Umbilic Diffraction Research Articles

Near-zone transmission caustic of a hanging water drop.

A water drop hanging from a house siding board after a rain shower is near-normally illuminated by sunlight either shortly after sunrise or before sunset. A focusing caustic consisting of a bright V-shape or U-shape with a small bright elliptical shape immediately above it is frequently seen on the next lower siding board. In addition, there are two broad regions of illumination immediately above the caustic, fanning out to the upper left and upper right. This complicated pattern, composed of a bright V-shape or U-shape, and the bottom half of the small bright elliptical-shape immediately above it, is caused by the hyperbolic umbilic diffraction caustic near the condition of maximum focus. This can be observed because, by a stroke of good fortune, the distance between the lower edge of a siding board and the flat portion of the next siding board beneath it is nearly equal to the paraxial focal distance of the caustic. Blocking off the light incident on the top, bottom, left side, and right side of the drop was used to determine the portion of the drop responsible for different parts of the caustic. The results were found to match the predictions for the hyperbolic umbilic caustic.

Hyperbolic umbilic caustics from oblate water drops with tilted illumination: Observations

Various groups have reported observations of hyperbolic umbilic diffraction catastrophe patterns in the far-field scattering by oblate acoustically levitated drops with symmetric illumination. In observations of that type the drop's symmetry axis is vertical and the illuminating light beam (typically an expanded laser beam) travels horizontally. In the research summarized here, scattering patterns in the primary rainbow region and drop measurements were recorded with vertically tilted laser beam illumination having a grazing angle as large as 4 degrees. The findings from these observations may be summarized as follows: (a) It remains possible to adjust the drop aspect ratio (diameter/height) = D/H so as to produce a V-shaped hyperbolic umbilic focal section (HUFS) in the far-field scattering. (b) The shift in the required D/H was typically an increase of less than 1% and was quadratic in the tilt. (c) The apex of the V-shaped HUFS was shifted vertically by an amount proportional to the tilt with a coefficient close to unity. The levitated drops had negligible up-down asymmetry. Our method of investigation should be useful for other generalized rainbows with tilted illumination.

Stable and unstable Airy-related caustics and beams

Optical beams with an underlying caustic structure are stable under perturbation if the caustics belong to the catastrophe-theory classification; otherwise they are unstable. The original Airy beam in two spatial dimensions, with its curved caustic, is stable in this sense. But the separable Airy-product beam in three-dimensions is unstable: under separability-breaking perturbation, it unfolds into the hyperbolic umbilic diffraction catastrophe, which is stable. By including initial phase factors, a variety of new exact solutions of the paraxial wave equation can be generated, corresponding to Pearcey and higher-catastrophe beams with stable caustics, and with the associated diffraction catastrophes appearing in their canonical forms or as deformations of these.

Evaluation of cuspoid and umbilic diffraction catastrophes of codimension four

The evaluation of the two diffraction catastrophes of codimension four, namely, the butterfly and the parabolic umbilic, is here proposed by means of a simple computational approach developed in the past to characterize the whole hierarchy of the structurally stable diffraction patterns produced by optical diffraction in three-dimensional space. In particular, after expanding the phase integral representations of butterfly and parabolic umbilic in terms of (slowly) convergent power series, the retrieving action of the Weniger transformation on them is investigated through several numerical experiments. We believe that the methodology and the results presented here could also be of help for the dissemination of catastrophe optics to the widest scientific audience.

Axial and focal-plane diffraction catastrophe integrals

Exact expressions in terms of Bessel functions are found for some of the diffraction catastrophe integrals that decorate caustics in optics and mechanics. These are the axial and focal-plane sections of the elliptic and hyperbolic umbilic diffraction catastrophes, and symmetric elliptic and hyperbolic unfoldings of the X9 diffraction catastrophes. These representations reveal unexpected relations between the integrals.

On the numerical evaluation of umbilic diffraction catastrophes

A simple computational approach is proposed for the evaluation of umbilic diffraction catastrophes which, together with cuspoids, describe the whole hierarchy of the structurally stable diffraction patterns that can be produced by optical diffraction. In this paper, after expanding the double integral representations of hyperbolic and elliptic umbilics as convergent power series, the action of the Weniger transformation on them is studied. Exact expressions for the "on-axis" umbilic field have also been found, which extend previously published results to complex values of the control parameter. Numerical experiments aimed at giving evidence of the effectiveness and implementative ease of the approach are eventually presented.

