Lens

For other uses, seeLens (disambiguation).

Alensis a transmissiveopticaldevice that focuses or disperses alight beamby means ofrefraction.Asimple lensconsists of a single piece oftransparent material,while acompound lensconsists of several simple lenses (elements), usually arranged along a commonaxis.Lenses are made from materials such asglassorplasticand areground,polished,ormoldedto 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 asmicrowavelenses,electron lenses,acoustic lenses,orexplosive lenses.

Aburning apparatusconsisting of twobiconvex lens

Lenses are used in various imaging devices such astelescopes,binoculars,andcameras.They are also used as visual aids inglassesto correct defects of vision such asmyopiaandhypermetropia.

History

See also:History of opticsandCamera lens

Light being refracted by a spherical glass container full of water.Roger Bacon,13th century

Lens forLSST,a planned sky surveying telescope

The wordlenscomes 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.

Some scholars argue that the archeological evidence indicates that there was widespread use of lenses in antiquity, spanning several millennia.The so-calledNimrud lensis 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.Others have suggested that certainEgyptian hieroglyphsdepict "simple glass meniscal lenses".

The oldest certain reference to the use of lenses is fromAristophanes' playThe Clouds(424 BCE) mentioning a burning-glass. Pliny the Elder(1st century) confirms that burning-glasses were known in the Roman period. Pliny also has the earliest known reference to the use of acorrective lenswhen he mentions thatNerowas said to watch thegladiatorialgames using anemerald(presumablyconcaveto correct fornearsightedness,though the reference is vague).Both Pliny andSeneca the Younger(3 BC–65 AD) described the magnifying effect of a glass globe filled with water.

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'sOpticsbecame available in Latin translation in the 12th century (Eugenius of Palermo1154). Between the 11th and 13th century "reading stones"were invented. These were primitive plano-convex lenses initially made by cutting a glass sphere in half. The medieval (11th or 12th century) rock crystalVisby lensesmay or may not have been intended for use as burning glasses.

Spectacleswere invented as an improvement of the "reading stones" of the high medieval period in Northern Italy in the second half of the 13th century.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,and later in the spectacle-making centres in both theNetherlandsandGermany. 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).The practical development and experimentation with lenses led to the invention of the compoundoptical microscopearound 1595, and therefracting telescopein 1608, both of which appeared in the spectacle-making centres in theNetherlands.

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.Optical theory onrefractionand experimentation was showing no single-element lens could bring all colours to a focus. This led to the invention of the compoundachromatic lensbyChester Moore HallinEnglandin 1733, an invention also claimed by fellow EnglishmanJohn Dollondin 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.

Construction of simple lenses

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 theaxisof 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.

Toricor sphero-cylindrical lenses have surfaces with two different radii of curvature in two orthogonal planes. They have a differentfocal powerin different meridians. This forms anastigmaticlens. An example is eyeglass lenses that are used to correctastigmatismin someone's eye.

Types of simple 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 twoconcavesurfaces isbiconcave(or justconcave). If one of the surfaces is flat, the lens isplano-convexorplano-concavedepending on the curvature of the other surface. A lens with one convex and one concave side isconvex-concaveormeniscus.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, acollimatedbeam of light passing through the lens converges to a spot (afocus) behind the lens. In this case, the lens is called apositiveorconverginglens. For athin lensin air, the distance from the lens to the spot is thefocal lengthof the lens, which is commonly represented byfin diagrams and equations. Anextended hemispherical lensis 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 glasstypes, its focal point lies close to the ball's surface. Because of the ball's curvature extremes compared to the lens size,optical aberrationis much worse than thin lenses, with the notable exception ofchromatic aberration.

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 anegativeordiverginglens. 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

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 meniscuslens 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 meniscuslens 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 lenswith 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

Simulation of refraction at spherical surface atDesmos

For a single refraction for a circular boundary, the relation between object and its image in theparaxial approximationis given by

$${\frac {n_{1}}{u}}+{\frac {n_{2}}{v}}={\frac {n_{2}-n_{1}}{R}}$$

whereRis the radius of the spherical surface,n₂is the refractive index of the material of the surface,n₁is the refractive index of medium (the medium other than the spherical surface material),${\textstyle u}$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), and${\textstyle v}$is the on-axis image distance from the line. Due to paraxial approximation where the line ofhis close to the vertex of the spherical surface meeting the optical axis on the left,${\textstyle u}$and${\textstyle v}$are also considered distances with respect to the vertex.

