Diffraction Catastrophe Research Articles

Cusp diffraction catastrophe from spheroids: generalized rainbows and inverse scattering

The angular location of the recently discovered cusp pattern in the far-field scattering from an oblate spheroid is calculated as a function of the aspect ratio D/H. The calculation assumes the diameter D >> lambda and is limited to illumination perpendicular to the short axis of the spheroid. It agrees with observations for water drops in the range 1.22 < D/H < 1.37 with D ~ 1 mm.

Phase-space asymptotic analysis of wave propagation in homogeneous dispersive and dissipative media

Arising naturally in the study of one-dimensional pulse propagation in homogeneous dispersive and/or dissipative media are certain classes of oscillatory and/or diffusion integrals which encompass the canonical diffraction catastrophe integrals due originally to Thorn and Arnold. This is especially evident within the framework of a phase-space asymptotic analysis. Depending on the order of approximation of the exact solution beyond the Liouville or "first-order quasiparticle" limit, one recognizes caustic-like structures smoothed over by hyperdiffusion. Asymptotic series for these structures, which essentially define new basic functions, have been derived, but will not be presented here. Only the salient features of these structures will be reviewed briefly, and they will be illustrated by means of several simple canonical problems. Also, their applicability to other physical areas (e.g., wave propagation in deterministically and/or randomly varying channels, diffraction, etc.) will be pointed out.

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.]

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.]

Generalized Curvilinear Diffraction Gratings: V. Diffraction catastrophes

This paper is the last in a series [1–4] of five papers on the diffraction and imaging properties of generalized curvilinear diffraction gratings. Here we show how the bifurcation properties of two-dimensional potential functions may be studied via a diffraction grating representation. In particular, we derive the grating functions appropriate to Thom's seven elementary catastrophes. We then present a detailed study of the cusp grating and discuss the behaviour of its point source image diffraction pattern (IDP) in the context of catastrophe theory. In this way, we show how catastrophe theory allows for a simple geometric interpretation of discontinuous type image diffraction patterns.

The Double-cusp Unfolding of the 0 X 9 Diffraction Catastrophe

We define the double-cusp unfolding to be that four-dimensional unfolding of the codimension-8 unimodal singularity 0X9 for which the generating function is separable into a sum of two cusp catastrophe generating functions, one in each of the two state variables. We have made an experimental and theoretical study of the double-cusp unfolding, ψdc, of the 0X9 diffraction catastrophe by photographing plane sections of ψdc produced in light refracted by the water surface of a ripple tank, and comparing them with computer simulations from numerical evaluations of the diffraction integral. Experiment and theory are in excellent agreement. Computation shows that the intensity maximum is not at the geometrical focus, and we prove this result analytically. The product structure of ψdc has direct consequences for the wavefront dislocations, which are two orthogonal families of parallel straight lines. We also report observations of a five-dimensional unfolding of the 0X9 diffraction catastrophe, which we relate to...

Diffraction, Asymptotics and Catastrophes

First the method of stationary phase is used to develop various semiuniform asymptotic techniques that can be used to calculate the field in regions in which diffraction catastrophes occur. Three types of catastrophes are treated, namely those that take place at a shadow boundary, at a caustic and at a cusp. Then the diffraction of a spherical wave by a circular aperture is considered, and it is shown how the use of standard diffraction theories leads to diffraction integrals that can be evaluated by means of the semi-uniform asymptotic techniques developed for the three diffraction catastrophes mentioned above. Finally, it is pointed out how the asymptotic results provide new physical insight into the diffraction process, as well as offering great savings in computing time.

The Stokes set of the cusp diffraction catastrophe

Stokes and anti-Stokes lines are familiar in the asymptotic approximation of functions of a complex variable. The author generalises this notion and define the Stokes and anti-Stokes sets of a complex function of many (possibly complex) variables, defined by a diffraction-type integral. They are subsets of the Maxwell set of catastrophe theory, extended to complex variables. On the Stokes set the number of complex stationary points contributing to the integral changes by one, whereas on the caustic the number of real stationary points changes by two. Knowledge of the location of the Stokes set is essential to perform a full stationary phase analysis of a diffraction integral, and it imposes a constraint upon the positions of wavefront dislocations. For the canonical cusp diffraction catastrophe the author finds the explicit equation of the Stokes set, which is a broadened mirror image of the cusp caustic.

Phase-space projection identities for diffraction catastrophes

For each standard 'diffraction catastrophe' wavefunction psi , describing the interference pattern near a stable caustic, the authors derive two nonlinear identities. These relate the intensity mod psi mod 2 to an integral over the wavefunction psi corresponding to the same catastrophe, or to a less singular one. The identities are interpreted in terms of projections of Wigner functions from phase space on to coordinate space.

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.

On the analysis of diffraction catastrophes

The most important diffraction catastrophes (caustics) occurring in two-dimensional diffraction are analysed: the fold (elementary caustic), the cusp, the hyperbolic and the elliptic umbilic catastrophe. Special emphasis is given to the inverse problem: what information can be obtained from an observed diffraction catastrophe pattern? It is shown that the critical lines (rainbow lines) in the object plane and the behaviour of the phase function in their neighbourhood can be determined approximately from the characteristic vectors and the intensities of the main peaks in the pattern of these diffraction catastrophes.