From one- to two-dimensional solitons in the Ginzburg-Landau model of lasers with frequency-selective feedback

Abstract

We use the cubic complex Ginzburg-Landau equation coupled to a dissipative linear equation as a model of lasers with an external frequency-selective feedback. It is known that the feedback can stabilize the one-dimensional (1D) self-localized mode. We aim to extend the analysis to 2D stripe-shaped and vortex solitons. The radius of the vortices increases linearly with their topological charge, m, therefore the flat-stripe soliton may be interpreted as the vortex with m = ∞, while vortex solitons can be realized as stripes bent into rings. The results for the vortex solitons are applicable to a broad class of physical systems. There is a qualitative agreement between our results and those recently reported for models with saturable nonlinearity.

A PT -Symmetric Dual-Core System with the Sine-Gordon Nonlinearity and Derivative Coupling

“…The transition to the continuum limit as per Equation (7) transforms the last term in Equation (9) into the last term in the Hamiltonian density corresponding to Equation (3), with β ∼ − sin θ, while term h (v n+1 − v n ) in Equation (9), similar to its above-mentioned counterpart a (v n+1 − v n ) in Equation (8), becomes a full derivative, ψ x , in the continuum limit; hence, it may be dropped from the Hamiltonian. Furthermore, the term (sin θ) au n may be absorbed into the definition of the inner-chain FK Hamiltonian.…”

“…This possibility was first proposed in the context of nonlinear fiber optics in [1,2]; see also a review in [3]. More recently, a similar scheme was elaborated for the application of gain and the stabilization of solitons in plasmonics [4][5][6], as well as for the creation of stable two-dimensional dissipative solitons and solitary vortices in dual laser cavities [7,8]. Commonly-adopted models of dual-core nonlinear waveguides are based on linearly-coupled systems of nonlinear Schrödinger (NLS) equations, which include gain and loss terms [3].…”

“…Lastly, the above considerations demonstrate that the "polarization angle", θ, in the dual-FK chain (see Figure 1) determines the relative strength of the two couplings: The energy of the local two-particle coupling between the chains, which gives rise to terms ∼ in Equations (1) and (2), is determined by squared lengths of the red links; see Equation (8). The total energy of the three-particle coupling, which generates terms ∼ β in Equations (1) and (2), is proportional to the combined area of the shaded triangles; see Equation (9).…”

“…To derive the cross-derivative-coupling term in the Hamiltonian induced by the TPI, we note that the area of the triangle, confined by the links whose lengths are given by Equation (8), is:…”

Adler synchronization of spatial laser solitons pinned by defects

“…The interaction and locking phenomena which we observe in a semiconductor laser with feedback are well captured in a simple generic model consisting of a cubic CGL equation where solitons are stabilized by coupling to a linear filter equation [27]:…”