### 2. REVIEW OF STOKES PARAMETERS

Polarized light is conventionally described in terms of the Stokes parameters, which are presented in any optics text. Consider a nearly monochromatic plane electromagnetic wave propogating in the z-direction; nearly monochromatic here means that its frequency components are closely distributed around its mean frequency 0. The components of the wave's electric field vector at a given point in space can be written as

 (1)

The requirement that the wave is nearly monochromatic guarantees that the amplitudes ax and ay and the phase angles x and y will vary slowly relative to the inverse frequency of the wave. If some correlation exists between the two components in Eq. (1), then the wave is polarized.

The Stokes parameters are defined as the following time averages:

 (2) (3) (4) (5)

The averages are over times long compared to the inverse frequency of the wave. The parameter I gives the intensity of the radiation which is always positive and is equivalent to the temperature for blackbody radiation. The other three parameters define the polarization state of the wave and can have either sign. Unpolarized radiation, or ``natural light,'' is described by Q = U = V = 0.

The parameters I and V are physical observables independent of the coordinate system, but Q and U depend on the orientation of the x and y axes. If a given wave is described by the parameters Q and U for a certain orientation of the coordinate system, then after a rotation of the x - y plane through an angle , it is straightforward to verify that the same wave is now described by the parameters

 (6)

From this transformation it is easy to see that the quantity Q2 + U2 is invariant under rotation of the axes, and the angle

 (7)

transforms to - under a rotation by and thus defines a constant orientation, which physically is parallel to the electric field of the wave. The Stokes parameters are a useful description of polarization because they are additive for incoherent superposition of radiation; note this is not true for the magnitude or orientation of polarization.

While polarization has a magnitude and an orientation, it is not a vector quantity because the orientation does not have a direction, describing only the plane in which the electric field of the wave oscillates. Mathematically, the Stokes parameters are identical for an axis rotation through an angle of , whereas for a vector, such a rotation would lead to an inverted vector and a full rotation through 2 is required to return to the same situation. The transformation law in Eq. (6) is characteristic of the second-rank tensor

 (8)

which also corresponds to the quantum mechanical density matrix for an ensemble of photons (Kosowsky, 1996) (the matrix is 2 by 2 because the photon has two helicity states).