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Spherical Aberrations

Definition: optical aberrations resulting from spherical optical surfaces

German: sphärische Aberrationen

Categories: general optics, vision, displays and imaging

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Most optical lenses have spherical surfaces, because those can be most easily fabricated with high optical quality. That surface shape, however, is not ideal for imaging; the outer parts of the lens are then too strongly curved. This is most obvious when considering a ball lens. Figure 1 demonstrates this for a ball lens with 10 mm diameter and a refractive index of 1.515 (N-BK7 glass at 633 nm), which is used to focus parallel incoming light. The outer incoming rays are crossing the optical axis substantially sooner than the paraxial ones.

rays at ball lens
Figure 1: Focusing of light with a ball lens. While the paraxial rays have a focus position as indicated with the gray vertical line, the outer rays are more strongly refracted. Therefore, such lenses would lead to strong spherical aberrations when used for imaging.
rays at ball lens
Figure 2: Focusing of divergent light with a ball lens.

When using lenses with spherical surfaces for imaging applications, the explained effect leads to so-called spherical aberrations which can seriously degrade the image quality. Similarly, the use of spherical lenses for focusing or collimating laser beams leads to beam distortions.

In many cases, the aberration effects are far less extreme than those shown above for the ball lens, since the involved curvatures are not that strong.

Spherical Aberrations from Plane Plates

The problem of spherical aberrations can be generalized to all aberrations associated with a non-ideal radial dependence of phase changes. That can occur even for plane surfaces, e.g. of plane-parallel plates, when divergent or convergent light travels through such a plate. This is essentially because the law of refraction contains the sine rather than the tangent function, which would be required to avoid spherical aberrations.

An example case in shown in Figure 3.

spherical aberrations from a plate
Figure 3: When convergent rays travel through a plate, they do not meet any longer in a common focal point.

Quantification of Spherical Aberrations

The strength of spherical aberrations of an optical system or an optical component such as a lens is often quantified by plotting the deviation of the longitudinal position of the image focal point as a function of the transverse offset of incident rays. Often, one exchanges the coordinate axes, so that the resulting plot corresponds more closely to a horizontal optical axis. The mentioned position error may scale with the square of the transverse beam coordinate, but in cases where the spherical aberrations are partly compensated (see below), that compensation may work for a particular horizontal offset but not so well for other offsets.

Reducing Spherical Aberrations

Spherical aberrations can be reduced in different ways:

  • The simplest method is to restrict the area of the incoming light with an optical aperture. That way, one can prevent that the outer regions, where spherical aberrations are most extreme, contribute to the image. However, that implies a reduced light throughput.
  • One can use aspheric lenses, which have modified surface shapes such that spherical aberrations are avoided.
  • One can use a combination of spherical lenses designed such that spherical aberrations are well compensated. This method is frequently used in photographic objectives, for example.

To some extent, one can also reduce spherical aberrations by choosing an appropriate type of lens, depending on the required configuration (see Figure 3):

  • For imaging a small spot to a spot of equal size, the symmetric biconvex lens is well suited. However, it is even better to use two plano-convex lenses in combination, with the flat surfaces on the outer sides.
  • For an asymmetric application, such as focusing a collimating beam or collimating a strongly divergent beam, a plano-convex lens can be more appropriate. The best solution would actually be an asymmetric lens with optimized curvature radii on both sides, but a plano-convex lens is often close enough. It must be oriented such that the curved surface is on the side of the collimated beam. Both lens surfaces then contribute to the focusing action.
optimum lens types for refocusing and collimation
Figure 4: Recommended lens types for refocusing and collimation. The middle case for symmetric refocusing is better than the first one in terms of aberrations, but may have higher losses due to two additional optical surfaces.

Generally, lenses should be used such that both surfaces contribute similarly to the focusing action.

The development of improved optical fabrication methods for aspheric optics has led to their increased use, allowing manufacturers to make high-performance objectives with fewer lenses – which can also result in improved light throughput. Note, however, that other kinds of optical aberrations can then still occur.

See also: optical aberrations, lenses
and other articles in the categories general optics, vision, displays and imaging

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