Geometric phase in optics and angular momentum of light

Abstract

Physical mechanism for the geometric phase in terms of angular momentum exchange is elucidated. It is argued that geometric phase arising out of the cyclic changes in the transverse mode space of Gaussian light beams is a manifestation of the cycles in the momentum space of the light. Apparent non-conservation of orbital angular momentum in the spontaneous parametric down conversion for the classical light beams is proposed to be related with the geometric phase.2 Phase singularities in light beams had been an active area of research, however prior to 1992 [1] their relationship with a well defined orbital angular momentum (OAM) of electromagnetic (EM) waves was not clear. Paraxial wave solutions of the source free Maxwell equations with helical wavefronts were shown to possess OAM in the units of ž (Planck constant/2π) in [1], and a nice early exposition of the phase singularities and their generation using computer-generated holograms can be found in [2]. For a recent review we refer to [3]. On the other hand, stimulated by the Berry phase in quantum mechanics [4], its optical analogues i.e. Pancharatnam phase in the polarization state space (geometrically represented by the Poincare sphere) and Rytov-Vladimirskii phase in the wave-vector space (geometrically represented by the sphere of directions in the momentum space) were rediscovered. There exists a considerable debate on the question whether GP in optics is a classical or a quantum phenomenon, see [5] and also the review [6], however the issue of the physical origin of GP was addressed in [5]. It was suggested that spin and orbital parts of the angular momentum were responsible for the Pancharatnam and Rytov-Vladimirskii-ChiaoWu (RVCW) phases respectively. [Note that Chiao and Wu proposed the spin redirection phase [7] similar to the earlier work of Rytov-Vladimirskii, hence the name RVCW phase].Earlier Jiao et al [8] considered both phases simultaneously, and used the geometry of a generalized Poincare sphere to suggest that the angular momentum exchange of light with optical elements was common origin of both the phases. Later the analyses by van Enk [9] and Banerjee [10] supported the hypothesis [5] that angular momentum exchange was a physical mechanism for the GP. In [9] van Enk proposed that geometric phase arising out of the cycles in the mode space of Gaussian light beams was a new phase. A Poincare-sphere equivalent for OAM states was proposed by Padgett and Courtial [11], and rotational frequency shift in a rotating mode converter for OAM bearing beams was interpreted in terms of this GP. In a recent paper, Galvez et al [12] claim to have made first direct measurement of GP in the mode space. The meaning of angular momentum exchange remains obscure in the cited works though we clearly related it with the angular momentum holonomy in [5], and gave a plausibility argument for the gauge theoretic approach for the Pancharatnam phase in [13].In this paper we address three issues. Since the light beams with OAM are now well Mathematically, GP...

Angular momentum and geometrical phases in tight-focused circularly polarized plane waves

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Quadrature Squeezing and Geometric-Phase Oscillations in Nano-Optics

“…From Figure 3B, we can more clearly see the relation between the frequency of the geometric-phase oscillation and ω. Evidently, as ω increases, the oscillation of γ G,β (t) becomes rapid. The geometric phase associated with light waves, except for its oscillation, was also studied by other research groups [33][34][35][36]. In particular, Kuratsuji [33] investigated the geometric phase of a polarized light described by the SU(2) coherent state using a different scheme based on the geometry of two interfering beams which were initially split from a source beam.…”

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Topological defects, geometric phases, and the angular momentum of light

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