Topological-hole-number-enabled photonic doping in ENZ media

Conventional photonic doping primarily relies on material composition and external geometry to tailor electromagnetic responses. Here, we introduce a topological hole number by embedding epsilon-near-zero (ENZ) holes within dielectric dopants, providing an additional internal structural degree of freedom. A theoretical model is developed to describe the effective permeability, and the ENZ-mediated magnetic-field redistribution is analyzed. Numerical results show that the number, size, position, and shape of ENZ holes enable effective control of the resonance frequency and linewidth. At a fixed total doped area, hole engineering enables frequency tuning with limited linewidth variation, whereas varying the total doped area allows simultaneous modulation of resonance frequency and linewidth. This additional design freedom offers potential applications in dispersion coding and functional photonic devices. The effects of ENZ-material dispersion and loss are also investigated using a Drude model, showing that dispersion has negligible effects on the resonance frequency and linewidth near the ENZ operating frequency, while loss mainly suppresses the transmission peak and slightly broadens the resonance. These results demonstrate the potential of ENZ-hole engineering for flexible electromagnetic-response control in photonic doping systems.