Diffractive Orbits Research Articles

Diffraction of Wigner functions

We describe the contribution of diffractive orbits to semiclassical approximations of Wigner function propagators. These contributions are based on diffractively scattered rays used in the geometrical theory of diffraction (GTD). They provide an extension of well-established approximations of Wigner-function propagators based on rays that propagate by specular reflection and refraction. The wider aim of this approach is to allow for diffractive mechanisms to be accounted for in Eulerian approaches to ray-tracing simulations. Such approaches propagate densities of rays rather than follow rays individually. They promise to be a more efficient means of performing ray-tracing simulations in complex environments with applications in, for example, planning of wireless signal coverage for mobile communication networks.

Diffractive Propagation on Conic Manifolds

In this survey, we review some applications and extensions of the author’s results with Richard Melrose on propagation of singularities for solutions to the wave equation on manifolds with conical singularities. These results mainly concern: the local decay of energy on noncompact manifolds with diffractive trapped orbits (joint work with Dean Baskin); singularities of the wave trace created by diffractive closed geodesics (joint work with G. Austin Ford); and the distribution of scattering resonances associated to such closed geodesics (joint work with Luc Hillairet).

The trace formula for a point scatterer on a compact hyperbolic surface

An exact trace formula for the perturbation of the Laplacian by a Dirac delta potential on a compact hyperbolic Riemann surface is derived. The formula can be considered an analogue of the Selberg trace formula. The difference of perturbed and unperturbed trace is expressed as an identity term plus a sum over combinations of diffractive orbits which visit the position of the potential.

Diffractive orbits in the length spectrum of a two-dimensional microwave cavity with a small scatterer

In a two-dimensional rectangular microwave cavity dressed with one pointlike scatterer, a semiclassical approach is used to analyze the spectrum in terms of periodic orbits and diffractive orbits. We show, both numerically and experimentally, how the latter can be accounted for in the so-called length spectrum, which is retrieved from two-point correlations of a finite-range frequency spectrum. Beyond its fundamental interest, this first experimental evidence of the role played by diffractive orbits in the spectrum of an actual cavity, can be the first step towards a technique to detect and track small defects in wave cavities.

Finite geometries and diffractive orbits in isospectral billiards

Several examples of pairs of isospectral planar domains have been produced in the two-dimensional Euclidean space by various methods. We show that all these examples rely on the symmetry between points and blocks in finite projective spaces; from the properties of these spaces, one can derive a relation between Green functions as well as a relation between diffractive orbits in isospectral billiards.

Low-rank perturbations and the spectral statistics of pseudointegrable billiards.

We present an efficient method to solve Schrödinger's equation for perturbations of low rank. The method is ideally suited for systems with short range interactions or quantum billiards. It involves a secular equation of low dimension, which directly returns the level counting function. For illustration, we calculate the number variance for two pseudointegrable quantum billiards: the barrier billiard and a right triangle billiard. In this way, we obtain precise estimates for the level compressibility in the semiclassical (high energy) limit. In both cases, our results confirm recent theoretical predictions, based on periodic orbit summation, disregarding diffractive orbits.

Observation of diffractive orbits in the spectrum of excited NO in a magnetic field

We investigate the experimental spectrum of excited NO molecules in the diamagnetic regime and develop a quantitative semiclassical framework to account for the results. We show the dynamics can be interpreted in terms of classical orbits provided that in addition to the geometric orbits, diffractive effects are appropriately taken into account. We also show how individual orbits can be extracted from the experimental signal and use this procedure to reveal the first experimental manifestation of inelastic diffractive orbits.

Nonclassical paths in the recurrence spectrum of diamagnetic atoms.

Using time-independent scattering matrices, we study how the effects of nonclassical paths on the recurrence spectra of diamagnetic atoms can be extracted from purely quantal calculations. This study reveals an intimate relationship between two types of nonclassical paths: exotic ghost orbits and diffractive orbits. This relationship proves to be a previously unrecognized reason for the success of semiclassical theories, such as closed-orbit theory, and permits a comprehensive reformulation of the semiclassical theory that elucidates its convergence properties.

