Spatially Periodic Inhomogeneous States in a Nonlinear Crystal with a Nonlinear Defect

Spatially periodic inhomogeneous stationary states are shown to exist near a thin defect layer with nonlinear properties separating nonlinear Kerr-type crystals. The contacts of nonlinear self-focusing and defocusing crystals have been analyzed. The spatial field distribution obeys a time-independent nonlinear Schrödinger equation with a nonlinear (relative to the field) potential modeling the thin defect layer with nonlinear properties. Both symmetric and asymmetric states relative to the defect plane are shown to exist. It has been established that new states emerge in a self-focusing crystal, whose existence is attributable to the defect nonlinearity and which do not emerge in the case of a linear defect. The dispersion relations defining the energy of spatially periodic inhomogeneous stationary states have been derived. The expressions for the energies of such states have been derived in an explicit analytical form in special cases. The conditions for the existence of periodic states and their localization, depending on the defect and medium characteristics, have been determined.     

Localization of Excitations near a Thin Structured Spacer between Linear and Nonlinear Crystals

It has been shown that localized and semi-localized stationary states exist near a thin structured defect layer between a linear medium and a Kerr nonlinear medium. Localized states are described by a monotonically decreasing amplitude of the field on the both sides of the interface between the media. Semilocalized states are characterized by the field that has the form of a standing wave in the linear medium and decreases monotonically in the nonlinear medium. Kerr media with self-focusing and defocusing are considered. The proposed model is described by a system of the linear and nonlinear Schrödinger equations with a specific potential simulating a thin structured defect layer. It has been shown that localized and semi-localized states exist in different energy ranges in the case of contact of the linear medium with the self-focusing medium. In the case of contact of the linear medium with the defocusing medium, two types of localized and semi-localized states differing in energy and field profile can exist in different energy ranges. In particular cases, expressions for energies of states of these types have been obtained and conditions of their applicability have been indicated.     

Nonlinear light propagation in the defect band of one-dimensional defective structures

In this paper we numerically investigate the nonlinear propagation of defect state in one-dimensional structures with defects. We investigate the nonlinear transmission spectra and the bistable response for defective structures with different index gradients. The results show that positive Kerr nonlinearity can suppress the Wannier-Stark localization. And the nonlinear response of defect states band exhibits an optical switch behavior, which may be applied to all-optical devices. And the gap solitons from these defect states are presented.     

Nonlinear wave processes in porous waterlike media containing a system of capillaries partially filled with a viscous liquid

A nonlinear dynamic state equation of waterlike porous material that contains a system of cylindrical capillaries partially filled with viscous liquid was received. It is shown that an acoustic nonlinearity of such media contains the relaxation elastic and inelastic components due to the nonlinear dependence of the capillary and viscous pressure in fluid on the capillary diameter. For the medium, theoretical study of such nonlinear phenomena as generation of the second harmonic and a difference frequency wave, self-demodulation of high-frequency pulses as well as the change in the propagation velocity and absorption coefficient of a test wave being under an action of static loading have been carried out. The frequency dependences of medium nonlinearity parameters for these processes were determined.     

Nonlinear lamb waves in a metal plate with defects

Experimental results on the propagation of finite-amplitude Lamb waves in a solid plate made of polycrystalline aluminum alloy with defects are presented. The Lamb waves are recorded and visualized using a scanning laser vibrometer. The dependences of the higher harmonic amplitudes, both averaged over the plate surface and measured at a point far from the defect and at the site of the defect, on the fundamental frequency amplitude are studied. A threshold character of the higher harmonic generation and a power-law behavior of their amplitudes are revealed, the latter feature being unconventional for the nonlinearity associated with the anharmonicity of the crystal lattice of the material. The possibility of locating the distribution of individual defects from the measured spatial distribution of structural nonlinearity in the material under study is experimentally demonstrated. The results of the experiments are explained in terms of the bilinear medium model.     

Localization of Excitations near a Thin Defect Layer with Nonlinear Properties,Separating Linear and Nonlinear Crystals

Technical Physics - It is shown that localized and quasi-local states exist near a thin defect layer with nonlinear properties, separating a linear medium from a Kerr-type nonlinear medium....     