Moving${\textstyle v}$toward the right infinity leads to the first or object focal length${\textstyle f_{0}}$for the spherical surface. Similarly,${\textstyle u}$toward the left infinity leads to the second or image focal length$f_{i}$.

$${\begin{aligned}f_{0}&={\frac {n_{1}}{n_{2}-n_{1}}}R,\\f_{i}&={\frac {n_{2}}{n_{2}-n_{1}}}R\end{aligned}}$$

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

The four cases of spherical refraction

ApplyingSnell's lawon the spherical surface,$n_{1}\sin i=n_{2}\sin r\,.$

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

$${\frac {n_{2}}{v}}+{\frac {n_{1}}{u}}={\frac {n_{2}-n_{1}}{R}}\,.$$

Lensmaker's equation

The position of the focus of a spherical lens depends on the radii of curvature of the two facets.

The (effective) focal length$f$of a spherical lens in air or vacuum for paraxial rays can be calculated from thelensmaker's equation:

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

• ${\textstyle \ n\ }$is therefractive indexof the lens material;
• ${\textstyle \ R_{1}\ }$is the (signed, seebelow)radius of curvatureof the lens surface closer to the light source;
• ${\textstyle \ R_{2}\ }$is the radius of curvature of the lens surface farther from the light source; and
• ${\textstyle \ d\ }$is the thickness of the lens (the distance along the lens axis between the twosurface vertices).

The focal length${\textstyle \ f\ }$is with respect to theprincipal planesof the lens, and the locations of the principal planes${\textstyle \ h_{1}\ }$and${\textstyle \ h_{2}\ }$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.

$$\ h_{1}=-\ {\frac {\ \left(n-1\right)f\ d~}{\ n\ R_{2}\ }}\$$$$\ h_{2}=-\ {\frac {\ \left(n-1\right)f\ d~}{\ n\ R_{1}\ }}\$$

The focal length$\ f\$is positive for converging lenses, and negative for diverging lenses. Thereciprocalof the focal length,${\textstyle \ {\tfrac {1}{\ f\ }}\,}$is theoptical powerof 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 theaberrationsare not the same in both directions.

Sign convention for radii of curvatureR₁andR₂

Main article:Radius of curvature (optics)

The signs of the lens' radii of curvature indicate whether the corresponding surfaces are convex or concave. Thesign conventionused to represent this varies,but in this article apositiveRindicates a surface's center of curvature is further along in the direction of the ray travel (right, in the accompanying diagrams), whilenegativeRmeans that rays reaching the surface have already passed the center of curvature. Consequently, for external lens surfaces as diagrammed above,R₁> 0andR₂< 0indicateconvexsurfaces (used to converge light in a positive lens), whileR₁< 0andR₂> 0indicateconcavesurfaces. 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

Sign convention for Gaussian lens equation

Parameter	Meaning	+ Sign	− Sign
so	The distance between an object and a lens.	Real object	Virtual object
si	The distance between an image and a lens.	Real image	Virtual image
f	The focal length of a lens.	Conversing lens	Diverging lens
yo	The height of an object from the optical axis.	Erect object	Inverted object
yi	The height of an image from the optical axis	Erect image	Inverted image
MT	The transverse magnification in imaging (=the ratio ofyitoyo).	Erect image	Inverted image

This convention seems to be mainly used for this article, although there is another convention such asCartesian sign conventionrequiring different lens equation forms.

Thin lens approximation

Ifdis small compared toR₁andR₂then thethin lensapproximation can be made. For a lens in air,f  is then given by

$$\ {\frac {1}{\ f\ }}\approx \left(n-1\right)\left[\ {\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}\ \right]~.$$

Derivation

A Diagram for a Spherical Lens Equation with Paraxial Rays.

