Spatial solitons in a medium with lumped amplification and dissipation

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)

A method is proposed to generate and study the dynamics of spatial light solitons in a birefringent medium with quadratic nonlinearity. Although no analytical expression for propagating solitons has been obtained, our numerical simulations show the existence of stable localized spatial solitons in the frequency forbidden band gap of the medium. The dynamics of these objects is quite rich and manifests for instance elastic reflections, or inelastic collisions where two solitons merge and propagate as a single solitary wave. We derive the dynamics of the slowly varying envelopes of the three fields (second harmonic pump and two-component signal) and study this new system theoretically. We show that it does present a threshold for nonlinear supratransmission that can be calculated from a series expansion approach with a very high accuracy. Specific physical implications of our theoretical predictions are illustrated on LiGaTe${}_{2}$ (LGT) crystals. Once irradiated by a cw laser beam of $10 \ensuremath{\mu}$m wavelength, at an incidence beyond the extinction angle, such crystals will transmit light, in the form of spatial solitons generated in the nonlinear regime above the nonlinear supratransmission threshold.

We elaborate two generic methods for producing two-dimensional (2D) spatial soliton arrays (SSAs) in the framework of the cubic–quintic (CQ) complex Ginzburg–Landau (CGL) model. The first approach deals with a broad beam launched into the dissipative nonlinear medium which is equipped with an imprinted grating of a sufficiently sharp form. The beam splits into a cluster of jets, each subsequently self-trapping into a stable soliton, if the power is sufficient. We consider two kinds of sharp gratings—‘raised-cosine’ (RC) and Kronig–Penney (KP) lattices—and two types of input beams: fundamental and vortical. By selecting appropriate parameters, this method makes it possible to create various types of soliton arrays, such as solid, annular (with single and double rings) and cross-shaped ones. The second method uses a ‘virtual lattice’, in the form of a periodic transverse phase modulation imprinted into the broad beam which is passed through an appropriate phase mask and then shone into a uniform nonlinear medium. Two different types of masks are considered; in the form of a ‘checkerboard’ or ‘tilings’. In these cases, broad fundamental and vortical beams may also evolve into stable SSAs if the beam power and spacing of the virtual phase lattice are large enough. By means of the latter technique, square-shaped, hexagonal and quasi-crystalline SSAs can be created.

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.

The effect of fluctuating parameters of quadratic nonlinear media on the dynamics of spatial solitons is analytically and numerically studied. We derive a system of equations for mean fields of both frequency components and show that fluctuations of the wave vector mismatch and the nonlinear coefficient induce effective linear and nonlinear losses, respectively. We find approximate analytical expressions for the damping rate of spatial solitons for both types of the random modulation and study the impact of the correlation length of the fluctuations. We show that the predictions of the mean field theory are in a good agreement with numerical simulations of the full stochastic system.

Photonic mesh lattices provide a versatile experimental platform for studying light propagation in a discrete environment. By using standard telecommunication equipment it is possible to influence the evolution of light by actively modulating its amplitude and phase. Fiber components thus offer a ready to use solution for creating a \( \mathbb{PT} \)-symmetric optical network. Here, we analyze two classes of \( \mathbb{PT} \)-symmetric networks: those with local \( \mathbb{PT} \) symmetry fulfilling the conditions of \( \mathbb{PT} \) symmetry in each time step but not globally and those with global \( \mathbb{PT} \) symmetry providing completely real eigenvalues. We investigate light propagation in these dissipative media in the linear and nonlinear regime and discuss nonlinear localization as well as the formation of soliton. Furthermore, Bloch oscillations are induced by an external phase modulation and are found to restore pseudo-Hermitian propagation.

We consider the expanded class of one- and two-dimensional spatial solitons that exist when the soliton is a compound entity composed of two orthogonal beams [1-3]. These solitons propagate in a homogeneous isotropic medium whose refractive index has an arbitrary dependence on intensity. As with classical solitons, their intensity profile remains axially uniform but, in addition, their polarization state now changes continuously with propagation. We refer to this new class of self-guided waves as ‘dynamic solitons’ because of their internal field dynamics. Dynamic solitons can have an arbitrary number of intensity peaks and thus exhibit novel intensity profiles. Their common salient property is that they are composed of two orthogonal beams, neither of which is, in general, a soliton on its own. Each beam is, however, a mode of the (axially uniform) linear optical waveguide induced by the dynamic soliton.

Self-trapping of light beams in nonlinear self-focusing Kerr media has been explored intensively during the last three decades. The self trapped beams, often in the form of spatial solitons, evolve from nonlinear changes in the refractive index (δn) of the material proportional to the intensity distribution of the light, when the confining effect of the refractive index exactly compensates for the defocusing effect of diffraction. Motivated by experimental results of self-focusing of light beams [1] in the presence of the optical Kerr effect, a theoretical solution for the resulting spatial solitons was found [2,3]. This was followed by demonstrations of self-trapping in vapors [4], liquids [5], and finally, in the last few years, spatial solitons in solid (glass) media [6]. However, the index changes needed for optical-Kerr spatial solitons require high intensities, often exceeding 1 MWatt/cm2. Several other materials, such as slightly-absorbing liquids, exhibit Kerr-like thermal nonlinearities at much lower power densities (Watts/cm2), typically with a negative δn. These materials enabled the observations of dark spatial solitons [7] (narrow dark “notches”) and optical vortex solitons [8,9] (dark solitons were also observed using optical-Kerr nonlinearities in ZnSe [10] at GWatt/cm2 power densities). The most distinct properties of all these Kerr-like solitons are (i) a strong intensity dependence and the existence of a power threshold, and (ii) stability of bright solitons only in a single transverse dimension (consequently, bright Kerr-type solitons can be observed only in a planar waveguide, otherwise the soliton becomes unstable and cannot be observed).KeywordsDark SolitonInput BeamBright SolitonSpatial SolitonPhotorefractive CrystalThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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.

