DIFFERENTIAL JONES MATRIX FOR CHOLESTERIC

Abstract

The formalism of the Jones matrices is a powerful theoretical tool widely used to calculate the parameters of the electric field of a light wave at the output of an optical system or medium. The Jones calculus is based on the linearity of the vector relation between the electric field of a light wave incident (E⇀0) on and exiting (E→) from an optical system or medium through the Jones matrix J, such that: E→=JE0→. In this approach, the J matrix describes an optical element as a whole and does not contain any information about its internal structure; for this reason it is called the integral Johns matrix (IJM). Although the IJM was introduced for optical systems with discrete elements, it is often used to model optically inhomogeneous media and appears to be even more popular in literature than the approach of the differential Jones matrix (DJM), which was specially developed by Jones for this type of problems.

In this paper we employ the DJM approach to the description of optical properties of a cholesteric liquid crystal (cholesteric). Cholesteric is a chiral nematic, whose director n↔ spontaneously twists around the axisZ⇀⊥n↔. Description of optical properties of the cholesteric was first performed by Mogen [Bulletin dela Societe FrancaiseMineralogie et Crystallographie. – 1911. – Vol. 34. – P. 71] using the method of the Maxwell differential equations in the framework of the model, according to which the azimuth of the diagonal tensor of dielectric permittivity varies linearly along the coordinate axisZ⇀⊥n↔. The first attempt to derive the Jones integral matrix for a twisted crystal was performed by Johns [J. Opt. Soc. Am. 1948. – Vol. 37. – P. 671]. An attempt to obtain the cholesteric IJM was made by Chandrasekar and Rao [Acta Cryst. – A24. – P. 445(1968)]. Results of both approaches appear to be applicable for the spectral range except the selective reflection band.

We show that the wave numbers of the eigenwaves propagating in the cholesteric are the eigenvalues and the electric field vectors of these eigenwaves are the eigenvectors of the cholesteric DJM. Taking advantage of this finding, we derived the cholesteric DJM in the local coordinate system for any light wavelength, including the spectral band of the selective reflection.

Key words: Jones matrix, differential Jones matrix, integral Johns matrix, cholesteric, selective reflection.