Fifth-order Nonlinearity Research Articles

Z-scan investigation of fifth-order optical nonlinearity induced by saturable-absorption from (TBA)2Ni(dmit)2: application for optical limiting

The Z-scan technique has been successfully used to determine the magnitudes and signs of the fifth-order optical nonlinearity induced by two-photon absorption. Here, it was used to investigate the fifth-order optical nonlinearity induced by saturable-absorption (SA) of a charge-transfer salt (TBA)2Ni(dmit)2 (TBA = tetra-n-butylammonium, dmit = 2-thioxo-1,3-dithiole-4,5-dithiolate) with 1064 nm pulses. Under excitation by these 1064 nm pulses, the titled molecule exhibits a typical SA effect at relatively low levels of intensity, while showing strong excited-state absorption (including the first and second excited-state) induced by SA at high intensity levels. Such an excited-state absorption is related to the fifth-order effect. The magnitudes and signs of the fifth- and third-order nonlinearities were estimated. Additionally, the optical limiting effect of this molecule in a dimethyl sulfoxide solution was also studied.

Analytic solution of the system of two coupled differential equations with the fifth-order non-linearity

In this paper an approximate method for solving a system of two coupled second-order non-linear differential equation with strong and weak non-linearity is developed. The strong non-linearities are of the third and of the fifth order. The suggested new method is partially based on the elliptic-Krylov–Bogolubov method and on the power series method developed for solving the weak non-linear differential equations. The one-frequency solution is considered for the case of certain relations between the equation parameters p and m, and initial conditions. As a special case the oscillatory motion of the physical system described with a fourth-order strong non-linear differential equation is considered. A wide analysis of the corresponding equation is done. The results of the suggested analytic procedure are compared with numerical ones. They show a good agreement.

Large high-order nonlinear phase shifts produced by χ(2) cascaded processes

Based on the deep understanding of χ (2) cascading nonlinearity, a configuration of phase-mismatched frequency tripling is proposed to produce high-order nonlinear phase shifts. In this process, induced effective fifth-order nonlinearity dominates, which is proportional to the square of input fundamental intensity in the limits of large phase-mismatch and negligible pump depletion in the first crystal. Compared with the conventional induced effective third-order nonlinearity, theoretical study and numerical simulations show that the induced fifth-order nonlinear phase shifts can be much larger in magnitude with a controllable sign and a richer saturable feature.

Third- and fifth-order optical nonlinearities in a new stilbazolium derivative

The third- and fifth-order optical nonlinear properties of a new stilbazolium derivative, trans-1-[p-(p-dimethylaminobenzyl-azo)-benzyl]-2-(N-methyl-4-pyridinium)-ethene iodide (DABAZO-PEI), were investigated with a Q-switched and mode-locked pulse laser with a duration of 35 ps at 1064 nm. Its linear optical properties were also discussed. DABAZO-PEI exhibits two-photon absorption and strong excited-state absorption induced by two-photon absorption. A positive third-order nonlinear refraction caused by bound electrons and negative excited-state refraction caused by the two-photon absorption-generated excited states were also observed. The magnitude and the sign of the real (refractive) and the imaginary (two-photon absorption) parts of the molecular third-order polarizability, and the excited-state absorption and refraction cross sections, were measured. A feature worth noting is the values of excited-state absorption and refraction cross sections. They are all as high as 10-17 cm2. Based on the excited-state absorption induced by the two-photon-absorption mechanism, a highly efficient optical-limiting behavior for incident fluence in the range of 0–10 J/cm2 was demonstrated in a 1-cm-path quartz cell containing DABAZO-PEI solution in dimethyl sulfoxide. The optical-limiting threshold was estimated to be as low as 5 mJ/cm2.

