A
lens is an
opticalOptics is the branch of physics which studies the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...
device with perfect or approximate
axial symmetryAxial symmetry is symmetry around an axis; an object is axially symmetric if its appearance is unchanged if rotated around some axis in a way.- See also :* Rotational symmetry has a more general discussion...
which
transmitsIn optics and spectroscopy, transmittance is the fraction of incident light at a specified wavelength that passes through a sample. A related term is absorptance, which is the fraction of light absorbed by a sample at a specified wavelength...
and
refractsRefraction is the change in direction of a wave due to a change in its velocity. This is most commonly observed when a wave passes from one medium to another...
lightLight is electromagnetic radiation, particularly radiation of a wavelength that is visible to the human eye ....
, converging or diverging the
beamA light beam or beam of light is a narrow projection of light energy radiating from a source into a beam. Sunlight is a natural example of a light beam when filtered through various mediums...
. A
simple lensIn optics, a simple lens or singlet lens is a lens consisting of a single simple element. Typical examples include a magnifying glass or a lens in a pair of simple reading glasses....
is a lens consisting of a single optical element. A
compound lens is an array of simple lenses (elements) with a common axis; the use of multiple elements allows more optical aberrations to be corrected than is possible with a single element. Manufactured lenses are typically made of
glassIn general Glass refers to a solid, brittle, transparent material, commonly used for windows, bottles, or eyewear. Examples of glassy materials include, but are not limited to, soda-lime glass, borosilicate glass, acrylic glass, sugar glass, Muscovy-glass, or aluminium oxynitride. The term glass...
or transparent
plasticPlastic is the general common term for a wide range of synthetic or semisynthetic organic amorphous solid materials suitable for the manufacture of industrial products...
. Elements which refract
electromagnetic radiationElectromagnetic radiation is a ubiquitous phenomenon that takes the form of self-propagating waves in a vacuum or in matter. It consists of electric and magnetic field components which oscillate in phase perpendicular to each other and perpendicular to the direction of energy propagation...
outside the visual spectrum are also called lenses: for instance, a
microwaveMicrowaves are electromagnetic waves with wavelengths ranging from as long as one meter to as short as one millimeter, or equivalently, with frequencies between 300MHz and 300 GHz. This is an extremely broad definition including both UHF and EHF , and various sources use different boundaries...
lens can be made from paraffin wax.
The variant spelling
lense is sometimes seen. While it is listed as an alternative spelling in some dictionaries, most mainstream dictionaries do not list it as acceptable.
History
The oldest lens artifact is the
Nimrud lensThe Nimrud lens is a 3000 year old piece of rock crystal, which was unearthed by Austen Henry Layard at the Assyrian palace of Nimrud. It may have been used as a magnifying glass, or as a burning-glass to start fires by concentrating sunlight...
, which is over three thousand years old, dating back to ancient
AssyriaAssyria was a civilization centered on the Upper Tigris river, in Mesopotamia , that came to rule regional empires a number of times in history. It was named for its original capital, the ancient city of Assur...
.
David BrewsterSir David Brewster FRS was a Scottish physicist, mathematician, astronomer, inventor, and writer.- Life and work :...
proposed that it may have been used as a
magnifying glassA magnifying glass is a convex lens which is used to produce a magnified image of an object. The lens is usually mounted in a frame with a handle ....
, or as a
burning-glassA burning glass or burning lens is a large convex lens that can concentrate the sun's rays onto a small area, heating up the area and thus resulting in ignition of the exposed surface. Burning mirrors achieve a similar effect by using reflecting surfaces to focus the light...
to start fires by concentrating sunlight. Assyrian craftsmen made intricate engravings, and could have used such a lens in their work. Another early reference to
magnificationMagnification is the process of enlarging something only in appearance, not in physical size. This enlargement is quantified by a calculated number also called magnification...
dates back to
ancient EgyptAncient Egypt was an ancient civilization of eastern North Africa, concentrated along the lower reaches of the Nile River in what is now the modern country of Egypt. The civilization coalesced around 3150 BC with the political unification of Upper and Lower Egypt under the first pharaoh, and...
ian
hieroglyphsEgyptian hieroglyphs was a formal writing system used by the ancient Egyptians that contained a combination of logographic and alphabetic elements. Egyptians used cursive hieroglyphs for religious literature on papyrus and wood...
in the 8th century BC, which depict "simple glass meniscal lenses".
The earliest written records of lenses date to Ancient Greece, with
AristophanesAristophanes , son of Philippus, of the deme Cydathenaus, was a prolific and much acclaimed comic playwright of ancient Athens. Eleven of his forty plays have come down to us virtually complete...
' play
The CloudsThe Clouds is a comedy written by the celebrated playwright Aristophanes lampooning intellectual fashions in classical Athens. It was originally produced at the City Dionysia in 423 BC and it was not well received, coming last of the three plays competing at the festival that year...
(424 BC) mentioning a burning-glass (a biconvex lens used to
focusIn geometrical optics, a focus, also called an image point, is the point where light rays originating from a point on the object converge. Although the focus is conceptually a point, physically the focus has a spatial extent, called the blur circle. This non-ideal focusing may be caused by...
the
sunThe Sun is the star at the center of the Solar System. The Earth and other matter orbit the Sun, which by itself accounts for about 99.86% of the Solar System's mass....
's rays to produce fire). The writings of
Pliny the ElderGaius Plinius Secundus , better known as Pliny the Elder, was an author, naturalist, and natural philosopher as well as naval and army commander of the early Roman Empire and personal friend of the emperor Vespasian...
