Nonlinear photonic crystals: II. Interaction classification for quadratic nonlinearities

Weakly nonlinear interactions between wavepackets in lossless periodic dielectric media are studied based on the classical nonlinear Maxwell equations. We consider nonlinear processes such that: (i) the amplitude of the wave component due to the nonlinearity does not exceed the amplitude of its linear component; (ii) the spatial range of a probing wavepacket is much smaller than the dimension of the medium sample, and it is not too small compared with the dimension of the primitive cell. These nonlinear processes are naturally described in terms of the Bloch modes and the dispersion relations of the underlying linear periodic medium. It turns out that only a few triads of modes have significant nonlinear interactions. They are singled out by the frequency and phase matching conditions and, as we show, by an additional selection rule: the group velocity matching condition. The latter condition is the most important selection rule for the nonlinear regimes. We give a complete quantitative classification of all possible significant interactions for quadratic nonlinearities. The classification is based on a universal system of indices reflecting the intensity of nonlinear interactions. The obtained classification points to the second harmonic generation as being one of the stronger nonlinear interactions, and often the strongest one.     

Nonlinear photonic crystals: II. Interaction classification for quadratic nonlinearities

Abstract

Weakly nonlinear interactions between wavepackets in lossless periodic dielectric media are studied based on the classical nonlinear Maxwell equations. We consider nonlinear processes such that: (i) the amplitude of the wave component due to the nonlinearity does not exceed the amplitude of its linear component; (ii) the spatial range of a probing wavepacket is much smaller than the dimension of the medium sample, and it is not too small compared with the dimension of the primitive cell. These nonlinear processes are naturally described in terms of the Bloch modes and the dispersion relations of the underlying linear periodic medium. It turns out that only a few triads of modes have significant nonlinear interactions. They are singled out by the frequency and phase matching conditions and, as we show, by an additional selection rule: the group velocity matching condition. The latter condition is the most important selection rule for the nonlinear regimes. We give a complete quantitative classification of all possible significant interactions for quadratic nonlinearities. The classification is based on a universal system of indices reflecting the intensity of nonlinear interactions. The obtained classification points to the second harmonic generation as being one of the stronger nonlinear interactions, and often the strongest one.     

Nonlinear photonic crystals: III. Cubic nonlinearity

Abstract

Weakly nonlinear interactions between wavepackets in a lossless periodic dielectric medium are studied based on the classical Maxwell equations with a cubic nonlinearity. We consider nonlinear processes such that: (i) the amplitude of the wave component due to the nonlinearity does not exceed the amplitude of its linear component; (ii) the spatial range of a probing wavepacket is much smaller than the dimension of the medium sample, and it is not too small compared with the dimension of the primitive cell. These nonlinear processes are naturally described in terms of the cubic interaction phase function based on the dispersion relations of the underlying linear periodic medium. It turns out that only a few quadruplets of modes have significant nonlinear interactions. They are singled out by a system of selection rules including the group velocity, frequency and phase matching conditions. It turns out that the intrinsic symmetries of the cubic interaction phase stemming from assumed inversion symmetry of the dispersion relations play a significant role in the cubic nonlinear interactions. We also study canonical forms of the cubic interaction phase leading to a complete quantitative classification of all possible significant cubic interactions. The classification is ultimately based on a universal system of indices reflecting the intensity of nonlinear interactions.     

Localization of intense electromagnetic waves in a relativistically hot plasma

We consider nonlinear interactions between intense short electromagnetic waves (EMWs) and a relativistically hot electron plasma that supports relativistic electron holes (REHs). It is shown that such EMW-REH interactions are governed by a coupled nonlinear system of equations composed of a nonlinear Schro dinger equation describing the dynamics of the EMWs and the Poisson-relativistic Vlasov system describing the dynamics of driven REHs. The present nonlinear system of equations admits both a linearly trapped discrete number of eigenmodes of the EMWs in a quasistationary REH and a modification of the REH by large-amplitude trapped EMWs. Computer simulations of the relativistic Vlasov and Maxwell-Poisson system of equations show complex interactions between REHs loaded with localized EMWs.     

