Induced Coherence and Stable Soliton Spiraling

Abstract

We develop a theory of soliton spiraling in a bulk nonlinear medium and reveal a new physical mechanism: periodic power exchange via induced coherence, which can lead to stable spiraling and the formation of dynamical two-soliton states. Our theory not only explains earlier observations, but provides a number of predictions which are also verified experimentally. Finally, we show theoretically and experimentally that soliton spiraling can be controled by the degree of mutual initial coherence.Self-guided optical beams (or spatial solitons) have attracted substantial research interest in the last three decades [1].Although interactions between twodimensional (2D) solitons in Kerr and non-Kerr media have been studied extensively, only the recent discoveries of stable three-dimensional (3D) solitons in different nonlinear bulk media In this Letter we develop, for the first time to our knowledge, a theory of soliton spiraling in a photorefractive-type nonlinear bulk medium. We derive an analytical model describing stable soliton spiraling and predict a number of new effects in soliton interactions, such as an induced coherence and control over 3D interactions, which we verify here experimentally, using experimental setup similar to that reported earlier [3]. Importantly, our analytical model and numerical simulations show that interacting-spiraling solitons conserve angular momentum. We believe that this result is a core foundation for future research on 3D soliton control, resembling the conservation of linear momentum in the interaction of more conventional (1+1)-dimensional solitons [6].We consider incoherent beam interaction in an isotropic saturable nonlinear medium described by two coupled normalized nonlinear Schrödinger (NLS) equationswhere u and w are the beam envelopes, z is the propagation distance; ∇ 2 ⊥ ≡ ∂ 2 /∂x 2 + ∂ 2 /∂y 2 accounts for the diffraction in the transverse (x, y) plane. This system, in the 2D case (i.e., for ∇ 2 ⊥ ≡ ∂ 2 /∂x 2 ), gives rise to incoherently-coupled soliton pairs [7] and to incoherent collisions [8] which have both been demonstrated with photorefractive screening solitons [9].We look for solitary waves of Eqs. (1) in the form u = U (r) exp (iβ u z), w = W (r) exp (iβ w z), where the envelopes U and W satisfy the equationsHere r ≡ x 2 + y 2 is the radial coordinate, and β u and β w are nonlinearity-induced shifts of the propagation constants. System (2) has two families of soliton solutions: U = G u (β u , r), W = 0 and U = 0, W = G w (β w , r), which can be found numerically by solving the equation, where α = {u, w}. These solutions can be characterised by the soliton powers P (β α ) ≡ 2π ∞ 0 G 2 α (β α , r) rdr. In addition to the one-component solitons, at β u = β v ≡ β there exists a family of two-component (vector) solitons defined as: U = G(β, r) cos θ, W = G(β, r) sin θ, where the variable θ characterises a power distribution between the components. Moving solitons of Eqs. (1) can be obtained by a well-known gauge transformation.

Stable vortex and dipole vector solitons in a saturable nonlinear medium

“…(27) only depends on the parameter s, not on p. We have checked numerically that this integral never vanishes for any s. Thus, the condition (27) is satisfied only when p = ±1. In this case, it follows from (13), (19) and (24) that w 1 = φ(r)e ±iθ , u 2 = u 20 (r), and w 3 = f (r)e ±iθ . We can continue the perturbation series expansions (14)- (16) to higher orders and find that all u 2n (n ≥ 0) corrections are only functions of r, and all w 2n+1 (n ≥ 0) corrections have the form g(r)e ±iθ .…”

Resonance- and phase-induced window sequences in vector-soliton collisions

“…Theoretically, vector-soliton collisions have been studied intensively in the past ten years. It has been shown that soliton transmission, reflection, trapping and spiraling can all occur [6,7]. More surprisingly, a fractal structure has been found where transmission, reflection and trapping occur in an intertwined, fractal manner [8].…”

“…Examples include pulse propagation in birefringent fibers and wavelength-divisionmultiplexed systems [1,2], operation of optical logic gates [3], and beam steering in Kerr or photo-refractive crystals [4][5][6]. Theoretically, vector-soliton collisions have been studied intensively in the past ten years.…”

Scattering of light by molecules of light

“…Sometimes the dipole excitation is so strong that the structure behaves as a pair of spiraling beams earlier analyzed in Ref. [8]. This is only one of many interesting phenomena observed in simulations which also include excitation of rotational modes by collision with a scalar soliton, annihilation or strong deflection of the incident soliton, etc.…”

“…We consider the propagation of two coherent light beams interacting incoherently in a saturable nonlinear medium. In the paraxial approximation, the beam mutual interaction can be described by a system of two coupled nonlinear Schrödinger (NLS) equations [3][4][5]8],…”