A Reply to the Comment by Mischa Megens, et al.

We develop a perturbation theory for the lifetime and emission intensity for isolated resonances in asymmetric resonant cavities. The inverse lifetime $\Gamma$ and the emission intensity $I\left(\theta \right)$ in the open system are expressed in terms of matrix elements of operators evaluated with eigenmodes of the closed resonator. These matrix elements are calculated in a semiclassical approximation which allows us to represent $\Gamma$ and $I\left(\theta \right)$ as sums over the contributions of rays which escape the resonator by refraction.

By using an order parameter description of the type-II second harmonic generation, we prove analytically that different 3D periodic solutions exist and have overlapping domains of stability. The stable periodic solutions are lamellae, 3D hexagons, and a body-centered cubic pattern. The coexistence of periodic and homogeneous solutions leads to localized structures such as light drops and cylinders which are found numerically.

We introduce and investigate a simple model to describe recent experiments by Douady and Daerr on flowing sand. The model reproduces experimentally observed compact avalanches, whose opening angle decreases linearly as a threshold is approached. On large scales the model exhibits a crossover from compact directed percolation to directed percolation; we predict similar behavior for the experimental system. We estimate the regime where “true” directed percolation morphology and exponents will be observed, providing the first experimental realization for this class of models.

Molecular dynamics simulations of the inelastic hard sphere model for granular media have been done to study the heat conduction between two parallel plates. The results show that Fourier's law is not valid and a new term proportional to the density gradient must be added to compute the heat flux. The new transport coefficient associated with the density gradient dependence has been measured vanishing in the case of elastic collisions.

We study the statics of an elastic sphere held in a rigid groove by gravity and friction, as determined by contact mechanics. As a function of the groove's opening angle and its tilt with respect to the vertical, we identify two regimes of static equilibrium for the ball. At large opening angle or small tilt, the ball rolls at both contacts as it is loaded. This is an analog of the “elastic” regime in the mechanics of granular media. At small opening angle or large tilt, the ball rolls at one contact and slides at the other as it is loaded, analogously with the “plastic” regime in the mechanics of granular media. In the elastic regime, the stress indeterminacy is resolved by the kinetics of the response to loading.

Turbulent transport of passive scalars by random wave fields is studied, with applications to statistics of chlorophyll concentration in the ocean. The existence of the small parameter $u_0/c_0$, where $u_0$ and $c_0$ are the characteristic particle velocity and wave phase speed, respectively, allows essentially exact calculations, and as such provides a rich testing ground for quantitative comparisons between theory and observation. General expressions are derived for the diffusion constant and mean drift velocity. It is shown that the spectrum of passive scalar fluctuations displays at least two distinct inertial range power laws even when the wave velocity field has only one.

We present numerical evidence which strongly suggests that irrational triangular billiards (all angles irrational with $\pi$) are mixing. Since these systems are known to have zero Kolmogorov-Sinai entropy, they may play an important role in understanding the statistical relaxation process.

A monochromatic point source, embedded in a three-dimensional disordered medium, is considered. The resulting intensity pattern exhibits a new type of long-range correlation. The range of these correlations is infinite and their magnitude, normalized to the average intensity, is of order $1/k_0\ell$, where $k_0$ and $\ell$ are the wave number and the mean free path, respectively.

We analyze multisoliton complexes and their dynamics in Kerr-like nonlinear media. The field in each of $M$ incoherently interacting components is calculated using an integrable set of coupled nonlinear Schrödinger equations. We obtain a general $N$-soliton solution describing propagation of multisoliton complexes and their collisions. The evolution of such higher-order soliton beams is determined by coherent and incoherent contributions from fundamental solitons. These internal interactions are revealed and illustrated by numerical examples. This theory can be applied to describe the properties of incoherent solitons in a photorefractive medium since these are, essentially, multisoliton complexes.

