Coherent light of wavelength λ illuminates a double slit S_1 - S_2...

The light from the double slits $S_1$ and $S_2$ travels a distance $D_1$ to reach the plane mirror.

Upon reflection, the plane mirror forms virtual images of the slits, let's call them $S_1'$ and $S_2'$. These virtual sources act as the coherent sources for the interference pattern on the screen.

The virtual sources $S_1'$ and $S_2'$ are formed at a distance $D_1$ behind the mirror. The separation between these virtual sources remains the same as the original slits, so $d' = d$.

The screen $\Sigma$ is located at a distance $D_2$ behind the plane of the double slits. Therefore, the distance from the mirror to the screen is $D_1 + D_2$.

The total effective distance $D$ from the virtual coherent sources $S_1'$ and $S_2'$ to the screen $\Sigma$ is the sum of the distance from the virtual sources to the mirror and the distance from the mirror to the screen:

$D = D_1 + (D_1 + D_2) = 2D_1 + D_2$

The fringe width $\beta$ of the interference pattern is given by the standard formula:

$\beta = \dfrac{\lambda D}{d}$

Substituting the effective distance $D$ into the formula, we get:

$\beta = \dfrac{\lambda(2D_1 + D_2)}{d}$