In a thin film, light reflecting off a higher-index boundary picks up a half-wave phase flip that can turn a constructive path difference into a dark reflected fringe. The arithmetic is exact only for the stated wavelength, film thickness, refractive index, and phase-flip bit in the ideal thin-film model. Real films add finite bandwidth, thickness variation, imperfect surfaces, angle spread, index dispersion, calibration, and measurement/source uncertainty.
A thin-film dark claim is accepted by optical path plus phase-flip bookkeeping. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
The film path is twice index times thickness
The reflected rays cross the film thickness twice. The helper records the optical path before applying the phase-flip bit.
2nt=2⋅1m=4m
The phase bit is part of the verdict
The same optical path can change verdict when the phase-flip bit changes. A thicker film is also rejected if it does not match the same dark row.
n222t1m1m2mpath4m4m8mflip101accepted100
A phase flip can make an integer film path dark
The reflected dark claim is accepted because the optical path and the phase-flip bit are both checked.