Differential and product Mueller matrix decompositions: A formal comparison

It is shown that the Mueller matrix logarithm and the Mueller matrix roots decompositions used for the extraction of the elementary polarization properties of a depolarizing medium, although being computationally different, are formally equivalent, being both based upon the differential representation of a continuously depolarizing medium. The common set of six elementary polarization properties provided by these two decompositions is generally different from that obtained from the various product decompositions summarized by the G-polar decomposition whereby the depolarization phenomenon is treated as being concentrated, and not uniformly distributed, within the medium. However, if the medium is weakly depolarizing, the two sets of elementary properties coincide to the first order in the depolarization and tend to the set of properties of the nondepolarizing estimate of the measured Mueller matrix obtained from its Cloude sum decomposition.