On the theory of Helmholtz solitons

This thesis is concerned with spatial optical solitons in (quasi-two \ndimensional) planar waveguides, where there is a single longitudinal and \n(effectively) a single transverse dimension, and whose symmetry the solutions \nrespect consistently. A key feature of Helmholtz soliton theory is that, by \nrecognizing the physical equivalence of the longitudinal and transverse dimensions \nin uniform media, it can access experimental contexts involving broad, moderately \nintense beams that propagate and interact at arbitrarily large angles. It thus provides \nan ideal platform for the systematic generalization of established paraxial results \n(restricted to applications involving vanishingly-small angles) to the finite-angle \ndomain. Helmholtz soliton theory is expected to play a fundamental role in the \ndesign of any futuristic integrated-optic device exploiting the propagation and \ninteraction of spatial soliton beams at oblique angles relative to a reference direction. \nExact analytical soliton solutions are derived for a variety of newly-proposed \nscalar and vector Non-Linear Helmholtz equations. These solutions are valid for a \nwide variety of media, such as some semiconductors, doped glasses and non-linear \npolymers. Different types of solution classes have been obtained, including \nhyperbolic (exponentially localized), algebraic (with power-law asymptotics), \namplitude-kink (where the intensity varies monotonically), and spatially-extended \n(such as trigonometric and cnoidal) waves. Exact analytical solutions have also been \nobtained in the presence of some higher-order effects - for example, gain/absorption \nand saturation of the non-linear refractive index. \nHelmholtz solitons are found to exhibit generic features (such as angular \nbeam broadening), and they reduce to their paraxial counterparts when an \nappropriate multiple limit (defining rigorously a paraxial beam) is enforced. Each new solution has been tested under a numerical perturbative analysis that examines \nits stability. Helmholtz solitons have been classified largely as robust attractors, in a \nnon-linear dynamical sense, and this stability is crucial if they are to be exploited \nsuccessfully in practical applications.