Rigorously aplanatic Descartes ovoids

Abstract

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.

Primary aberrations theory in optical systems composed of Cartesian refracting surfaces

“…As shown in [33], the primary Seidel aberrations are characterized for on-axis objects, by explicit expressions, for the cases of spherical, conic sections of revolution and aspherical refractive surfaces with axial symmetry. Cartesian surfaces are a special class of aspherical surfaces which, despite having important optical properties such as stigmatism and aplanatism [24,25], have no aberration theory in turn to them. The formulation of Cartesian surfaces gives as particular cases the spherical and conic section of revolution surfaces [23], therefore, the development of an expression for the primary aberrations of Cartesian surfaces should include those corresponding to them, as limiting cases.…”

“…Additionally, it has been proved that a subset of Descartes’s ovoids, under certain conditions, presents the property of aplanatism, allowing the design of coma-free imaging systems [24]. Also, four families of Descartes’s ovoids that strictly satisfy the Abbe sine condition have been identified [25]. Accordingly, the imaging systems derived from these families of surfaces present rigorous aplanatism, which increases the interest of these surfaces to be applied in the design of optical instruments with the highest demanding requirements, just as for spherical surfaces at their Young points in microscopy [26,27].…”