The spherical thin lens equation inparaxial approximationis derived here with respect to the right figure.The 1st spherical lens surface (which meets the optical axis at${\textstyle \ V_{1}\ }$as its vertex) images an on-axis object pointOto the virtual imageI,which can be described by the following equation,$$\ {\frac {\ n_{1}\ }{\ u\ }}+{\frac {\ n_{2}\ }{\ v'\ }}={\frac {\ n_{2}-n_{1}\ }{\ R_{1}\ }}~.$$For the imaging by second lens surface, by taking the above sign convention,${\textstyle \ u'=-v'+d\ }$and$$\ {\frac {n_{2}}{\ -v'+d\ }}+{\frac {\ n_{1}\ }{\ v\ }}={\frac {\ n_{1}-n_{2}\ }{\ R_{2}\ }}~.$$Adding these two equations yields$$\ {\frac {\ n_{1}\ }{u}}+{\frac {\ n_{1}\ }{v}}=\left(n_{2}-n_{1}\right)\left({\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}\right)+{\frac {\ n_{2}\ d\ }{\ \left(\ v'-d\ \right)\ v'\ }}~.$$For the thin lens approximation where$\ d\rightarrow 0\,$the 2nd term of the RHS (Right Hand Side) is gone, so

$$\ {\frac {\ n_{1}\ }{u}}+{\frac {\ n_{1}\ }{v}}=\left(n_{2}-n_{1}\right)\left({\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}\right)~.$$

The focal length$\ f\$of the thin lens is found by limiting$\ u\rightarrow -\infty \,$

$$\ {\frac {\ n_{1}\ }{\ f\ }}=\left(n_{2}-n_{1}\right)\left({\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}\right)\rightarrow {\frac {1}{\ f\ }}=\left({\frac {\ n_{2}\ }{\ n_{1}\ }}-1\right)\left({\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}\right)~.$$

So, the Gaussian thin lens equation is

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

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

$$\ {\frac {1}{\ f\ }}=\left(n-1\right)\left({\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}\right)\$$

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

Imaging properties

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 distanceffrom the lens. Conversely, apoint sourceof light placed at the focal point is converted into a collimated beam by the lens. These two cases are examples ofimageformation 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 distanceffrom the lens is called thefocal plane.

Lens equation

Forparaxial rays,if the distances from an object to a sphericalthin lens(a lens of negligible thickness) and from the lens to the image areS₁andS₂respectively, the distances are related by the (Gaussian)thin lens formula:

$${1 \over f}={1 \over S_{1}}+{1 \over S_{2}}\,.$$

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:

$$f^{2}=x_{1}x_{2}\,,$$

where$x_{1}=S_{1}-f$and$x_{2}=S_{2}-f\,.$${\textstyle x_{1}}$is positive if it is left to the front focal point${\textstyle F_{1}}$,and${\textstyle x_{2}}$is positive if it is right to the rear focal point${\textstyle F_{2}}$.Because${\textstyle f^{2}}$is positive, an object point and the corresponding imaging point made by a lens are always in opposite sides with respect to their respective focal points. (${\textstyle x_{1}}$and${\textstyle x_{2}}$are either positive or negative.)

This Newtonian form of the lens equation can be derived by using a similarity between trianglesP₁P_O1F₁andL₃L₂F₁and another similarity between trianglesL₁L₂F₂andP₂P₀₂F₂in the right figure. The similarities give the following equations and combining these results gives the Newtonian form of the lens equation.

$${\begin{array}{lcr}{\frac {y_{1}}{x_{1}}}={\frac {\left\vert y_{2}\right\vert }{f}}\\{\frac {y_{1}}{f}}={\frac {\left\vert y_{2}\right\vert }{x_{2}}}\end{array}}$$

A diagram of imaging with a single thick lens imaging.H₁andH₂are principal points whereprincipal planesof 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 imageof 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) if${\textstyle S_{1}}$,${\textstyle S_{2}}$,and${\textstyle f}$are with respect to theprincipal planesof the lens (${\textstyle f}$is theeffective focal lengthin this case).This is because of triangle similarities like the thin lens case above; similarity between trianglesP₁P_O1F₁andL₃H₁F₁and another similarity between trianglesL₁'H₂F₂andP₂P₀₂F₂in the right figure. If distancesS₁orS₂pass through amediumother than air or vacuum, then a more complicated analysis is required.

If an object is placed at a distanceS₁>ffrom a positive lens of focal lengthf,we will find an image at a distanceS₂according to this formula. If a screen is placed at a distanceS₂on 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 theretinaserves 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 ofS₁from the camera. Put another way, modifyingS₂causes objects at a differentS₁to come into perfect focus.

Virtual image formation using a positive lens as a magnifying glass.

In some cases,S₂is 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 theyappearto 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,S₁then 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 eyeto create areal imageon theretina.

