Optical switching between bistable soliton states: a theoretical review

Enns, Richard H.; Edmundson, D.; Rangnekar, S. S.; Kaplan, A. E.

doi:10.1007/bf00624674

Cited by 44 publications

(23 citation statements)

References 56 publications

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“…There are several available numerical methods for solving such a boundary-value problem and finding the pertur- bation growth rates, γ n [5,11,26]. We treated Eqs.…”

Section: The Linear-stability Analysismentioning

confidence: 99%

Symmetric and asymmetric solitons and vortices in linearly coupled two-dimensional waveguides with the cubic-quintic nonlinearity

Dror

Malomed

2011

Physica D: Nonlinear Phenomena

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It is well known that the two-dimensional (2D) nonlinear Schrödinger equation (NLSE) with the cubic-quintic (CQ) nonlinearity supports a family of stable fundamental solitons, as well as solitary vortices (alias vortex rings), which are stable for sufficiently large values of the norm. We study stationary localized modes in a symmetric linearly coupled system of two such equations, focusing on asymmetric states. The model may describe "optical bullets" in dual-core nonlinear optical waveguides (including spatiotemporal vortices that were not discussed before), or a BoseEinstein condensate (BEC) loaded into a "dual-pancake" trap. Each family of solutions in the single-component model has two different counterparts in the coupled system, one symmetric and one asymmetric. Similarly to the earlier studied coupled 1D system with the CQ nonlinearity, the present model features bifurcation loops, for fundamental and vortex solitons alike: with the increase of the total energy (norm), the symmetric solitons become unstable at a point of the direct bifurcation, which is followed, at larger values of the energy, by the reverse bifurcation restabilizing the symmetric solitons. However, on the contrary to the 1D system, both the direct and reverse bifurcation may be of the subcritical type, at sufficiently small values of the coupling constant, λ. Thus, the system demonstrates a double bistability for the fundamental solitons. The stability of the solitons is investigated via the computation of instability growth rates for small perturbations. Vortex rings, which we study for two values of the "spin", s = 1 and 2, may be subject to the azimuthal instability, like in the single-component model. In particular, complete destabilization of asymmetric vortices is demonstrated for a sufficiently strong linear coupling. With the decrease of λ, a region of stable asymmetric vortices appears, and a single region of bistability for the vortices is found. We also develop a quasi-analytical approach to the description of the bifurcations diagrams, based on the variational approximation. Splitting of asymmetric vortices, induced by the azimuthal instability, is studied by means of direct simulations. Interactions between initially quiescent solitons of different types are studied too. In particular, we confirm the prediction of the reversal of the sign of the interaction (attractive/repulsive for in-phase/out-of-phase pairs) for the solitons with the odd spin, s = 1, in comparison with the even values, s = 0 and 2.

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“…There are several available numerical methods for solving such a boundary-value problem and finding the pertur- bation growth rates, γ n [5,11,26]. We treated Eqs.…”

Section: The Linear-stability Analysismentioning

confidence: 99%

Symmetric and asymmetric solitons and vortices in linearly coupled two-dimensional waveguides with the cubic-quintic nonlinearity

Dror

Malomed

2011

Physica D: Nonlinear Phenomena

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“…They are ubiquitous objects in optical media [1] [3,7], quadratically nonlinear (χ (2) ) [2,8], and graded-index Kerr media [9]. While a fully localized STS in three dimensions (3D) has not yet been found in an experiment, 2D ones were observed in a bulk χ (2) medium [10].…”

