There are many ways to represent the polarization of electromagnetic radiation. A convenient one that is used in the description of CMB fluctuations was introduced by Stokes.

(a) Consider a monochromatic wave propagating along e_z with electric vector E = {E_x , E_y}e^iωt, where the components are complex numbers. Explain why this wave is completely polarized, and introduce the Stokes’ parameters (followingJackson, 1999)  

and

Sketch the behavior of the electric vector as the complex ratio r = E_y/E_x is varied, and hence associate Q, U, and V with different states of polarization.

(b) Derive the transformation laws for the Stokes’ parameters if we rotate the e_x, e_y directions about e_z through an angle ψ.

(c) Show that Q² + U² + V² = I² and that the polarization of the wave may be represented as a point on a sphere, which you should identify.

(d) Now suppose that the wave is polychromatic and partially polarized, and replace the definitions of I , Q, U, and V with the time averages (E_xE_x^∗), and so forth. Show that Q² + U² + V² 2, and give expressions for the degree of polarization and the associated position angle in terms of I , Q, U, and V.

Transcribed Image Text:

I = E₂E* + E₂E, Q = E.EE,E, U = 2R(EE),