Quaternion algebra for Stokes–Mueller formalism

Abstract

It is shown that the Stokes-Mueller formalism can be reformulated in terms of quaternions, and the quaternion approach is more suitable for the formalism of Mueller-Jones states that we have recently described. In terms of quaternions it can be shown that the vector and matrix states and the Jones matrix associated to nondepolarizing optical systems are different representations isomorphic to the same quaternion state, and this quaternion state turns out to be the rotator of the Stokes quaternion. It is also shown that the coherent linear combination of nondepolarizing optical media states and depolarization phenomena can be reformulated in terms of quaternion states.

Mueller matrix polarimetry of bianisotropic materials [Invited]

“…Besides allowing to calculate this non-depolarizing estimate, coherency matrices are essential for a deeper mathematical understanding of Mueller matrices [21]. For example, they are useful to model situations of partial coherence [22,23], to study depolarizing Mueller matrices from a statistical point of view [24] or to define a quaternion algebra for the Stokes-Mueller formalism [25]. A widely used method to quantify depolarization in experimental Mueller matrices is the depolarization index (DI) [26]:…”

Transforming nondepolarizing Mueller matrices into Jones matrices

“…A method for transforming non-depolarizing Mueller matrices into Jones matrices was given by Chipman [3]. In this note it is shown that transformation of non-depolarizing Mueller matrices into Jones matrices can also be accomplished by means of a four dimensional complex vector which is isomorphic to the Jones matrix [4][5][6].…”

Mueller Quaternion with Matrix Coefficients for Depolarizing and Nondepolarizing Media

“…In a recent work [3] it was shown that Z matrices and |h vectors are actually two different representations of the same quantity which is isomorphic to the h quaternion by observing that the Z matrix can be written as a linear combination of four basis matrices:…”

“…which is directly related to the covariance vector |h . It was also shown that the Jones matrix is also isomorphic to the quaternion h by writing the Jones matrix in terms of Pauli matrices [3]:…”

“…where M is a nondepolarizing Mueller matrix. In order to incorporate the depolarization effects into the formalism the following procedure was followed [3]:…”