Explicit Formulas for Relaxed Disarrangement Densities Arising from Structured Deformations

Structured deformations provide a multiscale geometry that captures the contributions at the macrolevel of both smooth geometrical changes and non-smooth geometrical changes (disarrangements) at submacroscopic levels. For each (first-order) structured deformation $(g,G)$ of a continuous body, the tensor field $G$ is known to be a measure of deformations without disarrangements, and $M:=\nabla g-G$ is known to be a measure of deformations due to disarrangements. The tensor fields $G$ and $M$ together deliver not only standard notions of plastic deformation, but $M$ and its curl deliver the Burgers vector field associated with closed curves in the body and the dislocation density field used in describing geometrical changes in bodies with defects. Recently, Owen and Paroni [13] evaluated explicitly some relaxed energy densities arising in Choksi and Fonseca's energetics of structured deformations [4] and thereby showed: (1) $(trM)^{+}$, the positive part of $trM$, is a volume density of disarrangements due to submacroscopic separations, (2) $(trM)^{-}$, the negative part of $trM$, is a volume density of disarrangements due to submacroscopic switches and interpenetrations, and (3) $|trM|$, the absolute value of $trM$, is a volume density of all three of these non-tangential disarrangements: separations, switches, and interpenetrations. The main contribution of the present research is to show that a different approach to the energetics of structured deformations, that due to Ba\'ia, Matias, and Santos [1], confirms the roles of $(trM)^{+}$, $(trM)^{-}$, and $|trM|$ established by Owen and Paroni. In doing so, we give an alternative, shorter proof of Owen and Paroni's results, and we establish additional explicit formulas for other measures of disarrangements.Comment: 17 pages; http://cvgmt.sns.it/paper/2776

High-Order-Mode Soliton Structures in Two-Dimensional Lattices with Defocusing Nonlinearity

While fundamental-mode discrete solitons have been demonstrated with both self-focusing and defocusing nonlinearity, high-order-mode localized states in waveguide lattices have been studied thus far only for the self-focusing case. In this paper, the existence and stability regimes of dipole, quadrupole and vortex soliton structures in two-dimensional lattices induced with a defocusing nonlinearity are examined by the theoretical and numerical analysis of a generic envelope nonlinear lattice model. In particular, we find that the stability of such high-order-mode solitons is quite different from that with self-focusing nonlinearity. As a simple example, a dipole (``twisted'') mode soliton which may be stable in the focusing case becomes unstable in the defocusing regime. Our results may be relevant to other two-dimensional defocusing periodic nonlinear systems such as Bose-Einstein condensates with a positive scattering length trapped in optical lattices.Comment: 14 pages, 10 figure

Effect of Nonlinearity on Adiabatic Evolution of Light

We investigate the effect of nonlinearity in a system described by an adiabatically evolving Hamiltonian. Experiments are conducted in a three-core waveguide structure that is adiabatically varying with distance, in analogy to the stimulated Raman adiabatic passage process in atomic physics. In the linear regime, the system exhibits an adiabatic power transfer between two waveguides which are not directly coupled, with negligible power recorded in the intermediate coupling waveguide. In the presence of nonlinearity the adiabatic light passage is found to critically depend on the excitation power. We show how this effect is related to the destruction of the dark state formed in this configuration

Realization of quantum walks with negligible decoherence in waveguide lattices

Quantum random walks are the quantum counterpart of classical random walks, and were recently studied in the context of quantum computation. Physical implementations of quantum walks have only been made in very small scale systems severely limited by decoherence. Here we show that the propagation of photons in waveguide lattices, which have been studied extensively in recent years, are essentially an implementation of quantum walks. Since waveguide lattices are easily constructed at large scales and display negligible decoherence, they can serve as an ideal and versatile experimental playground for the study of quantum walks and quantum algorithms. We experimentally observe quantum walks in large systems (similar to 100 sites) and confirm quantum walks effects which were studied theoretically, including ballistic propagation, disorder, and boundary related effects

New CMOS Compatible Platforms for Integrated Nonlinear Optical Signal Processing

Nonlinear photonic chips have succeeded in generating and processing signals all-optically with performance far superior to that possible electronically - particularly with respect to speed. Although silicon-on-insulator has been the leading platform for nonlinear optics, its high two-photon absorption at telecommunications wavelengths poses a fundamental limitation. This paper reviews some of the recent achievements in CMOS-compatible platforms for nonlinear optics, focusing on amorphous silicon and Hydex glass, highlighting their potential future impact as well as the challenges to achieving practical solutions for many key applications. These material systems have opened up many new capabilities such as on-chip optical frequency comb generation and ultrafast optical pulse generation and measurement.Comment: 8 pages, 10 figures 80 references. arXiv admin note: substantial text overlap with arXiv:1404.561

Phase-Insensitive Scattering of Terahertz Radiation

The nonlinear interaction between Near-Infrared (NIR) and Terahertz pulses is principally investigated as a means for the detection of radiation in the hardly accessible THz spectral region. Most studies have targeted second-order nonlinear processes, given their higher efficiencies, and only a limited number have addressed third-order nonlinear interactions, mainly investigating four-wave mixing in air for broadband THz detection. We have studied the nonlinear interaction between THz and NIR pulses in solid-state media (specifically diamond), and we show how the former can be frequency-shifted up to UV frequencies by the scattering from the nonlinear polarisation induced by the latter. Such UV emission differs from the well-known electric-field-induced second harmonic (EFISH) one, as it is generated via a phase-insensitive scattering, rather than a sum- or difference-frequency four-wave-mixing process

Design and Fabrication of Terahertz Metallic Gratings on a Two-Wire Waveguide

In this study, we present the design, fabrication and experimental characterization of waveguide-integrated gratings operating at THz frequencie

Dimension Reduction in the Context of Structured Deformations

In this paper we apply both the procedure of dimension reduction and the incorporation of structured deformations to a three-dimensional continuum in the form of a thinning domain. We apply the two processes one after the other, exchanging the order, and so obtain for each order both a relaxed bulk and a relaxed interfacial energy. Our implementation requires some substantial modifications of the two relaxation procedures. For the specific choice of an initial energy including only the surface term, we compute the energy densities explicitly and show that they are the same, independent of the order of the relaxation processes. Moreover, we compare our explicit results with those obtained when the limiting process of dimension reduction and of passage to the structured deformation is carried out at the same time. We finally show that, in a portion of the common domain of the relaxed energy densities, the simultaneous procedure gives an energy strictly lower than that obtained in the two-step relaxations

Hanbury Brown and Twiss Correlations of Anderson Localized Waves

When light waves propagate through disordered photonic lattices, they can eventually become localized due to multiple scattering effects. Here we show experimentally that while the evolution and localization of the photon density distribution is similar in the two cases of diagonal and off-diagonal disorder, the density-density correlation carries a distinct signature of the type of disorder. We show that these differences reflect a symmetry in the spectrum and eigenmodes that exists in off-diagonally disordered lattices but is absent in lattices with diagonal disorder.Comment: 4 pages, 3 figures, comments welcom