Higher Order Pancharatnam-Berry Phase and the Angular Momentum of Light

The first experimental demonstration of a new Pancharatnam-Berry phase for light beams with spatially inhomogeneous, or vector, states of polarization referred to as the higher-order Pancharatnam-Berry phase is presented. This new geometric phase is proportional to light’s total angular momentum, a sum of spin and higher dimensional orbital angular momentum, sharply contrasting the well-known Pancharatnam-Berry phase associated with the plane wave state of polarization of a spatially homogeneous light beam. The higher-order Pancharatnam-Berry phase is directly related to the rotational symmetry of a vortex-bearing electromagnetic field, associated with the rotational frequency shift of a light beam, and has implications in quantum information science as well as other physical systems such as electron vortex beams.

Figure 1

Experimental setup: A collimated (waist size $\sim 5\mathrm{mm}$) and linear polarized ${\mathrm{TEM}}_{00}$ mode from a He-Ne laser (632 nm, 5 mW) is converted into a good approximation of phase vortex-bearing ${\mathrm{LG}}_0^{\ell }$ mode using a reflective phase only spatial light modulator (SLM) (Holoeye LC-720) displaying a blazed fork grating which controls the OAM. A ${\mathrm{LG}}_0^{-\ell }$ mode (the reference beam) is generated by an odd number of reflections using a modified Mach-Zender interferometer comprised of two beam splitters (BSs) and one mirror (M). A HWP and quarter wave plate (QWP), which controls the SAM, are used to give the two beams orthogonal circular polarization. The two beams, given by $R_{\ell }$ and $L_{\ell }$, pass colinearly through two spin-orbit converters, SOC1 and SOC2, comprised of a HWP and $\pi \mathrm{CL}$ mode converter arranged in series. The $\pi \mathrm{CL}$ mode converters consist of two CLs ($f=5\mathrm{mm}$) spaced $2f$ apart. The physical transformations of the spin-orbit converters correspond to the path $ABA$ on the HOPS. The beams pass through a linear polarizer (P) and are imaged onto a CCD camera producing the interferograms described by Eq. (4). SOC2 is rotated an angle $\Delta \varphi$ which is equivalent to a $2\Delta \varphi$ equatorial rotation on the HOPS and results in a rotation of the lobes of the interferogram. (a) Fork grating displayed on SLM. (b) Intensity of ${\mathrm{LG}}_0^{\ell }$ mode. (c) Intensity of ${\mathrm{LG}}_0^{-\ell }$ mode. (d) Spin-orbit converters. (e) Cylic path on HOPS. (f) Rotating interferogram.

Figure 2

Experimental Data. (a) Table of interferograms for Eq. (4). Each row displays the observed interferogram lobe rotation for one full fringe shift ($m=1$) corresponding to the phase shift $2\gamma$ in steps of $\pi /2$ for a beam with TAM $J=(\ell +\sigma )$. The total SOC2 rotation $\Delta \phi$ is shown. (b) Theoretical (red) and experimental (blue) plot of Eq. 5 for $m=10$ fringe shifts. Crosses represent $\mathrm{sgn}(\ell )=\mathrm{sgn}(\sigma )$ and circles represent $\mathrm{sgn}(\ell )\ne \mathrm{sgn}(\sigma )$. Error bars are smaller than the markers.

Figure 3

Instantaneous field distributions, a snapshot in time, for circular polarized phase vortices of TAM $J=(\ell +\sigma )$. Red lines demarcate the field’s axis of symmetry.