The nanoparticle's conduction electrons are modeled with a Drude dielectric function ε(ω). Its dipole polarizability follows the quasistatic Clausius–Mossotti relation; the near-field enhancement and extinction spectrum both peak where Re(ε) ≈ −2εmedium.
The split-ring array is modeled as a driven LC oscillator: μ_eff(ω) = 1 − Fω²/(ω²−ω₀²+iωΓ) crosses negative just above its resonance ω₀, which is set by the ring size. A negative n_eff needs both ε_eff < 0 and μ_eff < 0 at the same wavelength.
ε(ω) = 1 − ωp² / (ω² + iωγ)
n_eff = sign(μ_eff)·√(ε_eff·μ_eff)