Conditions for the Physical Realizability of Polarization Matrices Characterizing Passive Systems

Abstract

We derive conditions for the physical realizability of polarization matrices characterizing passive systems or scattering media. By physically realizable, we mean that 0  g  1 where g ≡ (output intensity/input intensity). Using the singular-value decomposition of an arbitrary 2 × 2 complex-valued matrix, we prove that a Jones matrix T _J is physically realizable if 0  det T _J ⁺ T _J  1. Consequently singular Jones matrices (i.e. det T _J = 0) completely extinguish the output intensity irrespective of the input intensity because g ≡ 0. Corresponding results are obtained for Mueller-Jones matrices (the 4 × 4 real-valued matrices which are the four-dimensional representations of the two-dimensional 2 × 2 complex-valued Jones matrices). We also study the problem for general Mueller matrices; however because of their phenomenological character they do not admit of such criteria as do the Jones and Mueller-Jones matrices. This is because g now depends upon the matrix elements of the Mueller matrix and the input Stokes parameters; whereas for the Jones and Mueller-Jones matrices, g only depends upon the matrix elements. Finally we study the problem of relating the input and output mean randomness.