From Poynting vector to new degree of freedom of polarization

Abstract: Up till now, the Jones vector is, strictly speaking, only a notion about the state of polarization of plane electromagnetic waves though it is generally applied to paraxial fields approximately. Here we generalize it to non-paraxial fields. The same as the Jones vector for plane waves, the generalized Jones vector for non-paraxial fields is global in the sense that it does not depend on the field position. This is achieved by investigating the effect of the polarization on the Poynting vector in the non-paraxial superposition of four plane waves. Even more importantly, by doing so we find that in addition to the Jones vector, another degree of freedom, called the Stratton vector, is also needed to completely describe the state of polarization of non-paraxial fields. It is shown that the polarization described by the global Jones vector is dependent on the position. The position dependence of the polarization originates in the position dependence of the polarization bases. The Stratton vector specifies the way in which the polarization bases depend on the position. A general expression for the dependence of the Poynting vector on the Stratton and Jones vectors is also given.