Pancharatnam-Berry phase in electromagnetic double-pinhole interference

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.

Figure 1

Young's interference experiment with coherent vector fields. Stokes vectors ${\mathbf{S}}_m=(S_{0m},{\mathbf{P}}_m)$ contain the information about the intensities $S_{0m}$ and polarizations ${\mathbf{P}}_m$ of the electric fields at pinholes $m=1,2$, and $R_m$ is the distance from pinhole $m$ to the observation point. The pinhole separation in screen $\mathcal{A}$ is $d$ and the distance between $\mathcal{A}$ and the observation screen $\mathcal{B}$ is $D$. The enlargement illustrates the variation of the polarization state across one period in the interference pattern on $\mathcal{B}$.

Figure 2

Notations on the Poincaré sphere used in various steps of deriving the Pancharatnam-Berry geometric phase. The circle with the center point ${\mathbf{S}}_{\mathrm{o}}=({\widehat{\mathbf{P}}}_{\mathrm{a}}+{\widehat{\mathbf{P}}}_{\mathrm{b}})/2$ corresponds to the path of the Poincaré unit vector $\widehat{\mathbf{P}}\left(\gamma \right)$. The height of the spherical cap is $h$, while $\xi \left(\gamma \right)$ denotes the angle between the vectors ${\mathbf{S}}_{\mathrm{o}}$ and $\widehat{\mathbf{P}}\left(\gamma \right)$.

Figure 3

Pancharatnam-Berry phase within one period of the interference pattern on the observation screen as a function of the angle $\theta$ between the Poincaré vectors ${\mathbf{P}}_1$ and ${\mathbf{P}}_2$, and the angle $\phi =arctan[S_{01}/S_{02}]$ that describes the ratio of the intensities at the pinholes.

Figure 4

Experimental setup to control and measure the polarization states and intensities of the pinhole fields and to investigate their interference. Components are as follows: HeNe laser, BS1-2 beam splitters, LP1-3 linear polarizers, QWP1-2 quarter wave plates, M1-2 mirrors, PH1-2 pinholes, F1-2 lenses, D1-2 detectors, and PE1-4 circular polarizers. Insets show an image of the pinholes and an example of the fringe pattern.

Figure 5

Extraction of the parameter $|{\mu }_0|$ from the measured data. (a) Sample intensity profiles on the detector when both pinholes are open (solid black curve) and when only one is open (dashed green and dash-dotted red graphs). The dotted blue curve shows the (uncorrelated) sum of the intensities emanating from the pinholes. (b) Measured $|{\mu }_0|cos[{\alpha }_0-\gamma \left(x\right)]$ (black solid curve) and its sinusoidal fitting (dashed red curve). Horizontal lines indicate the possible extreme values.

Figure 6

Examples of the measured and theoretical values of the Pancharatnam-Berry phase. (a) Polarization states of the pinhole fields are fixed but the intensity ratio varies. Black objects (the solid line): $y$ and $x$ linearly polarized fields in pinholes 1 and 2, respectively. Red objects (the dashed line): circular and elliptical right-hand polarizations in pinholes 1 and 2, respectively. (b) Intensity ratio ($r\approx 0.49$) and the polarization state in pinhole 1 ($y$ linear) are fixed and the polarization state in pinhole 2 rotates. In all cases, circles and crosses correspond to the measured results obtained via Eqs. (13) and (21), while solid lines present the theoretical values calculated from Eq. (13).