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Lens

From Wikipedia, the free encyclopedia
(Redirected fromConvex lens)
Optical device which transmits and refracts light
Aburning apparatus consisting of twobiconvex lens
For other uses, seeLens (disambiguation).

Alens is a transmissiveoptical device that focuses or disperses alight beam by means ofrefraction. Asimple lens consists of a single piece oftransparent material, while acompound lens consists of several simple lenses (elements), usually arranged along a commonaxis. Lenses are made from materials such asglass orplastic and areground,polished, ormolded to the required shape. A lens can focus light to form animage, unlike aprism, which refracts light without focusing. Devices that similarly focus or disperse waves and radiation other than visible light are also called "lenses", such asmicrowave lenses,electron lenses,acoustic lenses, orexplosive lenses.

Lenses are used in various imaging devices such astelescopes,binoculars, andcameras. They are also used as visual aids inglasses to correct defects of vision such asmyopia andhypermetropia.

History

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This sectionneeds expansion with: history after 1823. You can help byadding missing information.(January 2012)
Light being refracted by a spherical glass container full of water.Roger Bacon, 13th century
Lens forLSST, a sky surveying telescope which had itsfirst light on 23 June 2025
See also:History of optics andCamera lens

The wordlens comes fromlēns, the Latin name of thelentil (a seed of a lentil plant), because a double-convex lens is lentil-shaped. The lentil also gives its name to ageometric figure.[a]

Some scholars argue that the archeological evidence indicates that there was widespread use of lenses in antiquity, spanning several millennia.[1] The so-calledNimrud lens is a rock crystal artifact dated to the 7th century BCE which may or may not have been used as a magnifying glass, or a burning glass.[2][3][4] Others have suggested that certainEgyptian hieroglyphs depict "simple glass meniscal lenses".[5][verification needed]

The oldest certain reference to the use of a lens as a burning-glass is fromAristophanes' playThe Clouds (424 BCE).[6]

Pliny the Elder (1st century) confirms that burning-glasses were known in the Roman period.[7]Pliny arguably referred to the use of acorrective lens when he mentions thatNero was said to watch thegladiatorial games using anemerald (possiblyconcave to correct fornearsightedness, though the reference is confusing and this interpretation has been disputed.[8][9]

In hisNaturales quaestiones,Seneca the Younger (3 BC–65 AD) wrote that "to those who see through water, all things are far larger; letters, however minute and obscure, are perceived larger and clearer through a glass sphere filled with water; fruits seem more shapely in form than they are, if they float in glass."[10]

Ptolemy (2nd century) wrote a book onOptics, which however survives only in the Latin translation of an incomplete and very poor Arabic translation. The book was, however, received by medieval scholars in the Islamic world, and commented upon byIbn Sahl (10th century), who was in turn improved upon byAlhazen (Book of Optics, 11th century). The Arabic translation of Ptolemy'sOptics became available in Latin translation in the 12th century (Eugenius of Palermo 1154).[clarification needed]

Between the 11th and 13th century "reading stones" were used. These were primitive plano-convex lenses initially made by cutting a glass sphere in half. The medieval (11th or 12th century) rock crystalVisby lenses may or may not have been intended for use as burning glasses.[11]

Spectacles were invented as an improvement of the "reading stones" of the high medieval period in Northern Italy in the second half of the 13th century.[12] This was the start of the optical industry of grinding and polishing lenses for spectacles, first in Venice and Florence in the late 13th century,[13] and later in the spectacle-making centres in both theNetherlands andGermany.[14]Spectacle makers created improved types of lenses for the correction of vision based more on empirical knowledge gained from observing the effects of the lenses (probably without the knowledge of the rudimentary optical theory of the day).[15][16] The practical development and experimentation with lenses led to the invention of the compoundoptical microscope around 1595, and therefracting telescope in 1608, both of which appeared in the spectacle-making centres in theNetherlands.[17][18]

Further information:History of the telescope

With the invention of the telescope and microscope there was a great deal of experimentation with lens shapes in the 17th and early 18th centuries by those trying to correct chromatic errors seen in lenses. Opticians tried to construct lenses of varying forms of curvature, wrongly assuming errors arose from defects in the spherical figure of their surfaces.[19] Optical theory onrefraction and experimentation was showing no single-element lens could bring all colours to a focus. This led to the invention of the compoundachromatic lens byChester Moore Hall inEngland in 1733, an invention also claimed by fellow EnglishmanJohn Dollond in a 1758 patent.

Developments in transatlantic commerce were the impetus for the construction of modern lighthouses in the 18th century, which utilize a combination of elevated sightlines, lighting sources, and lenses to provide navigational aid overseas. With maximal distance of visibility needed in lighthouses, conventional convex lenses would need to be significantly sized which would negatively affect the development of lighthouses in terms of cost, design, and implementation. Fresnel lens were developed that considered these constraints by featuring less material through their concentric annular sectioning. They were first fully implemented into a lighthouse in 1823.[20]

Construction of simple lenses

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Most lenses arespherical lenses: their two surfaces are parts of the surfaces of spheres. Each surface can beconvex (bulging outwards from the lens),concave (depressed into the lens), orplanar (flat). The line joining the centres of the spheres making up the lens surfaces is called theaxis 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.

Toric or sphero-cylindrical lenses have surfaces with two different radii of curvature in two orthogonal planes. They have a differentfocal power in different meridians. This forms anastigmatic lens. An example is eyeglass lenses that are used to correctastigmatism in someone's eye.

Types of simple lenses

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Types of lenses
Types of lenses

Lenses are classified by the curvature of the two optical surfaces. A lens isbiconvex (ordouble convex, or justconvex) if both surfaces areconvex. If both surfaces have the same radius of curvature, the lens isequiconvex. A lens with twoconcave surfaces isbiconcave (or justconcave). If one of the surfaces is flat, the lens isplano-convex orplano-concave depending on the curvature of the other surface. A lens with one convex and one concave side isconvex-concave ormeniscus. Convex-concave lenses are most commonly used incorrective lenses, since the shape minimizes some aberrations.

