Revealing topology with transformation optics

The study leverages transformation optics (TO) to reveal hidden symmetries in plasmonic metasurfaces and connect them to topological phases. Conventional topology analyses focus on periodic band structures in reciprocal space or real-space symmetry indicators; however, many structures possess symmetries that are not evident in real space. The authors propose using a virtual space provided by conformal mappings—where in-plane scalar potentials and magnetic fields are invariant—to analyze and design plasmonic systems. They investigate how singularities of the conformal map encode symmetry and can induce topological transitions, enabling the construction of topologically non-trivial metasurfaces and intuitive control over edge states. The work aims to provide an analytical framework where complex real-space systems are understood as lower-dimensional projections of simpler, spectrally equivalent structures in a transformed virtual space.

Transformation optics has been used for cloaks, subwavelength control, hidden dimensions, and broadband absorption, with key roles played by geometrical singularities and hidden symmetries. In topological photonics, topology of bands is characterized by invariants (e.g., Zak phase), initially in periodic crystals, later extended via real-space symmetry indicators to non-periodic systems (amorphous media, quasicrystals) supporting interface, corner, and disclination modes. Wannier functions provide a real-space characterization of topology. Prior TO work established spectral equivalence between plasmonic gratings and simple geometries via conformal maps, revealing hidden symmetries and enabling analytic designs. This paper builds on these to bridge TO with topological band analysis, focusing on how mapping singularities can tune topological invariants and edge states.

• System and mapping: The authors consider deeply subwavelength plasmonic metasurfaces generated via conformal mappings between a virtual space (w = u + iv) and real space (z = x + iy). An initial one-to-one mapping z = (A/2π) ln(w/w0 + i y0) transforms a uniform slab (between u0 and u0 + d) into a corrugated metasurface of period A, with parameters w0 controlling modulation depth and y0 chosen to keep the back side flat.
• Multi-valued mapping: To increase degrees of freedom, they introduce a multi-valued mapping z = (A/2π) ln[ 1/(e^{-w} + i y0) + a e^{i w} ], where a = δ + iδ. This creates two pairs of ln(0)/ln(∞) singularities and yields multiplicity: a single real-space metasurface corresponds to a coupled slab–eccentric-cylinder system in virtual space. The cylinder position depends on the complex parameter a0 = A exp(iθ) with amplitude δ and phase θ.
• Symmetry and degeneracy: Periodic boundaries in real space map to branch cuts in virtual space. At k_y = 0 (Γ point), the phase factor vanishes, eliminating virtual-space discontinuities and backscattering, leading to a two-fold degeneracy corresponding to band folding in the slab's dispersion.
• Topological transition design: Varying the sign of real a0 (θ = 0, ±π) shifts the ln(0) singularity position, controls mirror symmetry, and induces band inversion at Γ. Complex a0 breaks mirror symmetry and allows continuous evolution between phases while keeping the gap open.
• Calculations: Band structures are computed for transverse-magnetic (H_z) modes using a lossless Drude metal (plasma frequency ω_p = 2 eV) in vacuum. Γ-point eigenfrequencies are computed in both real and virtual spaces. A two-band Hamiltonian model based on shape deformation perturbation theory captures band inversion and yields Zak phases. Wannier functions are constructed with a smooth gauge over the Brillouin zone to track Wannier function centers (WFCs) versus θ. Edge states are analyzed via a supercell calculation (16 unit cells per metasurface) for a domain wall between two metasurfaces with a0 = 0.1 exp[i(θ0 ± Δθ)], and excited by an in-plane electric dipole at 45° to y.
• Virtual-space interpretation: The virtual-space cylinder perturbs slab modes. By inspecting slab eigenmodes (cos v and sin v profiles at u = u0), the sign of a0 sets cylinder position and relative phase, switching slab–cylinder interaction between repulsive and attractive, predicting band inversion without heavy computation.

