Impurity modes from impurity clusters in photonic band structures

A Green function calculation for the impurity modes of single impurities and clusters of impurities in a photonic band structure is given. Specifically, a truncated two-dimensional periodic dielectric medium composed from a square lattice array of dielectric rods placed perpendicularly between two parallel perfectly conducting plates is studied. Impurities are introduced into this system substitutionally by replacing one (single impurity case) or a number (cluster impurity case) of the rods in the periodic system by impurity rods. Exact expressions for the impurity mode frequencies are obtained in terms of the Green functions of the pure system. These expressions are then evaluated for the case of impurities with small cross-sectional areas using numerically determined Green functions. Linear frequency-independent, linear frequency-dependent and non-linear impurity dielectric media are treated.