We present an extension of the well-known method of "inverse iteration" for the standard eigenvalue problem to the nonlinear problem of finding dispersion relations for electromagnetic waves moving through a doubly-periodic structure. Numerical results are presented to illustrate the performance of the technique. A further improvement is described that allows an efficient "path following" algorithm where a curve of solutions is computed in (ω{sub}1, k{sub}(bloch)) space. We present dispersion relations calculated via this new method and compare the efficiency of this algorithm with that of more traditional methods.
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