Numerical method of studying nonlinear interactions between long waves and multiple short waves

Although the nonlinear interactions between a single short gravity wave and a long wave can be solved analytically, the solution is less tractable in more general cases involving multiple short waves. In this work we present a numerical method of studying nonlinear interactions between a long wave and multiple short harmonic waves in infinitely deep water. Specifically, this method is applied to the calculation of the temporal and spatial evolutions of the surface elevations in which a given long wave interacts with several short harmonic waves. Another important application of our method is to quantitatively analyse the nonlinear interactions between an arbitrary short wave train and another short wave train. From simulation results, we obtain that the mechanism for the nonlinear interactions between one short wave train and another short wave train (expressed as wave train 2) leads to the energy focusing of the other short wave train (expressed as wave train 3). This mechanism occurs on wave components with a narrow frequency bandwidth, whose frequencies are near that of wave train 3.     

Quasi-phasematching

The use of microstructured crystals in quasi-phasematched (QPM) nonlinear interactions has enabled operation of nonlinear devices in regimes inaccessible to conventional birefringently phasematched media. This review addresses basic aspects of the theory of QPM interactions, microstructured ferroelectrics and semiconductors for QPM, devices based on QPM media, and a series of techniques based on engineering of QPM gratings to tailor spatial and spectral response of QPM interactions. Because it is not possible in a brief review to do justice to the large body of results that have been obtained with QPM media over the past twenty years, the emphasis in this review will be on aspects of QPM interactions beyond their use simply as highly nonlinear alternatives to conventional birefringent media. To cite this article: D.S. Hum, M.M. Fejer, C. R. Physique 8 (2007).     

Rogue edge waves in the ocean

Theoretically possible rogue edge wave are studied over cylindrical bottom in the framework of nonlinear shallow water equations in a weakly nonlinear limit. The nonlinear mechanisms (nonlinear dispersion enhancement, modulation instability and multimodal interactions) of possible anomalous edge wave appearance are analyzed.     

Brownian particles with long- and short-range interactions

We develop the kinetic theory of Brownian particles with long- and short-range interactions. Since the particles are in contact with a thermal bath fixing the temperature T, they are described by the canonical ensemble. We consider both overdamped and inertial models. In the overdamped limit, the evolution of the spatial density is governed by the generalized mean field Smoluchowski equation including a mean field potential due to long-range interactions and a generically nonlinear barotropic pressure due to short-range interactions. This equation describes various physical systems such as self-gravitating Brownian particles (Smoluchowski-Poisson system), bacterial populations experiencing chemotaxis (Keller-Segel model) and colloidal particles with capillary interactions. We also take into account the inertia of the particles and derive corresponding kinetic and hydrodynamic equations generalizing the usual Kramers, Jeans, Euler and Cattaneo equations. For each model, we provide the corresponding form of free energy and establish the H-theorem and the virial theorem. Finally, we show that the same hydrodynamic equations are obtained in the context of nonlinear mean field Fokker-Planck equations associated with generalized thermodynamics. However, in that case, the nonlinear pressure is due to the bias in the transition probabilities from one state to the other leading to non-Boltzmannian distributions while in the former case the distribution is Boltzmannian but the nonlinear pressure arises from the two-body correlation function induced by the short-range potential of interaction. As a whole, our paper develops connections between the topics of long-range interactions, short-range interactions, nonlinear mean field Fokker-Planck equations and generalized thermodynamics. It also justifies from a kinetic theory based on microscopic processes, the basic equations that were introduced phenomenologically to describe self-gravitating Brownian particles, chemotaxis and colloidal suspensions with attractive interactions.     

Strongly nonlinear beats in the dynamics of an elastic system with a strong local stiffness nonlinearity: Analysis and identification

