Question #43221

A thin Gradient Index (GRIN) lens of thickness $t_0 = 2.0 \text{ cm}$ has a radially varying refractive index given by $n(r, \lambda) = n_0(\lambda) - \frac{1}{2} \alpha(\lambda) r^2$, where $r$ is the distance from the principal axis. The parameter $\alpha(\lambda)$ depends on the wavelength $\lambda$ as $\alpha(\lambda) = A + \frac{B}{\lambda^2}$, where $A = 100 \text{ m}^{-2}$ and $B = 2.0 \times 10^{-11}$. An opaque screen with two narrow slits is placed immediately after the GRIN lens. The slits are located at radial distances $r_1 = d$ and $r_2 = 2d$ on the same side of the principal axis, where $d = 1.0 \text{ mm}$. A plane wave containing two wavelengths, $\lambda_1 = 400 \text{ nm}$ and $\lambda_2 = 500 \text{ nm}$, is incident normally on the GRIN lens. An observation screen is placed parallel to the lens at a distance $D$ such that the central interference maximum for $\lambda_1$ forms exactly at the principal axis ($y=0$). Find the magnitude of the phase difference between the interfering waves of wavelength $\lambda_2$ at the principal axis ($y=0$) on the observation screen. (Assume $d \ll D$ and use the paraxial approximation).