In the course of developing an operator formalism in a previous paper we established commutation relations between pairs of (2×2) matrix operators representing various physical devices. These commutation relations were established under the stringent requirement that the outgoing Jones vector remains unchanged, independent of the order in which the operators operate on an arbitrary (quasimonochromatic) incoming field. In this paper we seek a wider class of operators that behave as though they commute, by imposing weaker requirements: (1) that the outgoing Jones vector remains unchanged only up to an arbitrary constant phase factor. In this case the outgoing coherency matrix (or the Stokes parameters), however, remains unchanged. Independent of this, we can impose a still weaker requirement: (2) that the outgoing coherency matrix remains unchanged only as far as the degree of polarization and the state of polarization is concerned but may be allowed to change in intensity. This paper deals with these weaker requirements. Pertinent examples of physical devices and their series combinations are also given.
A. S. MARATHAY, "Extension of the Commutation Relations in the Theory of Partial Polarization," J. Opt. Soc. Am. 56, 619-623 (1966)