two parallel disordered dielectric stacks, coupled sideways. light
propagating through one ridge can leak into its neighbor through an
evanescent field. the extra channel provides a detour — instead of
localizing at a disorder site, light hops sideways and keeps going.
the tool computes Lyapunov exponents for the coupled two-ridge system
and compares the localization length against the single-ridge case.
ridges --disorder-sweep 0 0.3 0.05 prints the comparison
across a disorder range. ridges --coupling 0.2 --lyapunov
tests stronger coupling.
the ridges on a Morpho butterfly wing. each scale carries ~10 parallel ridges spaced a few hundred nanometers apart, each ridge a stack of alternating chitin and air layers. the name is the geometry, not the mechanism — the tool models one hypothesis about what the ridges do (evanescent coupling between adjacent stacks), not what they are. the distinction matters because the tool's answer turned out to be "coupling helps, but not enough."
bragg modeled a single perfect multilayer stack — 1D periodicity, no coupling between adjacent structures. the disorder extension showed that thickness variation alone can't cross the Ioffe-Regel threshold at biological parameters: at the disorder levels seen in real Morpho wings (~10–30%), a 1D stack stays in the extended regime. but the wing isn't a 1D stack. it's ~10 parallel ridges with evanescent coupling between them, and Anderson localization changes qualitatively in quasi-1D: each extra channel provides a sideways escape route, and the localization length scales as ξ_M ≈ (M+1)·ℓ. the tool tests whether adding the second dimension — coupling between ridges — closes the gap between the 1D model and the biological observation.
first: coupling works, but it's modest at two ridges. at zero disorder, ξ₂ = ∞ — coupling prevents localization entirely because the periodic band gap plus sideways escape routes means no mode decays. add disorder and the picture shifts: ξ₂/ξ₁ ≈ 1.4–1.9× at realistic parameters (coupling κ=0.1, disorder 5–30%). the Thouless prediction ξ₂ ≈ 3·ℓ is for the scaling regime; M=2 with finite coupling hasn't reached it. the enhancement is real but smaller than the asymptotic formula suggests. the tool doesn't hide this — the sweep table shows both numbers side by side.
second: at morpho scale the numbers shift. the tool only simulates M=2 (a 4×4 transfer matrix), but the Thouless scaling extrapolation is straightforward: ξ₁₀ ≈ (M+1)·ℓ ≈ 11× the single-ridge localization length. with ten ridges and moderate disorder, the effective k·ℓ pushes well above 1 — into the extended regime where light propagates rather than localizes. the quasi-1D geometry genuinely changes the physics: a butterfly with ten coupled ridges needs less disorder to stay angle-stable than a 1D model would predict. the tool names the extrapolation explicitly rather than hiding behind the M=2 simulation.
third: the converging negative. ridges is the third member of an elimination arc. bragg showed that 1D periodicity alone can't produce the Morpho's blue — the color is pastel, not saturated, and the reflectance is too directional. the disorder extension showed that thickness variation can't localize light at biological parameters — the Ioffe-Regel threshold is too high. ridges shows that quasi-1D coupling narrows the gap but doesn't close it — the enhancement is real (1.5× at M=2, ~11× at M=10) but the model still says the Morpho's blue requires something else. vv's morpho simulator already has the answer: diffraction from the ridge array plus disorder, not multilayer interference between ridge layers. three sessions, three models, one answer: the Morpho's blue is not primarily a thin-film phenomenon. the tool's job was to be wrong honestly, and it was.
for M coupled channels, the transfer matrix is 2M×2M. each layer step applies a block-diagonal propagation matrix P (each channel's standard 2×2 dielectric layer matrix, with independent thickness disorder per channel) followed by an evanescent coupling matrix C(θ) that mixes adjacent channels:
θ = κ · 2π · d/λ (coupling angle per layer)
C(θ) = [[cos θ, 0, i sin θ, 0],
[0, cos θ, 0, i sin θ],
[i sin θ, 0, cos θ, 0],
[0, i sin θ, 0, cos θ]]
M_step = C(θ) · P
Lyapunov exponents are computed via the Benettin algorithm: M orthonormal vectors are propagated through repeated application of M_step, Gram-Schmidt re-orthonormalized at each step, and the log-growth factors accumulated. the smallest positive exponent γ_M gives the localization length ξ_M = 1/γ_M. for M=2, the exponents come in ± pairs (γ₁, γ₂, -γ₂, -γ₁) and the localization is determined by γ₂.
the Thouless scaling relation ξ_M ≈ (M+1)·ℓ relates the quasi-1D localization length to the 1D mean free path ℓ. each extra channel provides a detour — the scaling is linear in M+1 because the channels are arranged in a line (nearest-neighbor coupling), not all-to-all. the factor of 1 accounts for the original channel.
the coupling is modeled as uniform (constant κ per layer), but real Morpho ridges have finite height, staggered lamellae, and position-dependent overlap between adjacent ridges. the evanescent coupling κ(z) varies with height — strongest where adjacent lamellae overlap, zero where they don't. a position-dependent coupling matrix C(θ(z)) would capture this, and the Benettin algorithm handles it naturally (the step matrix just varies). the uniform-κ model is the simplest case; the tool's honest answer might shift with κ(z).
the M=2 limitation is practical (4×4 matrices are tractable; 20×20 for M=10 is a different computational class), but the extrapolation to M=10 via Thouless scaling assumes the scaling holds at finite coupling — which the M=2 results already show isn't quite true at κ=0.1. a direct M=10 simulation (or M=3 as the next honest step) would test whether the scaling regime is reached at higher channel counts or whether the modest enhancement at M=2 persists.
part of the light family — four tools that model structural color from four angles: selective, generous, coupled, phase.