No abstract

We show that every Mueller matrix, that is a real 4 x 4 matrix M which transforms Stokes vectors into Stokes vectors, may be factored as M = L2KLl where L1 and L2 are orthochronous proper Lorentz matrices and K is a canonical Mueller matrix having only two different forms, namely a diagonal form for type-I Mueller matrices and a non-diagonal form (with only one non-zero off-diagonal element) for type-I1 Mueller matrices. Using the general forms of Mueller matrices so derived, we then obtain the necessary and sufficient conditions for a Mueller matrix M to be Jones derived. These conditions for Jones derivability, unlike the Cloude conditions which are expressed in terms of the eigenvalues of the Hermitian coherency matrix T associated with M, characterize a Jones-derived matrix M through : he G eigenvalues and G eigenvectors of the real symmetric N matrix N = MGM associated with M. Appending the passivity conditions for a Mueller matrix onto these Jones-derivability conditions, we then arrive at an algebraic identification of the physically important class of passive Jones-derived Mueller matrices.

The classification of polarization elements, the polarization affecting optical devices that have a Jones-matrix representation according to the type of eigenvectors they possess, is given a new visit through the group-theoretical connection of polarization elements. The diattenuators and retarders are recognized as the elements corresponding to boosts and rotations, respectively. The structure of homogeneous elements other than diattenuators and retarders are identified by giving the quaternion corresponding to these elements. The set of degenerate polarization elements is identified with the so-called null elements of the Lorentz group. Singular polarization elements are examined in their more illustrative Mueller-matrix representation, and, finally the eigenstructure of a special class of singular Mueller matrices is studied.

We revisit the problem of identifying a Mueller matrix M through its N matrix N MGM where G is the familiar Minkowski matrix diag(1, -1, -1, -1) and the tilde denotes matrix transposition. Using the standard methods of reduction of symmetric matrices (tensors) to their canonical forms in Minkowski space, we then show that there exist only two algebraically distinct types of Mueller matrices, which we call types I and 11, and obtain the necessary and sufficient conditions for a Mueller matrix in terms of the eigenproperties of the associated N matrix. These conditions identify a Mueller matrix precisely and completely unlike the conditions derived earlier by Givens and Kostinski or by van der Mee. Observing that every Mueller matrix discussed hitherto in the literature is of the type I only, we construct examples of type-I1 Mueller matrices using the more familiar type-I (in fact pure Mueller) Mueller matrices. Further, we show that every G eigenvalue of an N matrix (see section 2 for a definition) is necessarily non-negative. Using this result, in an accompanying paper, we derive a general three-term factorization of a Mueller matrix which yields the general forms of Mueller and Jones-derived Mueller matrices and completely solves the problem of their algebraic structure.

The statistical ensemble formalism of Kim et al [J. Opt. Soc. Am. A4, 433 (1987)] offers a realistic model for characterizing the effect of stochastic nonimage-forming optical media on the state of polarization of transmitted light. With suitable choice of the Jones ensemble, various Mueller transformations-some of which are hitherto unknown-are deduced. It is observed that the ensemble approach is formally identical to the positive-operator-valued measures (POVMs) on the quantum density matrix. This observation, in combination with the recent suggestion by Ahnert and Payne [Phys. Rev. A71, 012330-1 (2005)]-in the context of generalized quantum measurement on single photon polarization states-that linear optics elements can be employed in setting up all possible POVMs enables us to propose a way of realizing different types of Mueller devices.