Backward-Wave Metamaterial Line (2D)

Negative-index metamaterials are famous for bending light "the wrong way," but the deeper signature is that the energy in the wave travels opposite to the direction its crests appear to move. This 2D simulation reads that signature the way an engineer designing a real backward-wave line would: off the dispersion diagram. The same periodic LC ladder model as the 3D version drives it — swap which element is series and which is shunt and the dispersion relation inverts — but here the physics is shown as three linked, honestly-2D graphs: a live carrier·envelope waveform, the ω(β) dispersion curve itself with a tangent line whose slope is computed by numerical differentiation rather than a pre-derived formula, and a scrolling spacetime chart that plots the actual worldlines of the phase crest and the energy packet. Tune the operating wavenumber and line type to watch the two worldlines split apart and lean opposite directions in the backward-wave regime, with the numeric derivative checked live against the closed-form one.