(23–79) also show that burning-glasses were known to the
Roman EmpireThe Roman Empire was the post-Republican phase of the ancient Roman civilization, characterised by an autocratic form of government and large territorial holdings in Europe and around the Mediterranean. The term is used to describe the Roman state during and after the time of the first emperor,...
, and mentions what is arguably the earliest use of a
corrective lensA corrective lens is a lens worn in front of the eye, mainly used to treat myopia, hyperopia, astigmatism, and presbyopia. Glasses or "spectacles" are worn on the face a short distance in front of the eye. Contact lenses are worn directly on the surface of the eye...
:
NeroNero Claudius Caesar Augustus Germanicus , born Lucius Domitius Ahenobarbus, also called Nero Claudius Caesar Drusus Germanicus, was the fifth and last Roman emperor of the Julio-Claudian dynasty. Nero was adopted by his great uncle Claudius to become heir to the throne...
was said to watch the
gladiatorial gamesA Gladiator was an armed combatant who entertained audiences in the Roman Republic and Roman Empire in violent confrontations with other gladiators, wild animals, and condemned criminals. Some gladiators were volunteers who risked their legal and social standing and their lives by appearing in the...
using an
emeraldEmeralds are a variety of the mineral beryl colored green by trace amounts of chromium and sometimes vanadium. Beryl has a hardness of 7.5 - 8 on the 10 point Mohs scale of mineral hardness. Most emeralds are highly included, so their toughness is classified as generally poor...
(presumably concave to correct for
myopiaMyopia , also called nearsightedness or shortsightedness, is a refractive defect of the eye in which collimated light produces image focus in front of the retina when accommodation is relaxed....
, though the reference is vague). Both Pliny and
Seneca the YoungerLucius Annaeus Seneca was a Roman Stoic philosopher, statesman, dramatist, and in one work humorist, of the Silver Age of Latin literature. He was tutor and later advisor to emperor Nero...
(3 BC–65) described the magnifying effect of a glass globe filled with
waterWater is an ubiquitous chemical substance that is composed of hydrogen and oxygen and is essential for all known forms of life.In typical usage, water refers only to its liquid form or state, but the substance also has a solid state, ice, and a gaseous state, water vapor or steam. Water covers 71%...
.
The word
lens comes from the Latin name of the
lentilThe lentil or daal or dal , considered a type of pulse, is a bushy annual plant of the legume family, grown for its lens-shaped seeds...
, because a double-convex lens is lentil-shaped. The genus of the lentil plant is
LensThe genus Lens of the legume family Fabaceae contains four species of small, erect or climbing herbs with pinnate leaves and small inconspicuous white flowers and small flattened pods...
, and the most commonly eaten species is
Lens culinaris. The lentil plant also gives its name to a
geometric figureIn geometry, a lens is a convex shape comprising two circular arcs, joined at their endpoints. If the arcs have equal radii, it is called a symmetric lens.The corresponding concave shape is the lune...
.
The
Iranian physicistPhysics in medieval Islam included experimental physics, mathematical physics and theoretical physics.The fields of physics that were studied by Muslim scientists during this time also included optics and magnetism , mechanics , and astrophysics .These studies...
and
mathematicianIn the history of mathematics, mathematics in medieval Islam, sometimes termed Islamic mathematics, is the mathematics developed in the Islamic world between 622 and 1600, during what is known as the Islamic Golden Age...
Ibn SahlThis article is about the physicist. For the physician, see Ali ibn Sahl Rabban al-Tabari. For the poet, see Ibn Sahl of Sevilla.Ibn Sahl was a Muslim mathematician, physicist and optics engineer of the Islamic Golden Age associated with the Abbasid court of Baghdad...
(c.940–c.1000) used what is now known as
Snell's lawIn optics and physics, Snell's law , is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through a boundary between two different isotropic media, such as water and glass...
to calculate the shape of lenses. Ibn al-Haytham (965–1038), known in the
WestThe Western world, also known as the West and the Occident , is a term that can have multiple meanings depending on its context...
as
Alhazen, wrote the first major optical treatise, the
Book of OpticsThe Book of Optics was a seven-volume treatise on optics, physics, mathematics, anatomy and psychology written by the Iraqi Muslim scientist, Ibn al-Haytham , from 1011 to 1021, when he was under house arrest in Cairo, Egypt.The book...
, which contained the earliest historical proof of a
magnifying deviceA magnifying glass is a convex lens which is used to produce a magnified image of an object. The lens is usually mounted in a frame with a handle ....
, a convex lens forming a magnified image. The book was translated into Latin in the 12th century, and became the standard textbook in the field and influenced many other writers.
Excavations at the
VikingA Viking is one of the Norse explorers, warriors, merchants, and pirates who raided and colonized wide areas of Europe from the late eighth to the early eleventh century. These Norsemen used their famed longships to travel as far east as Constantinople and the Volga River in Russia, and as far...
harbour town of Fröjel,
Gotland' is a county, province, municipality and diocese of Sweden and the largest island in the Baltic Sea. At 3,140 square kilometers in area, it makes up less than one percent of Sweden's total land area. The region also includes the small islands of Fårö and Gotska Sandön to the north, and the tiny...
,
SwedenSweden , officially the Kingdom of Sweden , is a Nordic country on the Scandinavian Peninsula in Northern Europe...
discovered in 1999 the rock crystal
Visby lensesThe Visby lenses are a collection of lens-shaped manufactured objects made of rock crystal found in several Viking hoards on the island of Gotland, Sweden, dating from the 11th or 12th century...
, produced by turning on pole-lathes at Fröjel in the 11th to 12th century, with an imaging quality comparable to that of 1950s
aspheric lensAn aspheric lens or asphere is a lens whose surfaces have a profile that is neither a portion of a sphere nor of a circular cylinder. In photography, a lens assembly that includes an aspheric element is often called an aspherical lens....
es. The Viking lenses concentrate sunlight enough to ignite fires.
