Topological photonics in nanoscaled systems with far field radiation and polarization singularities

The gist The topological photonics framework needs to be extended by a more comprehensive approach that properly treats such nanophotonic intrinsic effects and, in particular, the interplay of polarization and far-field radiation.

Introduction to Topological Photonics

Topology is a powerful framework for controlling and manipulating light, minimizing detrimental perturbations on the photonic properties Topology is a powerful framework for controlling and manipulating light, minimizing detrimental perturbations on the photonic properties. Nanophotonics is characterized by subtle phenomena that are not usually considered in other topological models’ realizations, such as nonlocality, strong field confinement, and light radiating to the farfield continuum. The ability to engineer matter at the nanoscale is a research field that has revolutionized science and technology. Nanophotonic systems offer unprecedented ways of controlling light at room temperature with remarkable features, from ultrafast dynamics in the order of femtoseconds and chip-scale integration to band dispersion control and light confinement into volumes beyond the diffraction limit [1–3]. Topology emerges as a promising research avenue to protect systems from disorder and imperfections.

Topological Concepts in Photonics

Topological phase transitions happen when the band gap closes and reopens, during which the bands become degenerate and the wavefunctions undergo a singular change in their topology, similar to piercing a sphere to transform it into a torus. Dirac points are a fundamental type of band degeneracy, around which topological properties, such as non-zero Berry phases, are often found. Weyl points in 3D and exceptional points in non-Hermitian systems with gain and loss have received great attention in photonics as a playground for topological phenomena. Chiral effects are another interesting aspect of topological implementations, where breaking in-plane symmetries with chiral structures might lift mode degeneracies and generate nonzero local Berry curvatures in the photonic bands.

Band Theory and Bulk Topology

A photonic band is formed by a refractive-index landscape ε(r) = ε(r+a), where a defines the photonic crystal periodicity. Applying Bloch’s theorem, both the electric field and the dielectric function are expanded in Fourier modes using the basis of the reciprocal lattice vectors G as ε(r) = ∑G εGe iG·r, and E(r) = ∑ G e i(k+G)·rψG(k)EG(k)⟩. The problem can be simplified to a standard eigenvalue problem by applying perturbation theory, with the components of the effective Hamiltonian being Hˆeff(k) G,G′ ≈ ωGδG,G' + pG′ ·pG∆εG−G′. From the eigenfunction ψ(k)⟩ of the effective Hamiltonian in Eq. (5), the Berry connection is defined as A(k) = i⟨ψ(k)∇kψ(k)⟩. The curl of Eq. (6) yields another gauge-invariant quantity, namely the Berry curvature B(k) = ∇k ×A(k). For 2D systems, the only nonzero component of the Berry curvature is Bz, and its integral over the entire Brillouin zone is the Chern number C = 1/2π R BZ z / k a topological invariant of the integer quantum Hall effect.

Non-Hermitian Systems and Bound States in Continuum (BICs)

Recent years have seen a growing interest in the interplay of topology and non-Hermiticity. Non-Hermitian photonics describes systems with optical loss and gain, where band gaps can be classified into two main categories: line gaps and point gaps. Bound states in the continuum (BICs) are localized modes embedded in dispersive bands appearing as light polarization singularities decoupled from the free-space radiation. BICs are protected from radiation decay by the existence of a quantized topological charge that tells how often the light polarization vector winds around the BIC’s momentum, namely its vortex center. The topological charge q of a symmetry-protected mode can be derived as qirrep = 1− n / 2π arg(εirrep), where εirrep is the character of that representation under a 2π/n rotation and is taken from the Cn character table.

Far-Field Topology and Polarization Singularities

The electric far field can be defined as EF ∝ Rˆψ⟩ = ∑ G cG pG, where Rˆ is a N × 2 matrix containing N (2 components) column vectors of the polarization directions pG, and ψ⟩ = (… cG …) T is the N components eigenmode of the effective Hamiltonian in Eq. (15). The far-field Berry connection is defined analogously to the bulk Berry connection in Eq. (6) as Af(kx, ky) = i⟨E∇kE⟩ = −cos2 θ / 2 ∂kx φ ˆkx +∂ky φ ˆky. The corresponding far-field Berry curvature is Bf(kx, ky) = ∇k ×Af(kx, ky)·ˆkz = 1/2 sinθ ∂kx / ∂φ∂ky φ −∂ky / ∂θ∂kx φ. This formula has been widely used to extract the Berry curvature in exciton-polariton planar microcavities [84, 85, 130, 131]. The polarization vortex of charge q is a source of −πq Berry phase, a feature that has been utilized to obtain optical spin-Hall effect from the topological charge of BICs [135, 136].