Anegativelens 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≫fas 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 distanceS₁to 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)beforethe 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,expressingS₂in terms ofS₁by the lens equation (or expressingS₁in terms ofS₂), and equating the derivative ofLwith respect toS₁(orS₂) to zero. (Note thatLhas no limit in increasing so its extremum is only the minimum, at which the derivate ofLis 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 imagemay also be viewed when put on a screen.

Magnification

The linearmagnificationof an imaging system using a single lens is given by

$$M=-{\frac {S_{2}}{S_{1}}}={\frac {f}{f-S_{1}}}\ =-{\frac {f}{x_{1}}}$$

whereMis 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 ifMis negative, as it is for real images, the image is upside-down with respect to the object. For virtual imagesMis 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 magnificationMis 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 lensesorwide-angle lensesaccording 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 cmfocal length, held20 cmfrom the eye and5 cmfrom the object, produces a virtual image at infinity of infinite linear size:M= ∞.But theangular magnificationis 5, meaning that the object appears 5 times larger to the eye than without the lens. When taking a picture of themoonusing a camera with a50 mmlens, one is not concerned with the linear magnificationM≈−50 mm/380000km=−1.3×10⁻¹⁰.Rather, the plate scale of the camera is about1°/mm,from which one can conclude that the0.5 mmimage 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=fandM= −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, sincediffractionplaces 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 lengthfare shown in red. Selected rays are shown for lettersE,IandKin 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 pointfdetermines the image size while other rays help to locate the image location.

Table for thin lens imaging properties

Images of Real Objects Formed by Thin Lenses

Lens Type	Object Location	Image Type	Image Location	Lateral Image Orientation	Image Magnification	Remark
Converging lens (or positive lens)	$\infty >S_{1}>2f$	Real (rays converging to each image point)	$f<S_{2}<2f$	Inverted (opposite to the object orientation)	Diminished
Converging lens	$S_{1}=2f$	Real	$S_{2}=2f$	Inverted	Same size
Converging lens	$f<S_{1}<2f$	Real	$\infty >S_{2}>2f$	Inverted	Magnified
Converging lens	$S_{1}=f$		$\pm \infty$
Converging lens	$S_{1}<f$	Virtual (rays apparently diverging from each image point)	$\vert S_{2}\vert >S_{1}$	Erect (same to the object orientation)	Magnified	As 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)	Anywhere	Virtual	$\vert S_{2}\vert <\vert f\vert,S_{1}>\vert S_{2}\vert$	Erect	Diminished

Aberrations

Main article:Optical aberration

Lenses do not form perfect images, and always introduce some degree of distortion oraberrationthat 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

Main article:Spherical aberration

Spherical aberrationoccurs 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

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 lengthfare focused at a point with distanceftanθfrom 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 calledbestformlenses.

Chromatic aberration

Main article:Chromatic aberration

Chromatic aberrationis caused by thedispersionof the lens material—the variation of itsrefractive index,n,with the wavelength of light. Since, fromthe formulae above,fis 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. Anapochromatis 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

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

Aperture diffraction

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

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

Simple lenses are subject to theoptical aberrationsdiscussed 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 lensis 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.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. For multi-lens imaging, 3rd lens surface (the front surface of the 2nd lens) can image the image made by the 2nd surface, and 4th surface (the back surface of the 2nd lens) can also image 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 as${\textstyle s_{o1}}$and${\textstyle s_{o2}}$,and the image distances as and${\textstyle s_{i1}}$and${\textstyle s_{i2}}$.If the lenses are thin, each satisfies the thin lens formula

$${\frac {1}{f_{j}}}={\frac {1}{s_{oj}}}+{\frac {1}{s_{ij}}},$$

If the distance between the two lenses is$d$,then${\textstyle s_{o2}=d-s_{i1}}$.(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.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.,$s_{i1}=d-f_{2}$.So, the thin lens formula for the 1st lens becomes

$${\frac {1}{f_{1}}}={\frac {1}{FFD}}+{\frac {1}{d-f_{2}}}\rightarrow FFD={\frac {f_{1}(d-f_{2})}{d-(f_{1}+f_{2})}}.$$

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

$${\frac {1}{f_{2}}}={\frac {1}{BFD}}+{\frac {1}{d-f_{1}}}\rightarrow BFD={\frac {f_{2}(d-f_{1})}{d-(f_{1}+f_{2})}}.$$

A simplest case is where thin lenses are placed in contact ($d=0$). Then the combined focal lengthfof the lenses is given by

$${\frac {1}{f}}={\frac {1}{f_{1}}}+{\frac {1}{f_{2}}}\,.$$

Since1/fis 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

$${\frac {1}{f}}=\sum _{k=1}^{N}{\frac {1}{f_{k}}}$$

where${\textstyle N}$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 by $${\frac {1}{f}}={\frac {1}{f_{1}}}+{\frac {1}{f_{2}}}-{\frac {d}{f_{1}f_{2}}}\,.$$

Asdtends to zero, the focal length of the system tends to the value offgiven for thin lenses in contact. It can be shown that the same formula works for thick lenses ifdis taken as the distance between their principal planes.