Widespread use of lenses did not occur until the use of
reading stoneA reading stone was an approximately hemispherical transparent object placed on top of text to magnify the letters so that people with presbyopia could read the text more easily. Reading stones were among the earliest common uses of lenses....
s in the 11th century and the invention of spectacles, probably in
ItalyItaly , officially the Italian Republic , is a country located on the Italian Peninsula in Southern Europe and on the two largest islands in the Mediterranean Sea, Sicily and Sardinia. Italy shares its northern, Alpine boundary with France, Switzerland, Austria and Slovenia...
in the 1280s. Scholars have noted that spectacles were invented not long after the translation of al-Haytham's book into Latin, but it is not clear what role, if any, the optical theory of the time played in the discovery.
Nicholas of CusaNicholas of Kues , also referred to as Nicolaus Cusanus and Nicholas of Cusa, was a cardinal of the Catholic Church from Germany , a philosopher, jurist, mathematician, and an astronomer. He is widely considered one of the great geniuses and polymaths of the 15th century...
is believed to have been the first to discover the benefits of concave lenses for the treatment of
myopiaMyopia , also called nearsightedness or shortsightedness, is a refractive defect of the eye in which collimated light produces image focus in front of the retina when accommodation is relaxed....
in 1451.
The
Abbe sine conditionThe Abbe sine condition is a condition that must be fulfilled by a lens or other optical system in order for it to produce sharp images of off-axis as well as on-axis objects...
, due to Ernst Abbe (1860s), is a condition that must be fulfilled by a lens or other optical system in order for it to produce sharp images of off-axis as well as on-axis objects. It revolutionized the design of optical instruments such as microscopes, and helped to establish the
Carl ZeissCarl Zeiss was an optician commonly known for the company he founded, Carl Zeiss Jena . Zeiss made contributions to lens manufacturing that have aided the modern production of lenses...
company as a leading supplier of optical instruments.
Construction of simple lenses
Most lenses are
spherical lenses: their two surfaces are parts of the surfaces of spheres, with the lens axis ideally perpendicular to both surfaces. Each surface can be
convex (bulging outwards from the lens),
concave (depressed into the lens), or
planar (flat). The line joining the centres of the spheres making up the lens surfaces is called the
axis of the lens. Typically the lens axis passes through the physical centre of the lens, because of the way they are manufactured. Lenses may be cut or ground after manufacturing to give them a different shape or size. The lens axis may then not pass through the physical centre of the lens.
ToricIn optics, a toric lens is a type of lens whose surface is a combination of a sphere and a cylinder. One surface of the lens is spherical, the other toroidal. Such lenses are commonly used for vision correction, in cases where there is astigmatism....
or sphero-cylindrical lenses have surfaces with two different radii of curvature in two orthogonal planes. They have a different focal power in different meridians. This is a form of deliberate
astigmatismAn optical system with astigmatism is one where rays that propagate in two perpendicular planes have different foci. If an optical system with astigmatism is used to form an image of a cross, the vertical and horizontal lines will be in sharp focus at two different distances...
.
More complex are
aspheric lensAn aspheric lens or asphere is a lens whose surfaces have a profile that is neither a portion of a sphere nor of a circular cylinder. In photography, a lens assembly that includes an aspheric element is often called an aspherical lens....
es. These are lenses where one or both surfaces have a shape that is neither spherical nor cylindrical. Such lenses can produce images with much less aberration than standard simple lenses.
Types of simple lenses
Lenses are classified by the curvature of the two optical surfaces. A lens is
biconvex (or
double convex, or just
convex) if both surfaces are convex. If both surfaces have the same radius of curvature, the lens is
equiconvex. A lens with two concave surfaces is
biconcave (or just
concave). If one of the surfaces is flat, the lens is
plano-convex or
plano-concave depending on the curvature of the other surface. A lens with one convex and one concave side is
convex-concave or
meniscus. It is this type of lens that is most commonly used in corrective lenses.
If the lens is biconvex or plano-convex, a collimated or parallel beam of light travelling parallel to the lens axis and passing through the lens will be converged (or
focused) to a spot on the axis, at a certain distance behind the lens (known as the
focal lengthThe focal length of an optical system is a measure of how strongly it converges or diverges light. For an optical system in air, it is the distance over which initially collimated rays are brought to a focus...
). In this case, the lens is called a
positive or
converging lens.
If the lens is biconcave or plano-concave, a collimated beam of light passing through the lens is diverged (spread); the lens is thus called a
negative or
diverging lens. The beam after passing through the lens appears to be emanating from a particular point on the axis in front of the lens; the distance from this point to the lens is also known as the focal length, although it is negative with respect to the focal length of a converging lens.
Convex-concave (meniscus) lenses can be either positive or negative, depending on the relative curvatures of the two surfaces. A
negative meniscus lens has a steeper concave surface and will be thinner at the centre than at the periphery. Conversely, a
positive meniscus lens has a steeper convex surface and will be thicker at the centre than at the periphery. An ideal
thin lensIn optics, a thin lens is a lens with a thickness that is negligible compared to the focal length of the lens...
with two surfaces of equal curvature would have zero
optical powerOptical power is the degree to which a lens, mirror, or other optical system converges or diverges light. It is equal to the reciprocal of the focal length of the device. The dioptre is the most common unit of measurement of optical power...
, meaning that it would neither converge nor diverge light. All real lenses have a nonzero thickness, however, which affects the optical power. To obtain exactly zero optical power, a meniscus lens must have slightly unequal curvatures to account for the effect of the lens' thickness.
