Backward-Wave Metamaterial Line

Line Type

Controls

Wavenumber βa 1.20 rad

Animation speed 1.0×

Wave-packet width σ 1.4

Live Readouts

ω (rad/s, norm.)

—

Direction

—

Phase velocity v_p

—

Group velocity v_g

—

v_p·v_g sign

—

How it works

A negative-index metamaterial can be modelled as a periodic LC ladder — the transmission-line theory used by Caloz & Itoh to design real backward-wave metamaterials. Swapping which element is series and which is shunt flips the sign relationship between phase and group velocity:

Right-handed line (series L, shunt C):
  ω(β) = ω_c · sin(βa/2)          — ordinary "forward" wave

Left-handed line (series C, shunt L, the dual circuit):
  ω(β) = ω_c / [2·sin(βa/2)]      — "backward" wave

v_p = ω/β   (phase velocity — how the crests move)
v_g = dω/dβ (group velocity — how energy/information moves)

For the right-handed line v_p and v_g are both positive: crests and the energy packet travel the same way, as in an everyday material with n > 0. For the left-handed line, v_g < 0 while v_p > 0 — the wavefronts appear to run backward relative to the direction the energy is actually flowing. This is exactly the Veselago backward-wave signature that split-ring-resonator / wire-array metamaterials reproduce at microwave and optical frequencies, giving rise to a negative effective refractive index.

Line type — toggles between the sin(βa/2) forward dispersion and the 1/sin(βa/2) backward dispersion.
Wavenumber βa — sets the operating point on the dispersion curve; ω, v_p and v_g update live from the exact formulas above.
Wave-packet width σ — narrows or widens the travelling Gaussian envelope so the energy packet's motion is easier to track separately from the fast carrier oscillation.
The bright marker riding the top rail tracks a fixed carrier phase (crest) at v_p; the marker on the bottom rail tracks the envelope peak at v_g. Watch them separate and move opposite ways in backward-wave mode.

Carrier / node voltage Phase marker (v_p) Envelope / energy marker (v_g)

⚠ Couldn't load the 3D engineThree.js failed to load from the CDN. Check your connection and reload.