Fluctuations of the Power of Coupled Modes

The behaviour of waves propagating in multimode waveguides can be described by coupled equations for the amplitudes of each mode.1 This description is rigorous, but has the disadvantage that the coupled wave equations usually cannot be solved. It has been shown that a much simpler description is possible if we limit our interest to knowledge about the average power carried by each mode. 2-4 Coupled equations for the average mode power have been derived and applied to the problem of wave propagation in multimode dielectric waveguides.4,5 However, the description of multimode waveguides in terms of average power is incomplete unless some information is available about the fluctuations of the actual power about the average value. With the help of the same perturbation approach that was used to derive the coupled power equations,4 we derive in this paper a differential equation for the variance of the power. The result of our perturbation theory is expressed in terms of the cross-correlation and the average power of the modes. In order to evaluate this expression, we need to make several assumptions. It has been shown in an earlier paper5 that the average power settles down to a steady-state distribution of power versus mode number that is 1793