Topics in Particle and Dispersion Science

Stokes vector

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The state of polarization and irradiance of a beam of electromagnetic radiation (ER) can be completely described by a set of four parameters referred to as the Stokes vector (for example, Hecht E 1987, Bohren CF and Huffman 1983). Each element of that vector has a unit of irradiance, i.e. [power length^-2], for example, W m^-2. The elements of the Stokes vector are defined as follows:

S0     =      I     =      <ElEl*> + <ErEr*>      S1          Q          <ElEl*> - <ErEr*>      S2          U          <ElEr*> + <ErEl*>      S3          V          i (<ElEr*> - <ErEl*>)

(1)

where two popular notations for the Stokes vector elements are shown on the left. Note that the index of 0 used for the top element of the stokes vector in the left-hand-side notation conflicts with the popular notation for the scattering matrix, whose top-left element is usually denoted M₁₁. In the above equation, <x> denotes the time average of x, E_l is the component of the electric field of the light wave parallel to a reference plane, E_r is the component of the electric field of the electromagnetic wave perpendicular to that reference plane, the asterisk denotes the complex conjugate, and i² = -1. The term "reference plane" implies immediately that the Stokes vector is defined relative to such a plane. If another plane, rotated by an angle, α about the beam direction, is later chosen as a reference plane, the Stokes vector must be transformed to reflect the new orientation of the reference plane (Fig. 1). Such transformation is necessary, for example, in Monte Carlo simulation of transfer of polarized ER in turbid media (for example, Bartel S and Hielscher 2000, Bohren CF and Huffman 1983).

This transformation uses a 4x4 rotation matrix, R(α), (for example, Bartel S and Hielscher 2000):

S'	= R(α) S
	=     1  0  0  0    S    0  cos(2α)  sin(2α)  0      0  -sin(2α)  cos(2α)  0      0  0  0  1	(2)

It can be seen that I, Q ² + U ², and V are invariant under such a transformation.

For an unpolarized beam <E_lE_l^*> = <E_rE_r^*> and <E_lE_r^*> = <E_rE_l^*> = 0. Hence, only the parameter I is nonzero. This is a particular case of a general relationship between elements of the Stokes vector:

I 2 ≥ Q 2 + U 2 + V 2

(3)

for a partially polarized beam. The equality applies to completely polarized ER beam. This leads to a definition of the degree of polarization, P.

If several incoherent ER beams (i.e. beams whose EM waves have no fixed phase relationships to each other) are superimposed, the Stokes vector of the combined beam is a sum of the Stokes vectors of the component beams, which follows from the fact that the elements of the Stokes vector are irradiances.

CITATION: Jonasz M. 2006. Stokes vector (www.tpdsci.com/Tpc/StoVec.php). In: Top. Part. Disp. Sci. (www.tpdsci.com).

HISTORY: Published: 22-Mar-2006 Modified: 10-Mar-2008 Peer-reviewed: 11-Mar-2008

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