Photonic Spin Hall Effect (PSHE)

• Photonic Spin Hall Effect (PSHE) is a spin–orbit coupling phenomenon where photons with opposite spins experience lateral shifts due to geometric phase effects.
• It manifests as both in-plane and transverse spin-dependent splitting in diverse systems such as topological insulators, graphene, and engineered metamaterials.
• PSHE offers high-precision sensing and metrological applications by amplifying subwavelength beam shifts, impacting quantum photonics and optoelectronic device design.

The photonic spin Hall effect (PSHE) is a spin–orbit coupling phenomenon in which photons of opposite spin (i.e., left- and right-circular polarization) experience spatially distinct trajectories upon propagation, reflection, or transmission in a structured optical environment. The effect results in spin-dependent beam splitting, typically manifested as nanometer- to micrometer-scale lateral displacements of the beam centroid. PSHE arises in contexts ranging from light–matter interaction at interfaces, artificially engineered polarization geometries, topological photonic systems, to anisotropic and non-Hermitian media. The underlying mechanisms often involve geometric phases such as the Pancharatnam–Berry phase, axion electrodynamics, or quantized electronic responses, linking the PSHE tightly to contemporary developments in topological photonics and precision nano-optics.

1. Spin–Orbit Coupling Mechanisms and Theoretical Frameworks

PSHE is rooted in the intrinsic spin–orbit interaction of light, where the polarization state couples to the spatial properties of the beam. At interfaces, this emerges from the constraint of transverse electromagnetic fields, leading to geometric phase accumulations that differ for opposite spins. Notably, interface-induced Rytov–Vladimirskii–Berry phases enter the Fresnel coefficients that describe optical reflection/refraction. In topological insulators (TIs), the presence of axion coupling modifies Maxwell’s equations according to:

$$\mathbf{D} = \epsilon \mathbf{E} + \frac{\alpha\Theta}{\pi}\, \mathbf{B}, \qquad \mathbf{H} = \frac{\mathbf{B}}{\mu} - \frac{\alpha\Theta}{\pi}\, \mathbf{E}$$

with the axion response entering through $$\mathcal{L}_\Theta = \frac{\alpha\Theta}{4\pi^2}\, \mathbf{E}\cdot\mathbf{B}$$. The generalized reflection matrix for such a system is:

$$R = \frac{1}{\Lambda} \begin{bmatrix} 1 - n^2 - (\alpha\Theta/\pi)^2 + n\xi_- & 2(\alpha\Theta/\pi) \ 2(\alpha\Theta/\pi) & -(1 - n^2 - (\alpha\Theta/\pi)^2) + n\xi_- \end{bmatrix}$$

where the cross-polarized elements (i.e., $r_{ps}, r_{sp}$) allow spin–orbit coupling only when the axion term is nonzero. For a spatially confined Gaussian beam, decomposition into angular spectra and Taylor expansion of the reflection coefficients leads to additional phase factors, e.g., $e^{\pm ik_x(i\eta - \theta)}$ and $e^{\pm ik_y \delta}$, corresponding to effective spin–orbit interactions producing in-plane and transverse spin-dependent splitting (1311.0556).

In other scenarios, as in geometric PSHE, the effect emerges not from an interface but from spatially varying polarization geometries, creating an effective Pancharatnam–Berry phase:

$$|\psi(\theta,\beta)\rangle = \cos\frac{\theta}{2}e^{i\beta/2}|R\rangle + \sin\frac{\theta}{2}e^{-i\beta/2}|L\rangle$$

with a local polarization rotation, e.g., $\beta = Qx + \beta_0$, inducing a phase gradient and spin-dependent momentum shift (Ling et al., 2014).

In materials with quantized electronic responses—such as graphene in the quantum Hall regime—the quantization of the Hall conductivity directly imparts a quantized Berry phase to the reflected beam, resulting in quantized PSHE step sizes (Cai et al., 2017).

2. Spin-Dependent Splitting: In-Plane vs. Transverse Shifts

The observed spin-dependent splittings of photonic beams are categorized into:

• In-plane (longitudinal) spin-dependent splitting: Along the direction parallel to the incidence/reflection plane, predominantly sensitive to modifications of the system’s symmetry, e.g., axion angle in TIs or quantum Hall phase in graphene (1311.0556, Cai et al., 2017).
• Transverse (Imbert–Fedorov) spin-dependent splitting: Perpendicular to the incidence/reflection plane, typically less sensitive—or even insensitive—to the topological or axionic parameters.

For instance, in TIs, the in-plane splitting vanishes in the absence of axion coupling ($\Theta=0$), while the transverse splitting is relatively impervious to $\Theta$ variations (1311.0556). In quantized PSHE in graphene, both the spatial and angular spin-dependent shifts are expressed as integer multiples of a quantized factor, with analytic formulae such as:

$$\langle x^H_{r\pm} \rangle = \pm 2(2n_c + 1)\, \operatorname{Sgn}[B]\,\frac{e^2}{4\pi\hbar\epsilon_0 c} \frac{1}{k_0} \operatorname{Re}[W^H]$$

where $n_c$ is the number of filled Landau levels (Cai et al., 2017). In general, the magnitude and direction of these shifts are tunable via system parameters such as axion coupling, gating fields, or incident polarization.

3. Polarization Structures and the Magneto-Optical Kerr Effect

PSHE has substantial impact on the polarization structure of light, especially in systems exhibiting the magneto-optical Kerr effect. Due to the cross-polarized reflection coefficients enabled by symmetry-breaking interactions (e.g., axion coupling or spin–orbit engineering), the reflected field comprises central and cross-polarized components whose interference leads to pronounced spatial polarization textures.

