Spherical aberration

A plano-convex lens, with radius of curvature $R$, index of refraction $n$, and thickness $t$, has a focal length $f$ that depends on the distance from the optical axis $h$ that the light rays pass through the lens. The variation of the focal length with distance off the axis is non-linear, and also depends on whether the parallel light enters the lens from the curved or flat side. For example, if parallel light is incident from the plane side, it images on axis at

$$s = R/(n-1) - h^2/(2 R)$$ (61)

behind the curved surface. The general form for spherical aberration is

$$\delta f = ah^2 / f$$ (62)

where $\delta f$ is the change in the focal length $f$. You can let $a=1$ here. For a plano-convex lens,

$$f= R / (n-1)$$ (63)