For a biconvex or plano-convex lens in a lower-index medium, acollimated beam of light passing through the lens converges to a spot (afocus) behind the lens. In this case, the lens is called apositive orconverging lens. For athin lens in air, the distance from the lens to the spot is thefocal length of the lens, which is commonly represented byf in diagrams and equations. Anextended hemispherical lens is a special type of plano-convex lens, in which the lens's curved surface is a full hemisphere and the lens is much thicker than the radius of curvature.

Another extreme case of a thick convex lens is aball lens, whose shape is completely round. When used in novelty photography it is often called a "lensball". A ball-shaped lens has the advantage of being omnidirectional, but for mostoptical glass types, its focal point lies close to the ball's surface. Because of the ball's curvature extremes compared to the lens size,optical aberration is much worse than thin lenses, with the notable exception ofchromatic aberration.

Biconvex lens
Biconvex lens

For a biconcave or plano-concave lens in a lower-index medium, a collimated beam of light passing through the lens is diverged (spread); the lens is thus called anegative ordiverging lens. The beam, after passing through the lens, appears to emanate from a particular point on the axis in front of the lens. For a thin lens in air, the distance from this point to the lens is the focal length, though it is negative with respect to the focal length of a converging lens.

Biconcave lens
Biconcave lens

The behavior reverses when a lens is placed in a medium with higher refractive index than the material of the lens. In this case a biconvex or plano-convex lens diverges light, and a biconcave or plano-concave one converges it.

Meniscus lenses: negative (top) and positive (bottom)

Convex-concave (meniscus) lenses can be either positive or negative, depending on the relative curvatures of the two surfaces. Anegative meniscus lens has a steeper concave surface (with a shorter radius than the convex surface) and is thinner at the centre than at the periphery. Conversely, apositive meniscus lens has a steeper convex surface (with a shorter radius than the concave surface) and is thicker at the centre than at the periphery.

An idealthin lens with two surfaces of equal curvature (also equal in the sign) would have zerooptical power (as its focal length becomes infinity as shown in thelensmaker's equation), meaning that it would neither converge nor diverge light. All real lenses have a nonzero thickness, however, which makes a real lens with identical curved surfaces slightly positive. To obtain exactly zero optical power, a meniscus lens must have slightly unequal curvatures to account for the effect of the lens' thickness.

For a spherical surface

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Simulation of refraction at spherical surface atDesmos

For a single refraction for a circular boundary, the relation between object and its image in theparaxial approximation is given by[21][22]

n1u+n2v=n2−n1R{\displaystyle {\frac {n_{1}}{u}}+{\frac {n_{2}}{v}}={\frac {n_{2}-n_{1}}{R}}}

whereR is the radius of the spherical surface,n2 is the refractive index of the material of the surface,n1 is the refractive index of medium (the medium other than the spherical surface material),u{\textstyle u} is the on-axis (on the optical axis) object distance from the line perpendicular to the axis toward the refraction point on the surface (which height ish), andv{\textstyle v} is the on-axis image distance from the line. Due to paraxial approximation where the line ofh is close to the vertex of the spherical surface meeting the optical axis on the left,u{\textstyle u} andv{\textstyle v} are also considered distances with respect to the vertex.

Movingv{\textstyle v} toward the right infinity leads to the first or object focal lengthf0{\textstyle f_{0}} for the spherical surface. Similarly,u{\textstyle u} toward the left infinity leads to the second or image focal lengthfi{\displaystyle f_{i}}.[23]

f0=n1n2−n1R,fi=n2n2−n1R{\displaystyle {\begin{aligned}f_{0}&={\frac {n_{1}}{n_{2}-n_{1}}}R,\\f_{i}&={\frac {n_{2}}{n_{2}-n_{1}}}R\end{aligned}}}

Applying this equation on the two spherical surfaces of a lens and approximating the lens thickness to zero (so a thin lens) leads to thelensmaker's formula.

Derivation

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The four cases of spherical refraction

ApplyingSnell's law on the spherical surface,n1sin⁡i=n2sin⁡r.{\displaystyle n_{1}\sin i=n_{2}\sin r\,.}

Also in the diagram,tan⁡(i−θ)=hutan⁡(θ−r)=hvsin⁡θ=hR{\displaystyle {\begin{aligned}\tan(i-\theta )&={\frac {h}{u}}\\\tan(\theta -r)&={\frac {h}{v}}\\\sin \theta &={\frac {h}{R}}\end{aligned}}}, and usingsmall angle approximation (paraxial approximation) and eliminatingi,r, andθ,

n2v+n1u=n2−n1R.{\displaystyle {\frac {n_{2}}{v}}+{\frac {n_{1}}{u}}={\frac {n_{2}-n_{1}}{R}}\,.}

Lensmaker's equation

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Simulation of the effect of lenses with different curvatures of the two facets on a collimated Gaussian beam.
The position of the focus of a spherical lens depends on the radii of curvature of the two facets.

The (effective) focal lengthf{\displaystyle f} of a spherical lens in air or vacuum for paraxial rays can be calculated from thelensmaker's equation:[24][25]

1 f =(n−1)[ 1 R1 −1 R2 + (n−1) d  n R1 R2  ] ,{\displaystyle {\frac {1}{\ f\ }}=\left(n-1\right)\left[\ {\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}+{\frac {\ \left(n-1\right)\ d~}{\ n\ R_{1}\ R_{2}\ }}\ \right]\ ,}where


The focal length f {\textstyle \ f\ } is with respect to theprincipal planes of the lens, and the locations of the principal planes h1 {\textstyle \ h_{1}\ } and h2 {\textstyle \ h_{2}\ } with respect to the respective lens vertices are given by the following formulas, where it is a positive value if it is right to the respective vertex.[25]

 h1=−  (n−1)f d  n R2  {\displaystyle \ h_{1}=-\ {\frac {\ \left(n-1\right)f\ d~}{\ n\ R_{2}\ }}\ } h2=−  (n−1)f d  n R1  {\displaystyle \ h_{2}=-\ {\frac {\ \left(n-1\right)f\ d~}{\ n\ R_{1}\ }}\ }

The focal length f {\displaystyle \ f\ } is positive for converging lenses, and negative for diverging lenses. Thereciprocal of the focal length, 1 f  ,{\textstyle \ {\tfrac {1}{\ f\ }}\ ,} is theoptical power of the lens. If the focal length is in metres, this gives the optical power indioptres (reciprocal 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. Other properties of the lens, such as theaberrations are not the same in both directions.

