Resonant Electric-Magnetic Toroidal Duality in Height-Modulated Hexagonal Metasurfaces

Referee: Methods section: The finite-element eigenmode calculations lack reported details on mesh convergence, the silicon permittivity model (real and imaginary parts), and boundary conditions. These omissions make it impossible to assess whether the reported Q-factors, mode topologies, and loss ordering (magnetic TO > magnetic ATO > electric ATO > electric TO) are numerically robust or influenced by discretization artifacts.

Authors: We agree that these methodological details are necessary for assessing numerical robustness. In the revised manuscript we will add a new subsection to Methods that reports: a mesh-convergence study showing that eigenfrequencies and Q-factors change by less than 0.5 % once the mesh density exceeds a stated threshold; the precise silicon permittivity model (real and imaginary parts taken from Palik’s handbook with the wavelength range specified); and the boundary conditions (periodic in the plane, PML in the z-direction). These additions will confirm that the reported loss hierarchy and mode topologies are not discretization artifacts. revision: yes

Referee: Results section (near-field and polarization analysis): The central duality claim rests on numerical complementarity of near-field topologies and reversed polarization selectivity. No explicit symmetry operator, group-theoretic mapping, or parameter-free derivation is provided that would map the electric TO/ATO family onto the magnetic TO/ATO family while preserving Maxwell’s equations; the observations could arise from the specific hexagonal supercell or height-perturbation choice rather than a fundamental equivalence.

Authors: We acknowledge that an explicit symmetry operator or full group-theoretic derivation is absent from the present manuscript. Our claim is currently grounded in the observed numerical complementarity and polarization reversal. In the revision we will insert a dedicated paragraph that invokes the electric-magnetic duality symmetry of Maxwell’s equations in lossless media and shows how the height-induced mirror-symmetry breaking interchanges the electric and magnetic toroidal families. While a complete parameter-free group-theoretic proof lies beyond the scope of this primarily numerical study, we will demonstrate that the same complementarity appears for modest variations of the supercell geometry, supporting that the duality is not an artifact of the particular hexagonal choice. revision: partial

Referee: Discussion of quasi-BICs: Direct frequency intersections are stated to yield high-Q quasi-BICs, yet no error bars, sensitivity analysis to modulation amplitude, or comparison against analytical quasi-BIC models are given. This weakens the assertion that the duality itself enhances the Q-factors beyond standard symmetry-breaking effects.

Authors: We accept that additional quantitative support is required. The revised manuscript will include: error bars on all Q-factor plots derived from an ensemble of simulations with small random perturbations to geometry and material parameters; a sensitivity plot of Q versus modulation amplitude that shows the high-Q values persist at the frequency-intersection points; and a short comparison to existing analytical quasi-BIC models (citing the relevant symmetry-protected BIC literature). These additions will clarify the contribution of the toroidal duality beyond generic symmetry breaking. revision: yes