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Journal of Modern Optics

Geometric phases from stacks of crystal plates

Michael Berry, S. Klein

Published Jan 1, 1996• 89 citations
Publisher

Abstract

With twisted stacks of N polarizers P or retarders R, the polarization of a light beam can be cycled around the Poincaré sphere on N similar arcs of great P or small R circles. We calculate the phase changes around these cycles (geometric for P; geometric + dynamical for R). In the continuum limit N → ∞ of a smoothly twisting medium, a P stack forces the light to follow its changing polarization, and the phase is the solid angle of the associated loop on the sphere; for an R stack, on the other hand, it is only in the adiabatic limit of slow twist (where the dynamical phase is large) that the geometric phase corresponds to that of the loop specified by the changing eigenpolarization of the medium. The predicted phase shifts are observed as fringe shifts in an interferometer for N=2, 3 and 4.

DOI
10.1080/09500349608232731

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With twisted stacks of N polarizers P or retarders R, the polarization of a light beam can be cycled around the Poincaré sphere on N similar arcs of great P or small R circles. We calculate the phase changes around these cycles (geometric for P; geometric + dynamical for R). In the continuum limit N → ∞ of a smoothly twisting medium, a P stack forces the light to follow its changing polarization, and the phase is the solid angle of the associated loop on the sphere; for an R stack, on the other hand, it is only in the adiabatic limit of slow twist (where the dynamical phase is large) that the geometric phase corresponds to that of the loop specified by the changing eigenpolarization of the medium. The predicted phase shifts are observed as fringe shifts in an interferometer for N=2, 3 and 4.
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.
The wave function of a quantum system may acquire an additional phase factor (the geometric phase), when its parameters undergo a cyclic change. An optical manifestation of the geometric phase is the change in the phase of a light wave produced when its state of polarization is taken around a circuit on the Poincare sphere. This phase change can be verified by interferometric measurements. We describe an interferometer in which the two beams traverse the same path in opposite directions, so that their optical paths are always equal, and the phase difference between them is varied by operating on the geometric phase. Measurements made with this interferometer, at a light level at which the probability of more than one photon being present simultaneously in the interferometer was negligibly small, confirmed that the interference effects observed due to the geometric phase persist down to the single photon level.
Geometric phase has historically been defined using closed cycles of polarization states, often derived using differential geometry on the Poincare sphere. Using the recently-developed wave model of geometric phase, we show that it is better to define geometric phase more generally, allowing every polarized wave to have a well-defined value at any point in its path. Using several example systems, we show how this approach provides more insight into the wave's behavior. Moreover, by tracking the continuous evolution of geometric phase as a wave propagates through an optical system, we encounter a natural explanation of why the conventional Poincare sphere solid angle method uses geodesic paths rather than physical paths of the polarization state.
We study two types of topological phases, i.e., Aharonov-Bohm phases, by means of optical interferometry. The first arises from a cycle of redirections of the photon spin and is closely related to the Aharonov-Anandan phase. The second arises from a cycle of polarization changes on the Poincare sphere and was discovered by Pancharatnam. Both of these phases, and their combined effects, have been investigated by a nonplanar Mach-Zehnder interferometer. The signs of the fringe shifts arising from these topological effects have been determined and agree with theory.