Objective: Determine the intensity distribution on the final screen by analyzing the two-stage interference process.

Step 1: Calculate Intensity incident on the second slits ($I_1$)

Consider the interference pattern formed by the first set of slits (separation $d_1$) at the location of the second set of slits (distance $D_1$). We calculate the intensity at the position of the top slit on the second cardboard, which is at a height $y = d_2/2$ from the optical axis.

The path difference $\Delta x$ between the waves from the two first slits reaching this point is:

$$ \Delta x = \frac{(\text{separation}) \times (\text{height})}{D} = \frac{d_1 \times (d_2/2)}{D_1} = \frac{d_1 d_2}{2 D_1} $$

The corresponding phase difference $\phi_1$ is:

$$ \phi_1 = \frac{2\pi}{\lambda} \Delta x = \frac{2\pi}{\lambda} \left( \frac{d_1 d_2}{2 D_1} \right) = \frac{\pi d_1 d_2}{\lambda D_1} $$

If $I_2$ is the intensity of the individual waves from the first slits, the resultant intensity $I_1$ at the second slit is:

$$ I_1 = I_2 + I_2 + 2\sqrt{I_2 I_2} \cos \phi_1 $$ $$ I_1 = 2 I_2 (1 + \cos \phi_1) = 2 I_2 \left( 2 \cos^2 \frac{\phi_1}{2} \right) = 4 I_2 \cos^2 \left( \frac{\pi d_1 d_2}{2 \lambda D_1} \right) $$

Step 2: Calculate Final Intensity on the Screen ($I$)

Now, the two slits on the second cardboard act as coherent sources with intensity $I_1$. They produce an interference pattern on the final screen at distance $D_2$. For a point at height $y$ on the final screen:

The phase difference $\phi_2$ is:

$$ \phi_2 = \frac{2\pi}{\lambda} \frac{d_2 y}{D_2} $$

The resultant intensity $I$ is:

$$ I = 4 I_1 \cos^2 \left( \frac{\phi_2}{2} \right) = 4 I_1 \cos^2 \left( \frac{\pi d_2 y}{\lambda D_2} \right) $$

Step 3: Combine the expressions

Substitute the expression for $I_1$ from Step 1 into the equation from Step 2:

$$ I = 4 \left[ 4 I_2 \cos^2 \left( \frac{\pi d_1 d_2}{2 \lambda D_1} \right) \right] \cos^2 \left( \frac{\pi d_2 y}{\lambda D_2} \right) $$

Let $I_0 = 16 I_2$ be the maximum possible intensity.

$$ I = I_0 \cos^2 \left( \frac{\pi d_1 d_2}{2 \lambda D_1} \right) \cos^2 \left( \frac{\pi d_2 y}{\lambda D_2} \right) $$

Note: The modulation factor $\cos^2(\dots)$ depends on the fixed geometry of the slits ($d_1, d_2, D_1$) and acts as a constant scaling factor for the intensity of the final fringe pattern.