We consider the propagation of electromagnetic waves in an optical device known as a 'directional coupler', which is widely used in telecommunication systems, This device consists of a pair of dielectric waveguides in close proximity. Such waveguides are strongly coupled and there is energy transfer between the waveguides. The propagation of light in such waveguides is paraxial and can be described by a parabolic differential equation closely resembling the Schrodinger equation for motion of a quantum particle in a two-dimensional time-dependent potential well. We exploit the quantum mechanical analogue of the optical system to write the propagator describing paraxial propagation as a path integral over optical paths. We use the Feynman-Kleinert variational procedure to calculate approximate expressions for the propagation constant and the field profile of the lowest-order mode of the waveguiding system. An approximate expression for the beat length of the system is also calculated in the case where the waveguides are strongly coupled. The results are found to be in better agreement with other theoretical calculations than are the results of a previous variational calculation.