Sign convention for radii of curvatureR1 andR2

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Main article:Radius of curvature (optics)

The signs of the lens' radii of curvature indicate whether the corresponding surfaces are convex or concave. Thesign convention used to represent this varies,[26] but in this article apositiveR indicates a surface's center of curvature is further along in the direction of the ray travel (right, in the accompanying diagrams), whilenegativeR means that rays reaching the surface have already passed the center of curvature. Consequently, for external lens surfaces as diagrammed above,R1 > 0 andR2 < 0 indicateconvex surfaces (used to converge light in a positive lens), whileR1 < 0 andR2 > 0 indicateconcave surfaces. The reciprocal of the radius of curvature is called thecurvature. A flat surface has zero curvature, and its radius of curvature isinfinite.

Sign convention for other parameters

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Sign convention for Gaussian lens equation[27]
ParameterMeaning+ Sign− Sign
soThe distance between an object and a lens.Real objectVirtual object
siThe distance between an image and a lens.Real imageVirtual image
fThe focal length of a lens.Converging lensDiverging lens
yoThe height of an object from the optical axis.Erect objectInverted object
yiThe height of an image from the optical axisErect imageInverted image
MTThe transverse magnification in imaging ( = the ratio ofyi toyo ).Erect imageInverted image

This convention is used in this article. Other conventions such as theCartesian sign convention change the form of the equations.

Thin lens approximation

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Ifd is small compared toR1 andR2 then thethin lens approximation can be made. For a lens in air,f  is then given by[28]

 1 f ≈(n−1)[ 1 R1 −1 R2  ] .{\displaystyle \ {\frac {1}{\ f\ }}\approx \left(n-1\right)\left[\ {\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}\ \right]~.}

Derivation

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A diagram for a spherical lens equation with paraxial rays

The spherical thin lens equation inparaxial approximation is derived here with respect to the right figure.[28] The 1st spherical lens surface (which meets the optical axis atV1{\textstyle V_{1}}as its vertex) images an on-axis object pointO to the virtual imageI', which can be described by the following equation,  n1  u + n2  v′ = n2−n1  R1  .{\displaystyle \ {\frac {\ n_{1}\ }{\ u\ }}+{\frac {\ n_{2}\ }{\ v'\ }}={\frac {\ n_{2}-n_{1}\ }{\ R_{1}\ }}~.}For the imaging by the second lens surface (I' as the object for this imaging), by taking the above sign convention,u′=−v′+d{\textstyle u'=-v'+d} and n2 −v′+d + n1  v = n1−n2  R2  .{\displaystyle \ {\frac {n_{2}}{\ -v'+d\ }}+{\frac {\ n_{1}\ }{\ v\ }}={\frac {\ n_{1}-n_{2}\ }{\ R_{2}\ }}~.} Adding these two equations yields  n1 u+ n1 v=(n2−n1)(1 R1 −1 R2 )+ n2 d  ( v′−d ) v′  .{\displaystyle \ {\frac {\ n_{1}\ }{u}}+{\frac {\ n_{1}\ }{v}}=\left(n_{2}-n_{1}\right)\left({\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}\right)+{\frac {\ n_{2}\ d\ }{\ \left(\ v'-d\ \right)\ v'\ }}~.} For the thin lens approximation whered→0,{\displaystyle d\rightarrow 0,} the 2nd term of the RHS (Right Hand Side) is gone, so

  n1 u+ n1 v=(n2−n1)(1 R1 −1 R2 ) .{\displaystyle \ {\frac {\ n_{1}\ }{u}}+{\frac {\ n_{1}\ }{v}}=\left(n_{2}-n_{1}\right)\left({\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}\right)~.}

The focal lengthf{\displaystyle f} of the thin lens is found by limitingu→−∞,{\displaystyle u\rightarrow -\infty ,}

  n1  f =(n2−n1)(1 R1 −1 R2 )→1 f =( n2  n1 −1)(1 R1 −1 R2 ) .{\displaystyle \ {\frac {\ n_{1}\ }{\ f\ }}=\left(n_{2}-n_{1}\right)\left({\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}\right)\rightarrow {\frac {1}{\ f\ }}=\left({\frac {\ n_{2}\ }{\ n_{1}\ }}-1\right)\left({\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}\right)~.}

So, the Gaussian thin lens equation is

 1 u +1 v =1 f  .{\displaystyle \ {\frac {1}{\ u\ }}+{\frac {1}{\ v\ }}={\frac {1}{\ f\ }}~.}

For the thin lens in air or vacuum wheren1=1{\textstyle n_{1}=1} can be assumed,f{\textstyle f} becomes

 1 f =(n−1)(1 R1 −1 R2 ) {\displaystyle \ {\frac {1}{\ f\ }}=\left(n-1\right)\left({\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}\right)\ }

where the subscript of 2 inn2{\textstyle n_{2}} is dropped.

Imaging properties

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As mentioned above, a positive or converging lens in air focuses a collimated beam travelling along the lens axis to a spot (known as thefocal point) at a distancef from the lens. Conversely, apoint source of light placed at the focal point is converted into a collimated beam by the lens. These two cases are examples ofimage 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 distancef from the lens is called thefocal plane.

