One possible formulation of a variational principle of the Fermat type for systems with time-dependent parameters is suggested. In a stationary case, it reduces to the Mopertui-Lagrange least-action principle. A class of Hamiltonians (dispersion relations) is indicated, for which the variational principle reduces to the Fermat principle in a general nonstationary case. Hamiltonians that are homogeneous functions of momenta are in this category. For the important case of nondispersive waves (corresponding to Hamiltonians being homogeneous function of momenta order 1) the Fermat principle fully determines the geometry of the rays. Equations relating the variation of signal frequency with the rate of change of propagation time are established.

A novel method is introduced to estimate surface coverage and the equation of state of insoluble surfactant on droplets, involving measurement of interfacial tension on a single parent drop and progressively subdivided generations of daughter drops. This has enabled quantitative decomposition of the dilution, tip-stretching, and Marangoni effects of surfactants on drop deformation. For a small viscosity ratio of 0.09, the Marangoni effect dominates, increasing first and then decreasing with surface coverage, the dilution effect is significant at high, and tip-stretching only at low surface coverage. For a viscosity ratio of 2.3, the dilution effect dominates, and neither Marangoni nor tip-stretching effects play an important role.

The ability to achieve near lossless coupling between a waveguide and a resonator is fundamental to many quantum-optical studies as well as to practical applications of such structures. The nature of loss at the junction is described by a figure of merit called ideality. It is shown here that under appropriate conditions ideality in excess of $99.97\%$ is possible using fiber-taper coupling to high-$Q$ silica microspheres. To verify this level of coupling, a technique is introduced that can both measure ideality over a range of coupling strengths and provide a practical diagnostic of parasitic coupling within the fiber-taper-waveguide junction.

Engineering of stop gaps between higher photonic bands provides an alternative to miniaturization of photonic crystals. Femtosecond laser microfabrication of highly correlated void channel polymer microstructures results in photonic crystals with large stop gaps and a multitude of higher-order gaps in the mid- and near-infrared spectral regions. The gap wavelengths obey Bragg’s law. Consistent with theory, varying the woodpile structure unit cell allows for tuning the number of higher-order gaps, and transitions from mere resonant Bragg scattering to stop band total reflection are observed.

We describe observations of the amplitude and phase of an electric field diffusing through a three-dimensional random medium, using terahertz time-domain spectroscopy. These measurements are spatially resolved with a resolution smaller than the speckle spot size and temporally resolved with a resolution better than one optical cycle. By computing correlation functions between fields measured at different positions and with different temporal delays, it is possible to obtain information about individual scattering events experienced by the diffusing field. This represents a new method for characterizing a multiply scattered wave.

A new type of synchronization, on-off collective imperfect phase synchronization, is found in a turbulent state. In the driver frame the nonlinear wave system can be transformed to a set of coupled oscillators moving in a potential related to the unstable steady wave. In “on” stages the oscillators in different spatial scales adjust themselves to collective imperfect phase synchronization, inducing strong bursts in the wave energy. The interspike intervals display a power-law distribution. In addition to the embedded saddle point, it is emphasized that the delocalization of the master mode also plays an important role in developing the on-off synchronization.

We report the surprising observation of directional tunneling escape from nearly spherical fused-silica optical resonators, in which most of the phase space is filled with nonchaotic regular trajectories. Experimental and theoretical studies of the dependence of the far-field emission pattern on both the degree of deformation and the excitation condition show that nonperturbative phase-space structures in the internal ray dynamics profoundly affect tunneling leakage of the whispering-gallery modes.

In this Letter we introduce a generalization of the Lorenz dynamical system using fractional derivatives. Thus, the system can have an effective noninteger dimension $\Sigma$ defined as a sum of the orders of all involved derivatives. We found that the system with $\Sigma <3$ can exhibit chaotic behavior. A striking finding is that there is a critical value of the effective dimension ${\Sigma }_{\mathrm{c}\mathrm{r}}$, under which the system undergoes a transition from chaotic dynamics to regular one.

A hybrid-aligned cell of the smectic $A$ liquid material 8CB gives two stable director configurations, one of which is periodic and gives strong diffraction of light. This photonic lattice director profile arises from “frustration” caused by conflicting constraints imposed by the boundary conditions and the constant amplitude smectic density wave. A model is proposed which accords well with the experiment results, predicting correctly the dependence of the periodicity on the cell thickness and reproducing optical polarization microscopy results.

We investigate theoretically and numerically the effect of polymer additives on two-dimensional turbulence by means of a viscoelastic model. We provide compelling evidence that, at vanishingly small concentrations, such that the polymers are passively transported, the probability distribution of polymer elongation has a power law tail: Its slope is related to the statistics of finite-time Lyapunov exponents of the flow, in quantitative agreement with theoretical predictions. We show that at finite concentrations and sufficiently large elasticity the polymers react on the flow with manifold consequences: Velocity fluctuations are drastically depleted, as observed in soap film experiments; the velocity statistics becomes strongly intermittent; the distribution of finite-time Lyapunov exponents shifts to lower values, signaling the reduction of Lagrangian chaos.

