Expression for Fringe Width

The position of the $n$th bright fringe above the central point $O$ is found by setting the path-difference condition for a maximum, $\Delta=n\lambda$, equal to the geometrical expression for path difference derived above:

The fringe width $\beta$ -- the centre-to-centre spacing between two consecutive bright (or, equally, two consecutive dark) fringes -- follows immediately as the difference between successive fringe positions:

$\beta=y_{n+1}-y_n=\frac{(n+1)\lambda D}{d}-\frac{n\lambda D}{d}=\frac{\lambda D}{d}$ …