In Young's double-slit experiment, let $β$ be the fringe width, and let $I_0$ be the intensity at the central bright fringe. At a distance x from the central bright fringe, the intensity will be

$I_0\, \cos^2(πx/β)$

$Δ=x\frac{d}{D}$

∴ Phase difference = $\phi=\frac{2π}{λ}Δ$

Let a = amplitude at the screen due to each slit.

∴ $I_0 = k(2a)^2 = 4ka^2$, where k is a constant.

For phase difference $\phi$, amplitude = $A = 2a \cos(\phi/2)$.

Intensity, $I = kA^2 = k(4a^2) \cos^2 (\phi/2) = I_0 \cos^2 (πΔ/λ)$

$=I_0\cos^2(\frac{π}{λ}.\frac{xd}{D})=I_0\,\cos^2(\frac{πx}{β})$