Polarization and Stokes ParametersInitial question: How can we measure astrophysical magnetic fields?In the last class we had some discussion about the polarization of a plane wave, butnow we need to go into it in more detail. Shu has more on this than Rybicki and Lightmando, so we’ll follow Shu.Let’s first consider a single, monochromatic, wave. Ask class: if the wave is propagatingin the z direction, what are the possible directions of linear polarization at any giveninstant? Since the wave is transverse, the linear polarization must be in the x-y plane. Wecould therefore break down the electric field into x and y components:E =ˆxEx+ˆyEy(1)whereˆx andˆy are unit vectors in the x and y directions. Ask class: is there a unique wayto define the x and y directions? No, in general there isn’t. That means that angles definedwith respect to a specific choice of the axes are to some extent arbitrary. However, if onesticks with a particular definition of axes, the differences in angles between different sourcescan be meaningful. It’s an important distinction to make.Ask class: have we exhausted the possible description of the polarization of amonochromatic wave? Specifically, since we can describe the electric field vector by aparticular combination of linear polarizations, will it stay with that combination forever?No, in fact there can be a time variation as well. If the frequency of the wave vector is ω,then we haveE =ˆxExcos(ωt − φx) +ˆyEycos(ωt − φy) . (2)Here φxand φyare the phases at time t = 0. Ask class: are these phases independentlymeaningful? Again, no. Picking a different zero for the time doesn’t change anythingphysically measurable, but it would change φxand φy. Ask class: Is there a combinationof these phases that is meaningful? Yes, the difference is independent of the particular zeroof time. Once again, it is important to keep these kinds of things straight; if you like, it’s aform of symmetry.Anyway, our general form of the electric field vector will trace out an ellipse over time.The major axis of the ellipse will have a tilt angle χ with respect to the x axis. We can thendefine new principal axes ˆx′and ˆy′along the axes of the ellipse, and write the electric fieldE =ˆx′E1cos ωt +ˆy′E2sin ωt . (3)There is also an axis ratio; we can identify E20≡ E21+ E22= E2x+ E2y, and define another“angle” β so that E1= E0cos β and E2= E0sin β. For a monochromatic wave we canthen define the Stokes parameters, which are four quantities quadratic in the electric fieldcomponents:I = E2x+ E2y= E20Q = E2x− E2y= E20cos 2β cos 2χU = 2ExEycos(φy− φx) = E20cos 2β sin 2χV = 2ExEysin(φy− φx) = E20sin 2β(4)Again we emphasize that this is for a monochromatic wave; we’ll get to what happens withsuperpositions of waves in a bit.Note that, as expected physically, only the phase difference ∆φ = φy− φxmattersrather than the independent phases. Note also that since the three parameters E0, β, and χdetermine the four parameters I, Q, U, and V , there must be a relation between the Stokesparameters. It’s I2= Q2+ U2+ V2(again, this only holds for 100% polarized waves). Onealso has the relationstan 2χ = U/Q, sin 2β = V/I . (5)Ask class: if V = 0, what does that mean for β? It means that β = 0 or ±π/2. Askclass: what does that tell us about the electric field? It means that it stays along one ofthe principal axes, meaning it’s linearly polarized. If instead Q = U = 0 (so that V = I),then β = ±π/4 and the axes are equal, so the electric field traces out a circle on the skyand the light is circularly polarized. It is called right-circularly polarized or left-circularlypolarized depending on whether β = π/4 or −π/4, but different conventions exist so bereally careful if the handedness matters to you!In a practical sense, we can’t really measure the electric field vector of monochromaticlight cycle by cycle. The frequencies are extremely high (107cycles per second even fordecameter radio waves), so we usually have to take a time average. In addition, a detectorwill almost always have some finite bandwidth, so in reality we’ll be averaging over anumber of different frequencies. Let’s denote averages over time and bandwidth by angularbrackets. Then what we really measure is¯I = hE2x+ E2yi = hE20i¯Q = hE2x− E2yi =¯I cos 2β cos 2χ¯U = 2hExEyi cos ∆φ =¯I cos 2β sin 2χ¯V = 2hExEyi sin ∆φ =¯I sin 2β(6)We’re still making the implicit assumption that the light is 100% elliptically polarized,otherwise the angles χ and β, as well as the phase difference ∆φ, would change over thebandwidth.But now let’s forego that assumption. Suppose we’re looking at a general source oflight. It will be a superposition of many different waves, which don’t necessarily have a fixedphase relation between themselves. Therefore, we can consider the total electric field to becomposed of many independent elliptically polarized waves, E =PnE(n). If we assume thatthe different streams add incoherently (like a random walk), then the Stokes parameters(which are quadratic) are the sums of squares instead of the square of sums, meaning that¯I =Xn¯I(n),¯Q =Xn¯Q(n),¯U =Xn¯U(n),¯V =Xn¯V(n). (7)This incoherent addition means that the relation I2= Q2+ U2+ V2no longer holds ingeneral, but is replaced by the inequalityI2≥ Q2+ U2+ V2. (8)In this case (which is the only one seen in practice), all four Stokes parameters areindependent and must be measured separately. We can then consider the light to be acombination of completely unpolarized light (with¯Qu=¯Uu=¯Vu= 0) and 100% ellipticallypolarized light in which¯Ip=³¯Q2p+¯U2p+¯V2p´1/2. (9)Then¯I =¯Ip+¯Iuand the fractional polarization is¯Ip/¯I.The introduction of polarization produces some mild complications in radiativetransfer. The key is to realize that one can think of the four Stokes parameters as differentcomponents of the electric field that propagate independently. Specifically, one can multiply|E|2by a factor that converts it into the specific intensity Iν, then do the same for theother Stokes parameters: Qν, Uν, and Vν. One can then think of the full specific intensityas a vector with these four quantities. A mild tweak used by Chandrasekhar is to defineI+ν=12(Iν+ Qν) and I−ν=12(Iν− Qν). These represent intensities of linear polarization intwo mutually orthogonal directions. The vector specific intensity is then~Iν= (I+ν,