Stokes parameters
The Stokes parameters are a set of values that describe the polarization state of electromagnetic radiation.
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.
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).
Below are shown some Stokes vectors for common states of polarization of light.
is the curve traced out by the electric field as a function of time in a fixed plane.
, 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 Stokes parameters are defined by[citation needed] where the subscripts refer to three different bases of the space of Jones vectors: the standard Cartesian basis (
The light can be viewed as a random variable taking values in the space C2 of Jones vectors
The opposite would be perfectly polarized light which, in addition, has a fixed, nonvarying amplitude—a pure sine curve.
In this case one may replace the brackets by absolute value bars, obtaining a well-defined quadratic map[citation needed] 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.
, they are 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:[6] However, we can define a total polarization intensity
are invariant, but With these properties, the Stokes parameters may be thought of as constituting three generalized intensities: where
[7] Using a linear polarizer and a quarter-wave plate, the following system of equations relating the Stokes parameters to measured intensity can be obtained:[8]
is the irradiance of the radiation at a point when the linear polarizer is rotated at an angle of
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 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 C2, whose boundary is the Bloch sphere.
The Jones vectors correspond to the underlying space C2, that is, the (unnormalized) pure states of the same system.
Note that the overall phase (i.e. the common phase factor between the two component waves on the two perpendicular polarization axes) 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.
It's easy to see that these states are the eigenvectors of Pauli matrices, and that the normalized Stokes parameters (U/I, V/I, Q/I) correspond to the coordinates of the Bloch vector (
Generally, a linear polarization at angle θ has a pure quantum state
; therefore, the transmittance of a linear polarizer/analyzer at angle θ for a mixed state light source with density matrix
With this configuration, if we perform the rotating analyzer method to measure the extinction ratio, we will be able to calculate
The transmittance of the resulting light through a linear polarizer (analyzer plate) along the horizontal axis can be calculated using the same Rodrigues' rotation formula and focusing on its components on
For purely circularly polarized light, T has a sinusoidal dependence on angle θ with a period of 180 degrees, and can reach absolute extinction where T=0.
Similarly, the effect of a half-wave plate rotated by angle θ is described by
, which transforms the density matrix to: The above expression demonstrates that if the original light is of pure linear polarization (i.e.
Such rotation of the linear polarization has a sinusoidal dependence on angle θ with a period of 90 degrees.
