Theory of Polarized Radiation

CHAPTER 2 THEORY OF POLARIZED RADIATION 2.1. Introduction The framework of classical physics will be used in the following four chapters to develop the theory of polarized radiative transfer in stellar atmospheres permeated by magnetic fields. Scattering and the Hanle effect will also be covered by the classical treatment. This allows us to establish the direct correspondence between the classical and quantum worlds, when the quantum field theory is developed in Chapters 7 and 8. The prior development of the classical theory is needed for a good understanding of the physical contents of the often abstruse formalism of quantum field theory. In the classical treatment the link between the macroscopic and microscopic properties of the medium is most conveniently described in terms of the macroscopic complex refractive index n, which is determined by the collective effect of the oscillating electric dipoles in the individual atoms. The atomic dipole moment is induced by the electric field of the electromagnetic radiation and is modified by the ambient magnetic field. The presence of the dipoles determines the dispersion relation for the electromagnetic waves, and this dispersion relation can be parametrized in terms of the refractive index n. The polarization of the electromagnetic waves can be treated using several different powerful tools, in particular the Jones vector, the coherency matrix, or the Stokes vector. Within these formalisms, the effect of the medium is generally described by non-commuting matrix operators, leading to matrix representations of the radiative transfer problem. The primary physical effects, derived from first principles within a classical framework, are most naturally expressed in terms of Jones vectors. The Jones formalism is however unable to describe a statistical ensemble of uncorrelated photons, which is needed to relate the theory to the world of observations. The statistical properties of the radiation field can be described by the coherency matrix and Stokes vector formalisms, which are equivalent to each other, and which can be derived from the Jones vectors. The Stokes formalism is directly related to the effect of measuring devices and is therefore the natural one to use for the interpretation of observations. The coherency matrix, which contains the equivalent polarization information, is the most natural representation for the radiation field in the quantum field theory, since it relates most directly to the creation and annihilation operators used there. A fourth way of describing polarized radiation is in terms of complex spherical vectors. It is needed when evaluating the interaction of the electromagnetic waves with the oscillating dipoles in a magnetic field, since the coupling between the 31 32 CHAPTER 2 dipole vector components caused by the v × B force of the magnetic field can be removed when transforming from a Cartesian to a spherical vector system. This formalism is most naturally introduced in the next chapter, where the interaction between radiation and matter is treated. 2.2. Maxwell's Equations The standard set of equations for the electromagnetic field can be written (cf. Jackson, 1975) r · D = ρc ; r · B = 0 ; @B r × E = − ; (2:1) @t @D r × H = j + : @t The first of these four equations expresses Coulomb's law after Gauss' theorem has been applied (ρc is the charge density), while the second expresses the absence of magnetic monopoles. The third equation represents Faraday's law, the fourth Ampère's law. The displacement current @D=@t is necessary to ensure that the continuity equation for the charge and current densities, @ρ c + r · j = 0 ; (2:2) @t is satisfied. In addition we need two equations relating D and B to E and H. We will only make use of the equations valid for vacuum, i.e., D = 0E ; (2:3) B = µ0H; which contain the universal constants −12 −1 0 = 8:854 × 10 farad m ; −7 −1 µ0 = 4π × 10 henry m ; representing the permittivity and permeability of the vacuum. Often in electromagnetic theory the macroscopic effects of a medium are described in terms of a dielectric constant and a magnetic permeability µ in Eq. (2.3). Such concepts however represent a phenomenological element of the theory, which hides the underlying physics and has led to physically inconsistent treatments in the past. We will therefore entirely refrain from introducing any or µ at all, to allow us to work from a rigorous version of Maxwell's theory. All the effects of matter will enter exclusively via the charge density ρc and the current density