Measurements of intensity fluctuations in a laser with a saturable absorber

Fluctuations of light intensity in a laser with an intracavity nonlinear absorber are studied experimentally. Measurements are made close to the tricritical point by varying the operating parameters of the laser. Experimental results are in agreement with the theoretical predictions based on the nonlinear oscillator model of the laser in which both the third- and fifth-order nonlinearities in the field amplitude are retained.

Role of femtosecond pulses in distinguishing third- and fifth-order nonlinearity for semiconductor-doped glasses

We report Z-scan experiments at a frequency below the band gap of semiconductor-doped glasses using 100 fs laser pulses. A large contribution from the positive third-order nonlinearity is observed in complete contrast with the picosecond z-scan experiments. The third- and fifth-order nonlinearity for semiconductor-doped glasses can be estimated from z-scans for ps and fs pulses.

Faraday resonance in two-dimensional standing gravity waves of finite depth

Faraday waves arise on the surface of a liquid in a container that is undergoing vertical periodic oscillations. The motion of two resonant modes with natural frequencies in the approximate ratio 3:1 is considered in this paper. We derive the nonlinear evolution equations of the dominant free-surface modes up to fifth-order in wave amplitude. It is found that the third-order evolution equations are integrable while the fifth-order evolution equations are chaotic, which shows that the fifth-order nonlinearity breaks the integrability of the third-order integrable system.

The high fifth-order nonlinearity in a new stilbazolium derivative: trans-1-[ p-( p-dimethylaminobenzyl-azo)-benzyl]-2-( N-methyl-4-pyridinium)-ethene iodide

The third- and fifth-order optical nonlinearities from the titled molecule were investigated using Z-scan technique. The titled molecule exhibits two-photon absorption (2PA) and 2PA-generated excited-state absorption (ESA). A very interesting feature is that the titled molecule shows a high fifth-order nonlinearity, being about one-order of magnitude larger than the reported organic molecules. Due to this strong ESA, a highly efficient optical limiting effect was observed.

Dynamics of the divergent dark and bright solitonic channels evolved from a dark solitonlike beam in a medium with competing third- and fifth-order nonlinearities

A theoretical study has been made of the dynamics of the divergent dark and bright solitonic channels that have evolved from a dark solitonlike beam in a self-defocusing medium with measurable fifth-order focusing nonlinearity. During perturbation the incident dark solitonlike beam with appropriate parameters is deformed and evolves into a dark solitonic channel, emitting additional radiation from which a bright companion channel is born. Global propagation becomes stable and falls into a pattern of the divergent dark and bright solitonic channels with desirable built-in mutual support by way of the background that typifies the counterbalance between diffraction of the solitonic channels and the joint effect of the competing nonlinearities. At this stage the energy redistribution among the solitonic channels and the background is retained up to a dynamic equilibrium, and the overall transverse momentum is conserved.

Analytic solution of the wave equation describing dispersion-free propagation of a femtosecond laser pulse in a medium with a cubic and fifth-order nonlinearity

In this paper an analytic solution of the wave equation describing the propagation of a femtosecond laser pulse with a Gaussian temporal profile in a nonlinear dispersion-free instant-response medium with a cubic and fifth-order nonlinearity is presented. The solution is obtained from the nonlinear wave equation by using the method of unidirectional waves, and takes into account the wave reflected at the input of the medium. In particular, the solution obtained describes propagation of a femtosecond laser pulse with a Gaussian temporal profile in a nonlinear dispersive-free instantaneous-response medium considered in the approximation of a weak nonlinearity.

