Geometric phase in beating of light waves

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.

Interference and polarization beating of independent arbitrarily polarized polychromatic optical waves

“…3(a)]. This fact has been verified by both analytical [27] and numerical calculations using Eqs. (10)- (13).…”

“…(4)- (6) are not valid, if the interfering waves are not independent. As an example, let us consider interference of a wave with its coherently frequencyshifted version, e.g., obtained via the Doppler effect [27]. It can be readily verified, by decomposing the waves into their frequency components, that the frequency-shifted version of a polychromatic field E 1 (t ) is simply equal to CE 1 (t )e i ωt , where C is a constant.…”

Quadrature Squeezing and Geometric-Phase Oscillations in Nano-Optics

“…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.…”

“…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.…”

“…In addition, the expectation values β|â † |β and β|(â † ) 2 |β are complex conjugates of the results of Equations (34) and (35), respectively. Using these relations we can easily confirm that the fluctuations of canonical variables defined as (∆y) β = [ β|ŷ 2 |β − ( β|ŷ|β ) 2 ] 1/2 where y = q, p are given by Equation (6) in the text.…”