Vortex-Induced Beam Shifts

• Vortex-induced beam shifts are spatial and angular displacements experienced by vortex beams carrying OAM when interacting with interfaces, governed by conservation laws and Berry phase effects.
• They generalize classical Goos–Hänchen and Imbert–Fedorov shifts, with enhanced sensitivities in layered, resonant, and topological media for precise sensing applications.
• These shifts bridge optics, acoustics, hydrodynamics, and topological studies by revealing fundamental wave-matter interactions and enabling advanced modulation and diagnostics.

Vortex-induced beam shifts are spatial and angular displacements experienced by wave beams carrying intrinsic orbital angular momentum (OAM) when they interact with interfaces, layered structures, or other structured environments. These shifts arise from the interplay between the beam’s phase topology (vortex structure), the conservation and conversion of linear and angular momenta, geometric (Berry) phases, and the specific properties (such as reflection, transmission, or refraction) of the encountered system. Such effects are well established in optics, acoustics, hydrodynamics, and topological media, and they play a decisive role in phenomena ranging from nonlinear wave refraction by fluid vortices to OAM-amplified beam shifts in multilayer structures and Weyl media.

1. Theoretical Foundations and General Framework

Vortex-induced beam shifts generalize and often enhance classical beam-shift effects—Goos–Hänchen (GH) and Imbert–Fedorov (IF)—to beams possessing intrinsic OAM (vortex beams, with topological charge $\ell \neq 0$). For paraxial waves incident on an interface, the field can be expanded in plane-wave components, and each undergoes angle- and polarization-dependent transformations, including amplitude (Fresnel) and geometric (Berry) phase factors (Bliokh et al., 2012).

In the angular spectrum representation, the centroid of a vortex beam after reflection/transmission is given by

$$\langle \mathbf{r}_\perp \rangle = \frac{\langle \tilde{\mathbf{E}} \mid i \partial_{\mathbf{k}_\perp} \mid \tilde{\mathbf{E}} \rangle}{\langle \tilde{\mathbf{E}} \mid \tilde{\mathbf{E}} \rangle}$$

where OAM generates vortex-induced modifications to both spatial and angular shifts. These expressions are connected directly to conservation laws for spin and orbital angular momentum, with the total extrinsic shift ensuring $J_z = L_z + S_z$ conservation.

For beams incident on stratified or topological media, additional physics arises from interference (Fabry–Pérot resonances), multiple scattering, or topological surface state contributions (Chattopadhyay et al., 2018, Lusk et al., 2016).

2. Vortex-Induced GH and IF Shifts at Planar Interfaces

The GH shift denotes a longitudinal (in-plane) displacement, while the IF shift is a transverse (out-of-plane) displacement occurring at interfaces. For vortex beams with OAM $\ell$, these shifts become:

• Spatial GH shift: $$\Delta X_a = \Delta X_a^{(\mathrm{GH})} + \ell \cdot (\tan \theta_a / k) \cdot \xi$$ (where $\xi$ depends on the interface and polarization),
• Spatial IF shift: $\Delta Y_a = -\ell / k \cdot [...] +$ spin–orbit and mixed $\ell$–$\sigma$ terms.

The OAM-dependent terms scale strictly with $|\ell|$, and for standard dielectric interfaces, the presence of OAM amplifies both spatial and angular beam shifts compared to purely spin-induced effects (Bliokh et al., 2012, Xiao et al., 2012).

Angular GH/IF shifts are proportional to $(|\ell|+1)$, leading to significant enhancements for high-charged vortex beams. Conservation of total angular momentum ($J_z$) strictly determines the precise value and direction of these shifts.

In left-handed material interfaces, spatial shifts remain unchanged compared to right-handed materials, while angular shifts are reversed in sign due to the reversal of the $z$-component linear momentum, succinctly explained by a unified momentum-conservation law (Xiao et al., 2012).

3. Beam Shifts in Multilayer and Resonant Structures

Layered structures and Fabry–Pérot resonators exhibit singular enhancements of vortex-induced beam shifts. For a sandwich structure (e.g., air/glass/air, or glass/air/glass), the OAM-dependent lateral shift (OIF) near a resonance diverges as

$$\Delta_{\mathrm{OIF}}(\ell) \sim \frac{A(\ell, \theta)}{\theta - \theta_{\mathrm{FPR}}}$$

where $\theta_{\mathrm{FPR}}$ denotes a Fabry–Pérot resonance angle, and the divergence arises due to vanishing denominator in the total reflection coefficient. In contrast, the longitudinal (GH) shift decays exponentially with decreasing layer thickness in total internal reflection regimes.

In practical terms, micron-scale centroid shifts are observed for beams with $|\ell|=1,2$ in standard glass/air/glass gaps of a few microns thickness, with the vortex-induced OIF component dominating over spin-induced contributions (Lusk et al., 2016).

