peculiar velocities in this region, strongly suggests that this structure is a main contributor to the observed flow.

Numerical methods applicable to laser refraction through a caustic

AbstractNumerical computation of laser beam refraction may involve difficulties associated with ray paths through a caustic, where first‐order amplitude values become infinite. We choose the special test case of a stratified gas, and compare the results obtained by numerical integration of the ray differential equations with both asymptotic results and direct numerical solution of the exact wave equation. A final section evaluates the numerical solution of a parabolic approximation to the Helmholtz equation, with particular reference to the effect of the caustic. The final conclusion is that, with appropriate safeguards, any of these methods can effectively deal with the presence of a caustic surface.

Wavefields near transverse cusp caustics produced by reflecting ultrasonic transients and tone bursts from curved surfaces

Abstract : Ultrasonic wavefields reflected from curved surfaces were studied in the vicinity of caustics, Acoustical and optical transverse cusp diffraction catastrophes produced by reflections from a curved metal surface in water were imaged by displaying the amplitude or intensity in an observation plane transverse to the general direction of propagation. The optical image was used to locate the cusp point in the observation plane. Acoustical diffraction patterns for sine waves, described by the Pearcey function, were calculated with the parameters determined by the experimental setup leaving no adjustable scaling parameters. The calculated and experimental acoustical diffraction patterns showed good agreement near the cusp point. The acoustical diffraction pattern showed the expected mirror symmetry about an axis. The pattern was shown to scale properly with frequency. The transverse cusp caustic separates space into a region with three rays and a region with one ray. Inside the caustic there are three rays, on the curve two of the rays merge and disappear leaving one ray outside. Transient signals reflected from curved surfaces exhibited the merging and disappearance of rays on the caustic. Relative arrival times for signals in calculated and recorded time traces agree well. The relation to the wavefront parameters of the temporal orientation of the travel time surface is discussed. The general shape of the travel time surface is that of the swallow tail caustic surface.

Aberrated foci in the time and frequency domains and experiments with acoustical diffraction catastrophes.

Properties of wave fields at and near foci are considered for situations in which the amplitude is limited by the shape of an initial wave front rather than its aperture or initial dimensions. Such aberrated foci are relevant to high-frequency scattering from curved surfaces and to understanding the response of source and receiver arrays. Caustics classified by generic catastrophes are considered such as produced by random phase perturbations or by aberrations of imaging systems [M. V. Berry and C. Upstill, Prog. Opt. 18, 258 (1980)]. Experiments were carried out in which ultrasonic transients and tone bursts were reflected from a curved metal sheet in water such that the resulting caustic surface was a cusp which opens up generally transverse to the propagation direction of the reflected wave front [C. K. Frederickson, Ph.D. dissertation, Washington State University (1991)]. The results illustrate several more general properties of transient responses and diffraction patterns near weak foci. For example, the singular part of the travel time surface describing the merging of echoes has a shape like that of the next higher cuspoid catastrophe: a swallowtail catastrophe. Conditions for neglecting the effects of apertures on catastrophe diffraction patterns are examined. [Work supported by ONR.]

Analysis of whispering-gallery superconducting dielectric resonator modes

The whispering-gallery (WG) modes of a superconducting dielectric resonator (SDR) based on a sapphire cylindrical dielectric resonator and a YBa2Cu3O7 − δ shielding cylinder have been studied. A method for the determination of the resonant frequencies and the maximum quality factor of such modes is presented. Calculations have shown that most of the mode energy could be confined between the caustic surface of the WG modes provided the dimensions of the SDR are properly selected, and a magnitude of 109 forQ of the SDR could be estimated. A phenomenal explanation is given to account for such outstanding microwave behavior.

Ray tracing and the effect of ray divergence and convergence on signal intensity

An extension of the technique of ray tracing is studied in which the variation of the signal intensity along the ray is calculated. This results from the divergence or convergence of neighbouring rays in a ray pencil. It is examined for the special case of plane stratification and applied to an anisotropic collisionless ionosphere. With the approximations of ray theory the signal intensity is infinite where a ray touches a caustic surface, and examples of this are given. The ray is said to be reflected where it is parallel to the strata. Here the signal intensity is bounded and continuous but there are singularities in some of the quantities used to calculate it. Methods of dealing with this are discussed.

Optimization of three- and four-element lens systems by minimizing the caustic surfaces

For an imaging optical system it is, in general, desirable to transform a collection of point sources of light into point images distributed over the focal plane with the appropriate magnification. In practice, this is achieved by varying the lens system parameters such that the spread of a bundle of rays from each object point over the image plane has been minimized. In this study, caustic surfaces are used to construct a merit function that describes the spread of the caustic surfaces from an ideal image point. This caustic merit function has been used to optimize a large collection of three- and four-element lens systems. The performance of the optimized lenses has been evaluated by comparing the rms blur circle radii vs field angle to that of similar lenses designed by conventional techniques. The average rms improvement has been calculated for optimized systems. Results indicate that minimizing the caustic merit function reduces the rms blur radii over the field of view and the total aberrations of the lens systems, particularly for systems with large apertures and wide fields of view.