Optical spatial dispersion in terms of Jones calculus

Traditionally, optical spatial dispersion (OSD) is defined as the dependence $\widehat{\varepsilon }\left(\vec{k}\right)$ of the dielectric permittivity tensor $\widehat{\varepsilon }$ on the light wave vector $\vec{k}$, similarly to the frequency ($\omega$) dispersion of the dielectric tensor $\varepsilon \left(\omega \right)$. We have developed an approach for the description of the OSD phenomena in the framework of Jones calculus. In Jones calculus the differential Jones matrix (DJM) $N$ is the generalization of the light wave-vector $\vec{k}$ in the same sense that $\vec{k}$ is the generalization of the light wave number $k$. The latter inspires us to expect that there must exist a way to describe the OSD phenomena in terms of the DJM. We show that such a relation between the OSD phenomena and Jones calculus indeed exists. To prove the latter we derive a general relation between the DJM and components of the dielectric permittivity tensor $\widehat{\varepsilon }$. We establish the relation of the DJM approach, proposed in this paper, to the traditional OSD approach of the gyration pseudotensor as well as to that developed by Mauguin for light propagation in cholesteric liquid crystals [M. C. Mauguin, Bull. Soc. Fr. Mineral. Crystallogr. N3, 71 (1911)]. We demonstrate that both the gyration pseudotensor and Mauguin's approach can be derived as particular cases of the proposed DJM approach. In our approach the integral Jones matrix (IJM) of the medium taking into account OSD is the product of the IJM without taking into account OSD by the correction IJM, which accounts for the OSD effects. In a general case, when all components of the OSD DJM $N^D$ are nonzero, the secular equation for the refractive indices of the eigenwaves is a quartic equation. The coefficient $a_3$ at the cubic term in the secular equation is nonzero only for nonzero OSD corrections to the average refractive index. For transparent crystals at nonzero OSD correction to the average refractive index and zero to all other correction parameters in $N^D$, the secular equation has two distinct real and two complex-conjugate roots. We assign the complex-conjugate roots to the forward and backward light scattering. Therefore, taking into account the OSD effect on the refractive index, the Jones calculus becomes capable of describing light scattering. The proposed Jones calculus approach is a general tool for taking into account OSD in optically inhomogeneous media, in which several or all OSD correction parameters are simultaneously nonzero, for example, in liquid-crystal cells with a spatially nonuniform director field, including those containing defects.