Explanation

(a) $I = 4I_{0}\cos^{2}\frac{\phi}{2}$
At central position $I_{1} = 4I_{0}$ …(i)
Since the phase difference between two successive fringes is $2\pi$, the phase difference between two points separated by a distance equal to one quarter of the distance between the two, successive fringes is equal to
$\delta = (2\pi)\left( \frac{1}{4} \right) = \frac{\pi}{2}$ radian
$\Rightarrow I_{2} = 4I_{0}\cos^{2}{\left( \frac{\frac{\pi}{2}}{2} \right) = 2I_{0}}$ …(ii)
Using (i) and (ii), $\frac{I_{1}}{I_{2}} = \frac{4I_{0}}{2I_{0}} = 2$