Question #43217

A transparent solid slab occupies the region $0 \leq x \leq L$ and $y \geq 0$ in air ($n_{\text{air}} = 1$). The refractive index of the slab varies spatially and depends on the wavelength $\lambda$ of light according to the relation: $n(y, \lambda) = \sqrt{n_0^2 + k(\lambda) y}$ where $n_0$ is a positive dimensionless constant and $k(\lambda)$ is a wavelength-dependent parameter with units of $\text{m}^{-1}$. A narrow beam of white light is incident normally on the face $x=0$ at the origin $(0,0)$, initially traveling along the positive $x$-axis. The beam propagates through the slab and exits into the air from the face $x=L$. Assuming $L$ is sufficiently small such that the angles made by the ray with the $x$-axis remain small ($\sin \theta \approx \tan \theta \approx \theta$), what is the angular dispersion $\Delta \theta = \theta_V - \theta_R$ between the violet and red components of the light after it completely exits the slab? (Given: $k_V$ and $k_R$ are the values of $k(\lambda)$ for violet and red light, respectively).