Lensmaker's equation
The focal length of a lens
in air can be calculated from the
lensmaker's equation:
where is the focal length of the lens, is the
refractive indexThe refractive index of a medium is a measure of how much the speed of light is reduced inside the medium. For example, typical soda-lime glass has a refractive index close to 1.5, which means that in glass, light travels at 1 / 1.5 = 2/3 the speed of light in a vacuum...
of the lens material, is the radius of curvature of the lens surface closest to the light source, is the radius of curvature of the lens surface farthest from the light source, and is the thickness of the lens (the distance along the lens axis between the two surface vertices).
Sign convention of lens radii R1 and R2
The signs of the lens' radii of curvature indicate whether the corresponding surfaces are convex or concave. The
sign conventionIn physics, a sign convention is a choice of the signs of a set of quantities, in a case where the choice of sign is arbitrary. "Arbitrary" here means that the same physical system can be correctly described using different choices for the signs, as long as one set of definitions is used...
used to represent this varies, but in this article if
R1 is positive the first surface is convex, and if
R1 is negative the surface is concave. The signs are reversed for the back surface of the lens: if
R2 is positive the surface is concave, and if
R2 is negative the surface is convex. If either radius is
infiniteInfinity refers to several distinct concepts – usually linked to the idea of "without end" – which arise in philosophy, mathematics, and theology...
, the corresponding surface is flat. With this convention the signs are determined by the shapes of the lens surfaces, and are independent of the direction in which light travels through the lens.
Thin lens equation
If
d is small compared to
R1 and
R2, then the
thin lensIn optics, a thin lens is a lens with a thickness that is negligible compared to the focal length of the lens...
approximation can be made. For a lens in air,
f is then given by
The focal length
f is positive for converging lenses, and negative for diverging lenses. The
reciprocalIn mathematics, a multiplicative inverse or reciprocal for a number x, denoted by
1⁄
x or x
−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of x is also called the reciprocal of x...
of the focal length, 1/
f, is the
optical powerOptical power is the degree to which a lens, mirror, or other optical system converges or diverges light. It is equal to the reciprocal of the focal length of the device. The dioptre is the most common unit of measurement of optical power...
of the lens. If the focal length is in metres, this gives the optical power in
dioptreA dioptre, or diopter, is a unit of measurement of the optical power of a lens or curved mirror, which is equal to the reciprocal of the focal length measured in metres . For example, a 3 dioptre lens brings parallel rays of light to focus at 1/3 metre...
s (inverse metres).
Lenses have the same focal length when light travels from the back to the front as when light goes from the front to the back, although other properties of the lens, such as the
aberrationsAberrations are departures of the performance of an optical system from the predictions of paraxial optics. Aberration leads to blurring of the image produced by an image-forming optical system. It occurs when light from one point of an object after transmission through the system does not converge...
are not necessarily the same in both directions.
Imaging properties
As mentioned above, a positive or converging lens in air will focus a collimated beam travelling along the lens axis to a spot (known as the
focal pointIn geometrical optics, a focus, also called an image point, is the point where light rays originating from a point on the object converge. Although the focus is conceptually a point, physically the focus has a spatial extent, called the blur circle. This non-ideal focusing may be caused by...
) at a distance
f from the lens. Conversely, a
point sourceA point source is a localised relatively-small source of something.Point source may also refer to:*Point source , a localised source of pollution*Point mass, a point source of gravitational field...
of light placed at the focal point will be converted into a collimated beam by the lens. These two cases are examples of
imageAn image is an artifact, for example a two-dimensional picture, that has a similar appearance to some subject—usually a physical object or a person....
formation in lenses. In the former case, an object at an infinite distance (as represented by a collimated beam of waves) is focused to an image at the focal point of the lens. In the latter, an object at the focal length distance from the lens is imaged at infinity. The plane perpendicular to the lens axis situated at a distance
f from the lens is called the
focal plane. (Note: In the figure below the image is actually larger than the object; this is a function of f and S1, described below)
If the distances from the object to the lens and from the lens to the image are
S1 and
S2 respectively, for a lens of negligible thickness, in air, the distances are related by the
thin lens formula:
.
This can also be put into the "Newtonian" form:
where and .
What this means is that, if an object is placed at a distance
S1 along the axis in front of a positive lens of focal length
f, a screen placed at a distance
S2 behind the lens will have a sharp image of the object projected onto it, as long as
S1 >
f (if the lens-to-screen distance
S2 is varied slightly, the image will become less sharp). This is the principle behind
photographyPhotography is the process, activity and art of creating still or moving pictures by recording radiation on a sensitive medium, such as a photographic film, or an electronic sensor...
and the
human eyeThe human eye is an organ which reacts to light for several purposes.As a conscious sense organ, the eye allows vision. Rod and cone cells in the retina allow conscious light perception and vision including color differentiation and the perception of depth...
. The image in this case is known as a
real imageIn optics, a real image is a representation of an object in which the perceived location is actually a point of convergence of the rays of light that make up the image. If a screen is placed in the plane of a real image the image will generally become visible on the screen...
.
Note that if
S1 <
f,
S2 becomes negative, the image is apparently positioned on the same side of the lens as the object. Although this kind of image, known as a
virtual imageIn optics, a virtual image is an image in which the outgoing rays from a point on the object never actually intersect at a point. A simple example is a flat mirror where the image of oneself is perceived at twice the distance from oneself to the mirror...
, cannot be projected on a screen, an observer looking through the lens will see the image in its apparent calculated position. A
magnifying glassA magnifying glass is a convex lens which is used to produce a magnified image of an object. The lens is usually mounted in a frame with a handle ....
creates this kind of image.
