Each period stacks a high-index layer (n₁) and a low-index layer (n₂), both cut to a quarter-wave at the design wavelength, d = λcenter/(4n), so a weak reflection from every interface returns in phase and adds up coherently. The panel below runs the exact 2×2 transfer-matrix method: every layer contributes a characteristic matrix built from cos δ and sin δ, where δ is its phase thickness, the matrices multiply in stack order, and the result combines with the optical admittances of air (n₀=1) and the substrate (ns=1.5) to give the transmission T at every wavelength from 300–1200 nm — no approximation, so it stays accurate for any N.
δ = (2π·n·d/λ)·cosθ
M(layer) = [[cosδ, i·sinδ/η], [i·η·sinδ, cosδ]]
M = M(layer N)···M(layer 1)
T = (η_s/η_0)·4η_0² / |η_0·M11 + η_0η_s·M12 + M21 + η_s·M22|²
The shaded band in the spectrum plot is the photonic bandgap — the range where T drops below 5%. More periods deepen and sharpen the gap without moving its center; a bigger n₁−n₂ contrast widens it; detuning the fill ratio away from 0.5 narrows or shifts it.
- Stack diagram — the actual layer sequence and physical thicknesses currently built.
- Spectrum plot — computed T(λ) across the visible/near-IR window, with the probe wavelength marked.
- Probe T / R — exact transmittance and reflectance (1−T, lossless stack) at the probe wavelength.