Weyl series for Aharonov-Bohm billiards

Following a conjecture of Berry and Howls (1994) concerning the geometric information contained within the high orders of Weyl series, we examine such series for the average spectral properties of two- and three-dimensional quantum ball billiards threaded by a single flux line at the centre. We adapt a Mellin-based scheme of Bordag et al (1996) to generate the Weyl series. It is shown that for a circular billiard, only a single Weyl series term is changed and thus the flux line only induces a simple constant shift in the average properties of the spectrum, although the fluctuations about this average will still be flux dependent. This implies that the late terms in the expansion are dominated by the diametrical periodic orbit of the unfluxed circle, rather than the shorter diffractive orbits encountering both the billiard boundary and the flux line. For a spherical billiard with flux the late terms suffer modifications which can be linked to diffractive orbits. The origins of the differences between the structure of the series are traced to the interaction of the geometry and symmetry breaking.

Rydberg molecules in external fields: a semiclassical analysis.

We analyze the spectra of simple Rydberg molecules in static fields within the framework of closed/periodic-orbit theories. We conclude that in addition to the usual classical orbits one must consider classically forbidden diffractive paths. Further, the molecule brings in a new type of "inelastic" diffractive trajectory in addition to the usual "elastic" diffractive orbits encountered in systems with point scatterers. The relative importance of inelastic versus elastic diffraction is quantified by merging the usual closed orbit theory framework with molecular quantum defect theory.

Atoms in parallel fields: analysis with diffractive periodic orbits

We show that fluctuations in the density of states of nonhydrogenic atoms in parallel fields are strongly influenced by diffractive periodic orbits. Unlike typical systems with a diffractive point scatterer, the atomic core of small atoms like lithium and helium is best understood as a combined geometric and diffractive scatterer. Each Gutzwiller (geometric) periodic orbit is paired with a diffractive orbit of the same action. We investigate, particularly, amplitudes for contributions from repetitions, and multiple scattering orbits. We find that periodic orbit repetitions are described by "hybrid" orbits, combining both diffractive and geometric core scatters, and that by including all possible permutations we can obtain excellent agreement between the semiclassical model and accurate fully quantal calculations. For high repetitions, we find even one-scatter diffractive contributions become of the same order as those of the geometric periodic orbit for repetition numbers n approximately Planck's over 2pi(-1/2). Although the contribution of individual diffractive orbits is suppressed by O(Planck's over 2pi(1/2)) relative to the geometric periodic orbits, the proliferation of diffractive orbits with increasing period means that the diffractive effect for the atom can persist in the Planck's over 2pi-->0 limit.

Influence of diffraction on the spectrum and wave functions of an open system

In this paper, we demonstrate the existence and significance of diffractive orbits in an open microwave billiard, both experimentally and theoretically. Orbits that diffract off a sharp edge of the system are found to have a strong influence on the transmission spectrum of the system, especially in the regime where there are no stable classical orbits. On resonance, the wave functions are influenced by both classical and diffractive orbits. Off resonance, the wave functions are determined by the constructive interference of multiple transient, nonperiodic orbits. Experimental, numerical, and semiclassical results are presented

Spectral statistics in chaotic systems with a point interaction

We consider quantum systems with a chaotic classical limit that are perturbed by a point-like scatterer. The spectral form factor K(tau) for these systems is evaluated semiclassically in terms of periodic and diffractive orbits. It is shown for order tau^2 and tau^3 that off-diagonal contributions to the form factor which involve diffractive orbits cancel exactly the diagonal contributions from diffractive orbits, implying that the perturbation by the scatterer does not change the spectral statistic. We further show that parametric spectral statistics for these systems are universal for small changes of the strength of the scatterer.

Diffractive corrections in the trace formula for polygonal billiards

We derive contributions to the trace formula for the spectral density accounting for the role of diffractive orbits in two-dimensional polygonal billiards. In polygons, diffraction typically occurs at the boundary of a family of trajectories. In this case the first diffractive correction to the contribution of the family to the periodic orbit expansion is of order of that of an isolated orbit, and gives the first sqrt[Planck's over 2pi] correction to the leading semiclassical term. Keller's geometrical theory of diffraction is inadequate for treating these corrections and we develop an alternative approximation based on Kirchhoff's theory. Numerical checks show that our procedure allows reduction of the typical semiclassical error by about two orders of magnitude. The method permits treatment of the related problem of flux-line diffraction with the same degree of accuracy.