The
magnificationMagnification is the process of enlarging something only in appearance, not in physical size. This enlargement is quantified by a calculated number also called magnification...
of the lens is given by:
,
where
M is the magnification factor; if |
M|>1, the image is larger than the object.
Notice the sign convention here shows that, if
M is negative, as it is for real images, the image is upside-down with respect to the object. For virtual images,
M is positive and the image is upright.
In the special case that
S1 = ∞, then
S2 =
f and
M = −
f / ∞ = 0. This corresponds to a collimated beam being focused to a single spot at the focal point. The size of the image in this case is not actually zero, since
diffractionDiffraction is normally taken to refer to various phenomena which occur when a wave encounters an obstacle. It is described as the apparent bending of waves around small obstacles and the spreading out of waves past small openings...
effects place a lower limit on the size of the image (see Rayleigh criterion).
The formulas above may also be used for negative (diverging) lens by using a negative focal length (
f), but for these lenses only virtual images can be formed.
For the case of lenses that are not thin, or for more complicated multi-lens optical systems, the same formulas can be used, but
S1 and
S2 are interpreted differently. If the system is in air or
vacuumIn everyday usage, vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty," but in reality, no volume of space can ever be perfectly empty...
,
S1 and
S2 are measured from the front and rear principal planes of the system, respectively. Imaging in media with an index of refraction greater than 1 is more complicated, and is beyond the scope of this article.
Aberrations
Lenses do not form perfect images, and there is always some degree of distortion or
aberration introduced by the lens which causes the image to be an imperfect replica of the object. Careful design of the lens system for a particular application ensures that the aberration is minimized. There are several different types of aberration which can affect image quality.
Spherical aberration
Spherical aberrationthumb|right|Spherical aberration. A perfect lens focuses all incoming rays to a point on the optic axis. A real lens with spherical surfaces suffers from spherical aberration: it focuses rays more tightly if they enter it far from the optic axis than if they enter closer to the axis. It therefore...
occurs because spherical surfaces are not the ideal shape with which to make a lens, but they are by far the simplest shape to which glass can be ground and polished and so are often used. Spherical aberration causes beams parallel to, but distant from, the lens axis to be focused in a slightly different place than beams close to the axis. This manifests itself as a blurring of the image. Lenses in which closer-to-ideal, non-spherical surfaces are used are called
aspheric lenses. These were formerly complex to make and often extremely expensive, but advances in technology have greatly reduced the manufacturing cost for such lenses. Spherical aberration can be minimised by careful choice of the curvature of the surfaces for a particular application: for instance, a plano-convex lens which is used to focus a collimated beam produces a sharper focal spot when used with the convex side towards the beam source.
Coma
Another type of aberration is
comaIn optics , the coma in an optical system refers to aberration inherent to certain optical designs or due to imperfection in the lens or other components which results in off-axis point sources such as stars appearing distorted. Specifically, coma is defined as a variation in magnification over...
, which derives its name from the
cometA comet is a Small Solar System Body that has coma and is bigger than a meteoroid. When close enough to the Sun, a comet exhibits a visible coma , and sometimes a tail, both because of the effects of solar radiation upon the comet's nucleus...
-like appearance of the aberrated image. Coma occurs when an object off the optical axis of the lens is imaged, where rays pass through the lens at an angle to the axis θ. Rays which pass through the centre of the lens of focal length
f are focused at a point with distance
f tan θ from the axis. Rays passing through the outer margins of the lens are focused at different points, either further from the axis (positive coma) or closer to the axis (negative coma). In general, a bundle of parallel rays passing through the lens at a fixed distance from the centre of the lens are focused to a ring-shaped image in the focal plane, known as a
comatic circle. The sum of all these circles results in a V-shaped or comet-like flare. As with spherical aberration, coma can be minimised (and in some cases eliminated) by choosing the curvature of the two lens surfaces to match the application. Lenses in which both spherical aberration and coma are minimised are called
bestform lenses.
Chromatic aberration
Chromatic aberrationIn optics, chromatic aberration is the failure of a lens to focus all colors to the same point. It occurs because lenses have a different refractive index for different wavelengths of light...
is caused by the
dispersionIn optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency, or alternatively when the group velocity depends on the frequency.Media having such a property are termed dispersive media...
of the lens material—the variation of its
refractive indexThe refractive index of a medium is a measure of how much the speed of light is reduced inside the medium. For example, typical soda-lime glass has a refractive index close to 1.5, which means that in glass, light travels at 1 / 1.5 = 2/3 the speed of light in a vacuum...
n with the wavelength of light. Since, from the formulae above,
f is dependent upon
n, it follows that different wavelengths of light will be focused to different positions. Chromatic aberration of a lens is seen as fringes of colour around the image. It can be minimised by using an
achromatic doubletAn achromatic lens or achromat is a lens that is designed to limit the effects of chromatic and spherical aberration. Achromatic lenses are corrected to bring two wavelengths into focus in the same plane....
(or
achromat) in which two materials with differing dispersion are bonded together to form a single lens. This reduces the amount of chromatic aberration over a certain range of wavelengths, though it does not produce perfect correction. The use of achromats was an important step in the development of the optical microscope. An
apochromat is a lens or lens system which has even better correction of chromatic aberration, combined with improved correction of spherical aberration. Apochromats are much more expensive than achromats.
Different lens materials may also be used to minimize chromatic aberration, such as specialized coatings or lenses made from the crystal
fluoriteFluorite is a halide mineral composed of calcium fluoride, CaF
2. It is an isometric mineral with a cubic habit, though octahedral and more complex isometric forms are not uncommon. Cubic crystals up to 20 cm across have been found at Dalnegorsk, Russia...
