Fishnet Metamaterial 2D: Negative-Index Ray Diagram & Dispersion

Drive Frequency

ω / ω_p 0.50

Fishnet Geometry & Loss

Hole-size resonance ω_LC/ω_p 0.50

Layer coupling strength F 0.60

Metal damping γ/ω_p 0.05

Effective Medium Response

Positive index (n > 0)

ε(ω) real / imag

1.00 / 0.00

μ(ω) real / imag

1.00 / 0.00

n(ω) = √(εμ)

1.00 + 0.00i

Refraction angle θ_t

18.4°

Power reflectance R = |(n−1)/(n+1)|²

0.0%

How it works

This is the 2D companion to the 3D fishnet-metamaterial simulator. It keeps exactly the same effective-medium physics — a Drude electric response from the perforated metal sheets and a Lorentzian magnetic response from the antiparallel current loops driven through the holes — but computes it independently as a flat ray diagram instead of a rendered 3D stack, and adds two live diagnostics the 3D scene never shows: a full ω-swept dispersion strip and the interface's power reflectance.

Electric (wire mesh), Drude model:
  ε(ω) = 1 − ωp² / (ω² + iωγ)

Magnetic (antiparallel-current loop), Lorentzian:
  μ(ω) = 1 − F·ω² / (ω² − ωLC² + iωγ)

Refractive index (causal branch, Im(n) ≥ 0):
  n(ω) = √(ε(ω)·μ(ω))

Power reflectance (normal-incidence form, used here as a
schematic approximation to the oblique interface):
  R = |(n − 1)/(n + 1)|²

Near ω_LC, Re(μ) dips below zero; combined with the already-negative Re(ε) below the plasma frequency ω_p, the product ε·μ can push Re(n) negative over a finite band — the same mechanism behind Smith's 2000 negative-index demonstration, scaled from microwave split-rings to the optical-frequency fishnet lattice.

ω/ω_p — the incident wave's frequency relative to the metal's plasma frequency; sweeps you through and past the negative-index band.
ω_LC/ω_p — set by the hole size and layer spacing; smaller holes and thinner spacers push this resonance higher.
Coupling strength F — how strongly the two metal sheets couple through the spacer; stronger coupling means a wider negative-μ band.
Damping γ — Ohmic loss in the metal; the fundamental limit on how deep and lossless the negative-index band can be.
When Re(n) < 0, the transmitted ray bends to the same side of the surface normal as the incident ray — drag the diagram or scroll to zoom in on the bend.
The dispersion strip below the ray diagram sweeps ω across the whole slider range at the current ω_LC, F and γ, plotting Re(n) and Im(n) with the negative-index band shaded — so you can see the whole band shift as you drag the geometry sliders, not just read one point of it.

Drag to pan, scroll to zoom

Incident ray Transmitted ray Reflected ray Fishnet layers