A detailed derivation of eigenvalue equation in two dimensional and three dimensional photonic crystals

A detailed derivation of eigenvalue equation in two dimensional and three dimensional photonic crystals is given by the plane-wave expansion method. Some mathematical formulas such as the rotation of vector, the gradient of scalar, the divergence of the vector, the vector triple product and the conversion between scalar and vector are employed. The eigenvalue equation in photonic crystals has become the important base for obtaining the band structure and the distribution of eigenmode.