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Physical Review A

Pancharatnam-Berry phase in electromagnetic double-pinhole interference

Antti Hannonen, Henri Partanen, Jani Tervo, Tero Setälä, Ari T. Friberg

Published May 20, 2019• 18 citations
Publisher

Abstract

Young's two-beam interference and the Pancharatnam-Berry geometric phase constitute two landmark results in optical physics. Employing the spectral interference law of electromagnetic waves, we establish a general expression for the Pancharatnam-Berry phase in Young's two-pinhole setup. We show that the phase is elegantly specified by the electric field Stokes parameters at the openings. The geometric phase can also be deduced from the interference pattern using Pancharatnam's definition of the generalized phase, as we demonstrate both numerically and experimentally. Our results imply a way to control the phase differences in two-beam electromagnetic interference by adjusting the input-field intensities and polarization states.

DOI
10.1103/physreva.99.053826

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According to Pancharatnam [S. Pancharatnam, in Geometric Phases in Physics, edited by A. Shapere and F. Wilczek (World Scientific, Singapore, 1989)], two light beams of differing elliptical polarization can be said to be ``in phase'' if their interference is maximally constructive. In the same spirit, the relative phase between two beams a and b is the amount by which the phase of beam b needs to be changed to put it in phase with beam a. Berry has established a simple relation between Pancharatnam's concept and a phase relation between two quantum-mechanical states: If two nonorthogonal states \ensuremath{\Vert}${\mathrm{\ensuremath{\psi}}}_{\mathit{a}}$〉 and \ensuremath{\Vert}${\mathrm{\ensuremath{\psi}}}_{\mathit{b}}$〉 are phased in such a way that \ensuremath{\Vert}${\mathrm{\ensuremath{\psi}}}_{\mathit{a}}$〉+\ensuremath{\Vert}${\mathrm{\ensuremath{\psi}}}_{\mathit{b}}$〉 has maximum norm, i.e., such that 〈${\mathrm{\ensuremath{\psi}}}_{\mathit{a}}$\ensuremath{\Vert}${\mathrm{\ensuremath{\psi}}}_{\mathit{b}}$〉 is real and positive, they can be said to be in phase or parallel, and in general the relative phase between them is the phase angle of 〈${\mathrm{\ensuremath{\psi}}}_{\mathit{a}}$\ensuremath{\Vert}${\mathrm{\ensuremath{\psi}}}_{\mathit{b}}$〉. This concept plays a role in the theory of the Abelian geometric phase. In the non-Abelian case, one has two n-dimensional subspaces instead of two single states, and instead of choosing the phase of each state one has to choose an orthonormal basis (including phase) in each subspace. We show that the concept of ``block diagonalization'' introduced by Cederbaum, Shirmer, and Meyer [J. Phys. A 22, 2427 (1989)] in the context of a molecular problem provides a natural generalization of the Pancharatnam-Berry phase relation.

