Spherical aberration

Spherical aberration occurs in optical systems using spherical lenses or mirrors, which are common because spherical shapes are simpler to manufacture. When light rays hit a spherical surface away from its center, they are bent either more or less than rays hitting near the center. This difference degrades the sharpness of the image. The effect was first described in the 11th century by Ibn al-Haytham in his work *Kitāb al-Manāẓir*.

A spherical lens has a point free of spherical aberration—called an aplanatic point—only at a lateral distance from the optical axis equal to the sphere's radius divided by the lens material's refractive index. Spherical aberration limits the focus quality of telescopes and other instruments. Because spherical surfaces are far easier to produce than aspherical ones, designers often use multiple spherical elements to correct the aberration rather than a single aspheric lens.

"Positive" spherical aberration means outer edge rays are bent more than an ideal lens would; "negative" aberration means they are bent less. The effect scales with the fourth power of the lens diameter and inversely with the third power of the focal length, making it much stronger in short focal ratio ("fast") lenses.

To correct spherical aberration, lens systems combine convex and concave elements, or use aspheric or aplanatic lenses. Correction is typically designed through numerical ray tracing. For simple designs, parameters can sometimes be calculated analytically. For a single spherical lens with given object distance *o*, image distance *i*, and refractive index *n*, spherical aberration can be minimized by adjusting the front and back radii *R₁* and *R₂* according to the formula (R₁ + R₂)/(R₁ – R₂) = [2(n² – 1)/(n + 2)] × [(i + o)/(i – o)], following the Cartesian sign convention (positive radius if the center of curvature is to the right of the surface, negative if to the left; object and image distances positive if to the right, negative if to the left).

For small telescopes with spherical mirrors and focal ratios shorter than f/10, light from a distant point source (like a star) does not focus at a single point: inner rays focus farther from the mirror than outer rays, preventing a sharp image. Thus, telescopes with focal ratios below f/10 typically use non-spherical mirrors or correcting lenses. Spherical aberration can be eliminated entirely by using lenses with aspheric surfaces. Descartes showed that lenses shaped as Cartesian ovals (rotated around the central axis) can perfectly focus light from a point on the axis or from infinity. In 2018, researchers derived a closed formula for a lens surface that eliminates spherical aberration, applicable when the other surface has any given shape.

Estimating the diameter of the focused spot due to spherical aberration is often done with ray optics, but this approach ignores light's wave nature, so results can be inaccurate due to interference effects. A simpler ray-optics formalism for thin lenses is the Coddington notation, which uses the lens's refractive index *n*, object distance *o*, image distance *i*, the distance *h* from the optical axis at which the outermost ray enters (half the clear aperture), the two lens radii *R₁* and *R₂*, and the focal length *f*.