Radiation transfer through the earth's atmosphere using the matrix operator method

The matrix operator method of solving radiation transfer problems is extended to include polarization. The reflection and transmission operators are presented with continuous variables. An explicit expression for the phase matrix for spherical particles is obtained from the Mie theory in a form applicable to the matrix operator method. The symmetry relations derived by Hovenier from symmetry arguments for the phase matrix are rigorously demonstrated. The symmetry relations for the reflection and transmission operator in both the homogenous and inhomogenous cases are rigorously proven from the symmetry properties of the phase matrix. The reflection and transmission opperators are Fourier decomposed in azimuth and the resulting equations are discretized, yielding the usual discrete equations of the matrix operator method. Intensity, polarization, and direction of polarization results are given for a conservative Rayleigh atmosphere with optical depths ranging from the very small to the semi-infinite limit. The position of the babinet, and Brewser/Arago neutral points are given as a function of optical depth for the reflected and transmitted radiation. The change in the direction of polarization from single scattering is given for the reflected radiation for various depths. The change in the direction of polarization increases with optical depth to depths slightly greater than one. The change then decreases to the semi-infinite limit