Path Difference, Phase Difference and Conditions for Maxima and Minima

Let $P$ be a point on the screen at a distance $y$ from the central point $O$, in a Young's double slit set-up with slit separation $d$ and screen distance $D\gg d$. Using this approximation, the two paths $S_1P$ and $S_2P$ are very nearly parallel, and simple geometry (dropping a perpendicular from $S_1$ onto $S_2P$) gives the path difference between them as

The corresponding phase difference, since a path difference of one full wavelength $\lambda$ corresponds to a phase difference of $2\pi$, is $\phi=(2\pi/\lambda)\Delta$.

If the two slits send out waves of equal amplitude and hence equal intensity $I_0$, the resultant intensity at $P$, found by adding the two waves as phasors and squaring, works out to

$I=4I_0\cos^2(\phi/2)$ …