We consider a linear cantilever beam attached to ground through a strongly nonlinear stiffness at its free boundary, and study its dynamics computationally by the assumed-modes method. The nonlinear stiffness of this system has no linear component, so it is essentially nonlinear and nonlinearizable. We find that the strong nonlinearity mostly affects the lower-frequency bending modes and gives rise to strongly nonlinear beat phenomena. Analysis of these beats proves that they are caused by internal resonance interactions of nonlinear normal modes (NNMs) of the system. These internal resonances are not of the classical type since they occur between bending modes whose linearized natural frequencies are not necessarily related by rational ratios; rather, they are due to the strong energy-dependence of the frequency of oscillation of the corresponding NNMs of the beam (arising from the strong local stiffness nonlinearity) and occur at energy ranges where the frequencies of these NNMs are rationally related. Nonlinear effects start at a different energy level for each mode. Lower modes are influenced at lower energies due to larger modal displacements than higher modes and thus, at certain energy levels, the NNMs become rationally related, which results in internal resonance. The internal resonances of NNMs are studied using a reduced order model of the beam system. Then, a nonlinear system identification method is developed, capable of identifying this type of strongly nonlinear modal interactions. It is based on an adaptive step-by-step application of empirical mode decomposition (EMD) to the measured time series, which makes it valid for multi-frequency beating signals. Our work extends an earlier nonlinear system identification approach developed for nearly mono-frequency (monochromatic) signals. The extended system identification method is applied to the identification of the strongly nonlinear dynamics of the considered cantilever beam with the local strong nonlinear stiffness at its free end.     

Observation of nonlinear optical interactions of ultralow levels of light in a tapered optical nanofiber embedded in a hot rubidium vapor

We report the observation of low-light level optical interactions in a tapered optical nanofiber (TNF) embedded in a hot rubidium vapor. The small optical mode area plays a significant role in the optical properties of the hot vapor Rb-TNF system, allowing nonlinear optical interactions with nW level powers even in the presence of transit-time dephasing rates much larger than the intrinsic linewidth. We demonstrate nonlinear absorption and V-type electromagnetically induced transparency with cw powers below 10 nW, comparable to the best results in any Rb-optical waveguide system. The good performance and flexibility of the Rb-TNF system makes it a very promising candidate for ultralow power resonant nonlinear optical applications.     

Self-localisation of dipolar Bose-Einstein condensates in leaking optical lattices

We investigate self-localisation of dipolar Bose-Einstein condensates (BECs) in 1D nonlinear lattices via boundary dissipation in a dissipative nonlinear Schrödinger model (DNLS) with nearest-neighbour dipole-dipole interactions (DDI). By including both contact interactions and DDIs, we observe that a rich variety of self-localised modes (i.e., single discrete breathers, moving breathers and multi-breathers) can exist in dipolar systems in optical lattices. Furthermore, we find that DDIs can suppress the formation of single discrete breathers and support the formation of multi-breathers. Our results show that including both contact interactions and DDIs may provide a way to experimentally obtain stationary multi-breathers in optical lattices via boundary dissipations.     

Nonlinear optical responses due to exciton-phonon interactions in strongly coupled exciton-phonon systems

The excitonic nonlinear optical responses due to exciton-phonon interactions in strongly coupled exciton-phonon systems are investigated theoretically. It is shown that the influence of exciton-phonon interactions on the nonlinear optical absorptions and Kerr nonlinear coefficients is significant as the signal field frequency detuning from the exciton frequency approaches to the optical phonon frequency. How to manipulate the nonlinear optical responses by using the control fields is also presented.Received: 16 March 2004, Published online: 4 May 2004PACS: 42.50.Gy Effects of atomic coherence on propagation, absorption, and amplification of light; electromagnetically induced transparency and absorption - 42.50.Hz Strong-field excitation of optical transitions in quantum systems; multiphoton processes; dynamic Stark shift - 42.25.Bs Wave propagation, transmission and absorption - 72.80.Le Polymers; organic compounds (including organic semiconductors)     

Generalized pohlmeyer transformation — Some nonlinear systems with exponential interactions

The Pohlmeyer transformation between the O(3) nonlinear σ-model and the sine-Gordon equation is generalized to the case of the chiral model on a Kähler manifold. As a result nonlinear systems with the exponential interactions are obtained. Solutions of these systems are derived from solutions of the initial chiral model.     

Stability and phase transition of localized modes in Bose–Einstein condensates with both two- and three-body interactions

We investigate the stability and phase transition of localized modes in Bose–Einstein Condensates (BECs) in an optical lattice with the discrete nonlinear Schrödinger model by considering both two- and three-body interactions. We find that there are three types of localized modes, bright discrete breather (DB), discrete kink (DK), and multi-breather (MUB). Moreover, both two- and three-body on-site repulsive interactions can stabilize DB, while on-site attractive three-body interactions destabilize it. There is a critical value for the three-body interaction with which both DK and MUB become the most stable ones. We give analytically the energy thresholds for the destabilization of localized states and find that they are unstable (stable) when the total energy of the system is higher (lower) than the thresholds. The stability and dynamics characters of DB and MUB are general for extended lattice systems. Our result is useful for the blocking, filtering, and transfer of the norm in nonlinear lattices for BECs with both two- and three-body interactions.     

