Incidence-independent adjacent-fringe wavelength ($\lambda =2a|\sin {\theta }_{\max }-\sin {\theta }_{\min }|$)

In oblique-incidence two-slit interference, the incident wave contributes a constant phase offset while changing observation angle changes the outgoing optical path difference by $a\Delta \sin \theta$. Subtracting the phase conditions of an adjacent minimum and maximum cancels the unknown incidence angle and order and gives the displayed wavelength. Nonadjacent extrema need their intervening phase-order difference; angular spacing should not be replaced by a small-angle formula unless justified.

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.