We analyze nonlinear wave propagation and cascaded self-focusing due to second-harmonic generation in Fibbonacci optical superlattices and introduce a novel concept of nonlinear physics, the quasiperiodic soliton, which describes spatially localized self-trapping of a quasiperiodic wave. We point out a link between the quasiperiodic soliton and partially incoherent spatial solitary waves recently generated experimentally.

We study the passage of monochromatic, cw laser light through a cell of homeotropically aligned nematic liquid crystal at a slightly oblique angle, polarized perpendicular to the plane of incidence. Experiments in this geometry have revealed complex, time-dependent dynamics of the director motion. We derive a model for the director dynamics that shows both periodic and chaotic behavior at different light intensities as observed in the experiments. It follows an uncommon route to chaos via gluing bifurcations which has not yet been observed in any real physical system.

We show that a combined medium with fast nonlinear response and plasma dispersion law can provide compression of an initially quasimonochromatic pulse below its carrier period. Solitonlike propagation for pulses with duration as short as one field oscillation is predicted. We propose that these phenomena can be observed in modern high-intensity guiding systems.

We report the observation of optical Bloch oscillations in waveguide arrays. The required linear variation of the propagation constant across the thermo-optic polymer array was obtained by applying a temperature gradient. Bloch oscillations manifesting themselves as transverse oscillations of the propagating light beam can be attributed to the existence of localized states (Wannier-Stark states) with equidistant eigenvalue spacing (Wannier-Stark ladder). The period and amplitude of the oscillations can be controlled by varying the temperature gradient.

We experimentally demonstrate the occurrence of optical Bloch oscillations in a waveguide array with linearly growing effective index of the individual guides. We monitored the output profiles for varying propagation lengths and observed a periodic transverse motion of the field and a complete recovery of the initial excitation. The action of the focusing nonlinearity leads to a loss of recovery, symmetry breaking, and power-induced beam spreading.

Direct measurements of the velocity in turbulent convection of ${\mathrm{SF}}_6$ near its gas-liquid critical point by light scattering on the critical density fluctuations were conducted. The temperature, velocity, and cross frequency power spectra in a wide range of the Rayleigh and Prandtl numbers show scaling behavior with indices, which are rather close to the Bolgiano-Obukhov scaling in the wave number domain. The statistics of the velocity fluctuations remain Gaussian up to the Reynolds numbers of ${10}^5$.

We establish that the entropy production rate of a classically chaotic Hamiltonian system coupled to the environment settles, after a transient, to a metastable value given by the sum of positive generalized Lyapunov exponents. A metastable steady state is generated in this process. This behavior also occurs in quantum systems close to the classical limit, where it leads to the restoration of quantum-classical correspondence in chaotic systems coupled to the environment.

Strange nonchaotic attractors (SNAs) can be created due to the collision of an invariant curve with itself. This novel “homoclinic” transition to SNAs occurs in quasiperiodically driven maps which derive from the discrete Schrödinger equation for a particle in a quasiperiodic potential. In the classical dynamics, there is a transition from torus attractors to SNAs, which, in the quantum system, is manifest as the localization transition. This equivalence provides new insight into a variety of properties of SNAs, including its fractal measure. Further, there is a symmetry breaking associated with the creation of SNAs which rigorously shows that the Lyapunov exponent is nonpositive. We show that these characteristics associated with the appearance of SNA are robust and occur in a large class of systems.

We report the first numerical analysis of the radiation force exerted on a subwavelength particle by an evanescent field localized near a subwavelength aperture. The radiation force is calculated by using the Maxwell stress tensor through the electric field distribution obtained with the finite-difference time-domain method. The result indicates that a particle moves towards the aperture. We found that if two particles exist the first particle is trapped and the second one is also attracted to the first one. The radiation force is found to be larger than the forces due to thermal fluctuations and to gravity.

High-energy electrons have been trapped in an intense, single-beam, ponderomotive-optical trap. Thomson-scattered light from the center of a tightly focused pulsed laser beam is enhanced when a novel trapping focus is used. The observed spatial distribution and energy dependence of the scattered light agree with the results of computer simulations for ordinary and trapping beams.