. This naturally occurring substance has the highest known
Abbe numberIn physics and optics, the Abbe number, also known as the V-number or constringence of a transparent material, is a measure of the material's dispersion in relation to the refractive index...
, indicating that the material has low dispersion.
Other types of aberration
Other kinds of aberration include
field curvature,
barrelIn geometric optics and cathode ray tube displays, distortion is a deviation from rectilinear projection, a projection in which straight lines in a scene remain straight in an image...
and
pincushion distortion, and
astigmatismAn optical system with astigmatism is one where rays that propagate in two perpendicular planes have different foci. If an optical system with astigmatism is used to form an image of a cross, the vertical and horizontal lines will be in sharp focus at two different distances...
.
Aperture diffraction
Even if a lens is designed to minimize or eliminate the aberrations described above, the image quality is still limited by the
diffractionDiffraction is normally taken to refer to various phenomena which occur when a wave encounters an obstacle. It is described as the apparent bending of waves around small obstacles and the spreading out of waves past small openings...
of light passing through the lens' finite
apertureIn optics, an aperture is a hole or an opening through which light travels. More specifically, the aperture of an optical system is the opening that determines the cone angle of a bundle of rays that come to a focus in the image plane. The aperture determines how collimated the admitted rays are,...
. A
diffraction-limitedThe resolution of an optical imaging system — a microscope, telescope, or camera — can be limited by factors such as imperfections in the lenses or misalignment. However, there is a fundamental maximum to the resolution of any optical system which is due to diffraction...
lens is one in which aberrations have been reduced to the point where the image quality is primarily limited by diffraction under the design conditions.
Compound lenses
Simple lenses are subject to the optical aberrations discussed above. In many cases these aberrations can be compensated for to a great extent by using a combination of simple lenses with complementary aberrations. A
compound lens is a collection of simple lenses of different shapes and made of materials of different refractive indices, arranged one after the other with a common axis.
The simplest case is where lenses are placed in contact: if the lenses of focal lengths
f1 and
f2 are "
thinIn optics, a thin lens is a lens with a thickness that is negligible compared to the focal length of the lens...
", the combined focal length
f of the lenses is given by
.
Since 1/
f is the power of a lens, it can be seen that the powers of thin lenses in contact are additive.
If two thin lenses are separated in air by some distance
d, the focal length for the combined system is given by
.
The distance from the second lens to the focal point of the combined lenses is called the
back focal length (BFL).
.
As
d tends to zero, the value of the BFL tends to the value of
f given for thin lenses in contact.
If the separation distance is equal to the sum of the focal lengths (
d =
f1+
f2), the combined focal length and BFL are infinite. This corresponds to a pair of lenses that transform a parallel (collimated) beam into another collimated beam. This type of system is called
afocal, since it produces no net convergence or divergence of the beam. Two lenses at this separation form the simplest type of
optical telescopeAn optical telescope is a telescope which is used to gather and focus light mainly from the visible part of the electromagnetic spectrum for directly viewing a magnified image for making a photograph, or collecting data through electronic image sensors....
. Although the system does not alter the divergence of a collimated beam, it does alter the width of the beam. The magnification of such a telescope is given by
,
which is the ratio of the input beam width to the output beam width. Note the sign convention: a telescope with two convex lenses (
f1 > 0,
f2 > 0) produces a negative magnification, indicating an inverted image. A convex plus a concave lens (
f1 > 0 >
f2) produces a positive magnification and the image is upright.
Uses of lenses
A single convex lens mounted in a frame with a handle or stand is a
magnifying glassA magnifying glass is a convex lens which is used to produce a magnified image of an object. The lens is usually mounted in a frame with a handle ....
.
Lenses are used as prosthetics for the correction of
visual impairmentVisual impairment or vision impairment is vision loss to such a degree as to qualify as a handicap through a significant limitation of visual capability resulting from either disease, trauma, or congenital or degenerative conditions that cannot be corrected by conventional means, such as...
s such as
myopiaMyopia , also called nearsightedness or shortsightedness, is a refractive defect of the eye in which collimated light produces image focus in front of the retina when accommodation is relaxed....
,
hyperopiaHyperopia, also known as farsightedness, longsightedness or hypermetropia, is a defect of vision caused by an imperfection in the eye , causing difficulty focusing on near objects, and in extreme cases causing a sufferer to be unable to focus on objects at any distance...
,
presbyopiaPresbyopia describes the condition where the eye exhibits a progressively diminished ability to focus on near objects with age...
, and
astigmatismAn optical system with astigmatism is one where rays that propagate in two perpendicular planes have different foci. If an optical system with astigmatism is used to form an image of a cross, the vertical and horizontal lines will be in sharp focus at two different distances...
. (See
corrective lensA corrective lens is a lens worn in front of the eye, mainly used to treat myopia, hyperopia, astigmatism, and presbyopia. Glasses or "spectacles" are worn on the face a short distance in front of the eye. Contact lenses are worn directly on the surface of the eye...
,
contact lensA contact lens is a corrective, cosmetic, or therapeutic lens usually placed on the cornea of the eye...
, eyeglasses.) Most lenses used for other purposes have strict
axial symmetryAxial symmetry is symmetry around an axis; an object is axially symmetric if its appearance is unchanged if rotated around some axis in a way.- See also :* Rotational symmetry has a more general discussion...
; eyeglass lenses are only approximately symmetric. They are usually shaped to fit in a roughly oval, not circular, frame; the optical centers are placed over the eyeballs; their curvature may not be axially symmetric to correct for
astigmatismAn optical system with astigmatism is one where rays that propagate in two perpendicular planes have different foci. If an optical system with astigmatism is used to form an image of a cross, the vertical and horizontal lines will be in sharp focus at two different distances...
