Structured-light control of axion electrodynamics in topological insulator scattering.

Topological insulators exhibit an axion-mediated magnetoelectric response that generates cross-polarised scattering channels strictly forbidden in conventional dielectrics, an optical fingerprint of the topological surface states themselves. Exploiting this fingerprint experimentally requires structured illumination capable of selectively amplifying the cross-polarised channel while suppressing background Mie scattering. Here we show that non-diffracting Lommel beams fulfil this role in a way that symmetric Bessel beams fundamentally cannot. We present the first theoretical treatment of polarised Lommel beam scattering by a topological insulator sphere, extending generalised Lorenz-Mie theory to incorporate the full topological magnetoelectric boundary conditions and deriving closed-form cross-polarised scattering coefficients as a function of the axion angle θ0. The central result is that the Lommel asymmetry parameter c provides continuous, tunable control over multipole excitation: by varying c, one preferentially drives the multipole orders that couple most strongly to the axion term, amplifying the cross-polarised signal while suppressing the co-polarized background, a capability absent in any cylindrically symmetric beam. At moderate axion coupling (θ0 = π), the cross-polarised-to-co-polarized intensity ratio reaches order 10⁻², well within the detection range of standard polarimetric instrumentation. The inversion protocol for recovering the axion angle shows that the phenomenon is no longer a computational study of a beam-material combination; instead, it is a non-contact optical metrology protocol for axion angle retrieval. The ratio of cross-polarised to co-polarised scattering intensity increases nonlinearly and monotonically with θ0, confirming an unambiguous optical signature of the topological magnetoelectric effect. Circularly polarised Lommel beams further reveal pronounced handedness-dependent scattering asymmetry arising from spin-orbit coupling at the surface states. These results establish a quantitative framework connecting structured light parameters, topological charge, asymmetry, cone angle, and polarisation, to axion electrodynamics, and identify spatial-light-modulator-generated Lommel illumination as a practical, non-contact route to optical characterisation of topological surface states in Bi₂Se₃ and Bi₂Te₃ nanoparticles.