Lens equation

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Forparaxial rays, if the distances from an object to a sphericalthin lens (a lens of negligible thickness) and from the lens to the image areS1 andS2 respectively, the distances are related by the (Gaussian)thin lens formula:[29][30][31][32]

1f=1S1+1S2.{\displaystyle {1 \over f}={1 \over S_{1}}+{1 \over S_{2}}\,.}

Single thin lens imaging with chief rays

The right figure shows how the image of an object point can be found by using three rays; the first ray parallelly incident on the lens and refracted toward the second focal point of it, the second ray crossingthe optical center of the lens (so its direction does not change), and the third ray toward the first focal point and refracted to the direction parallel to the optical axis. This is a simple ray tracing method easily used. Two rays among the three are sufficient to locate the image point. By moving the object along the optical axis, it is shown that the second ray determines the image size while other rays help to locate the image location.

The lens equation can also be put into the "Newtonian" form:[27]

f2=x1x2,{\displaystyle f^{2}=x_{1}x_{2}\,,}

wherex1=S1−f{\displaystyle x_{1}=S_{1}-f} andx2=S2−f.{\displaystyle x_{2}=S_{2}-f\,.}x1{\textstyle x_{1}} is positive if it is left to the front focal pointF1{\textstyle F_{1}}, andx2{\textstyle x_{2}} is positive if it is right to the rear focal pointF2{\textstyle F_{2}}. Becausef2{\textstyle f^{2}} is positive, an object point and the corresponding imaging point made by a lens are always in opposite sides with respect to their respective focal points. (x1{\textstyle x_{1}} andx2{\textstyle x_{2}} are either positive or negative.)

This Newtonian form of the lens equation can be derived by using a similarity between trianglesP1PO1F1 andL3L2F1 and another similarity between trianglesL1L2F2 andP2P02F2 in the right figure. The similarities give the following equations and combining these results gives the Newtonian form of the lens equation.

y1x1=|y2|fy1f=|y2|x2{\displaystyle {\begin{array}{lcr}{\frac {y_{1}}{x_{1}}}={\frac {\left\vert y_{2}\right\vert }{f}}\\{\frac {y_{1}}{f}}={\frac {\left\vert y_{2}\right\vert }{x_{2}}}\end{array}}}

A diagram of imaging with a single thick lens imaging.H1 andH2 are principal points whereprincipal planes of the thick lens cross the optical axis. If the object and image spaces are the same medium, then these points are alsonodal points.
A camera lens forms areal image of a distant object.

The above equations also hold for thick lenses (including a compound lens made by multiple lenses, that can be treated as a thick lens) in air or vacuum (which refractive index can be treated as 1) ifS1{\textstyle S_{1}},S2{\textstyle S_{2}}, andf{\textstyle f} are with respect to theprincipal planes of the lens (f{\textstyle f} is theeffective focal length in this case).[25] This is because of triangle similarities like the thin lens case above; similarity between trianglesP1PO1F1 andL3H1F1 and another similarity between trianglesL1'H2F2 andP2P02F2 in the right figure. If distancesS1 orS2 pass through amedium other than air or vacuum, then a more complicated analysis is required.

If an object is placed at a distanceS1 >f from a positive lens of focal lengthf, we will find an image at a distanceS2 according to this formula. If a screen is placed at a distanceS2 on the opposite side of the lens, an image is formed on it. This sort of image, which can be projected onto a screen orimage sensor, is known as areal image. This is the principle of thecamera, and also of thehuman eye, in which theretina serves as the image sensor.

The focusing adjustment of a camera adjustsS2, as using an image distance different from that required by this formula produces adefocused (fuzzy) image for an object at a distance ofS1 from the camera. Put another way, modifyingS2 causes objects at a differentS1 to come into perfect focus.

Virtual image formation using a positive lens as a magnifying glass.[33]

In some cases,S2 is negative, indicating that the image is formed on the opposite side of the lens from where those rays are being considered. Since the diverging light rays emanating from the lens never come into focus, and those rays are not physically present at the point where theyappear to form an image, this is called avirtual image. Unlike real images, a virtual image cannot be projected on a screen, but appears to an observer looking through the lens as if it were a real object at the location of that virtual image. Likewise, it appears to a subsequent lens as if it were an object at that location, so that second lens could again focus that light into a real image,S1 then being measured from the virtual image location behind the first lens to the second lens. This is exactly what the eye does when looking through amagnifying glass. The magnifying glass creates a (magnified) virtual image behind the magnifying glass, but those rays are then re-imaged by thelens of the eye to create areal image on theretina.

Anegative lens produces a demagnified virtual image.
ABarlow lens (B) reimages avirtual object (focus of red ray path) into a magnified real image (green rays at focus)

Using a positive lens of focal lengthf, a virtual image results whenS1 <f, the lens thus being used as a magnifying glass (rather than ifS1 ≫f as for a camera). Using a negative lens (f < 0) with areal object (S1 > 0) can only produce a virtual image (S2 < 0), according to the above formula. It is also possible for the object distanceS1 to be negative, in which case the lens sees a so-calledvirtual object. This happens when the lens is inserted into a converging beam (being focused by a previous lens)before the location of its real image. In that case even a negative lens can project a real image, as is done by aBarlow lens.

For a given lens with the focal lengthf, the minimum distance between an object and the real image is 4f (S1 =S2 = 2f). This is derived by lettingL =S1 +S2, expressingS2 in terms ofS1 by the lens equation (or expressingS1 in terms ofS2), and equating the derivative ofL with respect toS1 (orS2) to zero. (Note thatL has no limit in increasing so its extremum is only the minimum, at which the derivate ofL is zero.)

Real image of a lamp is projected onto a screen (inverted). Reflections of the lamp from both surfaces of the biconvex lens are visible.
A convex lens (f ≪S1) forming a real, inverted image (as the image formed by the objective lens of a telescope or binoculars) rather than the upright, virtual image as seen in amagnifying glass (f >S1). Thisreal image may also be viewed when put on a screen.

