# Aberration

In optics, the aberration is a flaw in the imaging properties of optical systems such as lenses or mirrors, that causes light to be blurred, distorted, or spread out over some region of space rather than focused to a point.

By aberration of an optical system is meant any deformation of the image with respect to the object due not to defects in the construction of the system, but to the general laws of reflection and refraction. It is possible to distinguish axial aberrations and extra-axial aberrations depending on whether the object is placed on the optical axis of the system or outside it. Axial aberrations are axial chromatic aberration and sphericity aberration, extraxial aberrations are astigmatism, coma, field curvature, distortion and extraxial chromatic aberration.

## Defocus aberration

Defocus aberration is a type of aberration in which an image is simply out of focus. This type of aberration generally occurs using optical microscopes, telescopes, or binoculars.

## Astigmatism

In optics, astigmatism is a lens or mirror defect in which the size and shape of an image vary for different points of focus. Light passing through different parts of an astigmatic lens, for example, is focused at different distances beyond the lens, so that the image of a point can appear variously as a short horizontal or vertical line or an ellipse. The best focus is a small circle known as the circle of least confusion.

## Chromatic aberration

Chromatic aberration (also called chromatic distortion and spherochromatism) is a defect in a lens in which the various colors of the spectrum are not brought to the same focus. It is caused by dispersion because the refractive index of the lens elements varies with the wavelength of light. In particular, the refractive index of most transparent materials decreases with increasing wavelength.

The result is fringing – the formation of a colored halo along boundaries that separate dark and bright parts of the image. This problem seriously affected the performance of refracting telescopes for centuries and was the reason that so many refractors were built with large focal ratios: longer focal length lenses show a less chromatic error.

A better solution came with the introduction of corrective elements, using at least two different types of glass, into a compound lens. An achromatic lens corrects for red and blue light, whereas an apochromatic lens corrects for at least red, blue, and green. Reflecting telescopes are free from this type of aberration.

Even in the case of images provided by an electronic optical system realized with electric and magnetic fields there can be chromatic aberration, aberration of aperture and aberration due to ionization of the residual gas inside the electronic optical system. Chromatic aberration is due to the fact that the beam of charged particles passing through the system is never perfectly monochromatic (monoenergetic) because the particles are always emitted from the source (cathode) with a statistical energy distribution, so the electronic optical system causes a dispersion of trajectories that gives rise to blurred images. Aperture aberration is due to the fact that trajectories are not perfectly paraxial, that is they form too large an angle with the optical axis. The third type of electronic aberration is caused by ionization by collision of residual gas which, by placing itself around the beam, concentrates it damagingly.

## Comatic aberration (or coma)

Comatic aberration (or coma), in an optical system refers to aberration inherent to certain optical designs or due to imperfection in the lens or other components in which rays of light, striking the objective away from the optical axis are not brought to focus in the same image plane. For example, stars appearing distorted, appearing to have a tail (coma) like a comet.

Specifically, the comatic aberration is defined as a variation in magnification over the entrance pupil. In refractive or diffractive optical systems, especially those imaging a wide spectral range, coma can be a function of wavelength, in which case it is a form of chromatic aberration.

## Distortion

In geometric optics, distortion is a type of optical aberration that happens when there is a variation in magnification across the visual field.

## Petzval field curvature

Petzval field curvature, named for Joseph Petzval, describes the optical aberration in which the focus changes from the center to the edge of the field of view. In the presence of astigmatism, this problem is compounded because there are two separate astigmatic focal surfaces.

Field curvature varies with the square of the field angle or the square of the image height. Therefore, by reducing the field angle by one-half, it is possible to reduce the blur from field curvature to a value of 0.25 of its original size. Positive lens elements usually have inward curving fields, and negative lenses have outward curving fields. Field curvature can thus be corrected to some extent by combining positive and negative lens elements. Lenses with virtually no field curvature are called flat-field lenses.

## Spherical aberration

Spherical aberration is a defect of spherical mirrors and some lenses and eyepieces that causes parallel light rays striking the element at different distances from the optical axis to be brought to a focus at different points along the axis. This deviation reduces the quality of images produced by optical systems.

There is no place at which all the rays come to a single focus and give a sharp image. Spherical aberration may be reduced by narrowing the lens so that rays do not pass through the edges and by combining lenses to cancel out the defects in each kind of lens.

It occurs in optical systems with symmetry of revolution (spherical mirrors, simple lenses, etc.) and is due to the fact that, if we imagine to divide the system into many circular areas, from these cones of light rays depart converging at different points of the optical axis which, if the object is placed at infinity, coincide with as many fires of the system. Of these, the focus for the rays coming from the area closest to the axis (paraxial zone) is called paraxial focus, the one for the area farthest from the axis (marginal zone) is called marginal focus.

The set of rays coming out of the optical system envelops a rotational ridged surface called caustic. If the marginal focus is closer to the system than the paraxial focus, the system is called undercorrected, otherwise it is called overcorrected. The distance between these two focuses, called the stick, is a measure of sphericity aberration (longitudinal aberration or aperture aberration because it depends on the aperture diameter of the system).

If we then cut the beam emerging from the optical system with a screen perpendicular to the axis, a circle is observed on the screen whose radius is minimum for an intermediate position between the two foci. The radius of this circle (circle of least confusion) is a measure of what is known as transverse sphericity aberration.

Sphericity aberration, which is present in practically all optical systems with the exception of plane mirrors and parabolic mirrors, can be partially corrected in spherical mirrors by interposing in the ray path a plane-convex lens (Schmidt plate) which tends to compensate the mirror aberration with its own.

To minimize the effects of sphericity aberration one can always diaphragm the affected optical system. A system in which sphericity aberration has been corrected is called aplanatic.

## Aberration of starlight

Aberration of starlight is the difference between the observed position of a star and its true direction; this is a combined result of the observer’s motion across the path of the incoming starlight and the finite speed of light. In other words, the aberration of starlight is an apparent shift of the position of a celestial object during the year because at the speed of light with which the photons travel we have to sum the speed of the Earth along its orbit.

$\vec{v}_{LE}=\vec{v}_{LS}+\vec{v}_{SE}$

We have:

$\sin{\theta}=\dfrac{v}{c}$

where c is the speed of light and v = 3 · 104 m/s is the speed of Earth. vLE is the velocity of light with respect to the Earth. vSE is the velocity of the star with respect to the Earth. vLS is the velocity of light with respect to the star.

The typical example is that of a man running in the rain, and he must point the umbrella forward to avoid getting wet. Replacing the umbrella with a telescope and photons to the raindrops, and we will have an almost perfect analogy: but there is a difference, and it is not trivial. Photons travel at the speed of light in any frame of reference, but this does not apply to raindrops!

There are three components of the aberration of starlight: annual aberration (up to 20.47″) caused by the Earth’s revolution around the Sun, diurnal aberration (up to 0.3″) caused by the Earth’s axial rotation, and the very small secular aberration caused by the motion of the solar system through space. Stars on the ecliptic appear to move and fro along a line of 41″; stars 90° from the ecliptic appear to trace out a circle of radius 20.5″; and stars in intermediate positions ellipses of major axis 41″.