j. THEORY OF POLARIZED RADIATION 33 2.3. The Poynting Flux An energy equation containing an expression for the radiative energy flux (the Poynting flux) can be obtained by multiplying in Eq. (2.1) Faraday's law by −H, Ampère's law by E, and adding them. This gives @D @B −H · (r × E) + E · (r × H) = j · E + E · + H · : (2:4) @t @t Since the left hand side can be written as −∇ · (E × H), we obtain @ 0 1 E2 + B2 + r · (E × H) = −j · E : (2:5) @t 2 2µ0 This represents a continuity equation for the energy. The first term on the left hand side is the time variation of the energy density of the electromagnetic field, the second term is the divergence of the energy flux E × H, while the right hand side describes the energy losses due to Joule heating by the electric currents. If we are dealing with complex vectors E and H, e.g. plane waves of the type E = jEj exp(−i!t + k · r), then a similar derivation gives @ 0 1 jEj2 + jBj2 + Re r · (E × H∗) = −Re j · E∗ : (2:6) @t 2 2µ0 Since here the energy term is formed from the amplitudes jEj and jBj of the oscillations, the time averaged energy terms for a harmonically oscillating field are 1 2 1 obtained if we multiply Eq. (2.6) by 2 (since hsin i = 2 ). The average radiative flux is thus Re E × H∗=2. 2.4. The Electromagnetic Wave Equation For the treatment of radiation problems it is convenient to reformulate Maxwell's equations in terms of a vector potential A and a scalar potential Φ, defined by B = r × A ; @A (2:7) E = −∇Φ − : @t These definitions assure that B is divergence free, and that E satisfies Faraday's law. The remaining two Maxwell equations relating D to the charge density ρc and H to the current density j can then be written (since the spatial and temporal derivatives commute) 2 @ r Φ + (r · A) = −ρ /0 ; @t c (2:8) @ @2A r × (r × A) = µ0j − 0µ0 rΦ + : @t @t2 34 CHAPTER 2 Making use of the vector identity r × (r × A) = r (r · A) − r2A ; (2:9) and defining −1=2 c = (0µ0) ; (2:10) the second of Eqs. (2.8) can be written 2 2 1 @ @ A r A − r (r · A) − rΦ + = −µ0j : (2:11) c2 @t @t2 c represents the velocity of the electromagnetic wave in vacuum (speed of light). The potentials are not uniquely determined by the above equations, but one has the freedom to choose a gauge, the effect of which vanishes when applying the differential operators to derive the values of the physical fields E and B. For radiation problems the natural gauge to choose is the so-called Coulomb or radiation gauge defined by r · A = 0 : (2:12) With this gauge condition Eqs. (2.8) and (2.11) become 2 r Φ = −ρc/0 ; 2 (2:13) 2 1 @ A 1 @ r A − = −µ0j + rΦ : c2 @t2 c2 @t The first of these two equations is a Poisson equation for the scalar potential Φ, implying an instantaneous response of Φ to fluctuations in the charge density, without any wave propagation. This apparent non-local behaviour of Φ is allowed, since Φ is not the observed physical quantity, and the corresponding relevant physical field E always has local behaviour (disturbances propagate with the speed of light). The second equation on the other hand is a wave equation for the vector potential A, with the sources on the right hand side. These sources also contain a Φ-term, which however does not give any contributions to the A waves, for the following reason. Let us consider the solution to the Poisson equation, 0 0 1 ρc(r ) dV Φ(r) = 0 : (2:14) 4π0 Z jr − r j At distances large as compared with the size of the charge region Φ(r) becomes proportional to 1=r with asymptotically vanishing deviations from spherical sym- metry. Time variations in the charge density may therefore, via the @rΦ/∂t term, contribute to A components parallel to rΦ, i.e., parallel to r. These varying components can however never lead to A waves propagating from or to the charge region, since the waves can only be transversal and thus perpendicular to r. The transverse nature of the waves is a direct consequence of the gauge condition (2.12). If we introduce plane waves THEORY OF POLARIZED RADIATION 35 −i!t+ik·r A(t; r) = A0(!) e ; (2:15) the gauge condition implies k · A = 0 ; (2:16) i.e., the waves must be transverse. The scalar potential Φ thus decouples from the wave propagation problem, and the electromagnetic waves become described by the single equation 2 2 1 @ A r A − = −µ0j : (2:17) c2 @t2 t Only the transverse component of j (marked by index t), i.e., the component of the current density that is perpendicular to the propagation direction given by the wave vector k, can serve as a source for the waves.

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