For a three dimensional plane wave analysis, the propagation of a single mode wave E(r)ei??t 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.