Bistable Helmholtz solitons in cubic-quintic materials

J. M. Christian, G. S. McDonald, and P. Chamorro-Posada

Phys. Rev. A 76, 033833 – Published 28 September 2007

Abstract

We propose a nonlinear Helmholtz equation for modeling the evolution of broad optical beams in media with a cubic-quintic intensity-dependent refractive index. This type of nonlinearity is appropriate for some semiconductor materials, glasses, and polymers. Exact analytical soliton solutions are presented that describe self-trapped nonparaxial beams propagating at any angle with respect to the reference direction. These spatially symmetric solutions are, to the best of our knowledge, the first bistable Helmholtz solitons to be derived. Accompanying conservation laws (both integral and particular forms) are also reported. Numerical simulations investigate the stability of the solitons, which appear to be remarkably robust against perturbations.

Received 24 July 2007

DOI:https://doi.org/10.1103/PhysRevA.76.033833

Authors & Affiliations

J. M. Christian and G. S. McDonald

Joule Physics Laboratory, School of Computing, Science and Engineering, Institute for Materials Research, University of Salford, Salford M5 4WT, United Kingdom

P. Chamorro-Posada

Departamento de Teoría de la Señal y Comunicaciones e Ingeniería Telemática, Universidad de Valladolid, ETSI Telecomunicación, Campus Miguel Delibes s/n, 47011 Valladolid, Spain

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 76, Iss. 3 — September 2007

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Schematic diagram illustrating the forward (FWD) and backward (BWD) solitons of Eqs. (4). The FWD and BWD solutions are essentially indistinguishable, as they must be, since whether evolution occurs in the “forward” or ‘backward’ direction depends solely upon the relative orientation of the observer’s coordinate axes with respect to the beam. $x$ and $z$ are the laboratory or physical coordinates (schematically represented in the same unscaled units). The angles marked in the figure denote the sense of the propagation angle $θ > 0$ .Reuse & Permissions
Figure 2
Schematic diagram (to scale) illustrating the angular beam broadening effect. Beams are plotted in the laboratory or unscaled $(x, z)$ frame, and the coordinates are in the same units (determined by the choice of the beam widths). Part (a) shows a quasiparaxial $(θ = 6 °)$ configuration, and part (b) shows a typical Helmholtz $(θ = 60 °)$ regime. In (a), propagation is constrained to be very nearly along the $z$ axis, in which case geometrical broadening is negligible. In (b), evolution at a finite angle can lead to an arbitrarily large increase in the beam width as $θ \to \pm 90 °$ .Reuse & Permissions
Figure 3
(a) Bistable solution families for several values of $ν$ , obtained by solving Eq. (7). For fixed $∣ α ∣$ and $ν$ , solitons residing on the two branches have different peak intensities $ρ_{0}$ , but the same full-width-half-maximum values. The dashed line, corresponding to $ν = 1$ , was presented in Ref. 17. (b) Dependence of the parameter $β$ on $ρ_{0}$ [see Eq. (4d)].Reuse & Permissions
Figure 4
Helmholtz soliton profiles for (a) upper-, and (b) lower-branch solutions with $ν = 1$ and $α = - 0.15$ . From Fig. 3b, the peak amplitudes are $\sqrt{ρ_{0}} \approx 2.03$ and $\sqrt{ρ_{0}} \approx 1.14$ , respectively. Propagation angles are $θ = 0 °$ (solid line), $θ = 30 °$ (dashed line), $θ = 45 °$ (dotted line), and $θ = 60 °$ (dot-dashed line). Solid lines coincide with exact paraxial solution (12), where no broadening is present.Reuse & Permissions
Figure 5
Reshaping oscillations in the peak amplitude $\sqrt{ρ_{0}} \equiv {∣ u ∣}_{m}$ of perturbed bistable solitons (4) with $α = - 0.15$ , lying on (a) upper, and (b) lower branches of the curve shown in Fig. 3b. Reshaping of the corresponding monostable soliton (where $α = + 0.15$ ) is shown in part (c). Solid lines: $θ = 15 °$ ; dashed lines: $θ = 30 °$ ; dot-dashed lines: $θ = 45 °$ .Reuse & Permissions
Figure 6
(a) Evolution of a weakly perturbed lower-branch soliton ( $α = - 0.2$ , $ρ_{0} \approx 1.68$ ) in $({∣ u ∣}_{m}, \partial_{ζ} {∣ u ∣}_{m}, ζ)$ space. (b) Projection of the orbit in part (a) onto the $({∣ u ∣}_{m}, \partial_{ζ} {∣ u ∣}_{m})$ plane. (c) Projection of the orbit in part (a) onto the $(ζ, {∣ u ∣}_{m})$ plane. The scale length of the reshaping oscillation is comparable with that shown in Fig. 5b (solid line).Reuse & Permissions

Physical Review A

covering atomic, molecular, and optical physics and quantum information