Nonlinear noise waves in soft biological tissues

The study of intense waves in soft biological tissues is necessary both for diagnostics and therapeutic aims. Tissue represents an inherited medium with frequency-dependent dissipative properties, in which waves are described by nonlinear integro-differential equations. The equations for such waves are well known. Their group analysis has been performed, and a number of exact solutions have been found. However, statistical problems for nonlinear waves in tissues have hardly been studied. As well, for medical applications, both intense noise waves and waves with fluctuating parameters can be used. In addition, statistical solutions are simpler in structure than regular solutions; they are useful for understanding the physics of processes. Below a general approach is described for solving nonlinear statistical problems applied to the considered mathematical models of biological tissues. We have calculated the dependences of the intensities of the narrowband noise harmonics on distance. For wideband noise, we have calculated the dependence of the spectral integral intensity on distance. In all cases, wave attenuation is determined both by the specific dissipative properties of the tissue and the nonlinearity of the medium.     

Nonlinear Optical Response of a Fibonacci Multilayer

By studying numerically the light transmitting through a finite nonlinear Fibonacci multilayer, we find the gap-soliton-like phenomenon. The Fibonacci multilayer contains a Kerrtype nonlinearity in one component. When the frequency of incident light is in a stop gap, increasing the incident light intensity will make the system switch from a lower transmission state to a higher transmission state. Transmissivity for some frequencies can reach unit, which shows that they correspond to transparent states. Distribution of the field intensity in the Fibonacci structure has a particular gap-soliton-like shape, which has the multi-carrier wave.     

Switching behavior of a nonlinear Mach-Zehnder interferometer: Saturating nonlinearity

This paper uses the Beam Propagation Method to investigate numerically the switching behavior of a Nonlinear Mach-Zehnder Interferometer (NMZI). A saturating-type nonlinearity has been considered for the present investigations. It is shown that the input versus output characteristics change drastically when a Kerr type nonlinear medium is replaced by a saturating type nonlinear medium. In contrast to an NMZI with Kerr nonlinearity, where only quantitative behavior changes with NMZI length, quantitative as well as qualitative behaviors change in the case of a saturating nonlinearity. We propose an all-optical stabilizer and MZI with stable “ON” and “OFF” states on the basis of our investigation.     

Theoretical Investigation of Light Transmission in a Slab Cavity via Kerr Nonlinearity of Carbon Nanotube Quantum Dot Nanostructure

In this paper, we discuss the transmission properties of weak probe laser field propagate through slab cavity with defect layer of carbon-nanotube quantum dot (CNT-QD) nanostructure. We show that due to spin-orbit coupling, the double electromagnetically induced transparency (EIT) windows appear and the giant Kerr nonlinearity of the intracavity medium can lead to manipulating of transmission coefficient of weak probe light. The thickness effect of defect layer medium has also been analyzed on transmission properties of probe laser field. Our proposed model may be useful for integrated photonics devices based on CNT-QD for applications in all-optical systems which require multiple EIT effect.     

Collisions of Single-Cycle and Subcycle Attosecond Light Pulses in a Nonlinear Resonant Medium

By numerically solving the system of Maxwell–Bloch equations, we have examined theoretically collisions of extremely short single-cycle and unipolar subcycle pulses in a nonlinear resonant medium under conditions that the light interacts coherently with the medium. The dynamics of the electric field of structures of light-induced polarization and inversion difference has been considered in the situation in which pulses are overlapped in the medium. We show that the states of the medium (to the right and to the left of the overlap region of the pulses) may differ. In particular, we show that polarization waves with different characteristics can exist in the regions of the medium that are located on opposite sides of the overlap region of the pulses. These waves travel in different directions and have different spatial frequencies.     

Entangled states of light during its propagation in a transparent medium with cubic nonlinearity

When two coherent waves propagate in a transparent medium with cubic nonlinearity, a nonclassical state appears which is characterized by the quantum correlation of the waves. On the basis of exact solutions to the problem, it was found that such states lead to violation of the separability criterion for continuous variables.     

Role of optical nonlinearity on transverse localization of light in a disordered one-dimensional optical waveguide lattice

We report a detailed numerical investigation on transverse localization of light in a 1D disordered lattice consisting of a large array of coupled waveguides in the presence of nonlinearity in the medium. Our study reveals that the presence of a focusing type of nonlinearity favors faster localization of light while a defocusing type of nonlinearity degrades the quality of localization. It is shown that presence of either of these could over-shadow localization of light unless the strength of disorder is sufficiently strong. Influence of the input beam width on propagation of light in such a disordered nonlinear medium has also been discussed. The present study should be useful in potential applications, in which one could exploit dominance of focusing nonlinearity on transverse localization of light in a disordered medium.     

