Showing papers by "Rasheed M. A. Azzam published in 1972"

Simplified Approach to the Propagation of Polarized Light in Anisotropic Media—Application to Liquid Crystals*

Abstract: When a single complex variable χ is used to specify the ellipse of polarization of light passing through an anisotropic medium, a first-order, second-degree, ordinary differential equation (Riccati’s equation) governs the evolution of χ with distance along the direction of propagation. In this differential equation the properties of the medium are represented by the elements nij of its N-matrix, which was first introduced by Jones. Its solution χ(z,χ0) represents a trajectory in the complex plane which is traversed as the distance z is increased, starting from an initial polarization χ0 at z = 0. A stereographic projection onto a tangent sphere produces the corresponding trajectory in the more-familiar Poincare-sphere representation. The function χ(z,χ0) has been determined for propagation along (i) an arbitrary direction in a homogeneous anisotropic medium, and (ii) the helical axis of a cholesteric liquid crystal. The solution in the first case provides a unified law that leads to all the rules for the use of the Poincare sphere. For axial propagation in a cholesteric liquid crystal, it is found that two orthogonal polarizations are privileged in that the axes of their ellipses are forced to remain in alignment with the principal axes of birefringence of the molecular planes. The general solution (that satisfies the conditions of propagation) shows that the ellipse of polarization never repeats itself. As to the two parameters of the ellipse, the ellipticity is shown to be periodic with periodicity shorter than the pitch of the helical structure and the azimuth is aperiodic.

Polarization Transfer Function of an Optical System as a Bilinear Transformation

Abstract: The polarization states of light incident on and emerging from an optical system are represented by complex numbers χ¯ and ξ¯, respectively, in two different planes. In this representation, the input–output transfer function ξ¯=f(χ¯) is a conformal bilinear transformation with coefficients given by the elements of the system’s Jones matrix. From the known properties of the bilinear transformation, important conclusions can be reached on the response of optical systems to incident light of all possible polarization forms. In addition, the analysis appears to have considerable potential in the synthesis of systems to effect a prescribed polarization transfer. As an example, the ellipsometer is analyzed by use of some of the ideas developed.

Ellipsometric Measurement of the Polarization Transfer Function of an Optical System

Abstract: Conventional ellipsometry is extended to determine the polarization transfer function of any optical system. Only three null measurements are needed to determine the three complex parameters that define the transfer function. More measurements can be taken to overdetermine the unknown parameters, to reduce the effect of the various sources of error in the ellipsometer. The conditons for compensation (the existence of a null) are defined. A system composed of a retardation plate and a mirror is measured to demonstrate the various aspects of the theory and to illustrate the application of the method. This technique makes it possible to characterize surfaces that exhibit anomalous cross-scattering effects.