On a new form of the transfer equation with applications to multiple scattering of polarized light

Abstract

The stationary transfer equation for polarized light in a plane-parallel medium is transformed into a new equivalent form. In this Q-form transfer equation the intensity vector of the radiation field is equated to an angular integral of its spatial derivative. Expansion in generalized spherical functions enables us to derive explicit expressions for the integral kernel, which become divergent in the case of conservative scattering. In this case, a definite Q-form is obtained by means of K- and F-integrals, which permit us to reduce the original conservative transfer equation to a nonconservative pseudo-transfer equation. The convenience of the Q-form transfer equation is demonstrated for radiative transfer problems in atmospheres with internal sources proportional to integer powers of the optical depth. Explicit formulae are derived for the powers 0 and 1, which yield the intensity vector of the internal radiation field in terms of the surface Green's function matrices. Moreover, the Q-form transfer equations are employed to derive a set of nonlinear integral relations for Green's function matrices. In particular, a new nonlinear equation is obtained, which relates the Green's function matrix of a finite homogeneous medium to its surface values. Thus, we generalize the classical nonlinear equations that are usually derived by using principles of invariance.