On the theory of Helmholtz solitons

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