Thin-Film Destructive Interference (Dark Reflection)

This is the same optical path difference as the bright case, 2nt in the film's own wavelength, but read the other way. Because only the top surface inverts its reflection, a whole number of wavelengths in the path leaves the two reflected waves exactly out of step, and that colour disappears from the reflection. Everything it disappears from, it appears in — the missing light is transmitted, not destroyed, which is why a soap film looks complementary in reflection and in transmission.

Anti-reflection coatings are this equation used deliberately, with one wrinkle: a magnesium fluoride layer (n = 1.38) on glass (n = 1.52) inverts at both surfaces, so the half-wave cancels out and the dark condition becomes 2nt = (m + ½)λ instead — the famous quarter-wave coating t = λ/4n. Count your inversions before you pick a condition; that single bookkeeping step is where most of the marks go.

Worked case: a soap film with n = 1.33 seen in 600 nm orange light goes dark in first order at t = (1)(600 nm)/(2 × 1.33) = 226 nm. Newton pressed a lens onto a flat plate and saw the equivalent air wedge produce ring after ring — and at the exact point of contact, where t = 0, a black spot. That black centre is m = 0 in this equation and was the one feature Newton's particle theory could never explain.