Prospects of Wannier functions in investigating photonic crystal all-optical devices for signal processing

Wannier functions derived from Bloch functions have been identified as an efficient means of analyzing the properties of photonic crystals in which localized functions have now opened the door for 2D and 3D structures containing defects to be investigated. In this paper, based on the Maxwell equations in diagonalized form and utilizing Bloch waves we have obtained an equivalent system of algebraic equations in eigenform. By establishing and exploiting several distinct properties of the resulting eigenpairs, we demonstrate an ability to construct Wannier functions associated with the simplest one-dimensional photonic structure. More importantly, the numerical investigation of the inner- and intra-band orthonormality conditions as well as Hilbert space partitioning features shows a capability for multi-resolution analysis that will make all-optical signal processing devices with photonic crystal structures feasible.