## Abstract

We have detected a spin-dependent displacement perpendicular to the refractive index gradient for photons passing through an air-glass interface. The effect is the photonic version of the spin Hall effect in electronic systems, indicating the universality of the effect for particles of different nature. Treating the effect as a weak measurement of the spin projection of the photons, we used a preselection and postselection technique on the spin state to enhance the original displacement by nearly four orders of magnitude, attaining sensitivity to displacements of ∼1 angstrom. The spin Hall effect can be used for manipulating photonic angular momentum states, and the measurement technique holds promise for precision metrology.

Hall effects, in general, are transport phenomena, in which an applied field on the particles results in a motion perpendicular to the field. Unlike the traditional Hall effect and its quantum versions, in which the effect depends on the electrical charge, the spin Hall effect is driven by the spin state of the particles. It was recently suggested (*1*, *2*) and observed (*3*) that, even in the absence of any scattering impurities, when an electric field is applied to a semiconductor, a dissipationless spin-dependent current perpendicular to the field can be generated. A photonic version of the effect—the spin Hall effect of light (SHEL)—was recently proposed (*4*) in which the spin-1 photons play the role of the spin-1/2 charges, and a refractive index gradient plays the role of the electric potential gradient.

We use an air-glass interface to demonstrate the SHEL, in which the transmitted beam of light splits by a fraction of the wavelength, upon refraction at the interface, into its two spin components (Fig. 1A): the component parallel (*s* = +1, right-circularly polarized) and antiparallel (*s* = –1, left-circularly polarized) to the central wave vector. This effect is different from (i) the previously measured (*5*) longitudinal Goos-Hänchen (*6*) and transverse Imbert-Fedorov (*7*, *8*) shifts in total internal reflection, which are described in terms of evanescent wave penetration, and (ii) the recently reported “optical spin Hall effect,” which deals with optically generated spin currents of exciton-polaritons in a semiconductor microcavity (*9*). The splitting in the SHEL, implied by angular momentum conservation, takes place as a result of an effective spin-orbit interaction. The same interaction also leads to other effects such as the optical Magnus effect (*10*, *11*), the fine-splitting of the energy levels of an optical resonator (*12*) [in which the interaction resembles the spin-orbit (Russell-Saunders) coupling of electrons in atoms], and the deviation of photons from the simple geodesic paths of general relativity (*13*).

The exact amount of the transverse displacements due to the SHEL at an air-glass interface has been the subject of a recent debate (*4*, *14*–*16*). Our theory and experimental results support the predictions of Bliokh and Bliokh (*15*, *16*); although the calculations of other researchers (*4*, *11*, *14*) are not incorrect, they contain rather unfavorable initial conditions [see supporting online material (SOM)]. One can obtain close estimates of the magnitude of the displacements using solely the conservation of the *z* component of the total (spin plus orbital) angular momentum resulting from the rotational symmetry. However, only in certain circumstances will the result be exact (*16*). [For a general review of light's angular momentum, see (*17*).]

We can describe the SHEL as a consequence of a geometric phase (Berry's phase) (*18*), which corresponds to the spin-orbit interaction. It is already known that photons, when guided by an optical fiber with torsion, acquire a geometric phase whose sign is determined by the spin state (*19*, *20*). When a photonic wave packet changes direction because of a spatial variation in the refractive index, the plane-wave components with different wave vectors experience different geometric phases, affecting the spatial profile and resulting in the SHEL.

For a paraxial beam, the transverse beam state (for the relevant direction *y* and its associated wave vector *k*_{y}) at the air side of the interface (Fig. 1A), including the spin state |*s*〉, can be written as |Ψ_{a}〉 = ∫*dy*Ψ(*y*)|*y*〉|*s*〉 = ∫*dk _{y}*Φ(

*k*)|

_{y}*k*〉|

_{y}*s*〉, with Φ(

*k*) being the Fourier transform of Ψ(

_{y}*y*). At the glass side of the interface, under the action of the geometric phases, the state becomes , with , indicating +δ and –δ shifts for the wave packets of the parallel and antiparallel spin states. Here, the term exp() represents a coupling between the spin and the transverse momentum of the photons.

