Spheric long-focal-depth lens: A comment

A singlet spherical lens with spherical aberration, which is a good approximation of a logarithmic axicon and gives constant axial intensity along the focal segment created behind the lens, is proposed. The singlet lens axicon is a backward-type axicon and the phase distribution is the paraxial version of the forward annular axicon. For the forward axicon, rays from the inner part of the axicon cross the optical axis before rays form the outer part, yielding similar angles and consequently fairly uniform width along the focal line. Numerical optimization performed to find a singlet lens that performs the same task as the logarithmic axicon results in a phase function that is identical to that of the forward logarithmic axicon. The studies have shown that it is possible to combine the positive focusing element and the negative spherical aberration required of a forward lens axicon in a single spherical-lens element.