Geometric phase: two triangles on the Poincaré sphere

Abstract

The method of obtaining the dynamic and geometric part of the phase introduced by a birefringent medium in two kinds of interferometric experiments is presented. The mathematical formulas for the phases obtained using Jones formalism are visualized with the specific triangles on the Poincaré sphere. Generally, these triangles are similar to those used in the Pancharatnam's original construction developed by Courtial to calculate the Pancharatnam geometric phase for the light passing through a single birefringent plate. In these graphical constructions following Courtial's idea, we used the points representing both of the birefringent medium's eigenvectors. This allowed the most intuitive explanation of the mechanism of dividing the whole phase shift introduced by the birefringent plate into two different parts: dynamical and geometrical. The considered constructions were used as a description of two simple experiments with a birefringent medium in a Mach-Zehnder interferometer and a polariscopic setup. The experimental verifications of our theoretical predictions should convince the reader of the correctness of the assumed model.

Interferometric characterization of the structured polarized light beam produced by the conical refraction phenomenon

“…The theoretical results obtained will help us to interpret the experiments in Sections 3 and 4. First thing to consider is that the phase-difference δ between two beams with a different polarization participating in interference is given by [28]:…”

“…In this case, the phase-difference is strictly given by the projection between the two SOPs and thus has a pure geometric origin. This is the so-called geometric phase, which is related to the PancharatnamBerry phase [28], and provides a means to establish when two beams with a different SOP can be considered to be in phase. This distinction between the geometric and non-geometric origin of the phase-difference is of relevance for the purpose of the paper.…”

“…Since the CR beam is not scalar, i.e. it shows inhomogenous polarization structure, special care is necessary to avoid misinterpretation of the results due to additional Pancharatnam-Berry geometric phases [28][29][30] arising from the different polarizations in the reference and sample beams. On its turn, measurement of the geometric phase can be accomplished by using interferometers or polarimeters [31].…”

Geometric phase of an elliptical polariscope

“…where T and K stand for some constant values (K has a meaning of the contrast for measurements where fringe patterns are applied; see, for example, the setup with the Wollaston compensator described in [11]). Let us note that the argument of the cosine function in Eq.…”

“…Courtial [10] showed that trajectory describing the process of light polarization state transformation can be presented on the Poincaré sphere in a more convenient way using the points representing the birefringent plate's eigenstates. In our previous works, we expanded Courtial's original idea using simultaneously two points representing both, fast and slow, wave plate's eigenwaves [11] and even enhanced this formalism in the case of dichroic media [12]. This allowed us to present the intuitive graphical interpretation of the geometric phase calculation on the Poincaré sphere for some polarization experiments with a wave plate placed inside the Mach-Zehnder interferometer and inside the linear polariscope.…”