.
SunglassesSunglasses or sun glasses are a form of protective eyewear that usually enclose or protect the eye pupil in order to prevent strong light, ultraviolet rays, and increasingly, blue light from penetrating...
lenses may be designed to attenuate light without refraction.
Other uses are in imaging systems such as
monocularA monocular is a modified refracting telescope used to magnify the images of distant objects by passing light through a series of lenses and prisms; the use of prisms results in a lightweight telescope...
s,
binocularsBinoculars, field glasses or binocular telescopes are a pair of identical or mirror-symmetrical telescopes mounted side-by-side and aligned to point accurately in the same direction, allowing the viewer to use both eyes with binocular vision when viewing distant objects. Most are sized to be held...
,
telescopeAn optical telescope is a telescope which is used to gather and focus light mainly from the visible part of the electromagnetic spectrum for directly viewing a magnified image for making a photograph, or collecting data through electronic image sensors....
s,
microscopeA microscope is an instrument to see objects too tiny for the naked eye. The science of investigating small objects using such an instrument is called microscopy. Microscopic means invisible to the eye unless aided by a microscope.-History:An early microscope was made in 1590 in Middelburg, The...
s,
camerathumb |right|Cameras from Large to Small, Film to Digital A camera is a device that records images, either as a still photograph or as moving images known as videos or movies...
s and
projectorA movie projector is an opto-mechanical device for displaying moving pictures by projecting them on a projection screen. Most of the optical and mechanical elements, except for the illumination and sound devices, are present in movie cameras.-Physiology:...
s. Some of these instruments produce a
virtual imageIn optics, a virtual image is an image in which the outgoing rays from a point on the object never actually intersect at a point. A simple example is a flat mirror where the image of oneself is perceived at twice the distance from oneself to the mirror...
when applied to the human eye; others produce a
real imageIn optics, a real image is a representation of an object in which the perceived location is actually a point of convergence of the rays of light that make up the image. If a screen is placed in the plane of a real image the image will generally become visible on the screen...
which can be captured on
photographic filmPhotographic film is a sheet of plastic coated with an emulsion containing light-sensitive silver halide salts with variable crystal sizes that determine the sensitivity, contrast and resolution of the film...
or an optical sensor, or can be viewed on a screen.
Convex lenses produce an image of an object at infinity at their focus; if the
sunThe Sun is the star at the center of the Solar System. The Earth and other matter orbit the Sun, which by itself accounts for about 99.86% of the Solar System's mass....
is imaged, much of the visible and infrared light incident on the lens is concentrated into the small image. A large lens will create enough intensity to burn a flammable object at the focal point. Since ignition can be achieved even with a poorly made lens, lenses have been used as
burning-glassA burning glass or burning lens is a large convex lens that can concentrate the sun's rays onto a small area, heating up the area and thus resulting in ignition of the exposed surface. Burning mirrors achieve a similar effect by using reflecting surfaces to focus the light...
es for at least 2400 years. A modern application is the use of relatively large lenses to concentrate solar energy on relatively small
photovoltaic cellA photovoltaic cell is a type of Photoelectric cell that uses the photovoltaic effect to generate electrical energy using the potential difference that arises between materials when the surface of the cell is exposed to electromagnetic radiation....
s, harvesting more energy without the need to use larger, more expensive, cells.
Radio astronomyRadio astronomy is a subfield of astronomy that studies celestial objects at radio frequencies. The initial detection of radio waves from an astronomical object was made in the 1930s, but subsequent advances have identified a number of different sources of radio emission...
and
radarRadar is an object detection system that uses electromagnetic waves to identify the range, altitude, direction, or speed of both moving and fixed objects such as aircraft, ships, motor vehicles, weather formations, and terrain. The term RADAR was coined in 1941 as an acronym for RAdio Detection And...
systems often use dielectric lenses, commonly called a lens antenna to refract
electromagnetic radiationElectromagnetic radiation is a ubiquitous phenomenon that takes the form of self-propagating waves in a vacuum or in matter. It consists of electric and magnetic field components which oscillate in phase perpendicular to each other and perpendicular to the direction of energy propagation...
into a collector antenna. The
Square Kilometre ArrayThe Square Kilometre Array is a radio telescope in development which will have a total collecting area of approximately one square kilometre. It will operate over a wide range of frequencies and its size will make it 50 times more sensitive than any other radio instrument...
radio telescopeA radio telescope is a form of directional radio antenna used in radio astronomy. The same types of antennas are also used in tracking and collecting data from satellites and space probes...
, scheduled to be operational by 2020
http://www.skatelescope.org/pages/page_genpub.htm, will employ such lenses to get a collection area nearly 30 times greater than any previous antenna.
See also
- Aberration in optical systems
Aberrations are departures of the performance of an optical system from the predictions of paraxial optics. Aberration leads to blurring of the image produced by an image-forming optical system. It occurs when light from one point of an object after transmission through the system does not converge...
- Anti-fog
Anti-fog agents, also known as anti-fogging agents and treatments, prevent the condensation of water on a surface in the form of small droplets which resemble fog. Anti-fog treatments are often used for transparent glass or plastic surfaces in optics, such as the lenses and mirrors found in...
ging treatment of optical surfaces
- Axicon
An axicon is a specialized type of lens which has a conical surface. An axicon images a point source into a line along the optic axis, or transforms a laser beam into a ring. It can be used to turn a Gaussian beam into an approximation to a Bessel beam, with greatly reduced diffraction....
- Back focal plane
- Bokeh
Bokeh is a photographic term referring to the out-of-focus areas of an image produced by a camera lens using a shallow depth of field. In and of itself, bokeh implies no aesthetic judgment or qualities, either good or bad. Bokeh meaning "blur", it is only a factual synonym...
