In this work, we return to Descartes’s idea to develop a formalism to construct rigorously stigmatic singlet lenses comprising two Cartesian surfaces. Optical systems are built using a considerable number of spherical surfaces, presenting in most cases spherical aberration. Wasermann and Wolf proposed eliminating spherical aberration and minimizing third-order coma by using two adjacent aspherical surfaces. That is why, using a parametric formulation for Cartesian ovals, we propose the design of singlet lenses where the condition of rigorous stigmatism is guaranteed for each surface, and therefore, strictly speaking, in the pair of stigmatic points, the lens becomes an optical system free of spherical aberration. This formulation is unified to both refractive and reflective optical surfaces. Therefore, within the framework of the theory of rigorously stigmatic optical systems, making use of Cartesian surfaces for the construction of stigmatic ovoid singlet lenses, we achieve the same functionality of optical systems involving a set of spherical lenses. These lenses have the advantage of being formulated according to a generalized shape factor associated with the Coddington shape factor, allowing an easy classification of these stigmatic lenses. The ideal imaging is carried out by applying an exact ray-tracing method through these ovoid singlet lenses.

The minimization of spherical and coma aberrations in optical imaging systems is currently accomplished through the use of corrective aspheric optical surfaces. In this work, we develop a new, to the best of our knowledge, theory for the design of rigorously aplanatic optical systems, considering as a starting point the rigorous stigmatism theory of optical systems composed of Cartesian surfaces. The main characteristic of these surfaces is their, a priori, zero spherical aberration. In a general parametric formulation for systems made up of a set of these surfaces, the Abbe sine condition is adapted to simultaneously obtain the stigmatism and aplanatism conditions. Thus, we achieved the design of optical systems that in theory are immune to both coma and spherical aberrations.

It is known that, besides being stigmatic, spherical refracting surfaces are aplanatic at their Young points since they satisfy the Abbe sine condition rigorously. The Abbe sine condition is commonly applied to different optical systems using numerical methods or optimization processes, obtaining a design of approximately aplanatic systems. Here, we found several families of Cartesian surfaces, whose sets of each of these families constitute exactly aplanatic systems free of spherical aberration and coma. So, studying the different types of systems, it is found that rigorous aplanatism occurs for objects and images on curved surfaces.

Cartesian ovals, also known as rigorously stigmatic surfaces, are the simplest optical systems capable of producing a perfect point image. Exist both implicit and explicit expressions to represent these surfaces, but they treat both refractive and reflective surfaces independently. Because of the complexity of explicit expressions, the ray-tracing techniques for these surfaces are implemented using third-party software. In this paper, we express Cartesian ovals as a degenerated superconic curve and get a new explicit formulation for Cartesian ovals capable of treating image formation using both object and image points, either real or virtual, and in this formulation can deal with both reflective and refractive rigorously stigmatic surfaces. Finally, using the resultant expressions and the vector Snell–Descartes Law, we propose a self-contained analytical ray-tracing technique for all these surfaces.

Monochromatic and chromatic aberrations are imaging defects mainly studied from a geometrical optics point of view. These defects are treated through optimization and minimization methods to achieve acceptable performance in optical imaging systems, where the correct choice of glass materials is one of the main challenges. The selection of glass materials is a complex issue that requires a large amount of computing power within sophisticated computational algorithms and enough professional experience in the area. However, in this work, we propose a new methodology to treat the chromatic and geometrical aberrations simultaneously by taking advantage of the relationship between form parameters of Cartesian surfaces and wavelength in the material. From this relationship, we obtain an achromatism principle that establishes the conditions for refracting systems to present a strictly achromatic stigmatism.