## Helmholtz solitons: a new angle in nonlinear optics

Christian, JM, McDonald, GS, Chamorro-Posada, P and Sanchez-Curto, J 2007, *Helmholtz solitons: a new angle in nonlinear optics* , in: Nonlinear Photonics Topical Meeting, 2nd – 6th September 2007, Quebec City, Canada.

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## Abstract

We rigorously show that nonparaxial optical beams are not necessarily ultranarrow [1-3]. While existing paraxial models constrain considerations of broad beams to vanishingly-small propagation angles relative to the longitudinal (i.e. z) direction, arbitrary-angle (Helmholtz) effects yield geometrical corrections of arbitrary order and new qualitative phenomena. Conventional nonparaxial beam models (relying on perturbative expansions in terms of a single narrow-beam parameter) also prove to be invalid for describing regimes involving non-trivial angular geometries. Helmholtz corrections to paraxial theory can easily exceed 100% and are found to be essential for capturing oblique beam propagation, interaction, and interface effects [1]. Diffraction in NLH models is fully two-dimensional, occurring symmetrically in the (x,z) coordinates which define the waveguide plane. The NLH equation possesses a large number of families of exact analytical solutions that are entirely physical [1-3]. This presentation will draw on, and report significant extensions of, three of our most recent papers [1-3], in the area of Helmholtz soliton theory. A specific focus of novelty will be on new solution families arising from a wide range of material nonlinearities, such as cubic-quintic and saturable refractive indices. We will report the first exact analytical bistable Helmholtz solitons and demonstrate through simulations that the solitons lying on both branches of the bistable curve are robust attractors. We will also highlight novel types of solutions, such as the discovery of highly stable “Helmholtz boundary solitons”. Furthermore, for the general case of a generic dispersive nonlinearity, we will prove that the elliptic character of our governing equation is essential for describing arbitrary-angle effects. Ellipticity leaves in tact both the physical and the mathematical stability of exact solution families, as should be expected. Maintaining the numerical stability of these solutions also proves to be a straightforward task, and analysis has established that perceived instability in such elliptic models is a purely numerical artefact [2]. Finally, we have recently discovered [3], a compact version of Snell’s Law for bright nonlinear beams that is valid for Kerr solitons refracting at an interface for arbitrary angles of incidence. We will report, for the first time, that this result extends consistently (in the same compact form) to dark Kerr solitons [4]. The term “nonparaxial” is routinely used to refer to an optical beam whose waist w0 is comparable to, or smaller than, its carrier wavelength λ. The condition λ/w0 ≡ ε ~ O(1) then defines the so-called ultranarrow-beam scenario. To analyse such sub-wavelength fields, one is obliged to consider a perturbative expansion of Maxwell’s equations in terms of the parameter-of-smallness ε [5]. When ε ~ O(1), nonlinear divergence can no longer be neglected and there may be strong coupling between the transverse and longitudinal components of the electric field. To O(ε^2), the dominant transverse field component is governed by a Nonlinear Schrödinger (NLS) equation, supplemented by a range of higher-order nonlinear diffractive correction terms. Helmholtz nonparaxiality is physically and mathematically distinct from ultranarrow-beam nonparaxiality. In Helmholtz soliton theory [1-4], optical beams are always assumed to be broad so that ε << O(1) is always fully satisfied. By omitting the slowly-varying envelope approximation, NLH models provide an excellent description of broad-beam evolution. The governing equation then possesses many types of exact analytical soliton solutions. Helmholtz nonparaxiality gives rise to a range of generic corrections to known paraxial solutions, and these corrections can exceed 100%. For example, the geometrical factor 2κV^2 = (tanθ)^2, where κ ~ ε2, V is the soliton velocity (in a scaled coordinate system) and θ is the beam’s propagation angle relative to the longitudinal (z) axis, is a key feature of Helmholtz solutions. It depends solely upon the propagation angle θ of the beam and can be of any order, even when κ << O(1). The origin of the 2κV^2 correction lies in retaining the full generality of the ∂^2/∂z^2 operator, and it is thus independent of the system nonlinearity. Neither paraxial nor conventional narrow-beam models can access arbitrary-angle regimes since their governing equations lack the required x–z symmetry. We have considered a range of refractive-index distributions n2(E), which are associated with the normalized nonlinearity functions, and have derived a number of exact analytical solutions. For example, a competing cubic-quintic nonlinearity permits the existence of hyperbolic soliton solutions. We have also recently derived a new class of boundary soliton solution [6]. Boundary solitons are of fundamental mathematical and physical interest. They expand the range of known exact analytical solutions of fully second-order non-integrable wave equations. Despite their spatially-extended structure, we have found that boundary solitons are remarkably stable against perturbations. These robust structures connect regions of finite- and zero-amplitude disturbance. This feature suggests that they also may be termed “edge solitons”, since they can act as natural nonlinear boundary waves at the outer limits of, e.g. optical, disturbance. Moreover, the full Helmholtz framework of their definition permits the application of these exact analytical solutions with any orientation in the waveguide plane. Finite-size effects tend to play a profound role in two-dimensional optical pattern formation and it seems plausible that our Helmholtz boundary solitons, or higher dimensional counterparts, may also find application in this important subject area. The behaviour of solitons at interfaces is an inherently angular problem, for which previous paraxial (i.e. NLS-type) analyses are restricted to bright solitons and vanishingly-small angles of incidence/reflection/refraction [7]. Helmholtz soliton theory has recently provided, for the first time, a framework for analysing the reflection and refraction properties of bright [1], and dark [4], solitons encountering an interface at arbitrary angles. Our study is based on an NLH model that is generalized to describe the evolution of solitons at the interface between two Kerr focusing (defocusing) media. Bright (dark) soliton solutions in each media are phase-matched at the interface, thus relating soliton transverse velocities in each medium. By exploiting the geometrical relation between velocities and propagation angles, we have discovered a remarkably compact Helmholtz generalization of Snell’s law that governs Kerr soliton refraction. Excellent agreement has been found between analytical predictions and numerical simulations for a wide range of interface parameters. Our new Snell's law result, which is valid for different soliton types (bright and dark) and arbitrary angles, represents a new fundamental building block for studies involving the universal problem of soliton behaviour at the planar boundary of different nonlinear materials.