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confidence: 99%

Stable Spinning Optical Solitons in Three Dimensions

et al. 2002

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We introduce spatiotemporal spinning solitons (vortex tori) of the three-dimensional nonlinear Schrödinger equation with focusing cubic and defocusing quintic nonlinearities. The first ever found completely stable spatiotemporal vortex solitons are demonstrated. A general conclusion is that stable spinning solitons are possible as a result of competition between focusing and defocusing nonlinearities.Optical solitons (spatial, temporal, or spatiotemporal) are self-trapped light beams or pulses that are supported by a balance between diffraction and/or dispersion and various nonlinearities. They are ubiquitous objects in optical media [1] [3,7], quadratically nonlinear (χ (2) ) [2,8], and graded-index Kerr media [9]. While a fully localized STS in three dimensions (3D) has not yet been found in an experiment, 2D ones were observed in a bulk χ (2) medium [10]. The interplay of spatio-temporal coupling and nonlinearity may also play an important role in self-defocusing media [11].Spinning (vortex) solitons are also possible in optical media. Starting with the works [12], both delocalized ("dark") and localized ("bright") optical vortices in 2D were investigated [13,14,15]. In the 3D case they take the shape of a torus ("doughnut") [16,17]. However, the only previously known physical model which could support stable 3D vortex solitons is the Skyrme model [18], which has recently found a new important application to Bose-Einstein condensates (BEC) [19]. Our objective in this paper is to identify fundamental models of the nonlinear-Schrödinger (NLS) type in 3D that give rise to stable spinning solitons, as NLS models are much simpler and closer to more experimental situations, having applications to optics, BEC, plasmas, etc. (see below).For bright vortex solitons stability is a major issue as, unlike their zero-spin counterparts, the spinning solitons are prone to destabilization by azimuthal perturbations. In 2D models with χ (2) and saturable nonlinearities an azimuthal instability was revealed by simulations [14] and observed experimentally [15]. As a result, a soliton with spin 1 splits into two or three fragments, each being a moving zero-spin soliton. Simulations of the 3D spinning STS in the χ (2) model also demonstrates its instability-induced splitting into separating zero-spin solitons [17]. Nevertheless, the χ (2) nonlinearity acting in combination with the self-defocusing Kerr (χ (3) ) nonlinearity, gives rise to the first examples of stable spinning (ring-shaped) 2D solitons with spin s = 1 and 2 [20]. It should be stressed that all the 2D spinning solitons actually represent static spatial beams; on the contrary, 3D solitons are moving spatiotemporal ones, which are localized not only in the transverse plane, but also in the propagation coordinate, see below.A model which may support stable spinning solitons in 3D is the one with a cubic-quintic (CQ) nonlinearity, which (in terms of optics) assumes a nonlinear correction to the medium's refractive index in the form δn = n 2 I − n 4 I 2 , I being the lig...

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“…For bistable soli-tons to exist, the energy curve obtained when P is plotted against /3 must be, at least, U shaped. Because of stability considerations, however, nonlinearities which yield Nshaped energy curves are of particular interest from the viewpoint of optical switching with solitons [3] and other potential applications, such as all-optical logic operations. In general, the shape of the P (/3) curve can be deduced from the form of the nonlinearity.…”

Section: Introductionmentioning

confidence: 99%

“…It is well known (see [3] It is natural to ask next whether one could perform similar switching operations for BISOL2. Typically, when Ao is plotted against~0 the curve is C shaped.…”

Section: Introductionmentioning

confidence: 99%

Optical switching between bistable soliton states of the second kind

Enns

1993

Phys. Rev. A

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Optical switching between bistable soliton states of the second kind is demonstrated numerically when the arnplification of the input solitons is suitably adjusted. The numerical amplification study is supplemented by Anderson's [Phys. Rev. A 27, 3135 (1983)] variational approach, which sheds considerable light on the observed numerical behavior, although it is not able to predict switching.PACS number(s): 42.65.Pc, 42.50.Rh

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Optical switching between bistable soliton states: a theoretical review

Cited by 44 publications

References 56 publications

Symmetric and asymmetric solitons and vortices in linearly coupled two-dimensional waveguides with the cubic-quintic nonlinearity

Symmetric and asymmetric solitons and vortices in linearly coupled two-dimensional waveguides with the cubic-quintic nonlinearity

Stable Spinning Optical Solitons in Three Dimensions

Optical switching between bistable soliton states of the second kind

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