Nonlinear dynamics of complex hysteretic systems: Oscillator in a magnetic field

Complex hysteresis is a well-known phenomenon in many branches of science. The most prominent examples come from materials with a complex microscopic structure such as magnetic materials, shape-memory alloys, or, porous materials. Their hysteretic behavior is characterized by the existence of multiple internal system states for a given external parameter and by a non-local memory. The input-output behavior of such systems is well studied and in a standard phenomenological approach described by the so-called Preisach operator. What is not well understood, are situations, where such a hysteretic system is dynamically coupled to its environment. Since the hysteretic sub-system provides a complicated form of nonlinearity, one expects non-trivial, possibly chaotic behavior of the combined dynamical system. We study such a combined dynamical system with hysteretic nonlinearity. In this original contribution a simple differential-operator equation with hysteretic damping, which describes a magnetic pendulum is considered. We find, for instance, a fractal dependence of the asymptotic behavior as function of the starting values. The sensitivity of the system to perturbations is investigated by several methods, such as the 0–1 test for chaos and sub-Lyapunov exponents. The power spectral density is also calculated and compared with analytical results for simple input-output scenarios.     

Nonlinear model of a granulated medium containing viscous fluid layers and gas cavities

We analyze nonlinear oscillations and waves in a simple model of a granular medium containing inclusions in the form of fluid layers and gas cavities. We show that in such a medium, the velocity of one of the wave modes is low; therefore, the nonlinearity is high and the effects of interaction are more strongly expressed than usual.     

Axisymmetric Nonlinear Modulated Waves in a Cylindrical Shell

A quasi-hyperbolic equation is derived that simulates the axisymmetric propagation of bending waves in a cylindrical shell, which interacts with a nonlinearly elastic medium. With the correct asymptotic procedure, the study of a wave process reduces to analysis of a nonlinear Schrödinger equation. It is established that the development of modulation instability requires a “soft” nonlinearity of the medium surrounding the shell. Operating modes that allow the propagation of stable light envelope solitons are revealed.     

Nonlinear TE-polarized quasi-surface waves in a symmetric optical waveguide with a nonlinear core

A theory of nonlinear polarized symmetric quasi-surface waves in a symmetric planar structure with a nonlinear core and linear coatings has been proposed. The nonlinearity of the core is caused by exciton-photon interaction and optical exciton-biexciton conversion. The dispersion laws for propagating waves have been obtained and studied.     

Nonlinearity of solids with micro- and nanodefects and characteristic features of its macroscopic manifestations

The meaning of the experimentally measured nonlinear parameters of a medium is discussed. The difference in meaning between the local nonlinearity, which is measured in the vicinity of a single defect and depends on the size of the region of averaging, and the effective volume nonlinearity of the medium containing numerous defects is emphasized. The local nonlinearity arising at the tip of a crack is calculated; this non-linearity decreases with an increase in the region of second harmonic generation. The volume nonlinearity is calculated for a solid containing spherical cavities. The volume nonlinearity is also calculated for a medium containing infinitely thin cracks in the form of circular disks, which assume the shape of ellipsoids in the course of the crack opening. The nonlinear acoustic parameter is calculated with the use of the exact classical results of the theory of cracks.     

The role of power law nonlinearity in the discrete nonlinear Schrödinger equation on the formation of stationary localized states in the Cayley tree

We study the formation of stationary localized states using the discrete nonlinear Schr?dinger equation in a Cayley tree with connectivity K. Two cases, namely, a dimeric power law nonlinear impurity and a fully nonlinear system are considered. We introduce a transformation which reduces the Cayley tree into an one dimensional chain with a bond defect. The hopping matrix element between the impurity sites is reduced by . The transformed system is also shown to yield tight binding Green's function of the Cayley tree. The dimeric ansatz is used to find the reduced Hamiltonian of the system. Stationary localized states are found from the fixed point equations of the Hamiltonian of the reduced dynamical system. We discuss the existence of different kinds of localized states. We have also analyzed the formation of localized states in one dimensional system with a bond defect and nonlinearity which does not correspond to a Cayley tree. Stability of the states is discussed and stability diagram is presented for few cases. In all cases the total phase diagram for localized states have been presented. Received: 18 September 1997 / Revised: 31 October and 17 november 1997 / Accepted: 19 November 1997