Christian, JM, McDonald, GS, ChamorroPosada, P and SanchezCurto, 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 [13]. While existing paraxial models constrain considerations of broad beams to vanishinglysmall propagation angles relative to the longitudinal (i.e. z) direction, arbitraryangle (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 narrowbeam parameter) also prove to be invalid for describing regimes involving nontrivial 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 twodimensional, 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 [13]. This presentation will draw on, and report significant extensions of, three of our most recent papers [13], 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 cubicquintic 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 arbitraryangle 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 socalled ultranarrowbeam scenario. To analyse such subwavelength fields, one is obliged to consider a perturbative expansion of Maxwell’s equations in terms of the parameterofsmallness 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 higherorder nonlinear diffractive correction terms. Helmholtz nonparaxiality is physically and mathematically distinct from ultranarrowbeam nonparaxiality. In Helmholtz soliton theory [14], optical beams are always assumed to be broad so that ε << O(1) is always fully satisfied. By omitting the slowlyvarying envelope approximation, NLH models provide an excellent description of broadbeam 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 narrowbeam models can access arbitraryangle regimes since their governing equations lack the required x–z symmetry. We have considered a range of refractiveindex distributions n2(E), which are associated with the normalized nonlinearity functions, and have derived a number of exact analytical solutions. For example, a competing cubicquintic 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 secondorder nonintegrable wave equations. Despite their spatiallyextended structure, we have found that boundary solitons are remarkably stable against perturbations. These robust structures connect regions of finite and zeroamplitude 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. Finitesize effects tend to play a profound role in twodimensional 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. NLStype) analyses are restricted to bright solitons and vanishinglysmall 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 phasematched 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) 

Themes:  Energy Media, Digital Technology and the Creative Economy Subjects outside of the University Themes 
Schools:  Schools > School of Computing, Science and Engineering > Salford Innovation Research Centre (SIRC) 
Refereed:  Yes 
Depositing User:  JM Christian 
Date Deposited:  17 Oct 2011 11:51 
Last Modified:  21 Apr 2017 14:27 
References:  [1] P. ChamorroPosada 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, 15451560 (2007). [3] J. SánchezCurto et al., “Helmholtz solitons at nonlinear interfaces,” Opt. Lett. 32, 11261128 (2007). [4] P. ChamorroPosada and G. S. McDonald, “Helmholtz dark solitons,” Opt. Lett. 28, 825827 (2003). [5] S. Chi and Q. Guo, “Vector theory of selffocusing of an optical beam in Kerr media,” Opt. Lett. 20, 15981600 (1995). [6] L. Gagnon, “Exact travellingwave solutions for optical models based on the nonlinear cubicquintic Schrödinger equation,” JOSA A 9, 14771483 (1989). [7] A. B. Aceves et. al.,“Theory of lightbeam propagation at nonlinear interfaces. I. Equivalentparticle theory for a single interface,” Phys. Rev.A 39, 18091827 (1989). 
URI:  http://usir.salford.ac.uk/id/eprint/18439 
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