Optical aberration

In optics, aberration is a property of optical systems, such as lenses and mirrors, that causes the image created by the optical system to not be a faithful reproduction of the object being observed. Aberrations cause the image formed by a lens to be blurred, distorted in shape or have color fringing or other effects not seen in the object, with the nature of the distortion depending on the type of aberration.
Aberration can be defined as a departure of the performance of an optical system from the predictions of paraxial optics. In an imaging system, it occurs when light from one point of an object does not converge into a single point after transmission through the system. Aberrations occur because the simple paraxial theory is not a completely accurate model of the effect of an optical system on light, rather than due to flaws in the optical elements.
An image-forming optical system with aberration will produce an image which is not sharp. Makers of optical instruments need to correct optical systems to compensate for aberration.
Aberration can be analyzed with the techniques of geometrical optics. The articles on reflection, refraction and caustics discuss the general features of reflected and refracted rays.

Overview

With an ideal lens, light from any given point on an object would pass through the lens and come together at a single point in the image plane. Real lenses, even when they are perfectly made, do not however focus light exactly to a single point. These deviations from the idealized lens performance are called aberrations of the lens.
Aberrations fall into two classes: monochromatic and chromatic. Monochromatic aberrations are caused by the geometry of the lens or mirror and occur both when light is reflected and when it is refracted. They appear even when using monochromatic light, hence the name.
Chromatic aberrations are caused by dispersion, the variation of a lens's refractive index with wavelength. Because of dispersion, different wavelengths of light come to focus at different points. Chromatic aberration does not appear when monochromatic light is used.

Monochromatic aberrations

The most common monochromatic aberrations are:

Defocus
Spherical aberration
Coma
Astigmatism
Field curvature
Image distortion

Although defocus is technically the lowest-order of the optical aberrations, it is usually not considered as a lens aberration, since it can be corrected by moving the lens to bring the image plane to the optical focus of the lens.
In addition to these aberrations, piston and tilt are effects which shift the position of the focal point. Piston and tilt are not true optical aberrations, since when an otherwise perfect wavefront is altered by piston and tilt, it will still form a perfect, aberration-free image, only shifted to a different position.

Chromatic aberrations

Chromatic aberration occurs when different wavelengths are not focussed to the same point. Types of chromatic aberration are:

Axial chromatic aberration
Lateral chromatic aberration
Theory of monochromatic aberration

In a perfect optical system in the classical theory of optics, rays of light proceeding from any object point unite in an image point; and therefore the object space is reproduced in an image space. The introduction of simple auxiliary terms, due to Gauss, named the focal lengths and focal planes, permits the determination of the image of any object for any system. The Gaussian theory, however, is only true so long as the angles made by all rays with the optical axis are infinitely small, i.e., with infinitesimal objects, images and lenses; in practice these conditions may not be realized, and the images projected by uncorrected systems are, in general, ill-defined and often blurred if the aperture or field of view exceeds certain limits.
The investigations of James Clerk Maxwell and Ernst Abbe showed that the properties of these reproductions, i.e., the relative position and magnitude of the images, are not special properties of optical systems, but necessary consequences of the supposition of the reproduction of all points of a space in image points, and are independent of the manner in which the reproduction is effected. These authors showed, however, that no optical system can justify these suppositions, since they are contradictory to the fundamental laws of reflection and refraction. Consequently, the Gaussian theory only supplies a convenient method of approximating reality; realistic optical systems fall short of this unattainable ideal. Currently, all that can be accomplished is the projection of a single plane onto another plane; but even in this, aberrations always occurs and it may be unlikely that these will ever be entirely corrected.

