Relations between the Elements of the Phase Matrix for Scattering

Abstract

The process of scattering of radiation is usually characterized by a 4 × 4 transformation matrix which relates the radiation field vector incident on the scatterer to the scattered field vector. The nine relations between the 16 elements of this phase matrix for scattering are derived explicitly for the three most commonly used representations of the intensity vector, viz., Wolf's coherency matrix formalism, Chandrasekhar's and Stokes's representations. The invariance of these relations under the action of any optical train containing one or more elements characterized by their Jones representation is demonstrated. These relations should be useful in the theory of polarization optics. The same relations are also shown to hold after rotation of the axes of reference for the electric vectors in the incident and scattered beams. Since such a transformation is required in the formulation of the theory of radiative transfer, the relations derived here may find use in multiple scattering problems as well.

Fast computation of two-level circulant preconditioners

“…Consequently, we let πΦ 1 represent the net flux. Similarly, πΦ 2 , πΦ 3 and πΦ 4 correspond to Q, U and V . Substituting Eq.…”

“…(1.2)) to sets of four parameters with the same physical dimension. A beam of light which is exactly parallel may thus be described by Stokes parameters Φ 1 , Φ 2 , Φ 3 and Φ 4 such that πΦ 1 is the net flux and πΦ 2 , πΦ 3 and πΦ 4 are analogous to Q, U and V , respectively.…”

Radiative properties of particles in flames. Final technical report

“…Thus. there must be nine independent relations among the Sz3; as outlined by Abhyankar and Fymat (1969). Therefore, the scattering matrix for such a collection is merely the sum of the individual particle scattering matrices.…”

“…The earliest known work was carried out by Pritchard and Elliott (1960 On the theoretical side, much of the work in this area has been focussed on developing relations between the Mueller matrix elements for a single particle and modifying them for an arbitrary particle and a collection of such particles. Abhyankar and Fymat (1969) derived nine relations between the sixteen elements of the Mueller matrix for the three most commonly used representations of the intensity vector, viz. JVolf's coherency matrix formalism, Chandrasekhar's (1950) and Stokes' ( 1852) representations.…”

Radiative properties of char, fly-ash, and soot particles in coal flames. Technical progress report, second year, October 1994--December 1994

“…Only seven of them are independent corresponding to the four moduli I Sj I ( j = 1,2,3,4) and the three differences in phase between the Sj. Thus, there must be nine independent relations among the Sij; these are giv& by Abhyankar and Fymat (1969).…”