Two-dimensional quadratic solitons collision in KTP

Theoretical study of reorientation and torque of liquid crystal molecules under influence of external electric field and experimentally generation of spatial optical soliton beam and getting a sharp switching in chiral nematic liquid crystal

A summary is presented of the principal types of completely integrable partial differential equations having soliton solutions. Each type is derived from an appropriate physical model of an electromagnetic wave problem, with the intention to show how known mathematical results apply to a coherent class of physical problems in electromagnetic waves. The non-linear Schrodinger (NS) equation appears when the induced non-linear dielectric polarization is expanded in a series of powers of the electric field, only the linear and third-order polarizations are retained, and the temporal spectrum of the wave is a narrow band far removed from any resonance of the medium. The sine-Gordon equation appears from a similar optical model of propagation in a dielectric consisting of identical 2-level atomic systems, but resonance occurs between the carrier frequency of the wave and the transition frequency of the atoms. The Boussinesq and Korteweg– de Vries equations appear at different levels of approximation to a potential wave on a transmission line having a non-linear capacitance such that the charge stored is a non-linear function of the line potential. In all cases the evolution variable is the propagation distance; the transverse variable is time, but in the case of the NS equation it may alternatively be a spatial coordinate, giving rise to the possibility of spatial solitons as well as temporal solitons for NS-type problems. Two examples are derived of non-integrable Hamiltonian systems having spatial solitary waves, namely the second-order cascade interaction and vector spatial solitary waves of the third-order interaction, and a brief survey of the analytical solutions for the plane waves and solitary waves of these two types is presented. Finally, the addition of a second spatial dimension to the non-linear transmission line problem leads to the Kadomtsev–Petviashvili equations, and a further approximation for weakly modulated travelling waves leads to the Davey–Stewartson equations. Both of these completely integrable systems support combined spatial–temporal solitons.

We show analytically that bright and dark spatial self-similar waves can propagate in graded-index amplifiers exhibiting self-focusing or self-defocusing Kerr nonlinearities. The intensity profiles of the novel waves are identical with those of fundamental bright or dark spatial solitons supported by homogeneous passive waveguides with the same type of nonlinearity. Thus, we reveal a previously unnoticed connection between spatial solitons and self-similar waves. We also suggest that the discovered self-similar waves can be used in a promising scheme for the amplification and focusing of spatial solitons in future all-optical networks.

Spatial solitons can exist in various kinds of nonlinear optical resonators with and without amplification. In the past years different types of these localized structures such as vortices, bright, dark solitons, and phase solitons have been experimentally shown to exist. Many links appear to exist to fields different from optics, such as fluids, phase transitions, or particle physics. These spatial resonator solitons are bistable and due to their mobility suggest schemes of information processing not possible with the fixed bistable elements forming the basic ingredient of traditional electronic processing. The recent demonstration of existence and manipulation of spatial solitons in semiconductor microresonators represents a step in the direction of such optical parallel processing applications. We review some proof of principle soliton experiments on slow systems, and describe in more detail the experiments on semiconductor resonator solitons which are aimed at applications.

We present first-time calculations from the time-domain vector Maxwell's equations of spatial optical soliton propagation and mutual deflection, including carrier waves, in a 2-D homogeneous Kerr-type nonlinear dielectric. The nonlinear Schrodinger equation predicts that two co-propagating, in-phase spatial solitons remain bound to each other, executing a periodic separation. This disagrees with our new extensively tested finite-difference time-domain (FD-TD) solution of Maxwell's equations. FD-TD shows that co-propagating in-phase spatial solitons become unbound, i.e. diverge to arbitrarily large separations, if the ratio of soliton beamwidth to wavelength is order 1 or less. Not relying upon paraxial approximations or analogies to temporal soliton interactions, FD-TD appears to be a robust means of obtaining detailed models of the interaction of sub-picosecond pulsed light beams in nonlinear media directly in the space-time domain. >

An investigation of the propagation of optical beams and the main properties of spatial solitons in three-dimensional media with a local saturable nonlinear refractive-index change is presented. The fundamental bright-soliton solution is calculated from a first integral that describes a two-value solution branch. Although both solution branches are stable in the framework of linear stability theory, the dynamic of beam propagation shows that slightly perturbed initial soliton beams do not evolve to a perfect solitary beam but lead to periodic oscillations of the amplitude. The dependence of the long-living oscillations and the possible azimuthal-symmetry breaking with formation of filaments on the saturation parameter γ and the initial-beam parameters are studied in detail. The results are compared with experimental observations of two-dimensional photorefractive solitons. The interaction and the collision of two spatial solitons are investigated. Beam fusion can appear for parallel propagation as well for small collision angles and small phase differences for solitary beams of both solution branches. Furthermore, solitary-beam dragging with initially overlapping beams of different directions is studied.

Beam pointing control with spatial solitary waves in quadratic nonlinear media

Spatial solitary waves of the quadratic nonlinearity were discovered in the early 70's [1] and have been the subject of numerous investigations [2-5] due to the renewed emphasis placed on cascading for all-optical processing. [6-7] Such waves were recently demonstrated in bulk KTP crystals and Lithium Niobate planar waveguides, underlining the potential interest of such self-guiding fields for signal processing applications.[8-9] Among them, angular steering versus seed input phase [10] and versus power imbalance between input polarizations [11] have been reported in KTP. In this Communication, we elucidate some specific features of Type I and Type II second harmonic generation (SHG) interactions leading to the formation of 2+1 spatial solitonlike waves. Specifically, we demonstrate how a small angular tilt in the input fundamental frequency (FF) beam can provide pointing stabilization against peak power fluctuations when Type I (scalar) or Type II (vectorial) interactions are employed, while Type II processes can compensate random input angle perturbations in the weaker FF polarization component.