We study the effect of noncoherence on the onset of phase synchronization of two coupled chaotic oscillators. Depending on the coherence properties of oscillations characterized by the phase diffusion, three types of transitions to phase synchronization are found. For phase-coherent attractors this transition occurs shortly after one of the zero Lyapunov exponents becomes negative. At rather strong phase diffusion, phase locking manifests a strong degree of generalized synchronization, and occurs only after one positive Lyapunov exponent becomes negative. For intermediate phase diffusion, phase synchronization sets in via an interior crises of the hyperchaotic set.

Using a symmetry-based approach, we have designed polarization-independent waveguides in a 3D photonic crystal. A comprehensive series of numerical experiments, involving the propagation of pulsed signals through long straight waveguide sections and sharp bends, quantitatively evaluates the bend-transmission coefficient over the entire bandwidth of the corresponding guided modes. High ($\sim 95\%$) polarization-independent bend transmission is achieved within a certain frequency range.

Near-resonant conditions lead to a strong interaction between saddle node and modulational instabilities in diffractive optical frequency down-conversion systems. The study of such an interaction reveals a new type of stable phase-locked mixed-mode hexagonal structure below the lasing threshold. Without this interaction, hexagonal structures do not exist. The analytical expression of the threshold associated with these structures is derived. Numerical solutions of the governing equations are in close agreement with analytical predictions.

Monodisperse granular flows often develop regions with hexagonal close packing of particles. We investigate this effect in a system of inelastic hard spheres driven from below by a “thermal” plate. Molecular dynamics simulations show, in a wide range of parameters, a close-packed cluster supported by a low-density region. Surprisingly, the steady-state density profile, including the close-packed cluster part, is well described by a variant of Navier-Stokes granular hydrodynamics (NSGH). We suggest a simple explanation for the success of NSGH beyond the freezing point.

Small-world and scale-free networks are known to be more easily synchronized than regular lattices, which is usually attributed to the smaller network distance between oscillators. Surprisingly, we find that networks with a homogeneous distribution of connectivity are more synchronizable than heterogeneous ones, even though the average network distance is larger. We present numerical computations and analytical estimates on synchronizability of the network in terms of its heterogeneity parameters. Our results suggest that some degree of homogeneity is expected in naturally evolved structures, such as neural networks, where synchronizability is desirable.

Effects of interstitial air on the motions of a large intruder in a shaken granular bed are studied experimentally as a function of ambient air pressure, particle size of the bed, and the density of the intruder. It is found that the intruder always rises from the granular bed in the absence of air. However, the intruder can acquire both positive and negative buoyancy in the presence of air. Negative buoyancy can be observed only when both the density of the intruder and the particle size of the bed are small enough. This negative buoyancy can be explained by the unusual air pressure distribution found in the bed.

Experimental observation of a new mechanism of sandpile formation is reported. As a steady stream of dry sand is poured onto a horizontal surface, a pile forms which has a thin river of sand on one side flowing from the apex of the pile to the edge of its base. The river rotates about the pile, depositing a new layer of sand with each revolution, thereby causing the pile to grow. For small piles the river is steady and the pile formed is smooth. For larger piles, the river becomes intermittent and the surface of the pile becomes undulating. The essential features of the system that produce the phenomenon are discussed, and the robustness of the phenomena is demonstrated with experiments using different boundary conditions and sands.

Using a refractive index matching method, we investigate the trajectories of particles in three dimensional granular packing submitted to cyclic shear deformation. The particle motion observed during compaction is not diffusive but exhibits a transient cage effect, similar to the one observed in colloidal glasses. We precisely study the statistics of the step size between two successive cycles and observe that it is proportional to the shear amplitude. The link between the microscopic observations and the macroscopic evolution of the volume fraction during compaction is discussed.

We have observed for the first time stable spatial solitons in semiconductor optical amplifiers. Soliton destabilization due to the growth of background noise was suppressed by using patterned electrodes on the device. Numerical simulations fit very well with the experiment results. We show that it is possible to excite these solitons with about 60 mW input power.

We study the temperature dependence of diffuse reverberant ultrasound in elastic bodies. Transient wave forms are found to undergo an almost pure dilation of 0.0277% per degree, related to the temperature dependence of wave speeds. The wave forms also suffer a distortion that, we argue, depends on the rate of conversion between the dilatational ($P$) and shear ($S$) waves. Distortion is found to scale in a manner consistent with theoretical arguments but also appears to be a function of the degree of ray chaos in the body, indicating that the mixing rates are slower in more regular bodies.