The term astigmatism is used with various different (but related) meanings, which are explained in the following sections.
Astigmatism as a Kind of Optical Aberrations
Astigmatism is one of the classical types of optical aberrations which are experienced in imaging systems. It occurs, for example, when light hits a spherical optical lens or a spherically curved mirror under a substantial angle against the optical axis. Here, the focal length of the optical element is essentially dependent on the direction: it is reduced in the tangential plane and increased in the sagittal plane. As a result, perfect focusing of a circular laser beam, for example, is no more possible: the smallest beam radius for the tangential direction is reached sooner than for the sagittal direction.
Obviously, that phenomenon also leads to image distortions. It can be easily observed, for example, by viewing a small dot on paper through a magnifying glass which is substantially tilted against the normal direction. Depending on the focusing conditions, the image of a point may be elongated in one or the other direction.
In contrast to spherical aberrations, for example, astigmatism cannot be reduced simply by using an optical aperture. However, if a ray bundle also fills a large part of the lens area, the problem becomes worse and is in that case called coma.
Optical systems can often be corrected in terms of astigmatism by using an appropriate combination of different lenses, for example.
Astigmatism as a Defect of an Optical Lens
Astigmatism as explained in the previous section results for geometrical reasons even for perfectly spherical mirrors and lenses.
A similar kind of problem can result even for normal incidence if a lens or mirror has a different curvature in the vertical and horizontal direction. That occurs in many human eyes as a kind of refractive error; a substantial fraction of the human population suffers from that problem to some extent. This problem can be quantified in terms of the difference of dioptric powers between vertical and horizontal direction. Note that astigmatism of eyes can be even more complicated; for example, there can be an oblique astigmatism with modified directions of horizontal and vertical meridian or even irregular astigmatism.
Astigmatism of human eyes can be compensated with suitable prescription glasses or remedied with refractive eye surgery (see the article on medical lasers).
An extreme case would be a cylindrical lens or mirror, which is intentionally curved only in one direction, not having any dioptric power in the other direction.
Astigmatism of Light Beams
Astigmatism is also often understood as a property of a light beam – for example, a laser beam which propagated through a cylindrical lens. In simpler cases, astigmatism of a beam means that the focal points for the vertical and horizontal direction do not coincide. That problem may be corrected with an anamorphic prism pair, for example, or with cylindrical lenses or with tilted curved mirrors.
In laser technology, one can often consider the beam evolution separately and independently for the horizontal and vertical direction. This requires that any cylindrical lenses and the like are oriented such that they affect only either the horizontal or the vertical direction.
In other, more general cases, for example with a cylindrical lens rotated by 45°, the situation becomes substantially more complicated to describe mathematically, but that rarely occurs in optics and laser technology.
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