Floquet–Bloch solutions in a sawtooth photonic crystal

Band structure of a sawtooth photonic crystal for optical wave propagation along the axis of periodicity is investigated. Floquet-Bloch solutions are found and illustrated for the bandgaps, allowed bands, and bandedges of the crystal. Special attention is given to the cases where Floquet-Bloch solutions become periodic functions

Inequalities among eigenvalues of Sturm–Liouville problems

<p/> <p>There are well-known inequalities among the eigenvalues of Sturm&#8211;Liouville problems with periodic, semi-periodic, Dirichlet and Neumann boundary conditions. In this paper, for an arbitrary coupled self-adjoint boundary condition, we identify two separated boundary conditions corresponding to the Dirichlet and Neumann conditions in the classical case, and establish analogous inequalities. It is also well-known that the lowest periodic eigenvalue is simple; here we prove a similar result for the general case. Moreover, we show that the algebraic and geometric multiplicities of the eigenvalues of self-adjoint regular Sturm&#8211;Liouville problems with coupled boundary conditions are the same. An important step in our approach is to obtain a representation of the fundamental solutions for sufficiently negative values of the spectral parameter. Our approach yields the existence and boundedness from below of the eigenvalues of arbitrary self-adjoint regular Sturm&#8211;Liouville problems without using operator theory.</p