Absorption Signal of UV-Grade Fused Silica Contains Third- and Fifth-Order Nonlinearities

Absorption Signal of UV-Grade Fused Silica Contains Third- and Fifth-Order Nonlinearities

An adaptive multimodal approach to nonlinear sloshing in a rectangular tank

Two-dimensional nonlinear sloshing of an incompressible fluid with irrotational flow in a rectangular tank is analysed by a modal theory. Infinite tank roof height and no overturning waves are assumed. The modal theory is based on an infinite-dimensional system of nonlinear ordinary differential equations coupling generalized coordinates of the free surface and fluid motion associated with the amplitude response of natural modes. This modal system is asymptotically reduced to an infinite-dimensional system of ordinary differential equations with fifth-order polynomial nonlinearity by assuming sufficiently small fluid motion relative to fluid depth and tank breadth. When introducing inter-modal ordering, the system can be detuned and truncated to describe resonant sloshing in different domains of the excitation period. Resonant sloshing due to surge and pitch sinusoidal excitation of the primary mode is considered. By assuming that each mode has only one main harmonic an adaptive procedure is proposed to describe direct and secondary resonant responses when Moiseyev-like relations do not agree with experiments, i.e. when the excitation amplitude is not very small, and the fluid depth is close to the critical depth or small. Adaptive procedures have been established for a wide range of excitation periods as long as the mean fluid depth h is larger than 0.24 times the tank breadth l. Steady-state results for wave elevation, horizontal force and pitch moment are experimentally validated except when heavy roof impact occurs. The analysis of small depth requires that many modes have primary order and that each mode may have more than one main harmonic. This is illustrated by an example for h/l = 0.173, where the previous model by Faltinsen et al. (2000) failed. The new model agrees well with experiments.

Nonlinear Seismic Effects in Soils: Numerical Simulation and Study

To study the effects of nonlinearity in the seismic response of soils, a numerical simulation of the propagation of vertically incident seismic waves in hor- izontal soil layers were performed. Shear noiselike and monochromatic seismic waves of various intensities were used as input signals. The behavior of soils was described by a nonlinear hysteretic model. To extract and study nonlinear compo- nents in the ground response, the nonlinear system identification method and analysis of higher-order spectra of oscillations on the surface were applied. Even for weak input signals, the response of the simulated soils contained a noticeable nonlinear component. An increase in the intensity of input signals led to increasing distortions of propagating signals, due to the generation and growth of combination-frequency harmonics. The results show that odd types of nonlinearity are most typical for soils, such as cubic and fifth-order nonlinearities, causing generation of the third and fifth higher harmonics of main frequencies of input signals. Nonlinearities of even types, such as quadratic, fourth-order, and sixth-order, concerned with asymmetry, or skew- ness, of oscillations (i.e., quasi-static deformations of the surface) are usually weak, except some special cases, in which a stress-strain relationship of a soil can be rep- resented by functions with noticeable even components. A weak nonlinearity results in an increase in high-frequency components, due to the generation of higher har- monics. In cases of strong nonlinearity, in which a decrease in amplification and in shear moduli become noticeable, changes in spectra of propagating signals achieve their maximum. As a result, input signals with arbitrary spectra are transformed into output signals with spectra of the type of E(f ) � f k , where k depends on the properties of the medium.

A numerical study on evolution of a dark soliton-like beam in self-defocusing media with measurable fifth-order focusing nonlinearity

This letter addresses a numerical study on evolution of a dark soliton-like beam in self-defocusing media with measurable fifth-order focusing nonlinearity. Modified by the competing nonlinearities the incident dark soliton-like beam of appropriate parameters evolves into a dark solitonic channel twinned with its bright companion born of the additional radiation. Stable propagation takes shape of the divergent dark and bright solitonic channels when energy redistribution among the solitonic channels and the background keeps up to a dynamical equilibrium.

Homoclinic and heteroclinic bifurcations in the non-linear dynamics of a beam resting on an elastic substrate

The non-linear dynamics of a slender “elastica”, fixed at its base and free at the top, resting on an elastic substrate, axially loaded and subjected to periodic excitation, has been analyzed. Taking into account the non-linear inertial terms, the single-mode dynamics of the systems is governed by a Duffing equation with fifth-order non-linearities. In the considered range of parameters, two qualitatively different phase portraits exist. When the axial load p is less than the Eulerian critical value, there are three centers and two saddles (with the related stable and unstable manifolds). After the pitchfork bifurcation, the two saddles and the middle center coalesce in an unique new saddle which has a pair of symmetric homoclinic solutions. Melnikov criteria on the chaotic dynamics of the system are derived on the basis of analytical expressions for the homoclinic and the heteroclinic orbits. They involve transverse intersections of the stable and unstable manifolds that represent the starting point for a subsequent route to a chaotic dynamics. Numerical simulations which aim to show some effects of the global bifurcations on the actual dynamics are presented.

