Overview of Mueller and Jones Calculi

From page 122, in Shurcliff, Polarized Light, Harvard University Press, 1966.

The Mueller calculus can handle problems involving depolarization. The Jones calculus cannot.
The Mueller calculus is based on a phenomenological foundation, hence does not depend on the validity of the electromagnetic theory. The Jones calculus is derived directly from that theory.
The Jones calculus permits one to preserve information as to absolute phase. The Mueller calculus does not; indeed it categorically pays no heed to phase.
The Jones calculus is well suited to the handling of problems involving the combining of two beams that are coherent. The Mueller calculus is unable to do so, except perhaps with great difficulty.
The Mueller calculus employs a vector (Stokes vector) whose first term indicates the intensity directly. The vector employed by the Jones calculus does not do so; to find the intensity one must obtain the sum of the squares of the elements.
The Jones matrices employ elements associated with amplitude transmittance. The Mueller matrix elements are associated with intensity transmittance.
The Jones calculus is well suited to problems involving a large number of similar devices that are arranged in series in a regular manner, and permits an investigator to arrive at an answer expressed explicitly in terms of n, the number of such devices; see Chapter 9. The Mueller calculus is not suited to such purpose.
The Jones matrix of a train of absorbing or nonabsorbing, non-depolarizing polarizers and retarders contains no redundant information: the matrix contains four elements comprising eight constants, and none of these constants is a function of any other. The Mueller matrix of such a train contains much redundancy: sixteen constants appear, and only seven of these are independent.
The Jones matrix of a birefringent dichroic device may be differentiated to reveal information as to the device’s intensive properties. The Mueller calculus — in practice — lacks this capability.