The geometrical optics approximation is used to describe the propagation of waves through waveguides whose dielectric constant depends quadratically on the distance x transverse to the waveguide axis. When this dielectric constant or equivalently the square of the index of refraction is also periodically modulated with distance z down the waveguide axis, parametric instability occurs. This causes the rays to diverge exponentially away from the axis. Within certain regions of parameter space governed by strength and spatial frequency of the perturbation, chaos occurs. With a suitable scaling of coordinates, a wide range of nonlinear dynamics is explored that covers a general class of physically relevant parameters. The problem is shown to have a particularly useful structure as a perturbed Hamiltonian system because of the simple form of the dynamics when modulation is absent.
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