On-axis diffraction-limited design of bi-parabolic singlet lenses

Abstract

A lens limited by diffraction and having two parabolic surfaces is presented. The knowledge of the following parameters: object distance, relative refractive index, lens thickness, and image distance, enables to analytically calculate the parabolic front and back surfaces. The conditions to obtain diffraction-limited images for these lenses and its maximal diameter are described. These bi-parabolic lenses can be easily manufactured at reduced costs and can be used for several commercial and industrial applications. The method to obtain such surfaces and a simple example validated by using Oslo® are described here.

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

“…The example shows that when applying Eqs. (1)(2)(3)(4)(5)(6) the resulting corrective surface works correctly as it is clearly depicted in Fig. 2(a).…”

“…They ignore field aberrations that can grow considerably as it is the case of coma and astigmatism. On the difficulty of maintaining spherical aberration below the diffraction limit in a rotationally symmetric lens, we refer to [4][5][6].…”

“…The set of equations (1)(2)(3)(4)(5) are completely free of rules' signs. This solution is also advantageously completely differentiable with respect to conjugated distances.…”

“…It is important to point out that the set of equations (1)(2)(3)(4)(5) can be applied for designing one-axis bi-aspheric, rotationally symmetric and spherical aberration-free lenses. Particular solutions in [5][6][7] are not robust nor are complete nor closed [8].…”

“…By substituting {x a  r a cos  } and {y a  r a sin } in Eqs. (1)(2)(3)(4)(5) it is possible to obtain a set of parametric equations in cylindrical coordinates. Given that the surface is rotationally symmetric, parameter  vanish in the simplification process by using the fact that cos 2  +sin 2  = 1.…”