STOKES PARAMETERS


A set of values used to specify the phase and polarization of radiation. There are several different notations. Let and form a right-handed orthogonal coordinate system with the propagation direction s. Write the total intensity as

(1)

For monochromatic light, and will be uniquely specified if the degree of linear polarization
(2)

is known, so either or (I, Q) is known.

Let

(3)
(4)

so
(5)

where a and b are the semimajor axes of the elliptically polarized light, and is the angle between and the major axis. Define
(6)
(7)

The Stokes parameters may then be specified by any four of I, Q, U, V, , and . Physically, there are only three parameters: the phase difference and semimajor and semiminor axes. Therefore, for monochromatic light, there must be an equation connecting the four parameters. It is
(8)

However, this equality becomes an inequality for quasi-monochromatic light. For unpolarized light, , so . The Stokes parameters may be measured in the lab by measuring intensities transmitted through combinations of horizontal, vertical, , and left- and right-circular polarizers.


The Stokes parameters can also be defined in terms of the electric fields of the radiation. A traveling plane wave is denoted by

(9)

For convenience, separate E into parallel and perpendicular components and re-express as
(10)
(11)
(12)
(13)

The Stokes parameters may then be defined as
(14)
(15)
(16)
(17)

where
(18)

For quasi-monochromatic light,
(19)
(20)
(21)
(22)

In this case, (8) becomes


(23)

so
(24)


The Stokes parameters are related to the ellipsometric parameters by

(25)
(26)
(27)
(28)

where
(29)
(30)

and is the clockwise angle between and the major axis. It also follows that
(31)




The degree of polarization is

(32)

degree of linear polarization is
(33)

and degree of circular polarization is
(34)


Rotating the electric field vector through an angle ,

 
  (35)

so
 
  (36)
 
  (37)
 
  (38)
 
  (39)

These give
(40)
 
   
   
  (41)
 
   
  (42)
 
   
  (43)

Summarizing,
(44)


This can be verified by rotating the polarization clockwise by an angle , equivalent to rotating the axis clockwise by . From (31), for the unrotated polarization,

(45)

so the rotated polarization parameters are
 
  (46)
 
  (47)

and the orientation of the rotated polarization ellipse is
(48)


Mueller Matrices