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Journal of the Optical Society of America A

Geometric phase: two triangles on the Poincaré sphere

Piotr Kurzynowski, Władysław A. Woźniak, Małgorzata Szarycz

Published Mar 1, 2011• 29 citations

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.

DOI: 10.1364/josaa.28.000475

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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.

Photon pairs produced by type-I and type-II parametric down-conversion are passed through a Michelson interferometer. Two quarter-wave plates in one arm of the interferometer allow variation of the polarization state of the photons. We investigate experimentally the geometric phase or Pancharatnam phase acquired by single photons and photon pairs dependent on the solid angle that is subtended by the circuit that represents the varying state of polarization on the Poincar\'e sphere. It is found that the geometric phase acquired by the pair depends on the initial polarization state of the two photons. If both photons are in equal states of linear polarization, we observe a doubling of the geometric phase compared to single photons; in the case of orthogonal states of linear polarization, the geometric phase is completely canceled. Our results show the role of Pancharatnam's phase in nonclassical two-photon interference phenomena and the interplay between the geometric phase and the dynamical phase in these phenomena.

Since Pancharatnam's 1956 discovery of optical geometric phase and Berry's 1984 discovery of geometric phase in quantum systems, researchers analyzing geometric phase have focused almost exclusively on algebraic approaches using the Jones calculus, or on spherical trigonometry approaches using the Poincaré sphere. The abstracted mathematics of the former and the abstracted geometry of the latter obscure the physical mechanism that generates geometric phase. We show that optical geometric phase derives entirely from the superposition of waves and the resulting shift in the location of the wave maximum. This wave-based model provides a way to visualize how geometric phase arises from relationships between waves, and from the transformations induced by optical elements. We also derive the relationship between the geometric phase of a wave by itself and the phase exhibited by an interferogram, and provide the conditions under which the two match one another.

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.

We present a novel interferometric arrangement that makes it possible to measure with great versatility geometric phases produced in polarization states of classical light. Our arrangement is robust against thermal and mechanical disturbances and can be set up in a Mach-Zehnder, a Michelson or a Sagnac configuration. We present results concerning the geometric phase as an extension of previous measurements of the Pancharatnam, or total phase. The geometric phase is obtained by compensating the dynamical contribution to the total phase, so as to extract out of it a purely geometric phase. This can be achieved over trajectories on the Poincare sphere that are not necessarily restricted to be great circles (geodesics). We thus demonstrate the feasibility of our method for dynamical extraction of the geometric contribution to the total phase, a prerequisite for building geometric quantum gates. Although our results correspond to polarization states of classical light, the same methodology could be applied in the case of polarization states of single photons.

In crystal optics the concept of the geometric phases can be demonstrated to the naked eye with comparatively simple apparatus. Pancharatnam's phase is described by an area on the Poincaré sphere which is enclosed by geodesic lines (great circles) while the concept of the Berry phase does not restrict the nature of the contour. With a simple experiment we demonstrate that Pancharatnam's phase is accompanied by a second phase which compensates the pure area effect for areas enclosed by small circles. This compensating phase changes sign at time reversal and, thus, exhibits the features of a geometric phase.

We use the recently developed wave model of geometric phase to track the continuous evolution of geometric phase as a wave propagates through optical elements and throughout an optical system. By working directly with the wave properties, we encounter a natural explanation of why the conventional Poincaré sphere solid angle method must use geodesic paths rather than the physical paths of the polarization state-the "geodesic rule"-and show that the existing rules for the solid angle algorithm are incomplete. Finally, we use the physical model to clarify the differences between the Pancharatnam connection and the geometric phase of a wave.