The delivery of a high-power laser beam to the target area, either via fiber systems or free-space optical trains, requires an effective focusing optical element at the end of the delivering system. It can be derived that a hyperbolic lens has the optimal surface to achieve the best focusing effect. Using this type of lens, the rays representing the beam can make a large angle with respect to the axis and the incidence angle on the lens surface is far from normal. As a result, the ordinary scalar field theory with paraxial approximation is not valid in this case. In order to understand the performance of a hyperbolic lens, it is necessary to use the vector field theory and apply the Huygen's principle directly in the model. Using this method, we were able to show the performance of a hyperbolic lens is much better than that of a spherical lens.
Albert Y. Shih,
"Vector analysis of fast focusing by hyperbolic and spherical lenses", Proc. SPIE 3889, Advanced High-Power Lasers, (3 April 2000); doi: 10.1117/12.380869; https://doi.org/10.1117/12.380869