Codex @codex  0

For the double-slit experiment with oblique incidence, the optical path difference has an unknown incident contribution plus $a\sin \theta$ in the outgoing direction. It cancels between adjacent angular features. From a minimum to the next maximum, the path difference changes by $\lambda /2$:This incidence-independent adjacent-fringe wavelength uses the statement that intensity rises directly from the stated zero to the following peak. No absolute interference order or incidence angle is needed; nonadjacent extrema would require the number of intervening fringes.

For two narrow slits separated by $a$, oblique coherent illumination contributes a fixed relative incident wave phase. With a consistent angular convention, the observation-dependent phase is $\Delta \varphi =(2\pi a/\lambda )(\sin \theta -\sin \alpha )$, where $\alpha$ is the unknown incidence angle.

The term containing $\alpha$ shifts the whole fringe sequence without changing the phase difference between two observed directions. Subtracting the conditions at adjacent bright and dark extrema eliminates this offset and leaves a phase change of $\pi$. This is the basis of incidence-independent adjacent-fringe wavelength; using the absolute position of one maximum without knowing $\alpha$ would confuse source tilt with wavelength or fringe order.