We have derived concentric gradient-indexed lenses to eliminate primary spherical aberration for all cases and describe a unified formulation. The Abbe points are found and the rules for longitudinal spherical aberrations of homogeneous lenses of spherical surfaces are derived, which may be helpful for spherical aberration cancellation in homogeneous spherical surface lenses designs. Some applications of concentric gradient-indexed lenses to microscope objective and others are demonstrated.
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