Item Type: | Conference or Workshop Item (Poster) |
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Themes: | Energy Media, Digital Technology and the Creative Economy Subjects outside of the University Themes |

Schools: | Colleges and Schools > College of Science & Technology > School of Computing, Science and Engineering > Materials & Physics Research Centre |

Refereed: | Yes |

Depositing User: | JM Christian |

Date Deposited: | 17 Oct 2011 12:51 |

Last Modified: | 20 Aug 2013 18:14 |

References: | [1] P. Chamorro-Posada and G. S. McDonald, “Spatial Kerr soliton collisions at arbitrary angles,” Phys. Rev. E 74, 036609 (2006). [2] J. M. Christian et al., “Helmholtz bright and boundary solitons,” J. Phys. A: Math. & Theor. 40, 1545-1560 (2007). [3] J. Sánchez-Curto et al., “Helmholtz solitons at nonlinear interfaces,” Opt. Lett. 32, 1126-1128 (2007). [4] P. Chamorro-Posada and G. S. McDonald, “Helmholtz dark solitons,” Opt. Lett. 28, 825-827 (2003). [5] S. Chi and Q. Guo, “Vector theory of self-focusing of an optical beam in Kerr media,” Opt. Lett. 20, 1598-1600 (1995). [6] L. Gagnon, “Exact travelling-wave solutions for optical models based on the nonlinear cubic-quintic Schrödinger equation,” JOSA A 9, 1477-1483 (1989). [7] A. B. Aceves et. al.,“Theory of light-beam propagation at nonlinear interfaces. I. Equivalent-particle theory for a single interface,” Phys. Rev.A 39, 1809-1827 (1989). |

URI: | http://usir.salford.ac.uk/id/eprint/18439 |

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