- Cardinal point (optics)
The cardinal points and the associated cardinal planes are a set of special points and planes in an optical system, which help in the analysis of its paraxial properties...
- Corrective lens
A corrective lens is a lens worn in front of the eye, mainly used to treat myopia, hyperopia, astigmatism, and presbyopia. Glasses or "spectacles" are worn on the face a short distance in front of the eye. Contact lenses are worn directly on the surface of the eye...
- Eyepiece
An eyepiece, or ocular lens, is a type of lens that is attached to a variety of optical devices such as telescopes and microscopes. It is so named because it is usually the lens that is closest to the eye when someone looks through the device. The objective lens or mirror collects light and brings...
- F-number
In optics, the f-number of an optical system expresses the diameter of the entrance pupil in terms of the focal length of the lens; in simpler terms, the f-number is the focal length divided by the "effective" aperture diameter...
- Fresnel lens
A Fresnel lens is a type of lens developed by French physicist Augustin-Jean Fresnel for lighthouses; a similar design had previously been proposed by Buffon and Condorcet as a way to make large burning lenses....
- Gradient index lens
- Gravitational lens
A gravitational lens is formed when the light from a very distant, bright source is "bent" around a massive object between the source object and the observer...
- History of lensmaking
- Lens (anatomy)
The lens is a transparent, biconvex structure in the eye that, along with the cornea, helps to refract light to be focused on the retina. The lens, by changing shape, functions to change the focal distance of the eye so that it can focus on objects at various distances, thus allowing a sharp real...
- List of lens designs
|
MicroscopeA microscope is an instrument to see objects too tiny for the naked eye. The science of investigating small objects using such an instrument is called microscopy. Microscopic means invisible to the eye unless aided by a microscope.-History:An early microscope was made in 1590 in Middelburg, The...
MicrolensMicrolenses are small lenses, generally with diameters less than a millimetre and often as small as 10 micrometres . The small sizes of the lenses means that a simple design can give good optical quality but sometimes unwanted effects arise due to optical diffraction at the small features...
Numerical apertureIn optics, the numerical aperture of an optical system is a dimensionless number that characterizes the range of angles over which the system can accept or emit light...
Optical coatingAn optical coating is one or more thin layers of material deposited on an optical component such as a lens or mirror, which alters the way in which the optic reflects and transmits light. One type of optical coating is an antireflection coating, which reduces unwanted reflections from surfaces, and...
s
Optical lens designOptical lens design refers to the calculation of lens construction parameters that will meet a set of performance requirements and constraints, including cost and schedule limitations....
Optical lenticular
Photochromic lens
Photographic lensA photographic lens is an optical lens or assembly of lenses used in conjunction with a camera body and mechanism to make images of objects either on photographic film or on other media capable of storing an image chemically or electronically.While in principle a simple convex lens will suffice,...
Prime lensIn film and photography, a prime lens is either a photographic lens whose focal length is fixed, as opposed to a zoom lens, or it is the primary lens in a combination lens system....
Prism (optics)In optics, a prism is a transparent optical element with flat, polished surfaces that refract light. The exact angles between the surfaces depend on the application. The traditional geometrical shape is that of a triangular prism with a triangular base and rectangular sides, and in colloquial use...
Ray tracingIn physics, ray tracing is a method for calculating the path of waves or particles through a system with regions of varying propagation velocity, absorption characteristics, and reflecting surfaces. Under these circumstances, wavefronts may bend, change direction, or reflect off surfaces,...
SuperlensA superlens is a lens which is capable of subwavelength imaging, allowing for magnification of near field rays. Conventional lenses have a resolution on the order of one wavelength due to the so-called diffraction limit. This limit makes it impossible to image very small objects, such as...
TelescopeA telescope is an instrument designed for the observation of remote objects by the collection of electromagnetic radiation. The first known practically functioning telescopes were invented in the Netherlands at the beginning of the 17th century...
Zoom lensA zoom lens is a mechanical assembly of lens elements with the ability to vary its focal length , as opposed to a fixed focal length lens...
|
External links
- Applied photographic optics Book
- Book- The properties of optical glass
- Handbook of Ceramics, Glasses, and Diamonds
- Optical glass construction
- History of Optics (audio mp3) by Simon Schaffer, Professor in History and Philosophy of Science at the University of Cambridge
The University of Cambridge , located in the City of Cambridge, Cambridgeshire, United Kingdom, is the second oldest university in the English-speaking world and the fourth oldest in Europe...
, Jim Bennett, Director of the Museum of the History of Science at the University of OxfordThe University of Oxford , located in the UK city of Oxford, is the oldest surviving university in the English-speaking world and is regarded as one of the world's leading academic institutions. Although the exact date of foundation remains unclear, there is evidence of teaching there as far back...
and Emily Winterburn, Curator of Astronomy at the National Maritime MuseumThe National Maritime Museum in Greenwich, England is the leading maritime museum of the United Kingdom and may be the largest museum of its kind in the world. The historic buildings forming part of the Maritime Greenwich World Heritage Site, it also incorporates the Royal Observatory, Greenwich,...
(recorded by the BBCThe British Broadcasting Corporation, usually referred to by its abbreviation as the "BBC", is the longest established and largest broadcaster in the world...
).
- a chapter from an online textbook on refraction and lenses
- Thin Spherical Lenses on Project PHYSNET.
- Lens article at digitalartform.com
- Article on Ancient Egyptian lenses
- picture of the Ninive rock crystal lens
- Do Sensors “Outresolve” Lenses?; on lens and sensor resolution interaction.
Simulations