Physical Review A•Aug 1, 1991•10 citations
Young's two-beam interference and the Pancharatnam-Berry geometric phase constitute two landmark results in optical physics. Employing the spectral interference law of electromagnetic waves, we establish a general expression for the Pancharatnam-Berry phase in Young's two-pinhole setup. We show that the phase is elegantly specified by the electric field Stokes parameters at the openings. The geometric phase can also be deduced from the interference pattern using Pancharatnam's definition of the generalized phase, as we demonstrate both numerically and experimentally. Our results imply a way to control the phase differences in two-beam electromagnetic interference by adjusting the input-field intensities and polarization states.
An electromagnetic Young's interference experiment is considered with arbitrary, spectrally partially coherent and partially polarized electric fields at the pinholes. Any correlations that exist at the openings lead to modulations of the Stokes parameters on the observation screen. The electromagnetic degree of coherence can be viewed, in direct analogy to the usual scalar case, as a measure of the modulation contrasts of not just the optical intensity but also the polarization state. The electromagnetic degree of coherence can be obtained experimentally via four visibility measurements using appropriate wave plates. If the field at the pinholes is fully polarized, a periodic variation of the polarization state appears on the observation screen, leading to a Pancharatnam–Berry phase. It is shown that this geometric phase has a straightforward but elegant relationship to the Stokes parameters at the pinholes.
Young’s dual-pinhole interference experiment with arbitrary fully correlated and polarized vector light fields leads to a Pancharatnam–Berry geometric phase that is related to the associated dynamical phase. We demonstrate theoretically and experimentally how the dynamical phase across the interference pattern can be deciphered from the total phase, thereby leaving only the geometric phase. Our results constitute the first genuine interferometric phase measurements that yield the Pancharatnam–Berry geometric phase in Young’s two-beam interference setup.
Abstract The geometric phase for classical electromagnetic light beams, in its original formulation as introduced by Pancharatnam, concerns fields experiencing cyclic, discrete in-phase polarization-state changes. A similar phase was later recognized by Berry to govern the behavior of adiabatic quantum systems, with consequent extensions to nonadiabatic and noncyclic evolutions of the quantum state. However, no optical counterpart for the noncyclic, adiabatic (continuous) evolution has been demonstrated. Here we employ a modified Young’s two-pinhole setup with controlled pinhole polarizations and intensities to generate on interference an arbitrary continuous spatial evolution of the polarization state, an optical analogue to the adiabatic case. The customized arrangement allows separating at any point the accumulated dynamical and geometric phases from the total phase, enabling a detailed study of the noncyclic Pancharatnam–Berry phase in a continuous transformation. Our theoretical and experimental results are in excellent agreement and consistent with the geodesic rule for noncyclic evolutions.
Abstract Beating is a simple physical phenomenon known for long in the context of sound waves but remained surprisingly unexplored for light waves. When two monochromatic optical beams of different frequencies and states of polarization interfere, the polarization state of the superposition field exhibits temporal periodic variation—polarization beating. In this work, we reveal a foundational and elegant phase structure underlying such polarization beating. We show that the phase difference over a single beating period decomposes into the Pancharatnam–Berry geometric phase and a dynamical phase of which the former depends exclusively on the intensities and polarization states of the interfering beams whereas the sum of the phases is determined solely by the beam frequencies. Varying the intensity and polarization characteristics of the beams, the relative contributions of the geometric and dynamical phases can be adjusted. The geometric phase inherent in polarization beating is governed by a compact expression containing only the Stokes parameters of the interfering waves and can alternatively be obtained from the individual beam intensities and the amplitude of the intensity beats. We demonstrate both approaches experimentally by using an interferometer with a fast detector and a specific polarimetric arrangement. Polarization beating has a unique character that the geometric and dynamical phases are entangled, i.e. variation in one unavoidably leads to a change in the other. Our work expands geometric phases into a new domain and offers important novel insight into the role of polarization in interference of electromagnetic waves.
According to Pancharatnam [S. Pancharatnam, in Geometric Phases in Physics, edited by A. Shapere and F. Wilczek (World Scientific, Singapore, 1989)], two light beams of differing elliptical polarization can be said to be ``in phase'' if their interference is maximally constructive. In the same spirit, the relative phase between two beams a and b is the amount by which the phase of beam b needs to be changed to put it in phase with beam a. Berry has established a simple relation between Pancharatnam's concept and a phase relation between two quantum-mechanical states: If two nonorthogonal states \ensuremath{\Vert}${\mathrm{\ensuremath{\psi}}}_{\mathit{a}}$〉 and \ensuremath{\Vert}${\mathrm{\ensuremath{\psi}}}_{\mathit{b}}$〉 are phased in such a way that \ensuremath{\Vert}${\mathrm{\ensuremath{\psi}}}_{\mathit{a}}$〉+\ensuremath{\Vert}${\mathrm{\ensuremath{\psi}}}_{\mathit{b}}$〉 has maximum norm, i.e., such that 〈${\mathrm{\ensuremath{\psi}}}_{\mathit{a}}$\ensuremath{\Vert}${\mathrm{\ensuremath{\psi}}}_{\mathit{b}}$〉 is real and positive, they can be said to be in phase or parallel, and in general the relative phase between them is the phase angle of 〈${\mathrm{\ensuremath{\psi}}}_{\mathit{a}}$\ensuremath{\Vert}${\mathrm{\ensuremath{\psi}}}_{\mathit{b}}$〉. This concept plays a role in the theory of the Abelian geometric phase. In the non-Abelian case, one has two n-dimensional subspaces instead of two single states, and instead of choosing the phase of each state one has to choose an orthonormal basis (including phase) in each subspace. We show that the concept of ``block diagonalization'' introduced by Cederbaum, Shirmer, and Meyer [J. Phys. A 22, 2427 (1989)] in the context of a molecular problem provides a natural generalization of the Pancharatnam-Berry phase relation.