186 Cosmic Microwave Background
yʹyxxʹ zʹβθφze–γʹγʹFigure 8.4The geometry used in the text for describing the polarization of an incoming unpo-
larized plane wave photon,훾′in the (푥′,푦′)-plane, which is Thomson scattering against an elec-
tron, and subsequently propagating as a polarized plane wave photon,훾,inthe푧-direction.
A well-known property of light is its two states ofpolarization. Unpolarized light
passing through a pair of polarizing sunglasses becomes vertically polarized. Unpolar-
ized light reflected from a wet street becomes horizontally polarized. The advantage
of polarizing sunglasses is that they block horizontally polarized light completely, let-
ting all the vertically polarized light through. Their effect on a beam of unpolarized
sunlight is to let, on average, every second photon through vertically polarized, and
to block every other photon as if it were horizontally polarized: it is absorbed in the
glass. Thus the intensity of light is also reduced to one-half.
Polarized and unpolarized light (or other electromagnetic radiation) can be
described by theStokes parameters, which are the time averages (over times much
longer than 1∕휈)
퐼≡⟨푎^2 푥⟩+⟨푎^2 푦⟩,푄≡⟨푎^2 푥⟩−⟨푎^2 푦⟩,
푈≡⟨ 2 푎푥푎푦cos(휃푥−휃푦)⟩,푉≡⟨ 2 푎푥푎푦sin(휃푥−휃푦)⟩.}
(8.26)
The parameter퐼gives the intensity of light, which is always positive definite. The
electromagnetic field is unpolarized if the two components in Equation (8.25) are
uncorrelated, which translates into the condition푄=푈=푉=0. If two components
in Equation (8.25) are correlated, they either describe light that islinearly polarized
along one direction in the (푥,푦)-plane, orcircularly polarizedin the plane. In the linear
case푈=0or푉=0, or both. Under a rotation of angle휙in the (푥,푦)-plane, the quantity
푄^2 +푈^2 is an invariant (Problem 2) and theorientationof the polarization
훼≡^1
2arctan(푈∕푄) (8.27)