Light scattering from laser-damaged speckled surfaces.

Scattering from an initially plane surface of a thin Plexiglas plate damaged by an impinging coherent light beam of a laser is considered. The morphology and the characteristic dimensions of the damaged speckles was studied during the evolution of the phenomenon by interferometry and the creation of diffraction patterns. The cusped interferogram created by superposition of evoluting two initial diffraction patterns, created during successive damage steps, forms caustics corresponding to hyperbolic umbilic catastrophes. An experimental study of the evolution of the damage phenomenon clarifies the mechanisms of elastic and plastic deformations of the affected zone.

Multicusp caustics formed from reflections of warped surfaces.

The optical method of caustics developed mainly for studying singularities in stress fields was extended to define quantitatively the slope variation of warped surfaces. The existing theory was concerned with the study of infinitesimal but abrupt variations of thickness in elastic and plastic stress fields containing stress singularities due to either loading or geometry. In this paper the theory of caustics was extended to study the warping of any surface due mainly to twisting loads- While the caustics developed in previous uses were generalized epicycloid surfaces with or without a single cusp line, in the cases studied in this paper multicusp surfaces were developed. The quantitative interrelationship between the shape and size of the caustic and the respective mode of twisting of the surface was established, and interesting properties of these surfaces were disclosed. Applications to twisted elliptic, triangular, and square elastic bars clearly illustrate the importance of the method.

Optically exact equations of caustics for slant cracks biaxially loaded.

The equations of caustics have been derived based on the exact theory of geometrical optics, and they have been adapted to the problem of a slanting internal crack in a disk under biaxial loading. A comparison of the exact caustics with those derived from the far-field theory used in the past in the applications showed negligible difference between the two theories. Thus, it is shown that the approximate theory of caustics used before for determining singular fields in mechanics is sufficiently accurate for engineering applications.

Generalization of the theory of far-field caustics by the catastrophe theory.

To generalize the theory of far-field caustics, three theorems and several corollaries are presented in this paper. Using the law of reflection and catastrophe theory we have established conditions to predict caustic patterns in a 3-D space, which were created from the reflection of a light beam from an analytically known surface. The general theory was readily reduced to the already known cases of diffraction, indicating the validity of the general theory. Experimental evidence in two simple cases of reflectors, consisting of triangular and rectangular membranes, corroborated the results of the theory.

Some further properties of caustics useful in mechanical applications.

The caustics created from reflections of light rays at the vicinity of stress singularities and other stress concentrations in an isotropic and elastic plate submitted to a plane-stress mode of deformation yield important information concerning the order of singularity of the stress intensity or concentration factor at these particular points of a stress field. All information was extracted from particular measurements of the longitudinal or transverse diameters of the caustics, based on basic properties of caustics. In this paper some further properties were found, which are very useful in evaluating the stress field creating the caustic, which make the respective measurements independent of the angle of orientation of the caustic and more reliable than previously.

Elliptic polarization represented by the Carter and Smith charts.

The graphical method using the Poincaré sphere for representing elliptically polarized light is based on the fact that the latitude and longitude of a point on the unit-radius sphere represent the ellipticity and the azimuth of the corresponding light ellipse. Stereographic projections of the unit-diameter Poincaré sphere either on a western plane tangent at the equator of the sphere or on an equatorial plane tangent to either of the poles lead to a representation of any elliptical polarization state on the so-called Carter charts, while charts derived from orthographic projections in the Poincaré sphere and consisting again of families of orthogonal circles traced inside either the equator or the principal meridian of the unit-radius Poincaré sphere yielding either ellipticity-azimuth or amplitude ratio-phase angle niveau lines constitute the Smith charts. Both types of charts consist of families of orthogonal circles. These families of circles, traced on either chart and corresponding to parametric families with different properties for the corresponding elliptically polarized light, yield an easy and accurate solution to the problem of graphical evaluation of the optical properties of the resulting polarization state when an elliptically polarized plane wave is passing through different optically active elements.

Distance measuring based on caustics.

