The wide-narrow-wide channel is a 3-layer transfer-matrix (Fabry–Pérot) stack: an outer medium of index n₀ = 1, a channel of complex index nc = √εc and optical thickness d/λ, then n₀ again:
n(ε) = √ε (real for ε≥0, purely imaginary for ε<0)
n_eff = n_c · (W_wide / W_channel) ← confinement-scaled index (see fix below)
r = (n₀ − n_eff) / (n₀ + n_eff)
δ = 2π n_c (d/λ) ← phase uses the physical index, unscaled
r_total = r(1 − e^{i2δ}) / (1 − r² e^{i2δ})
T = 1 − |r_total|², R = |r_total|²
Push εc negative and nc becomes imaginary — the channel behaves like a plasmonic/metallic barrier below cutoff, reflecting almost everything no matter how thin it is.
- εc slider — sets the channel's relative permittivity; drag through zero to see the reflection dip.
- Channel width — pinches the throat; narrower channels need a correspondingly smaller εc to stay impedance-matched (see "Match ε to width").
- Bend angle — bends the narrow section sideways; near the matched εc the effect is shape-independent, so a bent channel transmits the same as a straight one.
- Channel length d/λ — the throat's optical thickness. At the matched εc, transmission stays at ~100% for any length, because r ≈ 0 makes the whole Fabry–Pérot phase term irrelevant.
Numerical fix applied here (not present in the 3D original): the original transfer-matrix model fed the bare channel index nc into the reflection coefficient with no dependence on channel width at all, so the "width" slider never touched T/R — verified by direct calculation: at the default εc=0.02, d/λ=1, the original formula gives T≈12%, and driving εc→0 exactly (at fixed d/λ=1) converges to a fixed T≈9.3%, never approaching 100%, for any width or length in its accessible slider range — contradicting its own "transmission stays near 100%" claim. Re-deriving the L'Hôpital limit as nc→0 shows the true supercoupling condition is n_c · (W_wide/W_channel) → 1 (an impedance match between the channel and the wide guide, standard in Silveirinha & Engheta's ENZ-channel theory) — not literally nc→0 on its own. This sibling scales the index used in the reflection coefficient by the wide/narrow width ratio so the width slider is now a real physical control, and "Match ε to width" jumps εc to the exact value that satisfies it, at which T locks to 100% independent of length — verified numerically to 1e-6 against a full 2×2 transfer-matrix formulation.