Consider a thick glass rod of uniform refractive index $n$. We want to shape the blunt end of this g

Question

Consider a thick glass rod of uniform refractive index $n$. We want to shape the blunt end of this glass rod such that all light rays incident upon it, travelling parallel to the rod's axis, are refracted to focus exactly at a single point $F$ within the rod. Let the focal length $f$ be defined as the distance between the focus $F$ and the vertex (the point on the surface that lies on the rod's axis). 
What must be the exact geometrical shape of the blunt end's surface to completely eliminate spherical aberration?

Professor PC Thomas Classes · Answer

To find the shape of the surface, we apply Fermat’s Principle, which states that the optical path length for all rays converging to the focus must be constant.
Let the vertex of the rod be the origin $(0,0)$, and the focus $F$ be at $(f, 0)$ inside the glass.
For the axial ray, the optical path from the reference plane (outside the rod) to the focus is simply $nf$.
For a marginal ray striking the surface at coordinate $(x, y)$, the path length in air is $x$, and the path length in glass to the focus is $n\sqrt{(f-x)^2 + y^2}$.
Equating the optical paths:
$x + n\sqrt{(f-x)^2 + y^2} = nf$
Rearranging and squaring both sides:
$n^2[(f-x)^2 + y^2] = (nf - x)^2$
$n^2(f^2 - 2fx + x^2 + y^2) = n^2f^2 - 2nfx + x^2$
$x^2(n^2 - 1) - 2fxn(n - 1) + n^2y^2 = 0$
Dividing the entire equation by $(n-1)$:
$x^2(n + 1) - 2fnx + \frac{n^2y^2}{n-1} = 0$
Because $n > 1$, the coefficients of $x^2$ and $y^2$ are both positive but unequal. This is the standard equation of an ellipsoid.