Dislocation lines in the hyperbolic umbilic diffraction catastrophe

The three-dimensional pattern of the hyperbolic umbilic diffraction catastrophe is computed from an integral representation. A detailed description is given of the geometrical arrangement of the wave dislocation lines (optical vortices) on which the diffraction pattern is based. From a crossed grid of nodal lines in the focal plane, two bundles of dislocation lines spring out symmetrically into the regions of 4-wave interference. Each dislocation line then follows a chain of curved segments which approximate successive steps along lattice vectors in the space group Fmmm . The result is a bundle of helices of non-circular cross-section that gradually straighten out until, far from the focal plane, they become the dislocations of the Pearcey diffraction pattern for the cusp catastrophe. A new phenomenon is the multiple puncturing of the caustic surface by a series of helical dislocations.

Evolution of the hyperbolic umbilic diffraction pattern from Airy rings

If the diffraction pattern of the hyperbolic umbilic diffraction catastrophe is produced by an optical system of increasing aperture, it passes continuously from the two-dimensional system of Airy rings in the focal plane, made by a very small aperture, to the full three-dimensional pattern corresponding to infinite aperture. The paper studies this transition by examining the truncated diffraction integral and following the evolution of the wave dislocation lines (phase singularities) on which the pattern is based. The seed of the evolution from a two-dimensional to a three-dimensional pattern turns out to be already present asymptotically even for the smallest aperture: namely, a column of small dislocation rings very close to the axis that stream in procession towards the focal plane, and become dislocations lying in the Airy fringe surfaces that run parallel to the main fold caustic, only to disappear ultimately by retreat to infinity. The evolution into the final dislocation pattern takes place by a sequence of primitive local topological events, such as reconnection (hyperbolic interchange) and ring creation.

From Airy rings to the elliptic umbilic diffraction catastrophe

The universal three-dimensional diffraction patterns associated with generic optical caustics are built on the loci of their line zeros—features variously known as wave dislocations, phase singularities, or optical vortices. All such patterns may be regarded as natural evolutions from the Fraunhofer diffraction pattern of concentric Airy rings surrounding the point focus of a lens. This paper considers as an example the elliptic umbilic caustic; it describes in detail, from numerical computation, how the simple planar system of Airy rings can change continuously into the much more complicated three-dimensional dislocation pattern decorating the caustic. The change is accomplished by the repeated use of five elementary changes in topology, which are treated analytically.

Doubly-focused echos from spheres unfold into a hyperbolic umbilic diffraction catastrophe

An underwater spherical target resides in an Airy field formed by reflection off a curved surface representing the sea floor or sea surface. In prior work [B. Dzikowicz and P. L. Marston, J. Acoust. Soc Am. 110, 2778 (2001)] direct returns of a tone burst from the surface reflection focused toward the target were shown to have a dependence on the target position described by an Airy function. The return echo can also be focused again by the surface onto the source and receive transducer. This gives the square of an Airy function for the case of a point target. With a finite sized target (as in the experiment) this goes over to a hyperbolic umbilic catastrophe with symmetric arguments. The arguments of the hyperbolic umbilic function are derived from only the relative return times of a transient pulse. Experiment confirms the predicted merging of transient echoes in the time domain, as well as the hyperbolic umbilic diffraction integral amplitudes for a tone burst. This method would allow for the observation of a target at a greater distance in the presence of a focusing surface. [Research supported by ONR.]

Appearance of supernumeraries of the secondary rainbow in rain showers

The secondary rainbow scattering angle for spheroidal drops of water is virtually independent of aspect ratio for most visible wavelengths. For most solar heights the residual aspect-ratio dependence shifts the bow toward a smaller deviation angle if the drop size increases. These two facts explain why the supernumeraries of the secondary rainbow are never seen in rain showers. At high solar elevations the flattening of drops results in a shift of the secondary rainbow toward a larger deviation angle. It is shown that this shift is still large enough to cause the formation of the first supernumerary in red light. This red supernumerary of the secondary rainbow may be observable by eye in natural showers if a red filter is used to remove the obscuring contribution of shorter wavelengths to the light of the rainbow. For indices of refraction far from that of water, a strong aspect-ratio dependence of the secondary rainbow angle is shown to be present. Some possible implications of this for the formation of a hyperbolic umbilic diffraction catastrophe in the secondary rainbow pattern are indicated.

Hyperbolic umbilic diffraction catastrophe in the scattering from a penetrable spheroid at short wavelengths

There has been recent progress in understanding the scattering patterns from spheroids at long wavelengths. At short wavelengths, however, partial wave sums are difficult to evaluate, while geometric optics is not applicable near angular foci where the scattering is enhanced. Consequently, the discovery and classification of angular foci is important to the understanding of short wavelength scattering. A novel focus or “diffraction catastrophe” was recently observed in the light scattered from oblate spheroidal drops of water in air [P. L. Marston and E. H. Trinh, Nature 312, 529–531 (1984)]. Hyperbolic umbilic focal sections were characteristic of drops having D/H≃1.31, where the drop's diameter is D in the equatorial plane and is H along the axis of rotational symmetry. This critical axis ratio was subsequently calculated [J. F. Nye, Nature 312, 531–532 (1984)] to be related to the refractive index μ by (D/H)2 = 3μ2/4μ2 − 1). This expression is in agreement with the observed D/H≃1.31 for water drops having μ = 1.332. In the present research, this expression was confirmed using a general ray‐tracing formalism. Nontrivial diffraction catastrophes should also be present in the farfield acoustical scattering from nonspherical penetrable objects. [Work supported by ONR.]

