For a three dimensional plane wave analysis, the propagation of a single mode wave E(r)eiout of frequency ??=??(k) through a periodic dielectric structure can be expressed: ??? ??? ??? ??? E ( r , t ) = ?? ( r ) ?? 2 c 2 E ( r , t ) eqn . 1 r is a three dimensional position vector, k is the three dimensional wave vector. c is the vacuum velocity of light, and ??? is a dielectric function.