Magnification

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The linearmagnification of an imaging system using a single lens is given by

M=−S2S1=ff−S1 =−fx1{\displaystyle M=-{\frac {S_{2}}{S_{1}}}={\frac {f}{f-S_{1}}}\ =-{\frac {f}{x_{1}}}}

whereM is the magnification factor defined as the ratio of the size of an image compared to the size of the object. The sign convention here dictates that ifM is negative, as it is for real images, the image is upside-down with respect to the object. For virtual imagesM is positive, so the image is upright.

This magnification formula provides two easy ways to distinguish converging (f > 0) and diverging (f < 0) lenses: For an object very close to the lens (0 <S1 < |f|), a converging lens would form a magnified (bigger) virtual image, whereas a diverging lens would form a demagnified (smaller) image; For an object very far from the lens (S1 > |f| > 0), a converging lens would form an inverted image, whereas a diverging lens would form an upright image.

Linear magnificationM is not always the most useful measure of magnifying power. For instance, when characterizing a visual telescope or binoculars that produce only a virtual image, one would be more concerned with theangular magnification—which expresses how much larger a distant object appears through the telescope compared to the naked eye. In the case of a camera one would quote theplate scale, which compares the apparent (angular) size of a distant object to the size of the real image produced at the focus. The plate scale is the reciprocal of the focal length of the camera lens; lenses are categorized aslong-focus lenses orwide-angle lenses according to their focal lengths.

Using an inappropriate measurement of magnification can be formally correct but yield a meaningless number. For instance, using a magnifying glass of5 cm focal length, held20 cm from the eye and5 cm from the object, produces a virtual image at infinity of infinite linear size:M = ∞. But theangular magnification is 5, meaning that the object appears 5 times larger to the eye than without the lens. When taking a picture of themoon using a camera with a50 mm lens, one is not concerned with the linear magnificationM ≈−50 mm /380000 km =−1.3×10−10. Rather, the plate scale of the camera is about1°/mm, from which one can conclude that the0.5 mm image on the film corresponds to an angular size of the moon seen from earth of about 0.5°.

In the extreme case where an object is an infinite distance away,S1 = ∞,S2 =f andM = −f/∞ = 0, indicating that the object would be imaged to a single point in the focal plane. In fact, the diameter of the projected spot is not actually zero, sincediffraction places a lower limit on the size of thepoint spread function. This is called thediffraction limit.

Images of black letters in a thin convex lens of focal lengthf are shown in red. Selected rays are shown for lettersE,I andK in blue, green and orange, respectively.E (at2f) has an equal-size, real and inverted image;I (atf) has its image atinfinity; andK (atf/2) has a double-size, virtual and upright image. Note that the images of letters H, I, J, and i are located far away from the lens such that they are not shown here. What is also shown here that the ray that is parallelly incident on the lens and refracted toward the second focal pointf determines the image size while other rays help to locate the image location.

Table for thin lens imaging properties

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Images of Real Objects Formed by Thin Lenses[27]
Lens TypeObject LocationImage TypeImage LocationLateral Image OrientationImage MagnificationRemark
Converging lens (or positive lens)∞>S1>2f{\displaystyle \infty >S_{1}>2f}Real (rays converging to each image point)f<S2<2f{\displaystyle f<S_{2}<2f}Inverted (opposite to the object orientation)Diminished
Converging lensS1=2f{\displaystyle S_{1}=2f}RealS2=2f{\displaystyle S_{2}=2f}InvertedSame size
Converging lensf<S1<2f{\displaystyle f<S_{1}<2f}Real∞>S2>2f{\displaystyle \infty >S_{2}>2f}InvertedMagnified
Converging lensS1=f{\displaystyle S_{1}=f}±∞{\displaystyle \pm \infty }
Converging lensS1<f{\displaystyle S_{1}<f}Virtual (rays apparently diverging from each image point)|S2|>S1{\displaystyle \vert S_{2}\vert >S_{1}}Erect (same to the object orientation)MagnifiedAs an object moves to the lens, the virtual image also gets closer to the lens while the image size is reduced.
Diverging lens (or negative lens)AnywhereVirtual|S2|<|f|,S1>|S2|{\displaystyle \vert S_{2}\vert <\vert f\vert ,S_{1}>\vert S_{2}\vert }ErectDiminished

Aberrations

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Optical aberration
Main article:Optical aberration

Lenses do not form perfect images, and always introduce some degree of distortion oraberration that makes the image an imperfect replica of the object. Careful design of the lens system for a particular application minimizes the aberration. Several types of aberration affect image quality, including spherical aberration, coma, and chromatic aberration.

Spherical aberration

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Main article:Spherical aberration

Spherical aberration occurs because spherical surfaces are not the ideal shape for a lens, but are by far the simplest shape to which glass can beground and polished, and so are often used. Spherical aberration causes beams parallel to, but laterally 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. Spherical aberration can be minimised with normal lens shapes by carefully choosing the surface curvatures 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

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Main article:Coma (optics)

Coma, orcomatic aberration, derives its name from thecomet-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 axisθ. Rays that pass through the centre of a lens of focal lengthf are focused at a point with distanceftanθ 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 acomatic circle (see each circle of the image in the below figure). 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 calledbestform lenses.

Chromatic aberration

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Main article:Chromatic aberration

Chromatic aberration is caused by thedispersion of the lens material—the variation of itsrefractive index,n, with the wavelength of light. Since, fromthe formulae above,f is dependent uponn, it follows that light of different wavelengths is focused to different positions. Chromatic aberration of a lens is seen as fringes of colour around the image. It can be minimised by using anachromatic doublet (orachromat) 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. Anapochromat is a lens or lens system with even better chromatic aberration correction, combined with improved spherical aberration correction. Apochromats are much more expensive than achromats.

Different lens materials may also be used to minimise chromatic aberration, such as specialised coatings or lenses made from the crystalfluorite. This naturally occurring substance has the highest knownAbbe number, indicating that the material has low dispersion.

