Plane wave solutions of biaxial crystal in principal axis coordinate system

When a plane wave propagates along any direction except the direction of the optical axes in biaxial crystal, there are only two solutions which have different propagation constants and polarizations. The plane wave solutions of a biaxial crystal in principal coordinate system were derived and the spatial dispersion relation of each solution was obtained by using Lagrange multiplier method and the ellipsoid concept of refractive index of the biaxial crystal. Proceeding from the obtained solution of the plane wave, the propagation of the plane wave along the optical axis is discussed. The obtained result was applied in uniaxial crystal for verification.