Polarized light transfer above a reflecting surface

Abstract

The existence and uniqueness are established for the solution of the equation of transfer of polarized light in a homogeneous atmosphere of finite optical thickness, assuming reflection by the planetary surface. A general Lp‐space formulation is adopted. The boundary value problem is first written as a vector‐valued integral equation. Using monotonicity properties of the spectral radii of the integral operators involved as well as recent half‐range completeness results for kinetic equations with reflective boundary conditions, the present results follow as a corollary.

Stability of Stationary Transport Equations with Accretive Collision Operators

“…The operator A must be positive semidefinite, but B=I&A need not have any compactness properties. Moreover, the solution is sought in an extension of the domain D(T) of the operator T in the original Hilbert space H. The existence and uniqueness of the solution of the stationary equation of transfer of polarized radiation were proved using invariance of the positive cone of functions having their values in the positive cone of Stokes vectors under the operators \TQ \ and B, without using that Re A is positive semidefinite [21,23]. In [24] its unique solvability was established using the accretiveness of A, with the help of the Fredholm alternative applied to the convolution integral equation version of (0.1) (0.2).…”

Reflection and transmission of polarized light: The adding method for homogeneous atmospheres

“…In [24] he considered Eqs. (1)- (2) and corresponding equations for media of infinite optical thickness, while in [25] reflection by the planetary surface was incorporated. In both of these publications it is assumed that F(0) is a measurable matrix that leaves invariant the positive cone of vectors I = (I, Q, U, V) satisfying I>~/Q2 + U2 + V:>~0, (7) while the phase function al(O) is nonnegative and satisfies the conditions 3r> 1: f1_1 al(O) d(cos 0)= 2 (8) fl al(0) r d(cos 0) < oe.…”

“…• + S L + (~ -Sb+~ -S~ + )-1 S~ -Sb2-(25) S;+=S~+ + S~-(~-S-+S+-)-~ S~+S~ a,(26)S; -= Sb, -(~ --S L +S~ -)-1 SL_. (27)The adding method consists of calculating the operator Sb from the operators Sb~ and Sb~ using the series expansions (~-S~ 1 SL+)-'= ~ (S~-SL+) ", physical interpretation of giving the contributions of successive reflection by the interface between the two adjacent slabs.…”