at the article on coherence. The opposite would be perfectly polarized light which, in addition, has a fixed, nonvarying amplitude—a pure sine curve. This is represented by a random variable with only a single possible value, say . In this case one may replace the brackets by absolute value bars, obtaining a well-defined quadratic map from the Jones vectors to the corresponding Stokes vectors; more convenient forms are given below. The map takes its image in the cone defined by |I |2 = |Q |2 + |U |2 + |V |2, where the purity of the state satisfies p = 1 (see below). The next figure shows how the signs of the Stokes parameters are determined by the helicity and the orientation of the semi-major axis of the polarization ellipse. Representations in fixed bases In a fixed () basis, the Stokes parameters when using an increasing phase convention are while for , they are and for , they are Properties For purely monochromatic coherent radiation, it follows from the above equations that whereas for the whole (non-coherent) beam radiation, the Stokes parameters are defined as averaged quantities, and the previous equation becomes an inequality: However, we can define a total polarization intensity , so that where is the total polarization fraction. Let us define the complex intensity of linear polarization to be Under a rotation of the polarization ellipse, it can be shown that and are invariant, but With these properties, the Stokes parameters may be thought of as constituting three generalized intensities: where is the total intensity, is the intensity of circular polarization, and is the intensity of linear polarization. The total intensity of polarization is , and the orientation and sense of rotation are given by Since and , we have Relation to the polarization ellipse In terms of the parameters of the polarization ellipse, the Stokes parameters are Inverting the previous equation gives Relationship to Hermitian operators and quantum mixed states From a geometric and algebraic point of view, the Stokes parameters stand in one-to-one correspondence with the closed, convex, 4-real-dimensional cone of nonnegative Hermitian operators on the Hilbert space C2. The parameter I serves as the trace of the operator, whereas the entries of the matrix of the operator are simple linear functions of the four parameters I, Q, U, V, serving as coefficients in a linear combination of the Stokes operators. The eigenvalues and eigenvectors