A distance measuring technique based on caustics formed by illuminating an ellipsoid reflector with a point light source was developed. A study of the influence of the ellipticity of the mirror and the relative positions of the screen where the caustic is formed, the point source, and the mirror on the limits and accuracy of the range measured was undertaken. It was found that the accuracy in measuring a distance may be greater than the accuracy obtained by a direct measurement of the same distance by any conventional method used in measuring the diameter of the caustic on the screen.

Properties of caustics from conic reflectors. 1: Meridional rays.

General relations for the caustic surfaces obtained by illuminating an ellipsoid, paraboloid, or hyperboloid reflector by a point-light source, lying along the principal axis of the reflector, were derived. A thorough study of the evolution of caustics with the particular type of reflector, as well as with the relative position of the point-light source and the reflector, was undertaken. Interesting laws for the shape, position, and properties of the caustics were derived, depending on the shape of the particular reflector used, its aperture, and the relative position of the light source and the reflector. These properties may be useful for the creation of wide angle all-reflective multimirror systems designed for definite limits of distances and which, by adjacent pairs of reflectors, present only a limited and reduced amount of third-order aberrations.

Surface topography by caustics.

The optical method of caustics, initially developed for recording abrupt plate slopes created by singularities in elastic stress fields, was extended to incorporate the study of the general case of any type of surface. A universal technique, based on the general theory of caustics developed in this paper, was formulated to study the topography of any surface from its corresponding caustics obtained by illuminating the surface by a parallel, convergent, or divergent light beam. The special case of an axisymmetric mirror with elliptical cross section, whose ellipticity varies from zero to infinity, was studied extensively to show the potentialities of the technique developed. It was shown that the caustics obtained are very sensitive to the particular form of the surface considered. From the procedure developed in this paper it was concluded that the method of caustics can be successfully used to record the topography of any surface with large or infinitesimal slopes.

Optical stress rosette based on caustics.

The reflected method of caustics, which is based on geometric optics, was used to define the principal stress difference and the principal directions at the interior of any generalized plane stress field by defining the stress distribution around small perforations drilled in the plate. A monochromatic coherent light beam, reflected on or transmitted through the plate at the neighborhood of the perforation, deviated by different amounts because of the refractive index variation and the thickness variation due to loading. The deviated light rays, received on a reference plane parallel to the plate, were concentrated along a singular curve, which enveloped a large part of the reflected or traversing rays and created a caustic. The properties of this singular curve were studied and found to depend on the biaxial stress field existing at the place of the perforation, as well as on the mechanical and optical properties of the material and the geometry of the optical setup. The double-kidney shape of the caustic presented a maximum diameter D(max), which coincided with one of the axes of symmetry of the caustic. The other axis of symmetry, which passed through the cusps of the caustic, coincided with the axis of maximum principal stress, while the size of D(max) yielded the principal stress difference. Thus, the caustic constitutes a sensitive stress rosette, which defines the orientation and the size of the stress field.

Alignment, orientation, and range finding by diffraction gratings.

A simple method for attaining alignment of a target and parallelism between a plane of reference and a target is developed by using as reference planes diffraction gratings of a convenient pattern. Coherent and monochromatic light, emitted from a laser, illuminates the. target, which is either a cross slit or an annular aperture. The reference grating is placed at a distance from the target. The principal maxima created by the light passing through the target and the grating form distorted shapes of the target on the plane of the grating. Alignment can be achieved by displacing the grating in its plane to form a symmetric pattern of the (+/-m)-order principal maximum image of the target. The distance between the target and the reference grating, as well as the angle subtended by their planes, can be accurately estimated by the size and the distortion of the symmetric (+/-m)-order principal maximum image. The apparatus used was very simple, and its flexibility can be enhanced by using a series of amplitude gratings of different line frequencies. Furthermore, automated detection and scanning of the spectrum of the grating increase considerably the accuracy of the method.

Reflected shadow method for the study of constrained zones in cracked plates.

An optical method was developed for the study of constrained zones around cracks in transparent specimens made of birefringent materials under conditions of plane stress. The absolute retardations of light rays impinging normally at the plate, partly reflected from the back face, and twice refracted at the front face create a caustic when emerged from the plate, which has the shape of a generalized epicycloid, when it is projected on a screen far from the plate. The caustic defines the constrained zone surrounding a crack tip. The characteristic properties of the caustic were studied in relation to the straining mode of the plate.