An Analysis of Rugate Filter Behavior

An analysis is made of the performance of a Bragg filter of arbitrary modulation profile in terms of solutions to the wave equation for a single repeat unit. This approach has advantages over the standard multiplication of matrices corresponding to very thin but flat steps in that it allows for rapid calculations and analytic studies. Special attention is given to filters with sinusoidal profiles. The transfer matrix for the filter is expressed in terms of the wavefunction and its first derivative. In the case of sinusoidal modulation, the wavefunctions are Mathieu functions. In all cases, simple relationships exist between centerband frequencies and linewidths, and the basic characteristics of the filter, such as the mean index and modulation depth. The results of computer simulations used in parametric studies of filter performance are presented and compared with analytic predictions, especially those of Coupled Mode Theory. It is found that corrections to the simplified theory must be made to account for the finite width of the filter. One benefit of the adopted approach is that it gives directly the strength of the optical electric field as a function of position in the filter.