Question #43229

A telescope has an objective lens of diameter $D = 0.10 \text{ m}$ and focal length $f_0 = 2.0 \text{ m}$. A thin transparent glass disk of thickness $t = 5.0 \text{ mm}$ is placed immediately in front of the objective lens. The refractive index of the glass disk varies radially as $n(r) = n_0 - \beta r^2$, where $r$ is the radial distance from the optical axis, $n_0 = 1.50$, and $\beta = 50 \text{ m}^{-2}$. The telescope is used to observe a distant star emitting light of vacuum wavelength $\lambda = 500 \text{ nm}$. If the viewing screen is adjusted to the new focal plane to obtain the sharpest image, what is the linear diameter of the first dark ring (Airy disk) of the diffraction pattern?