We show that the optical modes of a periodic nanostructure with frequency dependent dielectric constant (i.e., a dispersive optical nanostructure), in general can be written as an ordinary eigenvalue problem of a "dielectric function operator", for each distinct symmetry representation of the periodic nanostructure. For a frequency dependence in the form of polynomial rational function, the problem translates to a polynomial eigenvalue equation in the frequency of the mode. The resulting

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... can be solved using the basis functions of a dielectric backbone structure, which has a frequency independent dielectric constant. Rapid convergence is achieved when the basis functions are selected to be the modes of a dielectric backbone structure that minimizes the frequency perturbation of the dielectric function of the optical nanostructure. In particular, using a two dimensional photonic crystal constructed with a polar crystal as an example, we demonstrate that, remarkable simple cubic equations are sufficient to obtain accurate descriptions of eigenfrequencies.