In double-prism (DPS) transmission experiments, the relative shift between beams of opposite OAM ($+\ell$ and $-\ell$) is measured: $$\Delta_\mathrm{rel} = 2 \, \Delta_{IF}(\ell) \propto 2\ell$$ The resonant structure of the DPS enables enhancements such that sensitivities to temperature (via expansion of the gap) or thickness reach $\lesssim 4$ nm and $\delta T \sim 0.045^\circ$C for $|\ell|=20$ (Zhu et al., 2023). In such geometries, vortex beams significantly outperform conventional spin-based displacement-sensing techniques.

4. Hydrodynamic and Nonlinear Vortex–Wave Interactions

In hydrodynamic systems, surface-wave beams are refracted by underlying fluid vortices, which themselves experience a transverse recoil (“vortex-induced beam shift”) (Humbert et al., 2017). Key control parameters include:

• Stokes drift velocity: $U_s = 2\pi a^2 f k$
• Dimensionless parameter: $S = U_s / U_0$ (ratio of wave Stokes drift to vortex speed)

Vortex center shift: $\Delta y \sim S$ for small $S$ (linear regime), saturates at $O(R)$ for $S \gg 1$; decrease in surface vorticity follows a $1/S$ decay law. At higher wave intensities, surface vorticity expulsion localizes vorticity deeper in the fluid, analogous to electromagnetic “skin effect.” Correspondingly, the nonlinear refraction angle scales as $\theta \sim 1/S$ for large $S$. The balance between Stokes drift advection and vortex self-advection underpins these effects.

This hydrodynamically-motivated beam shift mechanism is physically and mathematically analogous to OAM-induced optical shifts, establishing a deep connection across wave physics domains.

5. Topological and Spatiotemporal Manifestations

Vortex-induced shifts take on unique forms in topological media and pulsed beams:

• In Weyl materials, reflection at a gapped interface generates a “half-vortex” shift structure in 2D $k$-space, centered where the surface Fermi arc touches the Weyl cone. The shift diverges as $$|\Delta(\mathbf K_\perp)| \sim |\mathbf K_\perp - \mathbf K_{fa}|^{-1/2}$$ at the touching point, a direct signature of Fermi-arc topology and Weyl node chirality (Chattopadhyay et al., 2018).
• For spatiotemporal vortex pulses (STVPs), OAM-dependent time delays arise, controlled by spatial properties of Fresnel coefficients and the peculiar geometry of the STVP. These OAM-tunable delays can be both sub- and super-luminal (i.e., negative and positive, but not caused by material dispersion), and their magnitude is set by $|\ell|$, pulse parameters, and interface geometry (Mazanov et al., 2021).

These manifestations extend beam-shift phenomena into regimes probing bulk topology and ultrafast optical information processing.

6. Applications and Practical Considerations

Vortex-induced beam shifts serve as sensitive, tunable probes and modulators of interface properties, material structure, and topological features. Notable applications include:

• Sensing: Enhanced lateral shifts in resonant structures enable real-time monitoring of gap thickness, temperature, environmental gas composition, and microstructural deformations, with absolute sensitivities and resolutions unattainable by spin-only methods (Zhu et al., 2023).
• Topological photonics/electronics: Measurement of half-vortex shifts in Weyl systems offers a direct, bulk probe of surface Fermi-arc connectivity (Chattopadhyay et al., 2018).
• Hydrodynamics: Vortex-core displacement measurements quantify nonlinear wave–current interaction and can inform oceanographic wave–eddy coupling modeling (Humbert et al., 2017).
• Multiplexed information and delay lines: OAM control over time delay and spatial shifts facilitates new schemes in optical communications and spatiotemporal beam manipulation (Mazanov et al., 2021).

Experimental realization is routine with paraxial optics for $|\ell| \sim 1$–$20$ and beam waists $w_0 \sim 10$–$100$ µm, while metrological enhancements—via multilayer design or topology—enable application-specific optimization.

7. Physical Interpretations and Unified Principles

Vortex-induced beam shifts are governed by a unified set of principles:

• Conversion and conservation laws for spin and orbital angular momentum, at both interface and system-wide levels, rigorously determine the magnitude and directionality of beam centroid shifts.
• Geometric (Berry) phases accumulated in angular momentum and polarization space underlie the IF-type (Hall) shifts.
• Interference, resonance, and topology control the amplification, sign, and singularity structure of OAM-dependent shifts, with system-specific enhancements or suppressions.
• In physical terms, these shifts can be viewed as the redistribution of intrinsic OAM (attached to the vortex beam) into extrinsic OAM (encoded in beam trajectory or centroid), mediated by properties of the interface or structure.

This unification establishes vortex-induced shifts as both diagnostic tools and fundamental probes of wave–matter interaction, whether in optics, fluid dynamics, or topological media.