A new aplanatic front lens system for microscope immersion objectives is proposed. This system consists of a uniform-index spherical outer shell suitably completed by an immersion liquid of the same index and an inner bead glass with spherically symmetric refractive index distribution. Such a doublet is theoretically capable of the stigmatic and real imaging between two concentric spheres, the one confined within the outer shell area (immersion) and the other external to the system. The refractive index profiles of the inner bead glass that are required for perfect imaging have been calculated on the basis of rigorous formalism of geometrical
optics. The derived index variations turn out to be quite moderate and seem to be technologically available.
Jacek L. Sochacki,
Carlos C. Gomez-Reino,
"Stigmatic index-gradient doublet for the microscope immersion objectives: a design," Optical Engineering 30(1), (1 January 1991). https://doi.org/10.1117/12.55759