Limitations and Future Directions

The discrepancy between bulk and far-field topology suggests a broader issue about photonic systems operating in regimes that fall outside the assumptions of standard topological band theory. Topological charges of BICs can be exchanged during band inversions, suggesting they can be a good indicator for topological phase transitions. Looking ahead, promising directions include the use of structured light and interferometry to probe polarization vortices, and the reconstruction of non-Hermitian topological invariants from radiation patterns. Floquet theory is a well-established framework for treating quantum system under periodic time modulations, used to induce properties that are absent in their static counterpart. Synthetic dimensions might offer even greater advantages, such as enhanced field confinement, access to otherwise inaccessible topological regimes, or even create nonstandard hyperbolic or fractal geometries.

Concluding Remarks

Topological photonics is a powerful framework that has matured rapidly over the past decade, both in fundamental science and technological applications. Far-field radiation may be a rich source of topological structure and nonlocal effects, manifesting through BICs or circularly-polarized states. Theoretical approaches that incorporate generalized eigenvalue problems, full non-Hermitian Berry curvatures, and response functions from left eigenvectors may be fundamental for capturing the full topological picture. Ultimately, pushing topological photonics into the strong coupling regime might unlock entirely new classes of topological quasiparticles and robust quantum optical phenomena in the nanoscale domain.

Acknowledgments

Thanks to K. Arjas, A. Gianfrate, H.S. Nguyen, and D. Trypogeorgos for fruitful discussions. This work was supported by the Research Council of Finland under the Academy Research Fellowship project n. 13354165, and by the Italian Ministry of University and Research under the Rita LeviMontalcini program.

References

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1. System can predict topological invariants in far-field radiation by incorporating polarization singularities into standard topological band theory, as highlighted by We highlight the emerging role that polarization singularities might play in defining the topological invariants in the far field, which are not fully captured by bulk observables alone.

2. AI system can design photonic crystals with specific topological protection by modeling and optimizing non-Hermitian effective Hamiltonians derived from a lossy nature and coupling terms, enabling the creation of modes like Bound states in the continuum (BICs) protected from radiation decay by the existence of a quantized topological charge.

3. The system can perform inverse design for topological photonic structures by utilizing frameworks that include generalized eigenvalue problems, allowing it to explore regimes beyond standard theory to realize novel Dirac-like degeneracies unique to photonic bands.

4. The improved system can analyze experimental far-field data and infer bulk topological properties by applying the derived formulas for far-field Berry curvature, such as Bf(kx, ky) = 1/2 sinθ ∂kx/∂φ θ∂ky / (∂ky/∂θ ∂kx / φ), to map observed polarization textures onto topological charges.

5. The system can develop robust models for non-Hermitian photonics by reconstructing the left eigenvectors from measurable quantities like Stokes parameters, which are linked to the system’s response to external perturbations, providing deeper insights into the full non-Hermitian properties of the system.

Topology is a powerful framework for controlling and manipulating light, minimizing detrimental perturbations on the photonic properties. Combining nanophotonics with topological concepts presents opportunities for both fundamental physics and technological applications. Although most topological photonic realizations have been inspired by condensed-matter analogue models, new topological ideas have just begun to be realized at the nanoscale. Nanophotonics is characterized by subtle phenomena that are not usually considered in other topological models' realizations, such as nonlocality, strong field confinement, and light radiating to the far-field continuum. In this perspective, we will discuss how standard topological band theory for photonic crystals needs to be extended by a more comprehensive approach that properly treats such nanophotonic intrinsic effects and, in particular, the interplay of polarization and far-field radiation. We highlight the emerging role that polarization singularities might play in defining the topological invariants in the far field, which are not fully captured by bulk observables alone. We conclude by outlining a set of open questions and promising directions for exploring novel concepts in topological nanophotonics and shaping next-generation photonic devices.