In the past ten years it was proved both theoretically and experimentally that cubic spatial solitons can be exploited to achieve fast all-optical processing of optical data. Taking benefit of the quasi-instantaneous response of Kerr type nonlinearity transient waveguides were generated by spatial solitons into homogeneous nonlinear materials[1]. It was also possible to modify at ultrafast speed the geometry of the waveguides thanks to the various interactions that may occur between soliton beams: couplers for signal splitting or combining and switches were realized using parallel solitons coupling [2] or solitons collision and fusion [3]. Spatial soliton based optical processing look attractive for routing high data rate streams. In the case of Kerr type nonlinearity however, the envisaged devices need a planar waveguide structure to avoid two-dimensional transverse instability and the peak powers required to reach self-guiding are close to 1 kW, even with the use of material of high nonlinearity. It was recently shown theoretically first and then experimentaly [4] that spatial solitary waves propagation can take place also in crystals exhibiting a substantial quadratic nonlinearity. This opened up new hopes on the use of soliton in optical processing. It was demonstrated that quadratic spatial solitary waves (QSSW) are stable even in the case of bulk propagation (2D-QSSW) on the contrary to their cubic counterparts. Moreover the advent of new materials, some of them of organic type, with high nonlinear coefficient lets expect that the self-guiding power threshold could be decreased below the above mentionned value for cubic spatial solitons. As it happened for interactions and collisions between kerr-type solitons, those between QSSWs are expected to offer the same promising field of applications. The problem of QSSWs collisions has been in fact recently addressed; as long as one refers to type I interactions, the problem has been treated numerically for both the 1D and 2D cases [5], analytically and experimentally only for the 1D case [6]. On the other hand, if type II (or vectorial) processes are concerned, the problem of QSSWs interactions has been faced only numerically or analytically both in the 1D and 2D cases [7].

It is shown that the propagation of steady-state spatial solitons in a bismuth titanium oxide (Bi12TiO20) photorefractive crystal is strongly affected by the presence of concomitant natural optical activity. We assume that the Bi12TiO20 crystal is (110) cut and that the external bias field is along the [001¯] axis. Such orientation introduces self-focusing in one polarization of an optical wave front, whereas it has no effect whatsoever on the component orthogonal to it. Under these conditions, the equations that describe the coupled evolution of the two polarizations of an optical planar beam are derived. Our numerical study of the interaction dynamics shows that, even though the optical beam experiences periodic diffraction in one of the polarization components, it can still recover significantly at particular distances within the crystal. Moreover, it is initially capable of maintaining its solitonlike form over a certain distance of propagation. Relevant examples are provided.

It is known that optical beams can propagate unchanged in the form of self-guided waves or spatial optical solitons due to compensation of the beam diffraction by nonlinearity-induced change in the material refractive index [1]. Many of the properties of spatial optical solitons, including the stability analysis [2] and the inverse scattering technique [3], are unique for the Kerr materials when the nonlinear part of the refractive index nnl(I) depends linearly on the light intensity I, nnl(I) = n2I, where n2 is the so-called Kerr coefficient. However, since the first efforts to observe self-guidence and spatial solitons experimentally, in applications one deals with non-Kerr materials where the dependence nnl(I) is not linear, e.g. it saturation. Recently it has been also shown, theoretically and experimentally, that self-guided beams can be observed in crystals with a strong photorefractive effect [4] and also due to phase-matched parametric interactions in a χ(2) nonlinear medium [5,6]. In many cases, propagation of nonlinear waves in such non-Kerr materials are described by the nonlinear Schrödinger with a more general dependence of the refractive index on the beam intensity. The main purpose of this talk is to present a panoramic overview of different properties of solitary waves in nonKerr materials including the detailed discussion of the stability of (scalar and vector) bright and dark solitons and the soliton coupled states.

We demonstrate the generation of solitons in the photorefractive materials with strong optical activity and absorption. The general wave equations are processed by rotation and affine transformations in order to decouple the effects of photorefractivity and optical activity. Analytical solutions of these equations are found in the case of slowly varying envelope and low-intensity approximations. The solutions show the occurrence of breathing solitons in sillenite crystals with strong optical activity and absorption in a particular orientation with respect to the external electric field. The absorption is decreasing the soliton intensity with the propagation distance (which set a limit in the soliton channel length) and is increasing the breathing period.

Spatial soliton formation in the bulk single-crystal polydiacetylene para-toulene sulfonate has been studied experimentally. The unique nonlinearity of this material at 1600 nm leads to stable solitons below an intensity maximum, and for higher intensities the input beams form a ring and a soliton.

A generalization of self-guided beams (spatial solitons) is discussed. Each beam is characterized by an axially uniform intensity profile whose polarization state changes with propagation. The soliton therefore has internal dynamics. It is composed of two orthogonally polarized modes of the linear waveguide induced by the dynamic soliton which, in general, do not exist as solitons independently. Our insight and exact analytical results for the threshold nonlinearity are found from the elementary physics of linear optical waveguides via a nonlinear self-consistency procedure.