We present an analysis of propagation along practical, single-mode, matched-cladding and depressed-cladding slab waveguides and fibres using the bound modes of the entire core-cladding-coating-air cross-section, instead of the usual bound-radiation mode model of a finite core and unbounded cladding. Our model readily quantifies the spatial transient in terms of modal absorption and scattering in the coating, and also displays leaky mode behaviour, as well as the characteristic transmission dips associated with the fundamental mode beyond its cutoff wavelength in a depressed-cladding fibre.

We calculate exactly the two bound Floquet modes of a periodic linear waveguide induced in a medium by a second-order soliton of the nonlinear Schrödinger equation. The modes are degenerate at the writing frequency, having the same quasi-propagation constant, which suggests applications of our method to spectral filtering.

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We use multiscale perturbation theory in conjunction with the inverse scattering transform to study the interaction of a number of solitons of the cubic nonlinear Schrödinger equation under the influence of a small correction to the nonlinear potential. We assume that the solitons are all moving with the same velocity at the initial instant; this maximizes the effect each soliton has on the others as a consequence of the perturbation. Over the long time scales that we consider, the soliton amplitudes remain fixed, while their center of mass coordinates obey Newton's equations with a force law for which we present an integral formula. For the interaction of two solitons with a quintic perturbation term we present more details since symmetries -one related to the form of the perturbation and one related to the small number of particles involved -allow the problem to be reduced to a one-dimensional one with a single parameter, an effective mass. The main results include calculations of the binding energy and oscillation frequency of nearby solitons in the stable case when the perturbation is an attractive correction to the potential and of the asymptotic "ejection" velocity in the unstable case. Numerical experiments illustrate the accuracy of the perturbative calculations and indicate their range of validity.

We consider the steady propagation of a light beam in a planar waveguide whose width and depth are periodically modulated in the direction of propagation. Using methods of soliton theory, a class of periodic potentials is presented for which the complete set of Floquet solutions of the linear Schrodinger equation can be found exactly at a particular optical frequency. For potentials in this class, there are exactly two bound Floquet solutions at this frequency, and they are degenerate, having the same Floquet multiplier. We study analytically the behavior of the waveguide under small changes in the frequency and observe a breaking of the degeneracy in the Floquet multiplier at first order. We predict and observe numerically the disappearance of both bound states at second order. These results suggest applications to spectral filtering.