Using dynamical barriers to control the transmission of light through slowly varying photonic crystals.

We use semiclassical Hamiltonian optics to investigate the propagation of light rays through two-dimensional photonic crystals when slow spatial modulation of the lattice parameters induces mixed stable-chaotic ray dynamics. This modulation changes both the shape and frequency range of the allowed frequency bands, thereby bending the resulting semiclassical ray trajectories and confining them within particular regions of the crystal. The curved boundaries of these regions, combined with the bending of the orbits themselves, creates a hierarchy of stable and unstable chaotic trajectories in phase space. For certain lattice parameters and electromagnetic wave frequencies, islands of stable orbits act as a dynamical barrier, which separates the chaotic trajectories into two distinct regions of the crystal, thereby preventing the rays propagating through the structure. We show that changing the frequency of the electromagnetic wave strongly affects the distribution of stable and unstable orbits in both real and phase space. This switches the dynamical barriers on and off and thus modulates the transmission of rays through the crystal. We propose microwave analogs of the photonic crystals as a route to the experimental study of the transport effects that we predict.

Hydrogeophysical imaging of deposit heterogeneity and groundwater chemistry changes during DNAPL source zone bioremediation.

Robust characterization and monitoring of dense nonaqueous phase liquid (DNAPL) source zones is essential for designing effective remediation strategies, and for assessing the efficacy of treatment. In this study high-resolution cross-hole electrical resistivity tomography (ERT) was evaluated as a means of monitoring a field-scale in-situ bioremediation experiment, in which emulsified vegetable oil (EVO) electron donor was injected into a trichloroethene source zone. Baseline ERT scans delineated the geometry of the interface between the contaminated alluvial aquifer and the underlying mudstone bedrock, and also the extent of drilling-induced physical heterogeneity. Time-lapse ERT images revealed major preferential flow pathways in the source and plume zones, which were corroborated by multiple lines of evidence, including geochemical monitoring and hydraulic testing using high density multilevel sampler arrays within the geophysical imaging planes. These pathways were shown to control the spatial distribution of the injected EVO, and a bicarbonate buffer introduced into the cell for pH control. Resistivity signatures were observed within the preferential flow pathways that were consistent with elevated chloride levels, providing tentative evidence from ERT of the biodegradation of chlorinated solvents.

Chaotic electron diffusion through stochastic webs enhances current flow in superlattices.

Understanding how complex systems respond to change is of fundamental importance in the natural sciences. There is particular interest in systems whose classical newtonian motion becomes chaotic as an applied perturbation grows. The transition to chaos usually occurs by the gradual destruction of stable orbits in parameter space, in accordance with the Kolmogorov-Arnold-Moser (KAM) theorem--a cornerstone of nonlinear dynamics that explains, for example, gaps in the asteroid belt. By contrast, 'non-KAM' chaos switches on and off abruptly at critical values of the perturbation frequency. This type of dynamics has wide-ranging implications in the theory of plasma physics, tokamak fusion, turbulence, ion traps, and quasicrystals. Here we realize non-KAM chaos experimentally by exploiting the quantum properties of electrons in the periodic potential of a semiconductor superlattice with an applied voltage and magnetic field. The onset of chaos at discrete voltages is observed as a large increase in the current flow due to the creation of unbound electron orbits, which propagate through intricate web patterns in phase space. Non-KAM chaos therefore provides a mechanism for controlling the electrical conductivity of a condensed matter device: its extreme sensitivity could find applications in quantum electronics and photonics.

Photonic Bloch oscillations and Wannier-Stark ladders in exponentially chirped Bragg gratings.

The formation of photonic Bloch oscillations and Wannier-Stark ladders is demonstrated in an exponentially chirped one-dimensional Bragg grating. The photonic Bloch oscillations are investigated using Hamiltonian optics, and direct analogies are made with electron dynamics in periodic potentials. The results of transfer matrix calculations are presented, which show the existence of a photonic Wannier-Stark ladder that should be detectable in experiments.

Compact fourth-order finite difference method for solving differential equations.

We present a fourth-order finite difference (FD) method for solving two-dimensional partial differential equations. The FD operator uses a compact nine-point stencil on a regular square grid. Despite the regular grid, Dirichlet boundary conditions can be applied on an arbitrarily shaped boundary without resorting to the usual stepped approximation. We demonstrate the superior convergence of the method over second-order techniques by solving the Schrödinger equation for an electron in a semiconductor quantum dot with a smoothly varying potential which generates classically chaotic dynamics.

Effects of geometrical ray chaos on the electromagnetic eigenmodes of a gradient index optical cavity.

Electromagnetic analogies of quantum chaos are investigated in two-dimensional optical cavities which have reflective surfaces and contain a gradient refractive index medium. As the shape of the cavity is transformed continuously from a rectangle to a parallelogram, the geometrical ray paths undergo a transition from stable to chaotic dynamics. In the chaotic regime, the spectral statistics of the cavity are accurately described by random matrix theory. In addition, the electromagnetic mode spectrum of the cavity is modulated by both real and ghost periodic ray paths. These paths also "scar" the electric field intensity distributions of regular subsets of cavity eigenmodes.

Evolution of fractal patterns during a classical-quantum transition.

We investigate how fractals evolve into nonfractal behavior as the generation process is gradually suppressed. Fractals observed in the conductance of semiconductor billiards are of particular interest because the generation process is semiclassical and can be suppressed by transitions towards either fully classical or fully quantum-mechanical conduction. Investigating a range of billiards, we identify a "universal" behavior in the changeover from fractal to nonfractal conductance, which is described by a smooth evolution rather than deterioration in the fractal scaling properties.

Effects of stochastic webs on chaotic electron transport in semiconductor superlattices.

We investigate chaotic electron transport in the lowest miniband of a semiconductor superlattice with a tilted magnetic field. This experimentally accessible non-Kolmogorov-Arnol'd-Moser system involves only stationary electric and magnetic fields, but is dynamically equivalent to a time-dependent kicked harmonic oscillator. The onset of chaos strongly delocalizes the electron orbits, thus raising the electrical conductivity. When the cyclotron and Bloch frequencies are commensurate, the phase space is threaded by a stochastic web, which produces a further resonant increase in the conductivity.

Electromagnetic wave chaos in gradient refractive index optical cavities.

Electromagnetic wave chaos is investigated using two-dimensional optical cavities formed in a cylindrical gradient refractive index lens with reflective surfaces. When the planar ends of the lens are cut at an angle to its axis, the geometrical ray paths are chaotic. In this regime, the electromagnetic mode spectrum of the cavity is modulated by both real and ghost periodic ray paths, which also "scar" the electric field intensity distributions of many modes. When the cavity is coupled to waveguides, the eigenmodes generate complex series of resonant peaks in the electromagnetic transmission spectrum.