RADIATION, SCATTERING AND POLARIZATION PETER MESZAROS A beam of electromagnetic radiation can be characterized by its direction of propagation *, its frequency *, and its state of polarization. These remain constant as the radiation propagates in empty space, unless the beam interacts with matter that scatters it. In this case a fraction of the incident radiation changes its direction and its polarization, and in some situations also its frequency, in a manner that depends on the characteristics of the incident radiation and on the nature of the scattering centers. The scattering centers can be electrons, atoms, molecules, or solid dielectric grains or aggregates made up from these. POLARIZED RADIATION Polarized radiation can be defined as that for which the plane of oscillation of the electric and magnetic field vectors, E and B, remains constant during propagation, or else varies in a predictable way. The polarization vector * at a particular point along the propagation trajectory is defined as a unit vector along E at that point. The polarization can be either linear, if the tip of the electric vector E always oscillates in the same plane, circular if the tip of E moves in a circle in the plane perpendicular to the direction of motion, and elliptical when [E] is not constant during the rotation of the plane of polarization. In the latter two cases, the plane of polarization rotates at an angular frequency equal to that of the wave. One can define the Stokes parameters I,Q,U, and V, where I gives the total intensity of the radiation, Q and U give the orientation of the ellipse of polarization, and V gives the circularity, or the ratio of principal axes of the polarization ellipse. Monochromatic light is always 100% polarized, and one of the four Stokes parameters is redundant, because they satisfy the equality I*=Q*+U*+V*ù Realistic radiation sources usually have a frequency spread **, and in general such light is not 100% polarized. Measurements of the polarization over periods of time ******* would detect chaotic variations of the polarization state. However, for quasi-monochromatic light with *******, it is possible to define appropriately time-averaged values of the Stokes parameters, valid for measurement times less than the so-called coherence time of the radiation, ******.The meaning of the Stokes parameters is the same as before, the only difference being that in this case there is the possibility of having partially polarized light. This can be described as a mixture of 100% polarized light and completely unpolarized light. All four Stokes parameters are needed, because now *********, and the fourth number describes the ratio of polarized to total intensity. The equality holds for purely polarized light, whereas for purely unpolarized light one has Q=U=V=0. SINGLE SCATTERING AND POLARIZATION One of the earliest and best-known studies of radiation scattering is that of lord Rayleigh, who investigated this phenomenon in connection with the scattering of sunlight in the Earth's atmosphere. The purpose was to find out why the color of the sky is blue, and such a study also provided an explanation of why scattered light becomes polarized. Although this was done originally for small spherical dielectric particles whose dimension was small compared to the wavelength of the light, it is of wider applicability, being representative also of the scattering by free electrons and by non-resonant electrons bound in atoms. The change in the polarization caused by a scattering event is most easily illustrated for linearly polarized radiation that is scattered by cold electrons (Thomson scattering), in which case the frequency * remains unchanged to a first approximation. Consider an electromagnetic wave, or beam of radiation traveling initially in the direction * along the z axis (Fig. 1), which is completely linearly polarized in the *,z plane; that is, its electric vector ** oscillates in the * direction (because in vacuum, the E and B vectors of an electromagnetic wave must be perpendicular to *). The radiation scattered by a single electron into a direction making an angle ******* with the original direction will have an intensity characterized by the differential cross section for this state of polarization. This is just the ratio of the flux scattered in this direction to the incident flux, which in terms of the angle * between the original polarization vector *** and the scattered direction * is given by (*********)************, where **************** is the classical radius of the electron, and *** is the differential of solid angle along the * direction. The scattered radiation will also be linearly polarized in the x,y plane, with a polarization vector perpendicular to *. A similar expression will be valid for radiation incident along * but which is linearly polarized perpendicular to the plane of scattering, that is, with ** along the x axis. In this case, the differential cross section is the same as the previous one, but with *******, and the radiation scattered into the direction ** will be linearly polarized parallel to the original polarization **. To see how polarized light can be produced in a scattering medium, it is useful to consider what happens to initially completely unpolarized light that suffers scattering. Because unpolarized light can be represented as an equal mixture of light polarized along two mutually orthogonal directions, such as ** and **, the differential scattering cross section for unpolarized light will be given by an average of the polarized cross sections for ** with * and ** with **** that is, *******************************. The total scattering cross section is obtained from this by integrating over the differential of solid angle *** for all possible final directions, giving the well-known Thomson value, (*****)********. In the preceding equation, the two terms give the relative proportion of scattered intensities polarized along the two perpendicular directions, the first one having ** in the y,z plane of scattering, and the second having *** along x, perpendicular to the plane of scattering, for a particular angle of scattering *. The two intensities are in the ratio ****. Thus unpolarized light acquires after a single scattering a degree of linear polarization given by p=**************. This degree of linear polarization depends on the angle of scattering, ranging from a value of 0 (no net polarization) for viewing along the incident direction *=0 (because by symmetry all directions about * are equivalent), to a value of unity (100% linearly polarized) for viewing perpendicular to the incident wave, ****** (because the electron motion is in a plane which contains the viewing direction). The Thomson scattering law described previously is an example of the more general Rayleigh scattering function, applicable whenever the scattering particle radiates as a simple dipole. This occurs for scattering by atoms at frequencies away from any resonances and for