General formula to design a freeform singlet free of spherical aberration and astigmatism: comment

Abstract

In their article [Appl. Opt. 58, 1010-1015] González-Acuña et al claimed: "an analytical closed-form formula for the design of freeform lenses free of spherical aberration and astigmatism." However, as we show here, their formula can only be applied when the object and image are both real and the image is inversed, additionally, the refractive index in the object and image media is the same. Here, we present the complete solution of this particular formula.The formula presented by González-Acuña et al.[1] is used to calculate the second surface of a lens, capable to correct the image from spherical aberrations due to the first lens' surface. We have found that this general formula only works when the object distance f a <0 and the image distance f b >0 are both real. Nevertheless, their solution fails when one of the combined planes is virtual, i.e., when f a >0 or f b <0 as the required sign rules are not included in the normal ˆa n formula (Eq. 2). This formula is required to obtain 2 v and its correct direction for any virtual-object's point. A different case appears when the image point is virtual. Virtual images are common in car's front lights and lightings. In some thick lenses with positive magnification the rays can cross internally and Eq. (7) does not include the required signs rules. They have shown examples consisting exclusively of real objects and real images. The authors declare that: "the normal is perpendicular to the tangent plane of the input surface at the origin." and that: "We recall that a necessary condition for the validity of Eq. (7) is that the surface normal should be perpendicular to the tangent plane to the input surface at the origin." This is the definition of the normal and not one condition. However, the condition is that the normal at the origin must be on the optical axis (0,0) [0,0, 1] a    n , which restricts its use to a family of freeform surfaces. This is contradictory with the phrase on the arbitrarily of the surface freeform: "These equations may look cumbersome, but it is quite remarkable that they could be expressed in closed form for an arbitrary freeform input surface."The fact that "the validity of Eq. (7) also requires that the rays do not intersect each other inside the lens because, in this case, Ψ b overlaps itself, leaving from being homeomorphic with respect to Ψ a , and the vicinity of the neighborhoods is not preserved" shows that the solution is not complete and that it can only be used to design partially freeform lenses. As a matter of fact, as the sign's rule has not been taken into account, Eq. (7) only applies when the rays do not cross in the interior of the lens. Nevertheless, there are two different families of solution, one having a negative magnification and the second with positive magnification.