Newton’s rings

A symmetric glass–air–glass film at normal incidence, with thickness t=t₀+r²/(2R). One reflection phase reversal makes the contact center dark in reflection. Phase order q=2t/λ. Reflected minima and transmitted maxima have q=m; the other extrema have q=m+½. Index m starts at 0. The radius satisfies r²=(m+ε)λR−2Rt₀; negative r² means no physical ring at that index.

The intensity uses the lossless Airy-film model: surface reflectivity ρ=[(n_g−1)/(n_g+1)]², F=4ρ/(1−ρ)², reflection=Fsin²(πq)/(1+Fsin²(πq)), transmission=1−reflection. Outer lens/plate surfaces are ignored. Intensities refer to light entering the air-film system from glass. A transmitted minimum is not complete extinction. The image independently stretches contrast to make both views readable; the center value is the actual model intensity, not the stretched brightness.

Show rings computes a bounded raster with four subpixel samples. The parabolic gap is a small-radius approximation; pixel resolution limits fine rings. Clicking picks the nearest existing selected feature. Changing optical parameters requires a new image; selecting an existing ring within the current image updates its highlight and graph immediately. Ring samples are theoretical values, not real measurements.

Paste index; radius in millimetres to fit r²=S(m+ε)+b with a free intercept. λ=1000S/R in nm and t₀=−1000b/(2R) in nm, when r² is in mm². At least two different indices are needed. A negative fitted gap can indicate noise or inconsistent indexing, not physical overlap. RMSE is for squared radius, not wavelength uncertainty. Geometry/view changes clear active samples while pasted text remains. Typed inputs and pasted data text, including unfinished entries, stay in this browser. Images, sampled points and fits are temporary. Reset restores optical defaults while keeping pasted text. Monochromatic coherent light, no spectrum, absorption, roughness or full curved-ray model.

Compute the ring image. Ring selection then updates directly; render again after changing optics or display resolution.