Two topological phases in optics by means of a nonplanar Mach-Zehnder interferometer

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Abstract
Experiments on two of the recent manifestations of Berry’s phases in optics and their mutual influence are reported. The first is the phase that arises from a cycling in the directions of a beam of light, so that the tip of the spin vector of a photon in this beam traces out a closed curve on the sphere of spin directions. The second is Pancharatnam’s phase that arises from a cycling in the polarization states of the light while keeping the direction of the beam of light fixed, so that the tip of the Stokes vector traces out a closed curve on the Poincaré sphere. What happens when the two phases are combined in the same experiment is examined. We find that the two phases are additive if we use the notion of a generalized Poincaré sphere. The classical theory of the effect based on the Jones calculus is presented, as well as a general analysis of Berry’s phases for Maxwell’s equations. The signs of these phases have been determined experimentally, and they agree with theory.