 Unknown incidence offset in double-slit phase

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.