Observable consequences include:

• Kerr rotation of the central beam, quantified as $\tan\theta_K = r_{sp}/r_{pp}$ for a horizontally polarized incident beam (1311.0556).
• Double-peak intensity profiles in the cross-polarized channels, corresponding to the spatial separation of opposite circular polarization components.
• Rotation and symmetry changes in the local electric field (polarization arrows) as functions of system parameters.

Such polarization-resolved measurements permit retrieval of critical physical parameters such as the axion angle or Hall conductivity, and offer handles for controlling polarization in advanced photonic devices.

4. Amplification and Detection via Weak Measurement Schemes

Due to the typically nanometric magnitude of PSHE-induced beam shifts, detection frequently relies on weak measurement techniques. The standard protocol involves:

1. Preparing (pre-selecting) the beam in a particular polarization.
2. Allowing the beam to undergo spin–orbit coupling (e.g., reflection from an engineered interface).
3. Post-selecting in a nearly orthogonal polarization state—typically realized by a polarization analyzer set at a small angular deviation $\Delta$ from orthogonality.
4. The resulting interference between nearly orthogonal components produces a centroid shift amplified proportionally to the "weak value" $A_w$, with the net displacement:

$$\delta_w = A_w^{\text{mod}} \delta_+^x, \qquad A_w^{\text{mod}} = |A_w|F$$

where $F$ encompasses the free-propagation-induced amplification (1311.0556). Such techniques have been demonstrated to amplify otherwise undetectable displacements to the order of microns or millimeters (Cai et al., 2017, Zhou et al., 2014).

5. Routes to Enhanced and Controllable Photonic Spin Hall Effect

Several mechanisms for enhancement and control of the PSHE have been investigated:

• Topological insulators: Tuning the axion angle via surface magnetization or external fields enables routing of the in-plane spin separation (1311.0556).
• Hyperbolic metamaterials and subwavelength gratings: Deeply subwavelength metal–dielectric stacks with extreme anisotropy greatly increase PSHE, with observed shifts reaching hundreds of microns over nanometric thicknesses (Takayama et al., 2018).
• Engineered polarization geometry: Imprinting spatially varying polarization distributions (“metapolarization”) creates geometric PSHE, enabling flexible design of spin-dependent routing (Ling et al., 2014).
• Parity-time (PT) symmetric, non-Hermitian systems: By operational design at an exceptional point, PSHE can be sharply amplified or suppressed, with the transverse shift exhibiting sign-switching across the EP (Zhou et al., 2019).
• Dirac semimetals and optical Tamm states: The excitation of interface-localized Tamm plasmons with Dirac semimetals as active layers produces giant spin-dependent shifts, tunable by Fermi level, layer thickness, or spacer thickness (Yin et al., 2022).
• Rydberg atomic gases and EIT: Nonlocal third-order susceptibilities (from Rydberg-Rydberg interactions) under electromagnetically induced transparency provide dynamic, highly tunable PSHE with amplified shifts (Liu et al., 31 May 2025).
• Quantum-well tunneling: Fano-type quantum interference in asymmetric double quantum wells yields giant, controllable PSHE, further amplified by balanced gain/loss cavities (Badshah et al., 4 Sep 2025).
• Bulk twisted anisotropic media: The spatial profile of the geometric (Pancharatnam–Berry) phase, set by the local optical axis rotation, acts as a synthetic gauge field, routing left- and right-handed polarizations along mirror-symmetric paths (Jisha et al., 7 Jul 2025).

Direct experimental approaches are increasingly circumventing the earlier requirement for weak post-selection: multipolar engineering and “superscattering” in Mie scatterers with broken rotational symmetry enable direct, intensity-preserving observation of the spin Hall shift at the single-particle level (Khan et al., 4 Jul 2025).

6. Metrological and Sensing Applications

PSHE is recognized as a sensitive probe of interface, film, and quantum material properties. Applications include:

• Measurement of physical parameters such as: nanometal film thickness, number of graphene layers, axion coupling in topological insulators, and magneto-optical constants of magnetic films (Zhou et al., 2014).
• Quantum metrology: In low-frequency (THz) studies, the magnitude and dependence of PSHE shifts allow extraction of the quantum geometric capacitance of 2D insulators, providing access to the quantum metric of the electronic band structure (Fernández-Méndez et al., 4 May 2025).
• Precision sensing: The extraordinary sensitivity of the shift to system parameters (e.g., Fermi energy, external fields, cavity gain/loss, or interface conductivity) enables high-precision index sensing or environmental sensing in lab-on-chip systems.

7. Outlook and Future Directions

The ongoing evolution of PSHE research points to several key directions:

• Integration into optoelectronic and spintronic devices, particularly where dynamic, real-time control of polarization and beam trajectory are needed.
• Advanced metamaterials and non-Hermitian engineering to achieve enhanced spin–orbit effects, including robust, topologically protected light transport.
• Extension to new material and frequency regimes—including THz, mid-IR, and atomically thin quantum crystals—expanding the utility of PSHE as a diagnostic and information-processing modality.
• Use as a probe of electronic topology and quantum geometry, where optical shifts provide a non-invasive signature of band structure invariants.
• Device applications in quantum information, beam steering, and optical logic, leveraging the ability to route photons according to their intrinsic spin with subwavelength spatial resolution.

The photonic spin Hall effect embodies a confluence of advanced physical concepts—geometric phase, symmetry breaking, quantum topological responses, and strong light–matter interaction—and acts as a bridge between fundamental optics, materials science, and applied photonic engineering.