Press a curved lens onto flat glass — why does a bullseye of dark and bright rings appear, with a DARK centre, and what can the rings measure?
▶ Launch the interactive simulationModel the thin air wedge t(r)=t₀+r²/2R between a plano-convex lens (R=1 m) and a flat plate; combine the top and bottom reflections (with the π half-wave flip at the bottom surface) into the reflected intensity I=sin²(2πt/λ), locate the dark-ring radii from a noisy radial scan, and fit r_m² against the ring index m.
thin-film interference — dark rings at r_m²=mλR (a straight line of slope λR, rings crowding outward as √m) and, decisively, a DARK centre: with zero path difference the lone π flip at the bottom reflection makes r=0 destructive — the direct fingerprint of the half-wave shift. The slope weighs the wavelength of light with a ruler (recover λ≈589 nm given R), and lifting the lens by λ/2 slides one whole fringe past any point — counting fringes counts half-wavelengths of motion (interferometric metrology). The interference branch of the optics arc, after Brewster and Malus.