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Reflected intensity (display) along a radius from the contact point, per colour channel
Dark ring radius squared versus ring number, with the linear fit (slope = lambda R / n)
Air gap thickness versus radius: exact sphere, parabola and the dark-ring thicknesses
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How to use
Choose a preset or set your own radius of curvature, light source and gap liquid, then look at the circular eyepiece view. Try: (1) sodium light and count the dark rings in the field; (2) double R and check that the rings spread out by sqrt(2); (3) read two ring diameters with the Measure sliders and compare the R formula with the value you set (also the numerical checks below); (4) add 147 nm of dust and watch the centre turn bright; (5) pour water into the gap: rings shrink by 1/sqrt(1.333) and the picture dims; (6) switch to white light for the thin-film colours, and to transmitted light for the complementary pattern; (7) shift the contact point sideways. The cross-section panel shows the gap with an exaggerated vertical scale.
The optical table, stage, optical flat, lens in its ring, lamp housing with condenser, beamsplitter, measuring microscope with vernier scale, monitor, lens case and props were modelled and baked in Blender; Three.js r160 projects the same ring model onto the flat, the monitor and the section panel and animates the light beams. The ring pattern drawn on the flat is magnified (about 1.4 x the eyepiece field) so that the orders can be seen from the camera.
The science
The experiment. A plano-convex lens is laid with its curved face on an optical flat. Between them is a very thin wedge of air whose thickness grows with distance from the contact point. Light reflected from the bottom of the lens and from the top of the flat interferes: you see concentric dark and bright rings (Newton, 1717; the explanation by wave interference is Young's). Measuring the ring diameters gives the wavelength of the light, or the radius of curvature of the lens.
Gap geometry. For a sphere of radius R the gap at distance r from the contact point is exactly
d(r) = R - sqrt(R² - r²) ≈ r² / (2R)
(the difference between the two is below one part in 10&sup4; for the numbers here and is checked on this page).
Interference condition. The beam reflected at the lens surface (glass to gap, n_glass > n_gap) has no phase jump; the beam reflected at the flat (gap to glass) gets a half-wave jump. With a gap of refractive index n the optical path difference is 2 n d plus half a wavelength:
dark ring: 2 n d = m lambda bright ring: 2 n d = (m - 1/2) lambda => r_m = sqrt(m lambda R / n)
So the centre is dark (m = 0) and r² grows linearly with the ring number m. With a dust grain or film of thickness d0 at the contact point the gap is d0 + r²/2R, the rings are shifted, and a quarter-wave offset makes the centre bright.
Intensity. Two reflections of amplitude r_f = (n_glass - n)/(n_glass + n) each (Fresnel, normal incidence) interfere as
I(d) = 4 r_f² sin²(2 pi n d / lambda)
so in a liquid with n close to the glass index the reflections nearly cancel in strength (r_f → 0): the pattern fades, which is why the "Auto exposure" option exists. With glass 1.52 and air the maximum is 4 x 0.0426 = 17 % of the incident light; water cuts the brightness by roughly a factor of 10.
Measuring R without knowing where the centre is. Dust and lens flattening at the contact point make the centre unreliable, so use two rings, m and m + k, and the diameters D = 2r:
R = (D²_{m+k} - D²_m) n / (4 k lambda)
The unknown offset d0 cancels. The "Measure" sliders do exactly this on the ring radii found numerically from the model.
White light. Each wavelength has its own ring pattern. The page integrates the intensity over 390-710 nm with the CIE 1931 colour matching functions (analytic multi-lobe fit), converts to sRGB and normalises so that a flat spectrum is white. Rings of different colours drift out of step: the first dark ring is black, then purple, blue, green, yellow, red bands, and after about 8 orders the colours average to pale white.
Checks on this page: (1) R recovered from a linear fit of r² against m for the numerically located dark rings; (2) R recovered from two ring diameters read off the rendered image pixels (this exercises the colour look-up table and imaging path); (3) rings in a water-filled gap shrink by 1/sqrt(n); (4) the mean intensity over many orders is half the maximum; (5) exact sphere versus parabola.
Honest simplifications. Two-beam interference only (multiple reflections add a few per cent for air, more for high reflectivity); perfectly coherent, normal-incidence illumination of finite bandwidth treated as a single wavelength for monochrome sources (real sodium light is a close doublet, which makes the rings wash out near order 1000); no glass deformation (real lenses flatten slightly under load), no polarisation effects, no dispersion of the glass or the liquid; the transmitted view is the complement of the reflected one, whereas the real transmitted contrast is only about 8 % with air. In the 3D scene the lens curvature, the beams and the vertical scale of the gap section are exaggerated for the eye; the optics itself uses your chosen R.