The relativistic equations governing the nonlinear interaction of two light waves and a Langmuir wave are shown to admit two classes of solitary-wave solutions. Temporal solitary waves propagate at speeds greater than the speed of light and carry no information. Spatial solitary waves propagate at speeds less than the speed of light and do carry information. The properties of these waves are discussed and the spatial solitary waves are shown to be well suited to the beat-wave acceleration of particles.

Since the discovery of photorefractive spatial solitons [1, 2], these non-diffracting waves have been the subject of an intense research effort, because they are particularly interesting for building alloptical diodes, transistors, switches, and all-optical computers. Until now steady-state bright spatial solitons have been observed in BTO [3], SBN [4], InP:Fe [5], KNbO3 [6], KLTN [7], and BaTiO3 [8] crystals. Barium–calcium titanate (BCT) [9, 10] is a promising photorefractive material and an alternative to BaTiO3, which is much easier to grow and does not have any phase transition within the temperature range from 120 C to 100 C. This crystal also possesses slightly larger electrooptic coefficients r13 (20 pm/V) and r33 (130 pm/V) compared to BaTiO3. However, no studies of photorefractive spatial soliton formation have been reported so far in BCT crystals. In this contribution, we investigate photorefractive spatial soliton formation in iron-doped BCT crystals and observe, for the first time to our knowledge, the formation of steady-state two-dimensional (2D) screening solitons in this material. Samples of the congruently melting composition Ba0:23Ca0:77TiO3 are grown in the crystalgrowth laboratory of the Department of Physics at the University of Osnabr€uck [9, 10]. The dimension of our sample that is doped with 290 ppm Fe is 5 5 5 mm3. All surfaces are polished to optical quality. On both faces normal to the c-axis of the crystal, electrodes are prepared with silver paste. The light propagates along the a-axis in our experiment. Our experiments are conducted in a standard setup for soliton formation, in which an external electric field E0 is applied along the ferroelectric c-axis. A frequency-doubled Nd:YAG laser (l 1⁄4 532 nm) is used as the light source. Because of the low dark conductivity of our sample sd 6 10 14 W 1 cm 1 [9], a background illumination is necessary to tune the degree of saturation of the nonlinearity. This ordinarily polarized background light of the same wavelength 532 nm is made spatially incoherent by passing it through a rotating diffuser. The ratio r between the soliton light intensity (extraordinary polarization) and the background irradiance is r 5. The soliton intensity is always kept below 50 mW/cm2 to minimize photovoltaic self-defocusing. Figure 1 shows a characteristic result of 2D soliton formation. In this experiment, the beam intensity is I 30 mW/cm2. The beam diameter at the crystal’s input face is din 14 mm. Without an external electric field E0 applied, the beam width d increases during propagation due to both, phys. stat. sol. (a) 189, No. 1, R4–R5 (2002)

This work analyzes the dynamics of two-dimensional spatial solitons in dissipative graded index Kerr media, analytically and numerically. The dynamics of two-dimensional spatial solitons has been studied in a medium with lumped amplification. The presence of lumped amplification in the medium results in the formation of soliton like waves called similaritons. The spatial soliton dynamics has been studied in a dissipative medium. In a dissipative graded index Kerr medium, the balancing between the coefficients of constant dissipation and lumped amplification results in the stabilization of the beam. The beam dynamics has also been studied by varying amplification coefficients with propagation distance. Hyperbolic, linear, and exponential amplification profiles are considered. When amplification coefficient varies with propagation distance, the beam gets compressed. The beam compression is higher in the case of exponential amplification profile than that of hyperbolic and linear amplification profiles.

Numerical and experimental investigation of the interaction between incoherent spatial photorefractive solitons is performed. It is found that the interaction between two-dimensional solitons is different from the interaction between one-dimensional solitons. Whereas one-dimensional incoherent solitons display only attraction, two-dimensional solitons can display repulsion as well as attraction.

Laser generated soliton waveguides in photorefractive crystals

In this work we investigate the dynamics of a spatial soliton pulse under the presence of a linear Periodic Wave (PW), which dynamically induces a photonic lattice. We consider that propagation phenomena are governed by the well-known non-linear Schrodinger equation (NLSE), while Kerr-type non-linearity is in effect. Interaction phenomena are analyzed by forming a non-linear coupled differential equation system of the evolution of the soliton-beam parameters, which are the pulse amplitude, the transverse velocity, the mean position and the phase. The dynamical system governing the evolution of soliton parameters is derived by utilizing a quasi-particle approach based on the perturbed inverse scattering method. Direct numerical simulations of the NLS equation are shown to be in good agreement with the solution of the dynamical system, for a wide range of the parameters. The results show that efficient photon management, in terms of soliton control and beam steering, can occur for appropriate choices of the characteristics of the periodic lattice, which are the amplitude, the period, the pulse duration, the relative position with respect to the soliton beam in the transverse dimension and the initial transverse velocity.

The propagation of spatial solitons is systematically investigated in nonlocal nonlinear media with an imprinted transverse periodic modulation of the refractive index. Based on the variational principle and the infinitesimal approximation of Maclaurin series expansion, we obtain an analytical solution of such nonlocal spatial solitons and an interesting result that the critical power for such solitons propagation is smaller than that in uniform nonlocal self-focusing media. It is found that there exist thresholds in modulation period and lattice depth for such solitons. A stable spatial soliton propagation is maintained with proper adjustment of the modulation period and the lattice depth.