Stokes parameters
The Stokes parameters are a set of values that describe the polarization state of electromagnetic radiation. They were defined by George Gabriel Stokes in 1852, as a mathematically convenient alternative to the more common description of incoherent or partially polarized radiation in terms of its total intensity ("I"), (fractional) degree of polarization ("p"), and the shape parameters of the polarization ellipse. The effect of an optical system on the polarization of light can be determined by constructing the Stokes vector for the input light and applying Mueller calculus, to obtain the Stokes vector of the light leaving the  system. The original Stokes paper was discovered independently by Francis Perrin in 1942 and by Subrahamanyan Chandrasekhar in 1947, who named it as the Stokes parameters. Definitions. The relationship of the Stokes parameters "S", "S", "S", "S" to intensity and polarization ellipse parameters is shown in the equations below and the figure on the right. Here formula_2, formula_3 and formula_4 are the spherical coordinates of the three-dimensional vector of cartesian coordinates formula_5. formula_6 is the total intensity of the beam, and formula_7 is the degree of polarization, constrained by formula_8. The factor of two before formula_9 represents the fact  that any polarization ellipse is indistinguishable from one rotated by 180°, while the factor of two before formula_10 indicates that an ellipse is indistinguishable from one with the semi-axis lengths swapped accompanied by a 90° rotation. The phase information of the polarized light is not recorded in the Stokes parameters. The four Stokes parameters are sometimes denoted "I", "Q", "U" and "V", respectively. Given the Stokes parameters, one can solve for the spherical coordinates with the following equations: Stokes vectors. The Stokes parameters are often combined into a vector, known as the Stokes vector: The Stokes vector  spans the space of unpolarized, partially polarized, and fully polarized light. For comparison, the Jones vector only spans the space of fully polarized light, but is more useful for problems involving coherent light. The four Stokes parameters are not a preferred coordinate system of the space, but rather were chosen because they can be easily measured or calculated. Note that there is an ambiguous sign for the formula_13 component depending on the physical convention used. In practice, there are two separate conventions used, either defining the Stokes parameters when looking down the beam towards the source (opposite the direction of  light propagation) or looking down the beam away from the source (coincident with the direction of light propagation). These two conventions result in different signs for formula_13, and a convention must be chosen and adhered to. Examples. Below are shown some Stokes vectors for common states of polarization of light. Alternative explanation. A monochromatic plane wave is specified by its propagation vector, formula_15, and the complex amplitudes of the electric field, formula_16 and formula_17, in a basis formula_18. The pair formula_19 is called a Jones vector. Alternatively, one may specify the propagation vector, the phase, formula_20, and the polarization state , formula_21, where formula_21 is the curve traced out by the electric field as a function of time in a fixed plane. The most familiar polarization states are linear and circular, which are degenerate cases of the most general state, an ellipse. One way to describe polarization is by giving the semi-major and semi-minor axes of the polarization ellipse, its orientation, and the direction of rotation (See the above figure). The Stokes parameters formula_6, formula_24, formula_25, and formula_13, provide an alternative description of the polarization state which is experimentally convenient because each parameter corresponds to a sum or difference  of measurable intensities. The next figure shows examples of the Stokes parameters in degenerate states. Definitions. The Stokes parameters are defined by where the subscripts refer to three different bases of the space of Jones vectors: the standard Cartesian basis (formula_28), a Cartesian basis rotated by 45° (formula_29), and a circular basis (formula_30). The circular basis is defined so that formula_31, formula_32. The symbols ⟨⋅⟩ represent expectation values. The light can be viewed as a random variable taking values in the space "C" of Jones vectors formula_19. Any given measurement yields a specific wave (with a specific phase, polarization  ellipse, and magnitude), but it keeps flickering and wobbling between different outcomes. The expectation values are various averages of these outcomes. Intense, but unpolarized light will have "I" > 0 but "Q" = "U" = "V" = 0, reflecting that no polarization type predominates. A convincing waveform is depicted 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 formula_19. 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" | = |"Q" | + |"U" | + |"V" |, 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 (formula_28) basis, the Stokes parameters when using an "increasing phase convention  are while for formula_38, they are and for formula_40, 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 formula_44, so that where formula_46 is the total polarization fraction. Let us define the complex intensity of linear polarization to be Under a rotation formula_48 of the polarization ellipse, it can be shown that formula_6 and formula_13 are invariant, but With these properties, the Stokes 	Stokes parameters
14438703	parameters may be thought of as constituting three generalized intensities: where formula_6 is the total intensity, formula_54 is the intensity of circular polarization, and formula_55 is the intensity of linear polarization. The total intensity of polarization is formula_56, and the orientation and sense of rotation are given by Since formula_58 and formula_59, 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	Stokes parameters
14438704	-one correspondence with the closed, convex, 4-real-dimensional cone of nonnegative Hermitian operators on the Hilbert space C. 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 of the operator can be calculated from the polarization ellipse parameters "I", "p", "ψ", "χ". The Stokes parameters with "I" set equal to 1 (i.e. the trace 1 operators) are in one-to-one correspondence  with the closed unit 3-dimensional ball of mixed states (or density operators) of the quantum space C, whose boundary is the Bloch sphere. The Jones vectors correspond to the underlying space C, that is, the (unnormalized) pure states of the same system. Note that phase information is lost when passing from a pure state |φ⟩ to the corresponding mixed state |φ⟩⟨φ|, just as it is lost when passing from a Jones vector to the corresponding Stokes vector.