Question #43562

Two identical thin glass axicons (conical lenses) of refractive index $n_g = 1.4$ are placed coaxially. Each axicon has one flat surface and one conical surface. The thickness of each axicon is given by $t(y) = t_0 - \alpha y$, where $y \ge 0$ is the radial distance from the optical axis and $\alpha = 0.05 \text{ rad}$. They are arranged such that their conical surfaces face each other. The gap between the conical surfaces has a thickness $d = 2 \text{ mm}$ at the optical axis ($y=0$). This gap is filled with a fluid whose refractive index varies radially as $n(y) = n_0 - k y$, where $n_0 = 1.6$ and $k$ is a positive constant. The value of $k$ is specifically tuned such that the composite system acts as a perfect spherical lens for paraxial rays (i.e., all linear terms in the optical path length cancel out, eliminating conical aberration). What is the equivalent optical power of this composite system?