Generalized theory of interference, and its applications

Abstract

The superposition of two partially coherent but completely polarised beams is discussed. The formula for the intensity of the resultant beam is obtained from the interference formula for coherent beams by multiplying the third interference term by the degree of coherence γ (defined statistically). The states of the two given polarised beams and that of the resultant incompletely polarised beam may be characterised by respective vectors drawn from the centre of the Poincare sphere : the length of each vector and its orientation (i.e., point of intersection with the sphere) may be regarded as giving respectively the intensity and state of polarisation, of the polarised fraction of the corresponding beam. The vector for the resultant beam is obtained by adding to the sum of the two given vectors (which are directed towards points A and B), a third vector directed towards a point C″ on the Poincare sphere. This last vector which arises because of the interference of the beams, is specified in terms of the angles of the triangle ABC″, which is isoscles: the base angles A and B are both equal to the effective phase difference δ and the length of the vector is equal to\(2\gamma \sqrt {I_1 I_2 } \sin \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} \hat C''\).