Ray and wave aberrations revisited: a Huygens-like construction yields exact relations

Abstract

The aberrations of an optical system can be described in terms of the wave aberrations, defined as the departure from the ideal spherical wavefront; or the ray aberrations, which are in turn the deviations from the paraxial ray intersections measured in the image plane. The classical connection between the two descriptions is an approximation, the error of which has, so far, not been quantified analytically. We derive exact analytical equations for computing the wavefront surface, the aberrated ray directions, and the transverse ray aberrations in terms of the wave aberrations (OPD) and the reference sphere. We introduce precise conditions for a function to be an OPD function, show that every such function has an associated wavefront, and study the error arising from the classical approximation. We establish strict conditions for the error to be small. We illustrate our results with numerical simulations. Our results show that large numerical apertures and OPD functions with strong gradients yield larger approximation errors.

The determination of secondary ray-aberration coefficients for axis-symmetrical optical systems

“…(2a) and (2b) , have errors of a fifth order. Restrepo et al [1] found that Eqs. (2a) and (2b) thus yield large errors for systems with large numerical apertures.…”

“…where R reference and 𝜉 ′ are the radius and refractive index of the reference sphere, respectively. As discussed by Restrepo et al [1], Eqs. (16a) and (16b) inevitably induce errors as a result of their approximate nature.…”

“…(2a) and (2b) are obtained from the wavefront aberration function by using the following approximate partial derivatives [1] : where and are the radius and refractive index of the reference sphere, respectively. As discussed by Restrepo et al [1] , Eqs. (16a) and (16b) inevitably induce errors as a result of their approximate nature.…”

“…However, Eqs. (2a) and (2b) are obtained from the wavefront aberration function W(h 0 , 𝜌, 𝜙) by using the following approximate partial derivatives [1]:…”

“… The ray-aberrations in axis-symmetrical systems are conventionally derived from wavefront functions or characteristic functions using classical approximate partial derivatives. However, the resulting aberrations typically have fifth-order errors, as described by Restrepo et al (2017) [1] . Accordingly, in the present study, the secondary ray-aberration coefficients for object placed at finite distance are determined using the fifth-order Taylor series expansion of a skew ray.…”