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Physical Review A

Spin-redirection Berry phase with planar rays

Aymeric Braud, Renaud Gueroult

Published Mar 19, 2026• 2 citations
Publisher

Abstract

Geometric or Berry phases are fundamental manifestations that appear in many areas of physics. They arise from the geometry of the space describing the properties of multicomponent wave fields. An important example for electromagnetic waves is the spin-redirection Berry phase associated with the evolution of the spin direction. Because this effect has traditionally been studied in isotropic media where the spin is aligned with the ray trajectory, it has become commonly assumed that this spin-redirection Berry phase requires nonplanar rays. Here, we show that a spin-redirection phase can in fact arise along a planar ray if the spin evolves along the ray. We expose this effect through the singular example of a moving unmagnetized plasma, and demonstrate how this behavior can more generally arise from a finite transverse spin. In identifying this new spin-redirection mechanism, our work not only provides the tools to discover additional manifestations of spin-orbit interactions (SOIs) in nature, but also uncovers supplemental degrees of freedom to harness SOIs to control light.

DOI
10.1103/vkls-qpkg

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Geometric or Berry phases are fundamental manifestations that appear in many areas of physics. They arise from the geometry of the space describing the properties of multicomponent wave fields. An important example for electromagnetic waves is the spin-redirection Berry phase associated with the evolution of the spin direction. Because this effect has traditionally been studied in isotropic media where the spin is aligned with the ray trajectory, it has become commonly assumed that this spin-redirection Berry phase requires nonplanar rays. Here, we show that a spin-redirection phase can in fact arise along a planar ray if the spin evolves along the ray. We expose this effect through the singular example of a moving unmagnetized plasma, and demonstrate how this behavior can more generally arise from a finite transverse spin. In identifying this new spin-redirection mechanism, our work not only provides the tools to discover additional manifestations of spin-orbit interactions (SOIs) in nature, but also uncovers supplemental degrees of freedom to harness SOIs to control light.
Spin-orbit interactions (SOIs) of light are manifestations of coupling between components of light's angular momentum. They are at play in most basic optical processes, offering opportunities both to understand their fundamental origin and to control light in novel ways. Because SOIs become significant at subwavelength scale, they have largely been explored in the context of inhomogeneous materials exhibiting wavelength-scale structures, and notably metamaterials. Here we demonstrate that spin-orbit interactions can in fact analogously emerge in moving matter through the well-known light-dragging effects. SOIs in moving media are shown to manifest through a Berry phase induced by vorticity, which then leads to a rotation of the wave's polarization. In bringing together electrodynamics of moving media and SOIs of light, our work not only paves the way for the discovery of new fundamental effects but also uncovers novel means to harness SOIs to control light.
The wave superposition model of the geometric phase shows how the addition of waves creates a shift in the resulting wave position. While previous work focused on a basis of linearly polarized light waves and the Pancharatnam-Berry phase, we show how the spin-redirection phase (Rytov-Vladimirsky-Berry phase) can also be derived from the same approach of wave superposition, using rotating vectors to represent the superposing oscillations. The result is the first derivation of the spin-redirection phase using wave superposition. We illustrate this approach with two classic examples of the geometric phase of rotations in space: a system of three fold mirrors and the helically coiled fiber.
Vast literature on the experiments and mathematical formulations on the geometric phases signifies the importance of this subject. Physical mechanism for the origin of the geometric phases in optics was suggested in 1992 by the author in terms of the exchange of the angular momentum. Some of the literature has taken notice of it, however, the real import of the suggested mechanism and the angular momentum holonomy conjecture has remained elusive. The present contribution delineates the prevailing confusions and offers a thorough discussion on the nature of the light waves to resolve the conceptual issues. It is argued that at a fundamental level there exist only two types of geometric phases arising from (i) the geometry of the wave vector space, and (ii) the polarization state space. For the light beams of definite polarization spatial modes (HG or LG modes) the geometric phase has origin in the wave vector space that is different than the spin redirection phase for the polarized light. Angular momentum holonomy is proposed to be equivalent to the geometric phase in general.
In 1984 Berry addressed a quantum system undergoing a unitary and cyclic evolution under the action of a time-dependent Hamiltonian (M. V. Berry, 1984). The process was supposed to be adiabatic, meaning that the time scale of the system’s evolution was much shorter than the time scale of the changing Hamiltonian. Until Berry’s study, it was assumed that for a cyclic Hamiltonian the quantum state would acquire only so-called dynamical phases, deprived of physical meaning. Such phases could be eliminated by redefining the quantum state through a “gauge” transformation of the form |ψ〉 → eiα |ψ〉. However, Berry discovered that besides the dynamical, there was an additional phase that could not be “gauged away” and whose origin was geometric or topological. It depended on the path that |ψ〉 describes in the parameter space spanned by those parameters to which the Hamiltonian owed its time dependence. Berry’s discovery was the starting point for a great amount of investigations that brought to light topological aspects of both quantum and classical systems. Berry’s phase was soon recognized as a special case of more general phases that showed up when dealing with topological aspects of different systems. For example, the Aharonov-Bohm phase could be understood as a geometric phase. The rotation angle acquired by a parallel-transported vector after completing a closed loop in a gravitationally curved space-time region, is also a geometric, Berry-like phase. Another example is the precession of the plane of oscillation of a Foucault pendulum. Berry’s original formulation was directly applicable to the case of a spin-1/2 system evolving under the action of a slowly varying magnetic field that undergoes cyclic changes. A spin-1/2 system is a special case of a two-level system. Another instances are two-level atoms and polarized light, so that also in these cases we should expect to find geometric phases. In fact, the first experimental test of Berry’s phase was done using polarized, classical light (A. Tomita, 1986). Pancharatnam (S. Pancharatnam, 1956) anticipated Berry’s phase when he proposed, back in 1956, how to decide whether two polarization states are “in phase”. Pancharatnam’s prescription is an operational one, based upon observing whether the intensity of the interferogram formed by two polarized beams has maximal intensity. In that case, the two polarized beams are said to be “in phase”. Such a definition is analogous to the definition of distant parallelism in differential geometry. Polarized states can be subjected to different transformations which could be cyclic or not, adiabatic or not, unitary or not. The Pancharatnam-Berry Phase: Theoretical and Experimental Aspects
Abstract Using the hydrodynamical formalism of quantum mechanics for a Schrödinger spinning particle developed by Takabayashi, Vigier, and followers, which involves vortical flows, we propose a new geometrical interpretation of the pilot wave theory. The spinor wave in this interpretation represents an objectively real field, and the evolution of a material particle controlled by the wave is a manifestation of the geometry of space. We assume this field to have a geometrical nature, basing on the idea that the intrinsic angular momentum, the spin, modifies the geometry of the space, which becomes a manifold, represented as a vector bundle with a base formed by the translational coordinates and time, and the fiber of the bundle, specified at each point by the field of a tetrad eμae^a_{\mu}eμa​, forms from bilinear combinations of the spinor wave function. It has been shown that the spin vector rotates following the geodesic of the space with torsion, and the particle moves according to the geometrized guidance equation. This fact explains the self-action of the spinning particle. We show that the curvature and torsion of the spin vector line is determined by the space torsion of the absolute parallelism geometry.