J. Eur. Opt. Society-Rapid Publ. 2026, 22, 30 Ó The Author( s), published by EDP Sciences, 2026 https:// doi. org / 10.1051 / jeos / 2026025 Available online at: https:// jeos. edpsciences. org
EOSAM 2025 Guest editors: Omar El Gawhary, Stefan Witte, Ignacio Moreno
Journal of the European Optical Society-Rapid Publications
RESEARCH ARTICLE
Quasi-powers and primary aberrations of thin lenses in contact
Florian Bociort * Department of Imaging Physics, Faculty of Applied Sciences, TU Delft, 2628CJ Delft, Netherlands
Received 28 January 2026 / Accepted 14 March 2026
Abstract. This paper introduces a novel framework for analysing the aberrations of thin lenses, based on the concept of surface quasi-power. Using these surface variables, remarkably simple expressions have been derived for all primary aberrations of systems of thin lenses in contact. Apart from a constant term, primary aberrations become essentially sums of powers of the new variables. When the emphasis is on qualitative properties rather than on quantitative ones, then even in complex optical systems groups of lenses can be modelled as thin lenses in contact. Especially for spherical aberration, the simplicity of the new formalism helps explaining significant properties of the lens design landscape.
Keywords: Lens design, Aberration theory, Geometrical optics, Thin lenses.
1 Introduction
Thin-lens aberration theory is a foundational element of optical design, covered extensively in standard textbooks [ 1, 2 ]. The assertion that this venerable theory still has potential for significant new insights may therefore surprise many lens designers. By introducing for each lens surface a new variable, the quasi-power, we derive primary aberration expressions for multi-lens systems that are significantly simpler than traditional formulations. Simplicity facilitates insight, and the novel formalism can provide clear explanations for both established – but perhaps insufficiently understood – and recent findings.
In Section 2 we derive the new expression for the 3rdorder spherical aberration and show how the definition of“ quasi-powers” results naturally from the goal of simplifying the formalism. Section 3 is dedicated to the new type of surface variable, the quasi-power. In Section 4 other aberrations are discussed, and in Section 4.3 it is shown that all Seidel aberrations of thin lenses follow the same remarkably simple polynomial pattern when expressed in terms of quasi-powers. Section 5 provides several examples, one of which sheds light on a fundamental question in lens design – namely, why the lens design landscape exhibits such a large number of local minima.
2 Spherical aberration
Consider a rotationally symmetric lens group consisting of L thin lenses, all with the same refractive index n, inair
* Corresponding author: f. bociort @ tudelft. nl and in contact with each other, i. e. all axial distances between surfaces within the group are set to zero. The L thin lenses in contact either form a separate optical system or are part of a larger system. In this section we derive a new expression for the 3rd-order spherical aberration of this group of thin lenses.
2.1 Framework
The starting point of the derivation is the well-known formula for total Seidel spherical aberration S computed, using the heights h k and angles u k of the paraxially traced marginal ray, as a sum of the surface contributions S k of the 2L surfaces [ 1 ]
S ¼ X2L k¼1
S k ¼ X2L k¼1
�A 2 k h k
u kþ1
� u
k n kþ1 n k þ 8G k h 4 kð n kþ1 � n k Þ
Here, the refraction invariant A k is given by A k ¼ n k ðh k c k þ u k Þ ð2Þ
The angles u k are related to the surface powers P k and curvatures c k by the paraxial refraction formula
n kþ1 u kþ1 ¼ n k u k � h k P k ¼ n k u k � h k c k ðn kþ1 � n k Þ: ð3Þ
If surface k is aspheric and is described as a spherical surface plus a polynomial, then G k is the fourth-order radial coefficient appearing in the polynomial.
In the above formulas the angle u k and the refractive index n k are those before refraction at surface k, whereas the index k + 1 denotes the corresponding values after refrac-
ð1Þ
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