Application of Quaternions to Obtain Analytic Solutions to Systems of Polarization Components

Abstract

This paper describes the passage of light through a system of waveplates mathematically in terms of quaternions, an extension of the complex numbers, instead of the more usual Jones vectors and Jones matrices. Both the light beam and the waveplate are represented by a quaternion. It is possible to manipulate the quaternion expression more readily than the Jones matrix-vector expression; for example it can be inverted. The quaternion form of a waveplate is compactly related to its retardance and fast/slow axes, and the quaternion of a signal is closely related to its state of polarization (SOP), either expressed as a vector on the Poincar\'e sphere or as a polarization ellipse. The paper presents rules to decide if two optical signals are aligned or orthogonal in phase or in polarization from their quaternions, and presents the quaternion operations to change the phase or change the SOP. Several mathematical tools are identified, such as partial conjugation, to rearrange a quaternion expression, a tricky operation because multiplication does not commute. Put together, these advances let us understand how waveplates can act on a light beam to produce desired behavior. Finally, the quaternion math is put to work on two problems. A new endless optical phase shift system is designed out of waveplates. A prior solution to the problem used five waveplates, and in this paper the same task is done with only three waveplates. Also, failures of a polarization controller are studied, and found to be caused by singularities, which can occur frequently.