Other types of aberration

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Other kinds of aberration includefield curvature,barrelandpincushion distortion, andastigmatism.

Aperture diffraction

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Even if a lens is designed to minimize or eliminate the aberrations described above, the image quality is still limited by thediffraction of light passing through the lens' finiteaperture. Adiffraction-limited 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

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See also:Photographic lens,Doublet (lens),Triplet lens, andAchromatic lens

Simple lenses are subject to theoptical 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. Acompound 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.

In a multiple-lens system, if the purpose of the system is to image an object, then the system design can be such that each lens treats the image made by the previous lens as an object, and produces the new image of it, so the imaging is cascaded through the lenses.[34][35] As shownabove, the Gaussian lens equation for a spherical lens is derived such that the 2nd surface of the lens images the image made by the 1st lens surface. Following the same logic, in multi-lens imaging, the 3rd lens surface (the front surface of the 2nd lens) images the image made by the 2nd surface, and the 4th surface (the back surface of the 2nd lens) images the image made by the 3rd surface. This imaging cascade by each lens surface justifies the imaging cascade by each lens.

For a two-lens system the object distances of each lens can be denoted asso1{\textstyle s_{o1}} andso2{\textstyle s_{o2}}, and the image distances as andsi1{\textstyle s_{i1}} andsi2{\textstyle s_{i2}}. If the lenses are thin, each satisfies the thin lens formula

1fj=1soj+1sij,{\displaystyle {\frac {1}{f_{j}}}={\frac {1}{s_{oj}}}+{\frac {1}{s_{ij}}},}

If the distance between the two lenses isd{\displaystyle d}, thenso2=d−si1{\textstyle s_{o2}=d-s_{i1}}. (The 2nd lens images the image of the first lens.)

FFD (Front Focal Distance) is defined as the distance between the front (left) focal point of an optical system and its nearest optical surface vertex.[36] If an object is located at the front focal point of the system, then its image made by the system is located infinitely far way to the right (i.e., light rays from the object is collimated after the system). To do this, the image of the 1st lens is located at the focal point of the 2nd lens, i.e.,si1=d−f2{\displaystyle s_{i1}=d-f_{2}}. So, the thin lens formula for the 1st lens becomes[37]

1f1=1FFD+1d−f2→FFD=f1(d−f2)d−(f1+f2).{\displaystyle {\frac {1}{f_{1}}}={\frac {1}{FFD}}+{\frac {1}{d-f_{2}}}\rightarrow FFD={\frac {f_{1}(d-f_{2})}{d-(f_{1}+f_{2})}}.}

BFD (Back Focal Distance) is similarly defined as the distance between the back (right) focal point of an optical system and its nearest optical surface vertex. If an object is located infinitely far away from the system (to the left), then its image made by the system is located at the back focal point. In this case, the 1st lens images the object at its focal point. So, the thin lens formula for the 2nd lens becomes

1f2=1BFD+1d−f1→BFD=f2(d−f1)d−(f1+f2).{\displaystyle {\frac {1}{f_{2}}}={\frac {1}{BFD}}+{\frac {1}{d-f_{1}}}\rightarrow BFD={\frac {f_{2}(d-f_{1})}{d-(f_{1}+f_{2})}}.}

A simplest case is where thin lenses are placed in contact (d=0{\displaystyle d=0}). ThenFFD=BFD=f{\displaystyle FFD=BFD=f}, so the combined focal lengthf of the lenses is given by

1f=1f1+1f2.{\displaystyle {\frac {1}{f}}={\frac {1}{f_{1}}}+{\frac {1}{f_{2}}}\,.}

Since1/f is the power of a lens with focal lengthf, it can be seen that the powers of thin lenses in contact are additive. The general case of multiple thin lenses in contact is

1f=∑k=1N1fk{\displaystyle {\frac {1}{f}}=\sum _{k=1}^{N}{\frac {1}{f_{k}}}}

whereN{\textstyle N} is the number of lenses.

If two thin lenses are separated in air by some distanced, then the focal length for the combined system is given by1f=1f1+1f2−df1f2.{\displaystyle {\frac {1}{f}}={\frac {1}{f_{1}}}+{\frac {1}{f_{2}}}-{\frac {d}{f_{1}f_{2}}}\,.}

Asd tends to zero, the focal length of the system tends to the value off given for thin lenses in contact. It can be shown that the same formula works for thick lenses ifd is taken as the distance between their principal planes.[25]

If the separation distance between two lenses is equal to the sum of their focal lengths (d =f1 +f2), then the FFD and BFD are infinite. This corresponds to a pair of lenses that transforms a parallel (collimated) beam into another collimated beam. This type of system is called anafocal system, since it produces no net convergence or divergence of the beam. Two lenses at this separation form the simplest type ofoptical telescope. Although the system does not alter the divergence of a collimated beam, it does alter the (transverse) width of the beam. The magnification of such a telescope is given by

M=−f2f1,{\displaystyle M=-{\frac {f_{2}}{f_{1}}}\,,}

which is the ratio of the output beam width to the input 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. For further information on simple optical telescopes, seeRefracting telescope § Refracting telescope designs.

Non spherical types

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An aspheric biconvex lens.

Cylindrical lenses have curvature along only one axis. They are used to focus light into a line, or to convert the elliptical light from alaser diode into a round beam. They are also used in motion pictureanamorphic lenses.

Aspheric lenses have at least one surface that is neither spherical nor cylindrical. The more complicated shapes allow such lenses to form images with lessaberration than standard simple lenses, but they are more difficult and expensive to produce. These were formerly complex to make and often extremely expensive, but advances in technology have greatly reduced the manufacturing cost for such lenses.

Close-up view of a flatFresnel lens.

AFresnel lens has its optical surface broken up into narrow rings, allowing the lens to be much thinner and lighter than conventional lenses. Durable Fresnel lenses can be molded from plastic and are inexpensive.

Lenticular lenses are arrays ofmicrolenses that are used inlenticular printing to make images that have an illusion of depth or that change when viewed from different angles.

