Nastyshyn S. Matrix’s methods for the description of light propagation through distorted liquid crystalline mediums

The work is devoted to the description of oblique light propagation through the distorted liquid crystalline mediums, to the taking into account of optical spatial dispersion in terms of Jones matrix formalism including the cholesteric liquid crystal in the regime of selective light reflection as well as to the description of optical aberrations in confocal microscopy caused by refractive indexes mismatches of media in which light beam is propagating including the case of the distorted liquid crystalline medium having the spatially non-uniform refractive index. The most general approach for resolving the novel and classical problems in the field of optics is based on Maxwell’s differential equations. The differential Maxwell’s equations and the material equations create together the full equations system and it makes a capability for calculating the electrical field vector of the light wave propagating in the medium. The mentioned theory is based on the differential equation of the second-order that has four variables: three special coordinates and time. Significant simplification was achieved with Jones matrices formalism in which entering Jones vector of electrical field vector is related to exiting Jones vector of electrical field via a linear equation: where is the Jones matrix of the medium. One of the benefits of Jones matrices formalism is the analytical form of propagating and exiting Jones vector of electrical field vector. Jones matrix formalism was developed originally for the case of normal light propagation through the medium. In the literature there are attempts to build so-called extended Jones matrices for the description of oblique light propagation through distorted liquid crystalline media. But these results appear to be of too long and too complicated forms for analytical consideration. In this work we analyze the available extended Jones matrices and show that they can be simplified to a compact form via conventional trigonometric transformations. The approach of differential Jones matrices is employed to derive a compact form of Yeh’s extended Jones matrices [P. Yeh, J. Opt. Soc. Am. 72, 507 (1982)] for distorted liquid crystals. In this work the Jones matrix formalism is extended to account for the phenomena of optical spatial dispersion. We have established the relation of the differential Jones matrix approach, proposed in this paper, to the traditional optical spatial dispersion approach of the gyration pseudo-tensor as well as to that developed by Mauguin for light propagation in cholesteric liquid crystals. The ray tracing matrices approach is employed for description of optical aberrations in confocal microscopy of distorted liquid crystals. We have employed the ray tracing matrix approach for the description of focusing of Gaussian beam in the capillary gap filled with liquid. The relation between nominal focus position (NFP) and actual focus position (AFP) is derived. The obtained equation describes the so-called rescaling property caused by refractive index mismatch. According to this property, a layer of the thickness with the refractive index will be imaged by a confocal microscope with a dry objective as a layer that is times thinner than the air layer of the same thickness . We illustrate this property with the 3D fluorescent confocal microscopy image of a flat glass capillary partially filled with dye-doped glycerol scanned on two sides of its meniscus with air. We predict and experimentally confirm that due to the vertical refractive index mismatch rescaling a liquid crystalline layer with highly non-uniform director distribution should be imaged as a layer of non-uniform thickness, apparently appearing to be imaged with a rough upper (rear) surface. Our next step was to employ the ray tracing approach to the focusing of a paraxial Gaussian beam inside a spherical droplet. We find that for the on-axial case there is no focal shift for the probing beam, whether it is focused in the center or in the south pole of the droplet; for all other focus positions, the AFP is shifted toward the droplet center. Our expression relating NFP and AFP in the on-axial case is equivalent to Abbe’s invariant for refraction by a spherical surface.