Backward-Wave Metamaterial Line

Negative-index metamaterials are famous for bending light "the wrong way," but the deeper signature — the one that actually defines a negative-index medium — is that the energy in the wave travels opposite to the direction its crests appear to move. This simulator builds that effect from a real, textbook circuit model: a periodic LC ladder network. An ordinary line (series inductor, shunt capacitor) has phase velocity and group velocity pointing the same way, exactly like light in glass. Swap which element is in series and which is shunt — the dual, left-handed circuit used to engineer real backward-wave metamaterials — and the dispersion relation inverts: energy still flows forward while the wave crests visibly run backward. Tune the operating wavenumber and watch a carrier marker (phase velocity) and an energy-packet marker (group velocity) split apart and travel opposite directions along the chain, with both velocities computed live from the line's exact dispersion formula.