Bifocal lens has two or more, or a graduated, focal lengths ground into the lens.

Agradient index lens has flat optical surfaces, but has a radial or axial variation in index of refraction that causes light passing through the lens to be focused.

Anaxicon has aconical optical surface. It images apoint source into a linealong theoptic axis, or transforms a laser beam into a ring.[38]

Diffractive optical elements can function as lenses.

Superlenses are made fromnegative index metamaterials and claim to produce images at spatial resolutions exceeding thediffraction limit.[39] The first superlenses were made in 2004 using such ametamaterial for microwaves.[39] Improved versions have been made by other researchers.[40][41] As of 2014[update] the superlens has not yet been demonstrated atvisible or near-infrared wavelengths.[42]

A prototype flat ultrathin lens, with no curvature has been developed.[43]

Uses

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A watch with a plano-convex lens over the date indicator

A single convex lens mounted in a frame with a handle or stand is amagnifying glass.

Lenses are used asprosthetics for the correction ofrefractive errors such asmyopia,hypermetropia,presbyopia, andastigmatism. (Seecorrective lens,contact lens,eyeglasses,intraocular lens.) Most lenses used for other purposes have strictaxial symmetry; eyeglass lenses are only approximately symmetric. They are usually shaped to fit in a roughly oval, not circular, frame; the optical centres are placed over theeyeballs; their curvature may not be axially symmetric to correct forastigmatism.Sunglasses' lenses are designed to attenuate light; sunglass lenses that also correct visual impairments can be custom made.

Other uses are in imaging systems such asmonoculars,binoculars,telescopes,microscopes,cameras andprojectors. Some of these instruments produce avirtual image when applied to the human eye; others produce areal image that can be captured onphotographic film or anoptical sensor, or can be viewed on a screen. In these devices lenses are sometimes paired up withcurved mirrors to make acatadioptric system where the lens's spherical aberration corrects the opposite aberration in the mirror (such asSchmidt andmeniscus correctors).

Convex lenses produce an image of an object at infinity at their focus; if thesun is imaged, much of the visible and infrared light incident on the lens is concentrated into the small image. A large lens creates 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 asburning-glasses for at least 2400 years.[6] A modern application is the use of relatively large lenses toconcentrate solar energy on relatively smallphotovoltaic cells, harvesting more energy without the need to use larger and more expensive cells.

Radio astronomy andradar systems often usedielectric lenses, commonly called alens antenna to refractelectromagnetic radiation into a collector antenna.

Lenses can become scratched and abraded.Abrasion-resistant coatings are available to help control this.[44]

See also

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Notes

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  1. ^The variant spellinglense is sometimes seen. While it is listed as an alternative spelling in some dictionaries, most mainstream dictionaries do not list it as acceptable.

