The ratio of the intensity at the centre of a bright fringe to the intensity at a point onequarter of the fringwidth from the centre is

2

Two waves of a single source having amplitude A interfere. The resulting amplitude

$A_r^2 = A_1^2 + A_2^2 + 2A_1A \cos δ$

where $A_1 = A_2 = A$ and $δ$ = phase difference between the waves

$⇒ I_r = I_1 + I_2 + 2\sqrt{I_1I_2} \cos δ$

When the maxima occurs at the center, $δ=0$

$⇒I_{r_1} = 4I$  …(1)

Since the phase difference between two successive fringes is 2π, 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 $δ=(2π)(\frac{1}{4})=\frac{π}{2}$

$⇒I_{r_2} = 4I\cos^2(\frac{π/2}{2})=2I$   …(2)

Using (1) and (2)

$\frac{I_{r_1}}{I_{r_2}}=\frac{4I}{2I}=2$