Diffractive Orbits in an Open Microwave Billiard

We demonstrate experimentally the existence and significance of diffractive orbits in an open microwave billiard. Orbits that diffract off a sharp edge strongly influence the conduction spectrum, especially in the regime where there are no stable classical orbits. On resonance, the wave functions are influenced by both classical and diffractive orbits. Off resonance, the wave functions are determined by the constructive interference of multiple transient orbits. The experimental results agree well with quantum computations and with a semiclassical theory that incorporates diffraction.

Semiclassical treatment of diffraction in billiard systems with a flux line.

In billiard systems with a flux line, semiclassical approximations for the density of states contain contributions from periodic orbits as well as from diffractive orbits that are scattered on the flux line. We derive a semiclassical approximation for diffractive orbits that are scattered once on a flux line. This approximation is uniformly valid for all scattering angles. The diffractive contributions are necessary in order that semiclassical approximations are continuous if the position of the flux line is changed.

From chaotic to disordered systems - a periodic orbit approach

We apply periodic orbit theory to a quantum billiard on a torus with a variable number (N) of randomly distributed circular scatterers. Provided the scatterers are much smaller than the wavelength they may be regarded as sources of pure s-wave diffraction. The relevant part of the spectral determinant is due only to diffractive periodic orbits. We formulate this diffractive zeta function in terms of an transfer matrix, which is transformed to real form. The determinant is shown to reproduce the full density of states for generic configurations if . The zeros of the determinant are computed numerically. We study the statistics exhibited by these spectra. The numerical results suggest that the spectra tend to GOE statistics as the number of scatterers increases for typical members of the ensemble. A peculiar situation arises for configurations with four scatterers and kR tuned to , where the statistics appears to be perfectly Poissonian.

Diffraction in the semiclassical description of mesoscopic devices

In pseudo-integrable systems, diffractive scattering caused by wedges and impurities can be described within the framework of the Geometric Theory of Diffraction (GDT) in a way similar to the one used in the Periodic Orbit Theory of Diffraction (POTD). We derive formulas expressing the reflection and transition matrix elements for one and many diffractive points and apply it for impurity and wedge diffraction. Diffraction can cause backscattering in situations where usual semiclassical backscattering is absent causing an erodation of ideal conductance steps. The length of diffractive periodic orbits and diffractive loops can be detected in the power spectrum of the reflection matrix elements. The tail of the power spectrum shows ~1 l 1 2 decay due to impurity scattering and ~1 l 3 2 decay due to wedge scattering. We think this is a universal sign of the presence of diffractive scattering in pseudo-integrable waveguides.

Uniform approximation for diffractive contributions to the trace formula in billiard systems

We derive contributions to the trace formula for the spectral density accounting for the role of diffractive orbits in two-dimensional billiard systems with corners. This is achieved by using the exact Sommerfeld solution for the Green function of a wedge. We obtain a uniformly valid formula for single-diffractive orbits which interpolates between formerly separate approaches (the geometrical theory of diffraction and Gutzwiller's trace formula). It yields excellent numerical agreement with exact quantum results, also in cases where other methods fail.

Small disks and semiclassical resonances

We study the effect on quantum spectra of the existence of small circular disks in a billiard system. In the limit where the disk radii vanish there is no effect, however this limit is approached very slowly so that even very small radii have comparatively large effects. We include diffractive orbits which scatter off the small disks in the periodic orbit expansion. This situation is formally similar to edge diffraction except that the disk radii introduce a length scale in the problem such that for wave lengths smaller than the order of the disk radius we recover the usual semi-classical approximation; however, for wave lengths larger than the order of the disk radius there is a qualitatively different behaviour. We test the theory by successfully estimating the positions of scattering resonances in geometries consisting of three and four small disks.