On the selection of material for an aplanatic plano-convex asphere free of slope surface oscillations

Reliable manufacturing and high optical performance of plano-convex aspheric lenses necessitate an improved optical design characterized by an optimal refractive index of the lens material and the prevention of the slope surface shape oscillations, which provide an increased angular field of view and low sensitivity to misalignments. To formulate the criteria for the optimal refractive index, we propose an analytical representation of the optimal aspheric surface based on the Fermat’s principle in the form of parametric functions of coordinates, allowing the surface to be approximated by the elliptical conic section expanded with high-order aspheric deformation terms starting from the 6th power. There are determined the ranges of optimal refractive indices for materials of plano-convex aspheric lenses: n =1.605…1.665 for an aplanatic design with the Abbe sine condition fulfilled, ensuring low lens sensitivity of the lens to angular misalignment; n =1.735…1.835 for sharp imaging of elongated longitudinal objects in accordance with the Herschel’s condition.