Question #42871

In a modified Young's Double Slit Experiment, a monochromatic light of vacuum wavelength $\lambda$ is incident on two narrow slits $S_1$ and $S_2$ separated by a distance $d$. A screen is placed at a distance $D$ ($D \gg d$) from the slits. The region between the slits and the screen is filled with a transparent medium whose refractive index varies linearly along the transverse direction $y$ (parallel to the screen and perpendicular to the slits) as $n(y) = n_0 + \alpha y$, where $n_0$ and $\alpha$ are positive constants. The origin $y=0$ is on the central axis equidistant from the slits, with $S_1$ at $y = d/2$ and $S_2$ at $y = -d/2$. Assuming that light rays travel in straight lines from the slits to the screen and $d \ll y \ll D$, which of the following expressions gives the local fringe width $\beta(y)$ at a coordinate $y$ on the screen?