Each unit cell is a metallic split-ring resonator (SRR) — a conducting loop with a gap that behaves like a tiny LC circuit. Near its resonance the ring's induced current makes the lattice's effective magnetic permeability swing negative:
mu(f) = 1 − F·f² / (f² − f0²)
f0 = k / a (ring radius a sets the resonance)
When the cells' electric response (from an implicit wire backbone) is also negative, both permittivity and permeability are negative at once, and the effective refractive index becomes negative too:
n = − sqrt(|eps| · |mu(f)|), valid only where mu(f) < 0
sin(theta_t) = sin(theta_i) / n (signed Snell's law)
Because n is negative, the signed Snell's law flips the sign of the transmitted angle: the beam bends to the same side of the normal as the incident ray, instead of crossing to the opposite side the way an ordinary slab (n ≈ 1.5, shown as the pale reference block) always does. Outside the resonance band mu(f) is positive, the wave cannot propagate through the lattice, and the pulse train stops at the front face.
- Wave frequency — moves the drive frequency f relative to the ring's resonance f0; sweeping through the negative-mu band flips propagation on and off.
- Ring radius — sets f0 itself (bigger rings resonate lower), shifting where the negative-index band sits on the frequency slider.
- Lattice layers — thickness of the metamaterial slab; the internal path length (and so the sideways shift of the exit beam) scales with it.
- Reference beam — toggles a plain n≈1.5 slab alongside the metamaterial, refracting the ordinary way for direct comparison.
Real-world relevance: this SRR-based negative-index effect underlies proposed "superlenses" that beat the diffraction limit, and cloaking layers that steer electromagnetic waves around an object instead of scattering them.