Multimode Incoherent Spatial Solitons in Logarithmically Saturable Nonlinear Media

We show that multimode incoherent spatial solitons are possible in logarithmically saturable nonlinear media. The mode-occupancy function associated with these soliton states is found to obey a Poisson distribution. Our analysis indicates that two approaches, i.e., the dynamic coherent density description as well as static self-consistent multimode method lead to exactly the same results. Closed form solutions are obtained for s1 1 1dD as well as for s2 1 1dD circular and elliptical incoherent solitons. [S0031-9007(98)05547-1] To date, the nonlinear process of optical beam selffocusing has been investigated by means of coherent or laser sources [1]. Only very recently has the first experimental observation of incoherent self-trapping been reported in literature [2]. In this experiment, a monochromatic partially spatially incoherent light beam was found to self-trap in a biased photorefractive crystal. As in the case of coherent photorefractive spatial solitons, this was achieved via the saturable photorefractive drift nonlinearity [3,4]. More specifically, at a certain bias voltage the incoherent beam was found to self-trap whereas above or below this limit the beam underwent compression or expansion (diffraction), respectively. Even more importantly, the experiment suggested that incoherent spatial solitons may indeed be feasible in biased photorefractives. These results are by themselves of great significance since they shed new light on the complex dynamics of incoherent optical beams in a nonlinear environment. Shortly after, self-trapping of incoherent white light was also observed in the same material system [5]. In order to explain the results of Ref. [2], we have recently approached this problem by introducing a theory based on the so-called coherent density [6]. In this description, the underlying evolution model takes the form of a nonlinear Schrodinger-like integro-differential equation provided that at the origin the coherent density is appropriately scaled with respect to the angular power spectrum of the incoherent source. The results of this study were found to be in good agreement with the experimental data of Ref. [2]. Moreover, using this same approach it has also been shown that s1 1 1dD incoherent spatial solitons are in fact possible in saturable nonlinear media of the logarithmic type [7]. It is important to note, however, that, in general, the coherent density model is by nature better suited to describe dynamic evolution and thus it does not readily lend itself toward identifying stationary solutions such as incoherent spatial solitons. With the exception of the saturable logarithmic model (perhaps the only exactly soluble model in the coherent density description), the question of whether incoherent spatial solitons also exist or not in other saturable nonlinear systems (such as in biased photorefractives) cannot be easily addressed. In order to circumvent this problem, we have very recently developed an alternative theory which is essentially a self-consistent incoherent multimode approach [8]. In brief, in this procedure, an incoherent spatial soliton is sought, which intensitywise is a superposition of all the modes self-consistently guided in its nonlinearly induced waveguide [9]. By doing so, we have obtained s1 1 1dD photorefractive incoherent spatial solitons, the conditions necessary for their existence, as well as their coherence characteristics [8]. At first sight, the two theories [6,8] may seem to have very little in common. Thus it is important to ask whether the two approaches are mutually consistent or lead to the same results. Moreover, it is interesting to know whether s2 1 1dD incoherent spatial solitons are also possible in saturable nonlinear media. In this Letter we show that multimode incoherent spatial solitons are possible in logarithmically saturable nonlinear media. The mode-occupancy function of these soliton states is found to obey a Poisson distribution. Our analysis demonstrates that, in this case, the two theories, i.e., the dynamic coherent density description and the static self-consistent multimode method, lead to identical results. Two-dimensional circular and elliptical incoherent spatial solitons along with their characteristics are also obtained in closed form. Even though the saturable logarithmic nonlinearity differs from the photorefractive, it provides nevertheless a platform (perhaps the only platform) upon which the equivalence of the two previously mentioned approaches can be established in closed form. Finally, as recently argued by Snyder and Mitchell [10] and by Shen [11], simplifications of this sort in terms of “accessible” nonlinearities can provide valuable insight and still maintain the characteristic features of the underlying physical process.