In order to express the polarization states of the incident and refracted rays of a birefringent crystal more accurately by Jones vectors, with the aid of ellipsoid of wave normals and material equations, the electric fields of incident and refracted beams are considered as combination of o component and e component which can be decomposed into directions parallel and perpendicular to the incident plane. The relationship between electric components of incident and refracted waves is expressed by transmission coefficients from Fresnel equations. Considering the differences between the electric field vector and displacement vector of e ray, Fresnel equations are modified to meet the boundary conditions. Finally, Jones vectors for the electric fields of the incident and refracted rays are obtained. And numerical examples are given to show that there is difference between conventional and modified transmission coefficients, and the difference between the electric field vector and displacement vector of e ray cannot be neglected.