On the algebraic characterization of a Mueller matrix in polarization optics. II. Necessary and sufficient conditions for Jones-derived Mueller matrices

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.

Transmittance constraints in serial decompositions of depolarizing Mueller matrices: the arrow form of a Mueller matrix

“…A symmetric serial decomposition of a Mueller matrix M was developed by Ossikovski [36] that is closely related to the "normal form" dealt with previously by Xing [9], Sridhar and Simon [37], van der Mee [38], and Gopala Rao et al [39,40]:…”

From a nondepolarizing Mueller matrix to a depolarizing Mueller matrix

“…In this paper, we take advantage of the serial decomposition of a pure Mueller matrix and of the normal form [3][4][5][6] and symmetric decomposition [7] of a depolarizing Mueller matrix to obtain a constructive procedure for synthesizing a given arbitrary depolarizing Mueller matrix by identifying a reference pure Mueller matrix and varying its depolarizing properties through the correlative variation of the indices of polarimetric purity (hereafter IPP) defined by us in previous works [8].…”

“…Nevertheless, this is not a straightforward task because M Δ is required to be a Mueller matrix, which, in turn, entails the non-negativity of the coherency matrix C Δ associated with M Δ (Cloude's criterion [17,18]). Therefore, the following inequalities must be satisfied [19,6,20]:…”

Explicit algebraic characterization of Mueller matrices

“…The said statistical concept of a Mueller matrix entails more restrictive conditions (inequalities) than the very fact that a Mueller matrix is a linear transformation of Stokes vectors into Stokes vectors [2,[19][20][21][22][23][24][25][26] (not overpolarizing criterion, also called Stokes criterion). The names "Stokes matrices" [6,21,27] and "pre-Mueller matrices" [28,29] have been used for matrices satisfying the Stokes criterion, but several physical arguments have been pointed out to justify that only the subset of Stokes matrices that satisfy Cloude's criterion are physically realizable and can properly be called Mueller matrices [5,15,[23][24][25][26][27][28][29][30].…”

“…The names "Stokes matrices" [6,21,27] and "pre-Mueller matrices" [28,29] have been used for matrices satisfying the Stokes criterion, but several physical arguments have been pointed out to justify that only the subset of Stokes matrices that satisfy Cloude's criterion are physically realizable and can properly be called Mueller matrices [5,15,[23][24][25][26][27][28][29][30]. A recent interesting discussion on this key issue of polarization algebra in terms of Cloude's physical realizability of differential (local realizability) and integrated Mueller matrices can be found in [31].…”