In
optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of optical instruments, instruments that use or Photodetector, detect it. Optics usually describes t ...
, spherical aberration (SA) is a type of
aberration found in optical systems that have elements with
spherical surfaces. This phenomenon commonly affects
lenses
A lens is a transmissive optical device that focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a compound lens consists of several simple lenses (''elements''), ...
and
curved mirrors, as these components are often shaped in a spherical manner for ease of manufacturing. Light rays that strike a spherical surface off-centre are
refracted
In physics, refraction is the redirection of a wave as it passes from one medium to another. The redirection can be caused by the wave's change in speed or by a change in the medium. Refraction of light is the most commonly observed phenome ...
or
reflected
Reflection is the change in direction of a wavefront at an interface between two different media so that the wavefront returns into the medium from which it originated. Common examples include the reflection of light, sound and water waves. The ...
more or less than those that strike close to the centre. This deviation reduces the quality of images produced by optical systems. The effect of spherical aberration was first identified in the 11th century by
Ibn al-Haytham
Ḥasan Ibn al-Haytham (Latinization of names, Latinized as Alhazen; ; full name ; ) was a medieval Mathematics in medieval Islam, mathematician, Astronomy in the medieval Islamic world, astronomer, and Physics in the medieval Islamic world, p ...
who discussed it in his work
Kitāb al-Manāẓir.
Overview
A spherical lens has an
aplanatic point (i.e., no spherical aberration) only at a lateral distance from the optical axis that equals the radius of the spherical surface divided by the index of refraction of the lens material.
Spherical aberration makes the
focus
Focus (: foci or focuses) may refer to:
Arts
* Focus or Focus Festival, former name of the Adelaide Fringe arts festival in East Australia Film
*Focus (2001 film), ''Focus'' (2001 film), a 2001 film based on the Arthur Miller novel
*Focus (2015 ...
of
telescope
A telescope is a device used to observe distant objects by their emission, Absorption (electromagnetic radiation), absorption, or Reflection (physics), reflection of electromagnetic radiation. Originally, it was an optical instrument using len ...
s and other instruments less than ideal. This is an important effect, because spherical shapes are much easier to produce than aspherical ones. In many cases, it is cheaper to use multiple spherical elements to compensate for spherical aberration than it is to use a single
aspheric lens
An aspheric lens or asphere (often labeled ''ASPH'' on eye pieces) is a lens whose surface profiles are not portions of a sphere or cylinder. In photography, a lens assembly that includes an aspheric element is often called an aspherical lens.
...
.
"Positive" spherical aberration means rays near the outer edge of a lens are bent more than would be predicted for an ideal lens. "Negative" spherical aberration means such rays are bent less than would be predicted.
The effect is proportional to the fourth power of the diameter and inversely proportional to the third power of the focal length, so it is much more pronounced at short
focal ratio
An f-number is a measure of the light-gathering ability of an optical system such as a camera lens. It is calculated by dividing the system's focal length by the diameter of the entrance pupil ("clear aperture").Smith, Warren ''Modern Optical ...
s, i.e., "fast" lenses.
Longitudinal sections through a focused beam with negative (top row), zero (middle row), and positive spherical aberration (bottom row). The lens is to the left.
Correction
In lens systems, aberrations can be minimized using combinations of
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
and
concave lenses, or by using
aspheric lens
An aspheric lens or asphere (often labeled ''ASPH'' on eye pieces) is a lens whose surface profiles are not portions of a sphere or cylinder. In photography, a lens assembly that includes an aspheric element is often called an aspherical lens.
...
es or aplanatic lenses.
Lens systems with aberration correction are usually designed by numerical
ray tracing. For simple designs, one can sometimes analytically calculate parameters that minimize spherical aberration. For example, in a design consisting of a single lens with spherical surfaces and a given object distance , image distance , and refractive index , one can minimize spherical aberration by adjusting the radii of curvature and of the front and back surfaces of the lens such that
where the signs in this formula follow the Cartesian
sign convention, in which a radius of curvature is positive if the center of curvature is to the right of the surface and negative if it is to the left. Similarly the object and image distances are positive if the object or image is to the right of the lens and negative if they are to the left.