spherical dielectric particles. The phase function, that is, the angular and polarization dependence, remains the same as before, but the numerical coefficient in front becomes dependent on the wavelength of the radiation and the characteristics of the atoms or the particles. In the case of scattering by anisotropic particles, such as elongated grains or molecules, the scattering does not follow a simple dipole pattern, but is characterized by a differing polarizability along major axes of the scattering particles. The phase function is consequently more complicated, and in general it is no longer true that radiation scattered through an angle of ***** is 100% polarized. POLARIZATION BY MULTIPLE SCATTERINGS The examples discussed previously considered the polarization caused by a single scattering event. When radiation is scattered many times, such as when radiation escapes from a medium that has characteristic dimensions equivalent to many scattering mean free paths, the cumulative effect of the many individual scatterings does not lead to a simple coherent addition of the polarization caused by one scattering, because the successive angles of scattering are given by a random walk process. As a consequence, a significant fraction of the polarization achieved along one polarization direction ** in one scattering can be undone in the next scattering event, by reapportioning the radiation along a different polarization direction **. Knowing, however, the behavior for one scattering event allows one to calculate the degree of polarization of the radiation escaping from a scattering atmosphere of arbitrary dimensions, after a varying number of scatterings. In the general case of arbitrary elliptical polarization, one can define an intensity vector I=*(************) made up of the four Stokes parameters, which are a function of the angles * and * of propagation, and of the position * in the atmosphere. Here ** and ** are the intensities polarized in the plane of scattering and perpendicular to it, which can be used instead of the usual * and Q. The scattering is represented through a 4x4 scattering matrix corresponding to redistribution between the four Stokes parameters, ********, which is a function of the angles after and before scattering. Defining an optical depth ******** as in unpolarized radiative transfer (where ** and ** are the space density of scatterers and their angle-integrated total scattering cross section, respectively), the general transfer equation for polarized radiation is ***********************************. For a plane parallel atmosphere, one finds from the preceding equation that the components U=V=0. This is to be expected, because in this case there can be no * dependence, and there is no reason for circular polarization to be present. The previous system of four equations reduces to a simpler system of two equations with two unknowns: ***************************, where ***** is the cosine of the angle made by the radiation with the normal to the atmosphere surface. Solutions of this equation show that the degree of linear polarization p varies as a function of optical depth and angle of viewing. For infinite optical depth atmospheres, the linear polarization degree has a maximum value of 11.7% at **, or grazing viewing angles, with the electric vector parallel to the surface. The linear polarization decreases to 0 at **=0 (normal), because there one has no preferred direction. For lower optical depths, the direction of polarization remains initially the same, but the degree of polarization decreases, whereas for scattering optical depths of order less than a few, the direction of polarization switches to an electric vector perpendicular to the surface, and the degree of polarization reaches again maximum values of order 10%. An example for radiation sources uniformly distributed in depth through a finite plane parallel atmosphere of scattering depth ** is shown in Fig. 2. Notice that in the cold electron scattering approximation discussed previously, the intensity is grey, that is, independent of frequency. The polarization degree is obtained from considering all the escaping photons. However, the number of scatterings suffered by individual photons varies significantly about a mean value, given by **. It is possible to expand the transfer equation discussed previously, expressing the intensity as a sum of intensities made up of photons that have suffered none, one, two,... scatterings. If this is done, one finds that for low optical depth atmospheres the polarization of the photons that have been scattered much more than the average number of times can reach rather higher values than those mentioned previously. This is of interest when Comptonization is taken into account, that is, in the cumulative effects of the changes of frequency in individual scatterings, which are important in a hot plasma. In this Case, the scattering matrix for nonrelativistic thermal electrons at temperature T is given by **********************************. For other scatterers, such as atoms or spherical particles, the frequency dependence is also important. In the scattering by atoms (actually, by electrons bound in atoms with a characteristic binding frequency **), the total scattering cross section is given by ********, if the incident light has frequencies *****. This is the classical Rayleigh scattering regime, which was used to explain why the sky is blue (higher frequencies are scattered more efficiently), when not looking directly at the Sun, and why the Sun is red at sunset or sunrise. The degree of polarization is dependent on both the frequency and the angular distance from the Sun. Other, more complicated examples involve scattering by dielectric spheres (Mie scattering) and by grains of arbitrary shapes, which find applications in the scattering of light by interplanetary and interstellar dust grains. The latter case leads similarly to polarization of light because the anisotropic grains generally align themselves in response to the interstellar magnetic field. Additional Reading Chandrasekhar, S.(1960). Radiative Transfer. Dover, New York. Collins, G.W.,II.(1989). The Fundamentals of Stellar Astrophysics. W.H. Freeman, New York. Meszaros, P. and Bussard, R.W.(1986). The angle-dependent Compton redistribution function in x-ray sources. Ap. J. 306 238. Phillips, K.C. and Meszaros, P.(1986). Polarization and beaming of accretion disk radiation. Ap. J. 310 284. Rayleigh, Lord (1899). Scientific Papers, pp. 87, 104, 518. Cambridge University Press, Cambridge, U.K. Rybicki, G.B. and Lightman, A.P.(1979). Radiative Processes in Astrophysics. Wiley, New York. Sobolev, V.V.(1963). A Treatise on Radiative Transfer. Van Nostrand, New York. Sunyaev, R.A. and Titarchuk, L.G.(1985). Comptonization of low-frequency radiation in accretion disks: Angular distribution and polarization of hard radiation. Astron. Ap. 143 374. van de Hulst, H.C.(1981). Light Scattering by Small Particles. Dover, New York.