J. Eur. Opt. Society-Rapid Publ. 2026, 22, 3 Ó The Author( s), published by EDP Sciences, 2026 https:// doi. org / 10.1051 / jeos / 2025055 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
3D printed freeform micro-optics meets generated Jacobian equations based inverse designs
Carlos Jimenez 1, 2,* and Alois Herkommer 1, 2
1 Institute for Applied Optics( ITO), University of Stuttgart, 70569 Stuttgart, Germany 2 Research Center SCoPE, University of Stuttgart, 70569 Stuttgart, Germany
Received 1 September 2025 / Accepted 10 December 2025
Abstract. In this work, we explore the use of Generated Jacobian Equations( GJEs) for the inverse design of freeform micro-optical surfaces. We demonstrate the integration of this design technique into an applied pipeline targeting 3D printed freeform micro-optics fabricated via two-photon polymerization( 2PP). This work establishes a bridge between alternative design tools and cutting-edge micro-fabrication technologies, opening new opportunities in the field of engineered micro-optics. We demonstrate this via two representative examples, a round top-hat and a fringe pattern projection freeform lens.
Keywords: Inverse design, Freeform micro-optics, Two-photon polymerization.
1 Introduction
Freeform micro-optical components are central to the next generation of compact optical systems, enabling precise control over light propagation in ways that conventional optics cannot achieve. To this end, GJEs based inverse designs represent an interesting alternative to the realization of such freeform based micro-optical systems. However, such a design strategy has not been explored in combination with 2PP based micro-fabrication technologies. By combining the best from both fields, we highlight the potential for creating novel micro-optical elements tailored for specific optical transformations. In the following sections, we provide details behind the employed numerical implementation, the followed micro-fabrication process and the methodology used to extract relevant information on the smoothness associated to the computed freeform surface solutions.
2 Numerical implementation
Our implementation relies on the work presented in [ 1 ] for which an iterative algorithm is used to obtain freefrom optical surfaces via the use of a GJE. Such a design algorithm exploits the relationship between a generating function G and its unique inverse H, also referred to as a Hamilton’ s characteristic function. As in [ 1 ] we rely on an angular characteristic formulation to our problem, for which the source( x) and target( m) coordinates are both expressed in terms
* Corresponding author: jimenez @ ito. uni-stuttgart. de of stereographic values. For a system consisting of a point source with N interfaces prior to the sought freeform surface, the associated Hamilton’ s characteristic function can be expressed as
2jx � mj 2 Hðx; m; u ff Þ¼u 1 ðxÞ ð1 þjxj 2 Þð1 þjmj 2 Þ
"!# þ XN�1
2jy u i ðxÞ n i � 1 þ i�1 ðxÞ�mj 2 ð1 þjy i�1 ðxÞj 2 Þð1 þjmj 2 Þ i¼2
þ u ff
!
2jy n N � 1 þ N ðxÞ�mj 2 ð1 þjy N ðxÞj 2 Þð1 þjmj 2 Þ
with u 1, u i being the Euclidean distance functions between the N interfaces located before the freeform surface u ff, fy i�1 ðxÞg N�1 i¼2 represent the set of stereographic coordinates associated to the outgoing ray-directions( I) at interfaces i = 2,..., N�1 whiley N( x) corresponds to the stereographic coordinates acting as local source ones to the last term in equation( 1). Finally, u ff represents the Euclidean distance between the Nth interface and the freeform surface. To implement the algorithm, we make use of an open source finite element method( FEM) solver [ 2 ] which represents two main advantages. First, it allows the use of an unstructured grid to solve the required numerical problems enabling higher flexibility in terms of the source domain shape. Second, it allows the use of local basis functions to describe the solutions in terms of polynomials within each element. Complementary to this, we employ a
ð1Þ
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