Separation of variables in the computation of spectra in 2-D photonic crystals

The spectrum of the Maxwell operator corresponding to propagation of electr omagnetic waves in periodic dielectric media of square geometry is analyzed . In the 2-D case the Maxwell operator splits into two scalar operators cor responding to so-called TE and TM modes of propagating electromagnetic wave . In the present paper we show that both of these operators can be slightly m odified in such a way that separation of variables can be used. The eigenfu nctions and the spectra for the modified operators are then used to obtain Rayleigh-Ritz-type estimates of the spectrum of the Maxwell operator. ( See , for example, [Meade, Winn, and Joannopoulos, Photonic Crystals, Princeton University Press, 1995].).