Explicit Variational Solution of the Cubic-Quintic Nonlinear Schrödinger Equation for Pulse Propagation in Optical Fibers

With a variational method, we study the cubic-quintic nonlinear Schrödinger equation for pulse propagation in optical fibers. Anomalous dispersion that gives rise to fundamental bright solitary waves is considered. It is shown that within the specified limits, the present variational model demonstrates (for 1 picosecond pulse) good agreements with its analytical equivalent. It is proved that for negative coefficient of the fifth-order nonlinearity existence of the two-state solitary wave solution is a peculiarity of the variational formulation.

Nonlinear phase shift as a result of cascaded third-order processes

In centrosymmetric media, in addition to the third- and fifth-order nonlinearity, a cascaded ${\ensuremath{\chi}}^{(3)}:{\ensuremath{\chi}}^{(3)}$ nonlinearity contributes to the nonlinear refraction. The described effect takes place when the input beam is involved in nearly phase-matched third-harmonic generation. The nonlinear phase shift produced by this new cascaded nonlinearity is comparable to the nonlinear phase shift caused by the direct third-order nonlinearity ${\ensuremath{\chi}}^{(3)}.$ A new analytical approach, low depletion approximation, was developed and used to study the cascaded third-order processes in the conditions of not very high third-harmonic conversion coefficients. At higher conversion coefficients the effect was studied by numerical simulations. The presence of a seeded third-harmonic wave gives another possibility of controlling the additional phase shift of the fundamental wave. The conditions for stationary copropagation of the fundamental and the generated wave are presented.

Solitary wave propagation and bistability in the normal dispersion region of highly nonlinear optical fibres and waveguides

Properties of bright solitary wave solutions of the cubic-quintic nonlinear Schrödinger equation are considered in the normal dispersion region of optical fibres and waveguides with third- and fifth-order nonlinearities. The case of an optical fibre is concretely considered but other possible waveguide geometries are also discussed. For comparison, the properties of the black solitary wave existing in the same parameter range are also considered. All soliton parameters are presented in a unified way as functions of the normalised pulse energy. The possibility for bistable behaviour of the solitary waves is examined and a new type of solitary wave bistability (bright—dark) is introduced. The role of the geometry of the waveguide and the appropriate materials for experimental observation of such waves are also discussed.

Nonlinear Susceptibility of Strongly Nonlinear composites11The project supported partly by National Natural Science Foundation of China

By means of a variational approach, we study the effective response of strongly nonlinear composite media with a low concentration of cylindrical or coated cylindrical inclusions. In an attempt to consider the influence of nonlinear order to effective nonlinear response, we assume that every constituent of the composites obeys a current-field relation of the form . In dilute limit, we have presented some numerical results for the case of the third-order nonlinearity (i.e. ) and the fifth-order nonlinearity (i.e. ), and compared the effective response of the third-order nonlinearity with that of the fifth-order nonlinearity, some conclusions are drawn.

Nonlinear formation of dynamic holograms in a dye solution

A scheme for the formation and reconstruction, at the doubled frequency, of volume dynamic holograms is considered in the case of the fifth-order nonlinearity. The results are given of an experimental investigation of the diffraction of laser radiation by a transmission dynamic grating formed in a solution of polymethine dye No.3274U. A theoretical model is proposed for describing quadratic hologram formation and suitable for calculation of the energy efficiency of conversion of the radiation frequency.