Geometric Polarimetry—Part I: Spinors and Wave States

Abstract

A new formal approach for the representation of polarization states of coherent and partially coherent electromagnetic plane waves is presented. Its basis is a purely geometric construction for the normalized complex-analytic coherent wave as a generating line in the sphere of wave directions and whose Stokes vector is determined by the intersection with the conjugate generating line. The Poincaré sphere is now located in physical space, simply a coordination of the wave sphere, with its axis aligned with the wave vector. Algebraically, the generators representing coherent states are represented by spinors, and this is made consistent with the spinor-tensor representation of electromagnetic theory by means of an explicit reference spinor that we call the phase flag. As a faithful unified geometric representation, the new model provides improved formal tools for resolving many of the geometric difficulties and ambiguities that arise in the traditional formalism.Index Terms-Bivectors, covariant and contravariant spinors and tensors, geometry, phase flag, Poincaré sphere, state of polarization.

Geometric Polarimetry—Part II: The Antenna Height Spinor and the Bistatic Scattering Matrix

“…It might be said that is where problems began to arise in polarimetry because the two concepts are not formally consistent: Jones vectors effectively acquired a dual identity as both vectors and spinors, which are the carriers of the special unitary group, SU(2). In [21] this problem was resolved by identifying the descriptors that transform unitarily, and properly, as spinors, while Jones vectors remain properly as complex vectors. For any fixed wave propagation direction there is a 1:1 mapping between the two, which in some cases appears numerically trivial, but is inherently non-trivial, since spinors correspond geometrically to the complex generators of the Poincaré sphere, rather than to vectors.…”

“…Monostatic scattering, which involves greater symmetries, has yet further interesting features that can be revealed using geometric methods that space does not permit to be presented in the present paper. In the first paper [21] of the projected formal series on Geometric Polarimetry we undertook the most detailed investigation to date into how spinor algebra can be used in a rigorous way to represent electromagnetic polarization. This paper is a direct continuation, and the reader will be referred back to it for many of the fundamental developments that will be required here.…”

“…This paper is a direct continuation, and the reader will be referred back to it for many of the fundamental developments that will be required here. The most significant features to emerge in [21] were that spinors should be considered to represent complex coherent wave states geometrically not as vectors in the tradition of Jones vectors [22] but rather as generating lines on the Poincaré sphere via a direct construction in complex projective space. As we progress through the foundations of polarimetry it will become increasingly clearer that this concept underpins the inherent geometric connection between all polarimetric descriptors.…”

“…Whilst in [25] we deduced the spinor character of the antenna height as contravariant, space did not permit an explicit derivation. In [21] we concentrated exclusively on representations of wave state. Given that it has now been shown that wave states have covariant spinor representations, we are now in a position to give a similar derivation from first principles of the antenna height spinor.…”

“…The plan of the remainder of the paper is as follows. In section IV we develop a rigorous derivation of antenna height spinors using results from [21] and applying Schelkunoff's reaction theorem [31], [32]. In section VI, we develop the geometrical properties of the scattering matrix, seen as a bilinear form as in (1), while we devote section VIII to a modified definition of the Huynen polarization fork for the general bistatic case.…”