The Plano-Convex Hyperbolic Lens

One can avoid the focusing problems of the biconvex lens associated with its thickness by using a plano-convex lens. Since there is only one refracting surface one needs to double the curvature at the center of the lens so κ₀ = - 2/f where f is the focal length. One can also replace the parameter b by a dimensionless factor β = b/f. This gives the equation for the hyperbolic surface in a more useful form.


The focal point is then determined to an extremely high degree of precision. I encountered problems with the tool used to plot the ray paths due to an insufficient number of data points. The plot below shows the focal point zoomed a billion times and one still cannot observe any deviation from a point focus.


If my value for β is accurate then the dimensions for the focal region are on the order of 10^-15 centimeters which smaller than the diameter of an atom. So the limitations on the precision of the focus seems to depend on how well the lens surface can be shaped and diffraction effects.

I should also point out that I used n = 1 for the index of refraction for the ambient medium and n' = 1.5 for that of the lens. In general the parameter β depends on the two indexes.