The origin of this “spin-orbit” interaction term lies in the transverse nature of the photon polarization: The polarizations associated with the plane-wave components experience different rotations in order to satisfy the transversality after refraction. This is depicted pictorially in Fig. 1B with incoming horizontal polarization (|*H*〉) (along *x*_{I}). In the spin basis, this state corresponds to . In the lowest-order approximation, the change in the state after refraction is |*k _{y}*〉|

*H*〉→|

*k*〉(|

_{y}*H*〉 +

*k*δ|

_{y}*V*〉) = |

*k*〉|φ〉, with φ =

_{y}*k*δ << 1 and |

_{y}*V*〉 being vertical polarization. In the spin basis, , indicating the coupling exp().

As a result of the polarization-dependent Fresnel reflections at the interface, the opposite displacements of the two spin components actually depend on the input polarization state (see SOM for details). For |*H*〉 and |*V*〉 input polarizations, the displacements δ^{H} and δ^{V} are given by (Fig. 1C) (1) Here, θ_{I} and θ_{T} are, respectively, the central incident and transmitted angles related by Snell's law; *t*_{s} and *t*_{p} are the Fresnel transmission coefficients at θ_{I}; and λ is the wavelength of the light in the incident medium. In a continuously varying refractive index, the input polarization dependence disappears, and the motion can be formulated in terms of a particle moving in a vector potential in momentum space (*4*, *14*, *21*, *22*), along the same lines as with electronic systems (*23*, *24*).

For optical wavelengths, precise characterization of the displacements requires measurement sensitivities at the angstrom level. To achieve this sensitivity, we use a signal enhancement technique known from quantum weak measurements (*25*). In a quantum measurement, a property (observable **Â**) of a system is first coupled to a separate degree of freedom (the “meter”), and then the information about the state of the observable is read out from the meter. At the single-photon level, the SHEL is actually equivalent to a quantum measurement of the spin projection along the central propagation direction (observable , with eigenstates |+〉 and |–〉), with the transverse spatial distribution serving as the meter [similar to a Stern-Gerlach spin-projection measurement (*26*)]. However, the displacements generated by the SHEL here are much smaller than the width of the transverse distribution, resulting in a weak measurement: The meter states associated with different spin eigenstates overlap to a large extent. Therefore, the meter carries very little information about the state of the observable, leaving the initial state almost undisturbed. Although our experiment is at a classical level with a large number of photons in a quantum-mechanical coherent state, the results remain the same, with each photon behaving independently. Furthermore, in the paraxial regime, the dynamics of the transverse distribution are given by the Schrödinger equation with time replaced by path length, making the analysis identical to nonrelativistic quantum mechanics with an impulsive measurement interaction Hamiltonian *H*_{I} = *k*_{y}**Â**δ.

With the weak measurement taking place in between, the signal enhancement technique uses an appropriate preselection and postselection of the state of the observable to achieve an enhanced displacement in the meter distribution (*25*, *27*) (Fig. 2A). Given the preselected and postselected states |ψ_{1}〉 and |ψ_{2}〉, for sufficiently weak measurement strengths, the final position of the meter is proportional to the real part of the so-called “weak value” of the measured observable **Â** (2) which increases as the postselected state approaches being orthogonal to the preselected one. This effect was previously demonstrated (*28*) with real weak values, where a birefringent displacement of photons was enhanced by a factor of 20. The imaginary part of a weak value corresponds to a displacement in the momentum-space distribution of the meter, which with free evolution leads to the possibility of even larger enhancements. Furthermore, the order does not matter: The free evolution can take place first, followed by the weak measurement, but the final displacement will be identical (Fig. 2B). We describe the final displacement of the meter as the “modified weak value,” , where the factor *F* depends on the initial state of the meter and the amount of its free evolution before detection (see SOM).