References

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  1. ^Sines, George; Sakellarakis, Yannis A. (1987). "Lenses in antiquity".American Journal of Archaeology.91 (2):191–196.doi:10.2307/505216.JSTOR 505216.S2CID 191384703.
  2. ^Whitehouse, David (1 July 1999)."World's oldest telescope?".BBC News.Archived from the original on 1 February 2009. Retrieved10 May 2008.
  3. ^"The Nimrud lens/The Layard lens".Collection database. The British Museum.Archived from the original on 19 October 2012. Retrieved25 November 2012.
  4. ^D. Brewster (1852)."On an account of a rock-crystal lens and decomposed glass found in Niniveh".Die Fortschritte der Physik (in German). Deutsche Physikalische Gesellschaft. p. 355.
  5. ^Kriss, Timothy C.; Kriss, Vesna Martich (April 1998). "History of the Operating Microscope: From Magnifying Glass to Microneurosurgery".Neurosurgery.42 (4):899–907.doi:10.1097/00006123-199804000-00116.PMID 9574655.
  6. ^abAristophanes (22 January 2013) [First performed in 423 BC].The Clouds. Translated by Hickie, William James. Project Gutenberg. EBook #2562.[1]Archived 28 June 2017 at theWayback Machine
  7. ^Pliny the Elder,The Natural History (trans. John Bostock)Book XXXVII, Chap. 10Archived 4 October 2008 at theWayback Machine.
  8. ^Pliny the Elder,The Natural History (trans. John Bostock)Book XXXVII, Chap. 16Archived 28 September 2008 at theWayback Machine
  9. ^Woods, David (2006)."Pliny, Nero, and the 'Emerald' (NH 37,64)".Arctos.40:189–196.
  10. ^ReadAncientTexts.com.QUAESTIONES NATURALES: Liber I by Seneca | ReadAncientTexts.com. Retrieved13 January 2026.{{cite book}}:|website= ignored (help)
  11. ^Tilton, Buck (2005).The Complete Book of Fire: Building Campfires for Warmth, Light, Cooking, and Survival. Menasha Ridge Press. p. 25.ISBN 978-0-89732-633-9.
  12. ^Glick, Thomas F.; Steven John Livesey; Faith Wallis (2005).Medieval science, technology, and medicine: an encyclopedia. Routledge. p. 167.ISBN 978-0-415-96930-7.Archived from the original on 20 January 2023. Retrieved24 April 2011.
  13. ^Al Van Helden.The Galileo Project > Science > The TelescopeArchived 23 June 2004 at theWayback Machine. Galileo.rice.edu. Retrieved on 6 June 2012.
  14. ^Henry C. King (28 September 2003).The History of the Telescope. Courier Dover Publications. p. 27.ISBN 978-0-486-43265-6.Archived from the original on 2 July 2023. Retrieved6 June 2012.
  15. ^Paul S. Agutter; Denys N. Wheatley (12 December 2008).Thinking about Life: The History and Philosophy of Biology and Other Sciences. Springer. p. 17.ISBN 978-1-4020-8865-0. Retrieved6 June 2012.
  16. ^Vincent Ilardi (2007).Renaissance Vision from Spectacles to Telescopes. American Philosophical Society. p. 210.ISBN 978-0-87169-259-7. Retrieved6 June 2012.[permanent dead link]
  17. ^Microscopes: Time LineArchived 9 January 2010 at theWayback Machine, Nobel Foundation. Retrieved 3 April 2009
  18. ^Fred Watson (1 October 2007).Stargazer: The Life and Times of the Telescope. Allen & Unwin. p. 55.ISBN 978-1-74175-383-7. Retrieved6 June 2012.
  19. ^This paragraph is adapted from the 1888 edition of the Encyclopædia Britannica.
  20. ^Julia, Elton (18 July 2013)."A Light to Lighten our Darkenss: Lighthouse Optics and the Later Development of Fresnel's Revolutionary Refracting Lens 1780-1900".The International Journal for the History of Engineering & Technology.79 (2):72–76.doi:10.1179/175812109X449612 – via Taylor & Francis.
  21. ^"4.4: Spherical Refractors".Physics LibreTexts. 2 July 2019.Archived from the original on 26 November 2022. Retrieved2 July 2023.
  22. ^"Refraction at Spherical Surfaces".personal.math.ubc.ca.Archived from the original on 26 October 2021. Retrieved2 July 2023.
  23. ^Hecht, Eugene (2017). "5.2.2 Refraction at Spherical Surfaces".Optics (5th ed.). Pearson. p. 164.ISBN 978-1-292-09693-3.
  24. ^Greivenkamp 2004, p. 14
    Hecht 1987, § 6.1
  25. ^abcdHecht, Eugene (2017). "Chapter 6.1 Thick Lenses and Lens Systems".Optics (5th ed.). Pearson.ISBN 978-1-292-09693-3.
  26. ^"Rule sign for concave and convex lens?".Physics Stack Exchange. Retrieved27 October 2024.
  27. ^abcHecht, Eugene (2017). "Finite Imagery".Optics (5th ed.). Pearson.ISBN 978-1-292-09693-3.
  28. ^abHecht, Eugene (2017). "Thin-Lens Equations".Optics (5th ed.). Pearson.ISBN 978-1-292-09693-3.
  29. ^Nave, Carl R."Thin Lens Equation".Hyperphysics. Georgia State University.Archived from the original on 12 October 2000. Retrieved17 March 2015.
  30. ^Colwell, Catharine H."Resource Lesson: Thin Lens Equation".PhysicsLab.org.Archived from the original on 2 April 2015. Retrieved17 March 2015.
  31. ^"The Mathematics of Lenses".The Physics Classroom.Archived from the original on 10 March 2015. Retrieved17 March 2015.
  32. ^Tipler, Paul (1976).Physics. New York: Worth Publishers, Inc. p. 650.ISBN 087901041X.
  33. ^There are always 3 "easy rays". For the third ray in this case, seeFile:Lens3b third ray.svg.
  34. ^Hecht, Eugene (2017). "Thin-Lens Combinations".Optics (5th ed.). Pearson. p. 178.ISBN 978-1-292-09693-3.
  35. ^Vlasenko, Alexey (2011)."Lecture 9 Notes: 07 / 13 - Multiple-lens systems"(PDF).Physics 1C, Summer Session I, 2011 - University of California San Diego.Archived(PDF) from the original on 18 April 2024. Retrieved19 April 2024.
  36. ^Paschotta, Dr Rüdiger."focal distance".www.rp-photonics.com.doi:10.61835/6as.Archived from the original on 29 April 2024. Retrieved29 April 2024.
  37. ^Hecht, Eugene (2017). "Back and Front Focal Lengths".Optics (5th ed.). Pearson. p. 181.ISBN 978-1-292-09693-3.
  38. ^Proteep Mallik (2005)."The Axicon"(PDF). Archived fromthe original(PDF) on 23 November 2009. Retrieved22 November 2007.
  39. ^abGrbic, A.; Eleftheriades, G. V. (2004). "Overcoming the Diffraction Limit with a Planar Left-handed Transmission-line Lens".Physical Review Letters.92 (11) 117403.Bibcode:2004PhRvL..92k7403G.doi:10.1103/PhysRevLett.92.117403.PMID 15089166.
  40. ^Valentine, J.; et al. (2008). "Three-dimensional optical metamaterial with a negative refractive index".Nature.455 (7211):376–9.Bibcode:2008Natur.455..376V.doi:10.1038/nature07247.PMID 18690249.S2CID 4314138.
  41. ^Yao, Jie; Liu, Zhaowei; Liu, Yongmin; Wang, Yuan; Sun, Cheng; Bartal, Guy; Stacy, Angelica M.; Zhang, Xiang (15 August 2008). "Optical Negative Refraction in Bulk Metamaterials of Nanowires".Science.321 (5891): 930.Bibcode:2008Sci...321..930Y.CiteSeerX 10.1.1.716.4426.doi:10.1126/science.1157566.ISSN 0036-8075.PMID 18703734.S2CID 20978013.
  42. ^Nielsen, R.B.; Thoreson, M.D.; Chen, W.; Kristensen, A.; Hvam, J.M.; Shalaev, V. M.;Boltasseva, A. (2010)."Toward superlensing with metal–dielectric composites and multilayers"(PDF).Applied Physics B.100 (1): 93.Bibcode:2010ApPhB.100...93N.doi:10.1007/s00340-010-4065-z.S2CID 39903291. Archived fromthe original(PDF) on 9 March 2013.
  43. ^Patel, Prachi (2015)."Good-Bye to Curved Lens: New Lens Is Flat".Scientific American.312 (5): 22.doi:10.1038/scientificamerican0515-22b.PMID 26336702.Archived from the original on 19 May 2015. Retrieved16 May 2015.
  44. ^Schottner, G (May 2003). "Scratch and Abrasion Resistant Coatings on Plastic Lenses—State of the Art, Current Developments and Perspectives".Journal of Sol-Gel Science and Technology. Vol. 27. pp. 71–79.doi:10.1023/A:1022684011222.

Bibliography

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