Aberration of axial points (spherical aberration in the restricted sense)

Let be any optical system, rays proceeding from an axis point under an angle will unite in the axis point ; and those under an angle in the axis point. If there is refraction at a collective spherical surface, or through a thin positive lens, will lie in front of so long as the angle is greater than ; and conversely with a dispersive surface or lenses. The caustic, in the first case, resembles the sign '>' ; in the second '<'. If the angle is very small, is the Gaussian image; and is termed the longitudinal aberration, and the lateral aberration of the pencils with aperture. If the pencil with the angle is that of the maximum aberration of all the pencils transmitted, then in a plane perpendicular to the axis at there is a circular disk of confusion of radius, and in a parallel plane at another one of radius ; between these two is situated the disk of least confusion.
The largest opening of the pencils, which take part in the reproduction of, i.e., the angle, is generally determined by the margin of one of the lenses or by a hole in a thin plate placed between, before, or behind the lenses of the system. This hole is termed the stop or diaphragm; Abbe used the term aperture stop for both the hole and the limiting margin of the lens. The component of the system, situated between the aperture stop and the object, projects an image of the diaphragm, termed by Abbe the entrance pupil; the exit pupil is the image formed by the component, which is placed behind the aperture stop. All rays which issue from O and pass through the aperture stop also pass through the entrance and exit pupils, since these are images of the aperture stop. Since the maximum aperture of the pencils issuing from is the angle u subtended by the entrance pupil at this point, the magnitude of the aberration will be determined by the position and diameter of the entrance pupil. If the system be entirely behind the aperture stop, then this is itself the entrance pupil ; if entirely in front, it is the exit pupil.
If the object point be infinitely distant, all rays received by the first member of the system are parallel, and their intersections, after traversing the system, vary according to their perpendicular height of incidence, i.e. their distance from the axis. This distance replaces the angle in the preceding considerations; and the aperture, i.e., the radius of the entrance pupil, is its maximum value.

Aberration of elements, i.e. smallest objects at right angles to the axis

If rays issuing from are concurrent, it does not follow that points in a portion of a plane perpendicular at to the axis will be also concurrent, even if the part of the plane be very small. As the diameter of the lens increases, the neighboring point will be reproduced, but attended by aberrations comparable in magnitude to. These aberrations are avoided if, according to Abbe, the sine condition,, holds for all rays reproducing the point. If the object point is infinitely distant, and are to be replaced by and, the perpendicular heights of incidence; the sine condition then becomes. A system fulfilling this condition and free from spherical aberration is called aplanatic. This word was first used by Robert Blair to characterize a superior achromatism, and, subsequently, by many writers to denote freedom from spherical aberration as well.
Since the aberration increases with the distance of the ray from the center of the lens, the aberration increases as the lens diameter increases, and hence can be minimized by reducing the aperture, at the cost of also reducing the amount of light reaching the image plane.

Aberration of lateral object points (points beyond the axis) with narrow pencils — astigmatism

A point at a finite distance from the axis is, in general, even then not sharply reproduced if the pencil of rays issuing from it and traversing the system is made infinitely narrow by reducing the aperture stop; such a pencil consists of the rays which can pass from the object point through the now infinitely small entrance pupil. It is seen that the pencil does not meet the refracting or reflecting surface at right angles; therefore it is astigmatic. Naming the central ray passing through the entrance pupil the axis of the pencil or principal ray, it can be said: the rays of the pencil intersect, not in one point, but in two focal lines, which can be assumed to be at right angles to the principal ray; of these, one lies in the plane containing the principal ray and the axis of the system, i.e. in the first principal section or meridional section, and the other at right angles to it, i.e. in the second principal section or sagittal section. We receive, therefore, in no single intercepting plane behind the system, as, for example, a focusing screen, an image of the object point; on the other hand, in each of two planes lines and are separately formed, and in a plane between and a circle of least confusion. The interval, termed the astigmatic difference, increases, in general, with the angle made by the principal ray with the axis of the system, i.e. with the field of view. Two astigmatic image surfaces correspond to one object plane; and these are in contact at the axis point; on the one lie the focal lines of the first kind, on the other those of the second. Systems in which the two astigmatic surfaces coincide are termed anastigmatic or stigmatic.
Sir Isaac Newton was probably the discoverer of astigmation; the position of the astigmatic image lines was determined by Thomas Young; and the theory was developed by Allvar Gullstrand. A bibliography by P. Culmann is given in Moritz von Rohr's Die Bilderzeugung in optischen Instrumenten.