In our setup (Fig. 3), the SHEL takes place at the front surface of a variable angle prism (VAP) for various incidence angles θ_{I}, with the back surface adjusted to be at normal incidence to avoid secondary Hall shifts (because there is obviously no Hall shift at normal incidence). The VAP is constructed by attaching two BK7 round wedge prisms together with the surface tension of a thin layer of index-matching fluid and is mounted loosely to avoid any stress-induced birefringence. The prisms can rotate with respect to each other, and the entire assembly can rotate around three orthogonal axes, allowing the desired surface orientations. The enhancement effect is achieved by preselecting and postselecting the polarization states of the incoming photons in states |ψ_{1}〉 and |ψ_{2}〉, by the calcite polarizers P1 and P2, respectively (3)

With Δ << 1 being a small angle, the weak value of the spin component is given by . Lenses L1 and L2, respectively, focus and collimate the transverse spatial distribution of the incoming photons. The factor *F* in the modified weak value is determined by the transverse spatial state of the photons after lens L2 (see SOM) (4) Here, is the variance of the *y*-direction transverse distribution after lens L2, and *z*_{eff} = 125 ± 5 mm is the effective focal length of L2. The design of the VAP assures that the optical path length inside of the VAP remains the same for all incidence angles, so that *F* is always the same. We achieve a displacement at the position sensor by an amount that is times (∼10^{4} for our system) larger than the displacement caused by the SHEL alone, and we measure this by reading out the difference between the displacements for the two postselections |*V*±Δ〉.

In the results (Fig. 4), the dashed lines, which overlap with the solid lines up to 64°, represent the theory with the photons exiting the prism at normal incidence (Eq. 1). The solid lines also take into account the modifications (see SOM) arising from the fact that our VAP cannot satisfy the normal exiting condition beyond 64°. In order to stay in the linear response regime of both the enhancement technique and the position sensor, for incidence angles larger than 56°, the enhancement is adjusted (through the polarizer angle Δ) to be lower by a factor of 1.8: for θ_{I} ≤ 56°, for θ_{I} > 56°. With the value of *F* determined from a single-parameter fit to be 156 ± 2, all the data points lie on the theoretical curves (Fig. 4) with a SD of 3.5 Å from the theory (excluding the last two data points of curve |δ^{H}|, where there is a small, and as yet unexplained, discrepancy). By measuring , we can also experimentally determine the value of *F*, albeit less accurately; we find 157 ± 6, indicating the total enhancement factor to be for θ_{I} ≤ 56° and for θ_{I} > 56°. Figure 4 also shows the dependence of the displacements on the incident linear polarization (see SOM for the theory). For arbitrary incident polarization (other than |*H*〉 or |*V*〉), a kick in the *y* momentum is also expected; however, our measurements are not sensitive to this effect (see SOM).

The measurability of very small displacements is ultimately limited by the quantum noise of the light, because enough photons need to be collected to resolve the position of the transverse distribution (*29*). If the experiment is already quantum noise–limited, then (for a fixed number of photons entering the experimental setup) the enhancement due to preselection and postselection does not bring any gain in the measurement sensitivity: For any value of *F*, although the displacement is multiplied by a factor of , only a fraction of photons (|〈ψ_{1}|ψ_{2}〉|^{2} = sin^{2} Δ ≈ Δ^{2}) makes it through the postselection, which cancels the advantage of the enhancement. Nevertheless, most experiments are limited by technical issues [e.g., in our setup, by the laser pointing stability, the intensity saturation of the position sensor, or the unwanted (∼10-μm) displacements caused by rotating the polarizers]. In the experiment, we suppress all these technical issues by a factor of with respect to the signal and achieve a dc sensitivity/stability to displacements of ∼1 Å, without the need for vibration or air-fluctuation isolation. The upper limit to the enhancement comes from the achievable extinction ratio of the polarizers and from the electronic noise in the position sensor.

The overall sensitivity of our measurements can be increased by orders of magnitude, by incorporating standard signal modulation and lock-in detection techniques, and thus holds great promise for precision metrology. In fact, in principle, Hall displacements themselves can be made arbitrarily large, separating beams associated with different spin states (*4*, *21*) [and also with different orbital angular momentum states (*22*)]. In addition, the SHEL itself may become an advantageous metrological tool (e.g., for characterizing refractive index variations measured at subwavelength distances from the region of interest, because any Hall displacement accompanying a beam deflection is larger than the displacement caused by the deflection itself at such propagation distances).

**Supporting Online Material**

www.sciencemag.org/cgi/content/full/1152697/DC1

SOM Text

Fig. S1

References