Question #43126

A point charge $Q$ is placed on the principal axis of a thin plano-convex lens at a distance $u$ from its flat surface. The lens has a radius of curvature $R$ for its convex surface and a maximum central thickness $t_0$. The lens is made of an electro-optic material whose refractive index varies with the local electric field magnitude $E$ as $n(E) = n_0 + \alpha E^2$, where $n_0$ and $\alpha$ are positive constants. Assume $u \gg R_{\text{aperture}}$ (paraxial approximation) and that the electric field inside the lens is approximately the vacuum field produced by the point charge $Q$. The charge $Q$ also acts as a point source of monochromatic light. Let $E_0 = \frac{Q}{4\pi\epsilon_0 u^2}$ be the electric field magnitude at the pole of the flat surface. Which of the following expressions correctly represents the effective optical power $P$ of the lens for paraxial rays emitted by $Q$?