Influences of nonlinear interactions of polaritons in uniaxial crystals

We have analysed theoretically the polarization and dielectric constant of uniaxial ionic crystals,including the effect of the nonlinear interactions between ions.The spectrum of polaritons(coupling modes of photons and optical phonons) under nonlinear interactions was developed.A new branch of dispersion relations emerges in the original frequency gap.     

Localization phenomena in nonlinear Schrödinger equations with spatially inhomogeneous nonlinearities: Theory and applications to Bose-Einstein condensates

We study the properties of the ground state of nonlinear Schrödinger equations with spatially inhomogeneous interactions and show that it experiences a strong localization on the spatial region where the interactions vanish. At the same time, tunneling to regions with positive values of the interactions is strongly suppressed by the nonlinear interactions and as the number of particles is increased it saturates in the region of finite interaction values. The chemical potential has a cutoff value in these systems and thus takes values on a finite interval. The applicability of the phenomenon to Bose-Einstein condensates is discussed in detail.     

Nonuniversality aspects of nonlinear k(perpendicular) factorization for hard dijets

The origin of breaking of conventional linear k(perpendicular) factorization for hard processes in a nuclear environment is by now well established. The realization of the nonlinear nuclear k(perpendicular) factorization which emerges instead was found to change from one jet observable to another. Here we demonstrate how the pattern of nonlinear k(perpendicular) factorization, and especially the role of diffractive interactions, in the production of dijets off nuclei depends on the color properties of the underlying pQCD subprocess.     

Nonlinear photonic crystals: I. Quadratic nonlinearity

We develop a consistent mathematical theory of weakly nonlinear periodic dielectric media for the dimensions one, two and three. The theory is based on the Maxwell equations with classical quadratic and cubic constitutive relations. In particular, we give a complete classification of different nonlinear interactions between Floquet-Bloch modes based on indices which measure the strength of the interactions. The indices take on a small number of prescribed values which are collected in a table. The theory rests on the asymptotic analysis of oscillatory integrals describing the nonlinear interactions.     

Nanoscale nonlinear optical process: theoretical modeling of second-harmonic generation for both forbidden and allowed light

Based on a combination of the multiple-multipole method and nonlinear coupled-wave equations, a rigorous three-dimensional numerical simulation of nonlinear optical interactions between an optical near field and a nonlinear medium is performed, allowing us to study the dependence of second-harmonic (SH) near-field intensity on tip-sample distance and the polarization state of the incident fundamental wave. It is demonstrated that allowed and forbidden light make different contributions to the SH near-field intensity.     

Integrability Aspects and Soliton Solutions for a System Describing Ultrashort Pulse Propagation in an Inhomogeneous Multi-Component Medium

For the propagation of the ultrashort pulses in an inhomogeneousmulti-component nonlinear medium, a system of coupled equations isanalytically studied in this paper. Painlevé analysis shows thatthis system admits the Painlevé property under some constraints.By means of the Ablowitz-Kaup-Newell-Segur procedure, the Lax pairof this system is derived, and the Darboux transformation (DT) isconstructed with the help of the obtained Lax pair. With symboliccomputation, the soliton solutions are obtained by virtue of the DTalgorithm. Figures are plotted to illustrate the dynamical featuresof the soliton solutions. Characteristics of the solitonspropagating in an inhomogeneous multi-component nonlinear medium arediscussed: (i) Propagation of one soliton and two-peak soliton; (ii) Elastic interactions of the parabolic two solitons; (iii) Overlapphenomenon between two solitons; (iv) Collision of two head-onsolitons and two head-on two-peak solitons; (v) Two different typesof interactions of the three solitons; (vi) Decomposition phenomenonof one soliton into two solitons. The results might be useful in thestudy on the ultrashort-pulse propagation in the inhomogeneousmulti-component nonlinear media.