Covariant form of Sommerfeld's formula for spherical wave and application in fiber optics

It is observed that the Sommerfeld's formula for a progressive spherical wa ve is not covariant with respectro the rotation of the coordinate system. I t contains complex partial waves that are not physical. The standing wave r epresentation of the same formula is shown to be covariant and physical. It is applied to solve the boundary value problem of fiber optics. The radiat ion modes as well as the guided normal modes are obtained in closed forms. The intensity distribution in various modes is obtained. It is remarked tha t the mode of critical refraction is missing.