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

$$M=-{\frac {f_{2}}{f_{1}}}\,,$$

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

An aspheric biconvex lens.

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

Aspheric lenseshave at least one surface that is neither spherical nor cylindrical. The more complicated shapes allow such lenses to form images with lessaberrationthan 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 lenshas 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 lensesare arrays ofmicrolensesthat are used inlenticular printingto make images that have an illusion of depth or that change when viewed from different angles.

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

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

Anaxiconhas aconicaloptical surface. It images apoint sourceinto a linealongtheoptic axis,or transforms a laser beam into a ring.

Diffractive optical elementscan function as lenses.

Superlensesare made fromnegative index metamaterialsand claim to produce images at spatial resolutions exceeding thediffraction limit.The first superlenses were made in 2004 using such ametamaterialfor microwaves.Improved versions have been made by other researchers.As of 2014the superlens has not yet been demonstrated atvisibleor near-infraredwavelengths.

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

Uses

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 asprostheticsfor the correction ofrefractive errorssuch 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' lensesare 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,camerasandprojectors.Some of these instruments produce avirtual imagewhen applied to the human eye; others produce areal imagethat can be captured onphotographic filmor anoptical sensor,or can be viewed on a screen. In these devices lenses are sometimes paired up withcurved mirrorsto make acatadioptric systemwhere the lens's spherical aberration corrects the opposite aberration in the mirror (such asSchmidtandmeniscuscorrectors).

Convex lenses produce an image of an object at infinity at their focus; if thesunis 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-glassesfor at least 2400 years.A modern application is the use of relatively large lenses toconcentrate solar energyon relatively smallphotovoltaic cells,harvesting more energy without the need to use larger and more expensive cells.

Radio astronomyandradarsystems often usedielectric lenses,commonly called alens antennato refractelectromagnetic radiationinto a collector antenna.

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

See also

• Anti-foggingtreatment of optical surfaces
• Back focal plane
• Bokeh
• Cardinal point (optics)
• Caustic (optics)
• Eyepiece
• F-number
• Gravitational lens
• Lens (anatomy)
• List of lens designs
• Numerical aperture
• Optical coatings
• Optical lens design
• Photochromic lens
• Prism (optics)
• Ray tracing
• Ray transfer matrix analysis

Bibliography

• Hecht, Eugene(1987).Optics(2nd ed.). Addison Wesley.ISBN978-0-201-11609-0.Chapters 5 & 6.
• Hecht, Eugene (2002).Optics(4th ed.). Addison Wesley.ISBN978-0-321-18878-6.
• Greivenkamp, John E. (2004).Field Guide to Geometrical Optics.SPIE Field Guides vol.FG01.SPIE.ISBN978-0-8194-5294-8.

External links

• A chapter from an online textbook on refraction and lensesArchived17 December 2009 at theWayback Machine
• Thin Spherical LensesArchived13 March 2020 at theWayback Machine(.pdf) onProject PHYSNETArchived14 May 2017 at theWayback Machine.
• Lens article atdigitalartform
• Article onAncient Egyptian lensesArchived25 May 2022 at theWayback Machine
• FDTD Animation of Electromagnetic Propagation through Convex Lens (on- and off-axis) VideoonYouTube
• The Use of Magnifying Lenses in the Classical WorldArchived13 November 2017 at theWayback Machine
• Henker, Otto (1911)."Lens".Encyclopædia Britannica.Vol. 16 (11th ed.). pp.421–427.(with 21 diagrams)

Simulations

• Learning by SimulationsArchived21 January 2010 at theWayback Machine– Concave and Convex Lenses
• OpticalRayTracerArchived6 October 2010 at theWayback Machine– Open source lens simulator (downloadable java)
• Animations demonstrating lensArchived4 April 2012 at theWayback Machineby QED