J. Eur. Opt. Society-Rapid Publ. 22, 42( 2026) 407
Figure 1. Example of a saddle point generated by the augmentation of dimensionality of the optimisation landscape from a 1D merit function( left) to a 2D merit function( right).
into account aspherical surfaces were developed by F. Dai et al. [ 6 ] and freeform surfaces by B. Mao et al. [ 7 ]. In this article, we discuss the use of the saddle point construction method to automatically generate suitable starting points for spherical systems. We will first present the Saddle Point Construction Method( SPCM). It has been shown that the SPCM is a powerful method to generate new systems from already designed systems by increasing the number of lenses [ 8 – 11 ]. As a use case, we employ this for the automatic generation of achromatic cemented doublets using the spot size as a merit function. To reduce the number of generated doublets, we used a reduced glass map based on Principal Component Analysis( PCA) [ 12 ]. We compare the acquired results for an axial object field with those already published using a semi analytical approach. Thereafter, with an angular field of view of ± 2.5 °, we compare the resulting systems with catalogue achromatic cemented doublets. We conclude with the robustness and advantages of the presented method and its limits.
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The Saddle Point Construction method
In contrast to conventional optimization and deep learning methods, the Saddle Point Construction( SPC) method does not require a starting point or a training data set. This method was explored in order to be applied to optical design by Z. Hou [ 13 ]. Considering an N-dimensional optimization landscape, the saddle points are then stationary points surrounded by local minima and maxima in 1 or N � 1 directions forming a horse saddle( see Fig. 1). Indeed, saddle points lie between two basins of attraction [ 8 ].
The SPC method begins with an optimized optical system comprising N variables that minimize its merit function, which can initially consist of a single lens. To extend the system to N + 2 dimensions and generate a saddle point, a zero-thickness lens element with identical curvatures on both surfaces is introduced( referred to as a“ null” element). This“ null” element allows the addition of two additional variables, representing the curvatures of its surfaces, without altering the optical properties of the system or affecting the value of the merit function. A saddle point is formed for specific values of the surface curvature of the null lens element. A conventional saddle point detection algorithm [ 13 ] starts from a system for which the merit function is in a local minimum. Afterwards, the algorithm explores every direction around this minimum until it reaches a maximum in one or more other directions [ 14 ]. This mathematical approach to the saddle point construction method requires a properly optimized local minimum so that the residual gradient of the merit function approaches zero. However, it has been shown that the saddle point detection is not essential for optical design. An empirical approach starting from any local minimum enables one to generate saddle points while augmenting the number of dimensions of the problem( for instance, the number of surfaces). When the merit function is in a local minimum, adding variables will in most cases lead to a saddle point [ 15 ]. In this paper, we use the special version of the SPCM for rapid and simplified detection of the saddle point( see Fig. 2)[ 13 ]:
1. Start from a local minimum of the merit function( for instance, using a single lens).
2. Insert an element with the two surfaces having the same radii of curvature as the last surface of the previous lens, a zero thickness and the same material( called a“ null” element). For example, c3 = c4 = c2, with c3, c4 and c2 respectively the radii of curvatures of the first and second surfaces of the“ null” element and the second radius of curvature of the first element.
3. Vary the curvature c3 and c4 of the“ null” element by a small increment with c3 = c4 = c2 ±. Thus, two new systems are created.
4. Optimize the two systems with the default merit function given by Code V.
5. Incrementally increase the thickness of the“ null” element and of the airspace between the two elements, while the system is re-optimized after to find the two new minima induced by the detected saddle point.
6. Optimize the choice of glass of the second element.
The SPC special version leads to several solutions with different numbers of surfaces where the variable a corresponds to the number of added elements. These solutions are then summed up in a tree diagram which enables the