Stokes Parameters

The representation of polarized radiation described above can be cumbersome. At this point the concept of independent scattering is introduced. Independent scattering requires that interference of light scattered by different particles be undetectable (Hansen and Travis, 1974). The necessary condition for this to hold is that the scatterers must be separated by a few times their radius. To examine this, consider air at surface pressure, which has a number density, N, of about $2.5\times 10^{19}$ molecules$\cdot$cm^-3. The mean intermolecular distance is $(3/4\pi N)^{\frac{1}{3}}$ (Mihalas, 1978) or 4.5 nm which is a factor of 60 larger than the `radius' of a nitrogen molecule.

A consequence of independent scattering is that light may be described in the intensity domain and it will prove useful to adopt an alternate representation, known as Stokes parameters. The Stokes parameters are contained in a 4$\times$1 column matrix, called the Stokes vector,

$${\b I} = \left[ \begin{array}{c} I \\ Q \\ U \\ V \\ \en... ...ft[ \begin{array}{cccc} I & Q & U & V \end{array} \right]^{T}$$ (3.3)

where $[\hspace{1em}]^{T}$ denotes transpose. The Stokes vector will often be denoted as the transpose of a 1$\times$4 row matrix for the sake of brevity. Each element of the Stokes matrix can be written in terms of the time averages of the parallel and perpendicular components of the electric field,

$$<\cal{E}_{\parallel} \cal{E}_{\parallel}^* + \cal{E}_{\perp} \cal{E}_{\perp}^*>$$ I = (3.4)

$$<\cal{E}_{\parallel} \cal{E}_{\parallel}^* - \cal{E}_{\perp} \cal{E}_{\perp}^*>$$ Q = (3.5)

$$<\cal{E}_{\parallel} \cal{E}_{\perp}^* + \cal{E}_{\perp} \cal{E}_{\parallel}^*>$$ U = (3.6)

$$i<\cal{E}_{\parallel} \cal{E}_{\perp}^* - \cal{E}_{\perp} \cal{E}_{\parallel}^*>$$ V = (3.7)

where the asterisk represents the complex conjugate and <> identifies a time average. Each component has units of radiant intensity, or radiance (i.e. $\mu$W$\cdot$cm $^{-2} \cdot$nm $^{-1} \cdot$sr^-1). Radiance describes the amount of radiant energy passing normally through a differential area element in a direction confined to a differential solid angle element per unit time per unit wavelength interval. The Stokes representation is advantageous as all elements possess the same units and are real and measurable quantities. The Stokes elements for a single simple wave are related by I² = Q² + U² + V². After taking time averages, this identity becomes, $I^2 \geq Q^2 + U^2 + V^2$. The physical interpretation of each element is given below. The first, I, describes the total (polarized and unpolarized) radiance. If polarization is neglected, the Stokes vector reduces to the single element, I. The second, Q, is the radiance linearly polarized in the direction parallel or perpendicular to the reference plane. The third, U, is the radiance linearly polarized in the directions 45$^{\circ }$ to the reference plane, and the fourth, V, is the radiance circularly polarized.

Natural sunlight is an example of completely unpolarized light and so it could be represented by ${\b I}=[I,0,0,0]^T$. In general, however, light is partially polarized and so it consists of polarized and unpolarized components, I=I_pol + I_unpol. The degree of polarization, or fraction polarized, is defined as I_pol/I.

In terms of the Stokes components, the degree of polarization, P, is given by,

$$P = \frac{\sqrt{Q^2 + U^2 + V^2}}{I}$$ (3.8)

and the degree, of linear polarization, LP, is

$$LP = \frac{\sqrt{Q^2 + U^2}}{I}.$$ (3.9)

If U is much smaller than Q, an alternative definition is employed, LP=-Q/I, which preserves the sign of Q. Similarly, the degree of circular polarization, CP, is

$$CP = \frac{V}{I}$$ (3.10)

and the direction of polarization is given by the angle $\chi$,

$$\tan{2\chi}=\frac{U}{Q}.$$ (3.11)

The radiance components parallel and perpendicular to the plane of reference can also be expressed in terms of the Stokes parameters,

$$I_{\parallel} \frac{I + Q}{2}$$ = (3.12)

$$I_{\perp} \frac{I - Q}{2}.$$ = (3.13)