• Hidden degeneracy and its origin: Although the real-space metasurface lacks conventional symmetry-protected degeneracy, TO reveals a two-fold degeneracy at Γ arising from band folding of a uniform slab in virtual space; discontinuities vanish at Γ due to the Bloch phase factor.
• Multi-valued mapping controls topology: Introducing the multi-valued mapping with parameter a0 produces coupled slab–cylinder images in virtual space. The ln(0)/ln(∞) singularities correspond respectively to metasurface shaping and source positions. Varying a0 moves the cylinder and controls the band topology.
• Band inversion with real a0: For a0 = −0.1, 0, +0.1, the Γ-point degeneracy splits and a bandgap opens. Tracking mode parity shows an even/odd crossing at Γ as a0 changes sign, evidencing band inversion. A two-band Hamiltonian matches full-wave bands and provides Zak phases that confirm a topological transition between a0 < 0 and a0 > 0.
• Symmetry breaking with complex a0: Complex a0 breaks mirror symmetry and allows a continuous path between phases with the gap remaining open; Γ-point eigenfrequencies versus θ agree between real and virtual-space calculations.
• Predictive virtual-space framework: By inspecting trivial slab eigenmodes and the cylinder’s TO-determined position, the sign of the slab–cylinder interaction (repulsive vs attractive) predicts which band is higher/lower, pinpointing the critical transition at a0 = 0 without heavy computation.
• Wannier-center control via singularities: WFCs for upper and lower bands shift by one lattice constant as θ varies from −π to π. The WFC trajectories correlate with the v-position of the ln(0) singularity, demonstrating control of topological invariants through mapping singularities.
• Edge states and bulk–boundary: A domain wall between metasurfaces with different a0 hosts edge modes inside the bulk gap. Their frequencies traverse the gap as Δθ varies, and field profiles (S1, S2) localize at the interface under dipole excitation, confirming the bulk–boundary correspondence.
• Practical notes: Working frequency is set by the slab dispersion; topology is inferred intuitively from mode coupling in virtual space. Calculations used TM (H_z) polarization, Drude metal (ω_p = 2 eV), vacuum background, and supercells of 16 unit cells per side.

The findings demonstrate that transformation optics provides a powerful lens to analyze and design topological phases in plasmonic metasurfaces. By mapping a complex, corrugated real-space structure to a simpler virtual-space system (slab plus cylinder), hidden symmetries emerge that explain degeneracies and band behavior. The sign and position of conformal-map singularities govern band inversion and Zak phases, directly linking geometric features in virtual space to topological invariants in real space. This approach circumvents heavy numerical searches, as qualitative features (repulsive vs attractive coupling of slab and cylinder modes) predict the topology and the critical transition. The correlation between singularity trajectories and Wannier-function centers further cements the role of mapping singularities in controlling bulk topology, while edge-state observations validate the bulk–boundary correspondence. The methodology is broadly relevant to photonics and potentially other wave systems where TO applies.

The paper introduces a TO-based analytical framework where real-space plasmonic metasurfaces are understood as projections of multi-valued, spectrally equivalent virtual-space systems. Key contributions include: (1) revealing Γ-point degeneracies via virtual-space band folding; (2) demonstrating that conformal mapping singularities (through parameter a0) control band inversion and topological invariants; (3) predicting topology and critical points by inspecting slab mode coupling to a TO-induced cylinder, with minimal computation; (4) establishing control of Wannier-function centers via singularity positioning; and (5) confirming edge modes at domain walls consistent with bulk–boundary correspondence. Future directions include exploring other mapping singularities (e.g., higher-order roots), extending beyond 1D translationally invariant systems to 2D/3D photonic crystals by leveraging hidden rotational symmetries (e.g., ellipse–annulus equivalence), studying higher-order topological phenomena such as corner states, and translating the strategy to acoustics, elasticity, and spintronics.

The study focuses on 1D translationally invariant plasmonic metasurfaces and analyzes topology primarily through a specific class of conformal maps. The demonstrated control uses ln(0)/ln(∞) (and discussed square-root–type) singularities; higher-order singularities are proposed but not explored. Material modeling uses a lossless Drude metal and TM polarization, and experimental validation is not presented. Generalization to 2D/3D systems and other singularities is suggested but left for future work.