Polarizationpolarization 187transforms to훼−휙. Thus the orientation does not define a direction, it only refers the
polarization to the (푥,푦)-plane.
The photon is peculiar in lacking a longitudinal polarization state, and the polar-
ization is therefore not a vector in the (푥,푦)-plane; in fact it is a second-rank tensor.
This is connected to the fact that the photon is massless. Recall that the theory of
special relativity requires the photons to move with the speed of light in any frame.
Therefore they must be massless, otherwise one would be able to accelerate them to
higher speeds, or decelerate them to rest.
In a way, it appears as if there existed two kinds of photons. Physics has taken this
into account by introducing an internal property,spin. Thus, one can talk about the
two polarization states or about the two spin states of the photon.
Let us now turn to the Stokes parameters in Equation (8.26). The parameter퐼,
which describes the intensity of radiation, is, like푉, a physical observable indepen-
dent of the coordinate system. In contrast, the parameters푄and푈depend on the
orientation of the coordinate system. In the geometry of Figure 8.4, the coordinates
푥′,푦′define a plane wave of incoming radiation propagating in the푧′direction (primes
are used for unscattered quantities). The incoming photon훾′then Thomson scatters
against an electron and the outgoing photon훾continues as a plane wave in a new
direction,푧.
It follows from the definition of the Stokes parameters푄and푈that a rotation of
the푥′-and푦′-axes in the incoming plane by the angle휙transforms them into
푄(휙)=푄cos( 2 휙)+푈sin( 2 휙),푈(휙)=−푄sin( 2 휙)+푈cos( 2 휙). (8.28)We left it as an exercise (Problem 2) to demonstrate that푄^2 +푈^2 is invariant under
the rotation in Equation (8.28). It follows from this invariance that the polarization
푃is a second rank tensor of the form
푃=1
2
(
푄푈−i푉
푈+i푉 −푄)
. (8.29)
Thus the polarization is not a vector quantity with a direction unlike the electric field
vectorE.
Let us now see how Thomson scattering of the incoming, unpolarized radiation
generates linear polarization in the (푥,푦)-plane of the scattered radiation (we fol-
low closely the pedagogical review of A. Kosowsky [9]). The differential scattering
cross-section, defined as the radiated intensity퐼divided by the incoming intensity퐼′
per unit solid angle훺and cross-sectional area휎B, is given by
d휎T
d훺=퐼
퐼′
=
3 휎T
8 휋휎B
|푖′⋅i|^2 ≡퐾|푖′⋅i|^2. (8.30)Here휎Tis the total Thomson cross-section, the vectors푖′,iare the unit vectors in the
incoming and scattered radiation planes, respectively (cf. Figure 8.4), and we have
lumped all the constants into one constant,퐾. The Stokes parameters of the outgoing
radiation then depend solely on the nonvanishing incoming parameter퐼′,
퐼=퐾퐼′( 1 +cos^2 휃),푄=퐾퐼′sin^2 휃, 푈= 0 , (8.31)