Stable three-dimensional spatiotemporal solitons in a two-dimensional photonic lattice

Abstract

We investigate the existence and stability of three-dimensional spatiotemporal solitons in self-focusing cubic Kerr-type optical media with an imprinted two-dimensional harmonic transverse modulation of the refractive index. We demonstrate that two-dimensional photonic Kerr-type nonlinear lattices can support stable one-parameter families of three-dimensional spatiotemporal solitons provided that their energy is within a certain interval and the strength of the lattice potential, which is proportional to the refractive index modulation depth, is above a certain threshold value.

Stabilization of two-dimensional solitons and vortices against supercritical collapse by lattice potentials

“…In optics, it implies a periodic modulation of the refractive index in the transverse plane, as in photonic-crystal fibers), and in BEC it is induced by optical lattices (OLs). In the medium with the self-focusing (SF) χ (3) nonlinearity, the stabilization of fundamental and vortical 2D solitons (those with topological charges S = 0 and 1, respectively) under the action of the square-lattice potential was predicted in works [7] and [8] (see also works [9]; stable solitons were found too in photonic-crystal-fiber models [10]). The action of the stabilization mechanism can be summarized as follows.…”

Multidimensional semi-gap solitons in a periodic potential

“…Crossing the lower border of the existence domain (1) leads to disintegration of the localized state into linear Bloch waves (radiation) [19]. In the case of the attractive cubic nonlinearity (which corresponds to BEC where atomic collisions are characterized by a negative scattering length, while this is the case of the normal, self-focusing Kerr effect), 2D and 3D solitons can be stabilized not only by the potential lattice whose dimension is equal to that of the ambient space, but also by low-dimensional periodic potentials, whose dimension is smaller by one, i.e., 2D and 3D solitons can be stabilized by a quasi-1D [5,6] or quasi-2D [5,6,7] OL, respectively [in the former case, the qualitative estimate (1) for the width of the stability region at small ε is correct too]; however, 3D solitons cannot be stabilized by a quasi-1D lattice potential [5,6] [this is possible if the 1D potential is applied in combination with the Feshbach-resonance management, i.e., periodic reversal of the sign of the nonlinearity coefficient [20], or in combination with dispersion management, i.e., periodically alternating sign of the local GVD coefficient [21]]. Solitons can exist in such settings because the attractive nonlinearity provides for stable self-localization of the wave function in the free direction (one in which the low-dimensional potential does not act), essentially the same way as in the 1D NLS equation, and, simultaneously, the lattice stabilizes the soliton in the other directions (in the 3D model with the quasi-1D OL potential, the selflocalization in the transverse 2D subspace, where the potential does not act, is possible too, but the resulting soliton is unstable, the same way as the above-mentioned Townes soliton).…”

“…The physical models of this type emerge in the context of Bose-Einstein condensation (BEC) [2,3,4,5,6,7], where the periodic potential is created as an optical lattice (OL), i.e., interference pattern formed by coherent beams illuminating the condensate, and in nonlinear optics, where similar models apply to photonic crystals [8]. A different but allied setting is provided by a cylindrical OL ("Bessel lattice"), which can also support stable 2D [9] and 3D [10] solitons.…”

Matter-wave solitons in radially periodic potentials

“…A great challenge to the experiment is creation of multidimensional matter-wave solitons, as well as the making of spatiotemporal solitons in nonlinear optics [8]. In particular, multi-dimensional solitons trapped in a low-dimensional OL (i.e., 1D lattice in the 2D space [9], and 2D lattice in the 3D space [9,10]), have been predicted [9]. As these solitons keep their mobility in the free direction, the latter settings can be used to test head-on and tangential collisions between solitons.…”