Observation of the Geometric Spin Hall Effect of Light

The spin Hall effect of light (SHEL) is the photonic analogue of the spin Hall effect occurring for charge carriers in solid-state systems. This intriguing phenomenon manifests itself when a light beam refracts at an air-glass interface (conventional SHEL) or when it is projected onto an oblique plane, the latter effect being known as the geometric SHEL. It amounts to a polarization-dependent displacement perpendicular to the plane of incidence. In this work, we experimentally investigate the geometric SHEL for a light beam transmitted across an oblique polarizer. We find that the spatial intensity distribution of the transmitted beam depends on the incident state of polarization and its centroid undergoes a positional displacement exceeding one wavelength. This novel phenomenon is virtually independent from the material properties of the polarizer and, thus, reveals universal features of spin-orbit coupling.

Figure 1

Pictorial representation of the geometric SHEL. After the quarter-wave plate (QWP), the beam is circularly polarized and passes through a polarizer tilted by an angle $\theta$. The center of the intensity distribution of the transmitted beam appears shifted with respect to the axis of the incident beam. Such a shift can be measured by a quadrant detector put behind the polarizer. The reference frame $\{x^{\prime },y^{\prime },z^{\prime }\}$ is aligned with the polarizer surface.

Figure 2

The geometric SHEL at a polarizing interface. (a) The horizontal dark grey plane represents the polarizing interface with the Cartesian reference frame $\{x^{\prime },y^{\prime }\equiv y,z^{\prime }\}$ attached to it. The axis ${\widehat{\mathit{x}}}^{\prime }$ is taken parallel to the absorbing axis of the polarizing interface. $\theta \in [0,\pi /2)$ denotes the angle of incidence. The central wave vector ${\mathit{k}}_0$ of the incident beam defines the direction of the axis $\widehat{\mathit{z}}$ of the frame $\{x,y,z\}$ attached to the beam. It is instructive to study a wave vector $\mathit{k}$ in the $z\text{−}y$ plane, rotated by an angle $\delta \ll 1$ with respect to ${\mathit{k}}_0$. (b) The unit vector $\widehat{\mathit{a}}\perp \mathit{k}$, representing the direction of the absorbed component of the incident field, lies in the common plane of $\mathit{k}$ and ${\widehat{\mathit{x}}}^{\prime }$ (colored light brown in the figure) and can be obtained, in the first-order approximation, from the rotation by the angle $\delta \tan \theta$ around $\mathit{k}$, of the unit vector $\widehat{\mathit{x}}$.

Figure 3

Experimental setup: Simultaneous measurement of the incident state of polarization and the position of a light beam transmitted across a tilted polarizer. State preparation: The relative phase between horizontally and vertically polarized components is modulated using a polarizing beam splitter (PBS), a piezomirror ($M$), and quarter- and half-wave plates (QWP and HWP). The beam is spatially filtered using a single-mode fiber (SMF). Stokes measurement: The transmitted port of a beam splitter (BS) is used to monitor the state of polarization. We use the Stokes parameters $S_3$ to distinguish left- and right-hand circular polarization. Shift measurement: The beam is propagated across our sample, a tank containing a glass polarizer and an index-matching liquid, and its position is observed using a quadrant detector. An optional PBS ($A$) can be employed as an analyzer in front of the detector. The photocurrents $I_{+}$, $I_{-}$, $I_t$, and $I_b$ are amplified and digitally sampled for 1 s at 50 kHz.

Figure 4

Polarization-dependent beam shifts occurring at a tilted polarizer. Measurement data and theoretical predictions are shown. Both series of shift measurements were repeated five times. We report the mean value and standard deviation of the mean. The dashed blue line shows the theory for a perfect polarizer, while the solid lines were calculated from our phenomenological polarizer model whose empirical parameters ${\tau }_t^2$ and ${\tau }_a^2$ are determined by fitting the measured transmittance of the polarizer [see the Supplemental Material [33] and inset of Fig. 4 for further details]. (a) Overall displacement ${\mathrm{\Delta }}_T$ of the intensity barycenter after transmission across the polarizer. Inset: Measured transmittance of light beams across the polarizer as a function of the tilting angle $\theta$ and the incident state of polarization. The polarizer is aligned such that at normal incidence ($\theta =0°$), vertical ($V$) polarization is transmitted and horizontal ($H$) polarization is blocked. Details of the continuous fitting curves are given in the Supplemental Material [33]. (b) Displacement ${\mathrm{\Delta }}_y$ of solely the vertically polarized intensity component. The two measurements ${\mathrm{\Delta }}_T$ and ${\mathrm{\Delta }}_y$ differ due to the imperfect nature of our polarizer, as discussed in the Supplemental Material [33].

Figure 5

Theory for the conventional spin Hall effect of light compared to the measured geometric SHEL. A light beam transmitted across an interface between two media can undergo a transverse displacement known as the Imbert-Fedorov effect or the conventional SHEL. Here, we plot this displacement for a left-hand circularly polarized beam ($\sigma =+1$) for two different cases and compare this with the geometric SHEL studied in this work. (a),(b) Measured geometric SHEL $\frac{1}{2}{\mathrm{\Delta }}_T$ and $\frac{1}{2}{\mathrm{\Delta }}_y$, respectively, for the same configuration as in Fig. 4. (c) Geometric SHEL as predicted for an ideal polarizing interface. (d) Conventional SHEL occurring at an air-glass interface ($n_1=1$, $n_2=1.5$). (e) Conventional SHEL expected for the entrance face of our submerged polarizer ($n_1=n_L=1.521$, $n_2=n_G=1.517$).