Question #43184

In a Young's Double Slit Experiment, two parallel slits $S_1$ and $S_2$ of equal width $w$ and separation $d$ ($d \gg w$) are illuminated normally by a monochromatic light of wavelength $\lambda$. A transparent slab of thickness $t$ is placed directly in front of the upper slit $S_1$. The refractive index of the slab varies linearly across the slit width as $\mu(y) = \mu_0 + \alpha y$, where $y$ is the local transverse coordinate measured upwards from the center of $S_1$ ($-w/2 \le y \le w/2$). The lower slit $S_2$ is uncovered. It is observed that the zero-order interference maximum (where the phase difference between the waves from the centers of $S_1$ and $S_2$ is zero) has zero visibility because it exactly coincides with the first diffraction minimum of slit $S_1$. Assuming $\alpha > \frac{\mu_0 - 1}{d}$, which of the following expressions represents the gradient $\alpha$?