Evaluation of non-separable two-dimensional

In the treatment of reactive collisions by approximate methods such as the Distorted Wave Born Approximation, two-dimensional non-separable integrals are frequently encountered. In this article, we introduce the use of a two-dimensional canonical integral, the hyperbolic umbilic canonical diffraction function, on a model problem which leads to non-separable twodimensional Franck-Condon integrals. The identification of the parameters of the canonical function in terms of the physical parameters of the model is immediate in this case, and we find that the use of this function reproduces numerical quadrature results accurately with substantial savings in computing time. Extensions of the procedure to more general problems, anticipated by Child and Shapiro, are also discussed.

Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops

The characterization of short-wavelength scattering phenomena has been advanced by the study of diffraction catastrophes1,2. Examples include optical phenomena1–5 and molecular collisions6. The most familiar example of a diffraction catastrophe is the ordinary rainbow7–9. The angular variation of the scattering from a spherical drop locally corresponds to that of a fold-diffraction catastrophe1,2, a result most clearly seen with monochromatic illumination. We have studied the scattering from drops whose shape closely approximated that of an oblate spheroid with the short axis—the symmetry axis—vertical. The drops were illuminated by a horizontally propagating gaussian beam with a wavelength λ = 633 nm. These drops were observed to scatter in the horizontal rainbow region with patterns like those of hyperbolic–umbilic (classification D +4) diffraction catastrophes1,5,6. Visible D+4 diffraction patterns observed previously include light transmitted by frosted glass surfaces10 and by liquid lenses clinging to tilted glass plates1,4.

Light scattering from acoustically levitated spheroidal drops: Discovery of a new hyperbolic umbilic diffraction catastrophe

There has been recent progress in understanding the scattering patterns from penetrable spheroids at long wavelengths. At short wavelengths, however, partial‐wave sums are difficult to evaluate, while geometric optics is not applicable near angular loci where the scattering is enhanced. Consequently, the discovery and classification of angular foci is important to the understanding of short wavelength scattering. Our observation of a new focus or “diffraction catastrophe” in scattering from an oblate drop of water in air used an acoustic standing wave to levitate the drop and control its shape. The drop was illuminated by a horizontally propagating laser beam and its symmetry axis was vertical. The farfield scattering pattern was observed near the rainbow region. The two‐dimensional pattern was recognized to be that of a hyperbolic umbilic diffraction catastrophe, known because of other applications [M. V. Berry and C. Upstill, Prog. Opt. 18, 257–346 (1980)]. Changing the drop's aspect ratio corresponds to varying the “control parameter” C3. Strong focusing with C3 = 0 was observed. [P. Marston was supported in part by JPL and ONR.]

The elliptic umbilic diffraction catastrophe

We have made a detailed theoretical and experimental study of the three-dimensional diffraction pattern decorating the geometrical-optics caustic surface whose form is the elliptic umbilic catastrophe in Thom’s classification. This caustic has three sheets joined along three parabolic cusped edges (‘ribs’) which touch at one singular point (the ‘focus’). Experimentally, the diffraction catastrophe was studied in light refracted by a water droplet 'lens’ with triangular perimeter, and photographed in sections perpendicular to the symmetry axis of the pattern. Theoretically, the pattern was represented by a diffraction integral E(x,y,z) , which was studied numerically through computer simulations and analytically by the method of stationary phase. Particular attention was concentrated on the ‘dislocation lines’ where | E | vanishes, since these can be considered as a skeleton on which the whole diffraction pattern is built. Within the region bounded by the caustic surface the interference of four rays produces hexagonal diffraction maxima stacked in space like the atoms of a distorted crystal with space group R3m. The dislocation lines not too close to the ribs form hexagonally puckered rings. On receding from the focus and approaching the ribs, these rings approach one another and eventually join to form ‘hairpins’, each arm of which is a tightly wound sheared helix that develops asymptotically into one of the dislocations of the cusp diffraction catastrophe previously studied by Pearcey. Outside the caustic there are also helical dislocation lines, this time formed by interference involving a complex ray. There is close agreement, down to the finest details, between observation, exact computation of E(x,y,z) , and the four-wave stationary-phase approximation.