300 px, A point source as imaged by a system with negative (top row), zero (middle row), and positive spherical aberration (bottom row). The middle column shows the focused image, columns to the left show defocusing toward the inside, and columns to the right show defocusing toward the outside.
For small telescopes using spherical mirrors with
focal ratio
An f-number is a measure of the light-gathering ability of an optical system such as a camera lens. It is calculated by dividing the system's focal length by the diameter of the entrance pupil ("clear aperture").Smith, Warren ''Modern Optical ...
s shorter than , light from a distant point source (such as a
star
A star is a luminous spheroid of plasma (physics), plasma held together by Self-gravitation, self-gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night sk ...
) is not all focused at the same point. Particularly, light striking the inner part of the mirror focuses farther from the mirror than light striking the outer part. As a result, the image cannot be focused as sharply as if the aberration were not present. Because of spherical aberration, telescopes with focal ratio less than are usually made with non-spherical mirrors or with correcting lenses.
Spherical aberration can be eliminated by making lenses with an aspheric surface.
Descartes showed that lenses whose surfaces are well-chosen
Cartesian ovals (revolved around the central symmetry axis) can perfectly image light from a point on the axis or from infinity in the direction of the axis. Such a design yields completely aberration-free focusing of light from a distant source.
In 2018, researchers found a
closed formula for a lens surface that eliminates spherical aberration.
Their equation can be applied to specify a shape for one surface of a lens, where the other surface has any given shape.
Estimation of the aberrated spot diameter
Many ways to estimate the diameter of the focused spot due to spherical aberration are based on ray optics. Ray optics, however, does not consider that light is an electromagnetic wave. Therefore, the results can be wrong due to interference effects arisen from the wave nature of light.
Coddington notation
A rather simple formalism based on ray optics, which holds for thin lenses only, is the Coddington notation.
In the following, is the lens's refractive index, is the object distance, is the image distance, is the distance from the optical axis at which the outermost ray enters the lens, is the first lens radius, is the second lens radius, and is the lens's focal length. The distance can be understood as half of the clear aperture.
By using the Coddington factors for shape, , and position, ,
one can write the longitudinal spherical aberration as
[
If the focal length is very much larger than the longitudinal spherical aberration LSA, then the transverse spherical aberration, TSA, which corresponds to the diameter of the focal spot, is given by
]
See also
*Achromatic lens
An achromatic lens or achromat is a lens (optics), lens that is designed to limit the effects of chromatic aberration, chromatic and spherical aberration. Achromatic lenses are corrected to bring two wavelengths (typically red and blue) into ...
*Hubble Space Telescope
The Hubble Space Telescope (HST or Hubble) is a space telescope that was launched into low Earth orbit in 1990 and remains in operation. It was not the Orbiting Solar Observatory, first space telescope, but it is one of the largest and most ...
* Maksutov telescope
*Parabolic reflector
A parabolic (or paraboloid or paraboloidal) reflector (or dish or mirror) is a Mirror, reflective surface used to collect or project energy such as light, sound, or radio waves. Its shape is part of a circular paraboloid, that is, the surface ge ...
* Ritchey–Chrétien telescope
* Schmidt corrector plate
* Soft focus
References
External links
*
Spherical aberration
{{Webarchive, url=https://web.archive.org/web/20120723065201/http://toothwalker.org/optics/spherical.html , date=2012-07-23 '' at ''vanwalree.com'', PA van Walree, viewed 28 January 2007.
* http://www.telescope-optics.net/spherical1.htm
* Non-English articles on the Acuña-Romo equation: Spanish, German, Italian, Russian
Geometrical optics
Image defects
de:Abbildungsfehler#Sphärische Aberration