Lens (optics)
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/* Spherical aberration */ [[Fabrication and testing of optical components]]
[[Image:lenso.png|thumb|right|A lens.]]
[[Image:Lens and wavefronts.gif|right|frame|Lenses can be used to focus light.]]
A '''lens''' is an [[optics|optical]] device with perfect or approximate [[axial symmetry]] which [[transmittance|transmits]] and [[refraction|refracts]] [[light]], converging or diverging the beam. A [[simple lens]] is a lens consisting of a single optical element. A ''compound lens'' is an array of simple lenses (elements) with a common axis; the use of multiple elements allows more [[optical aberration]]s to be corrected than is possible with a single element. Manufactured lenses are typically made of [[glass]] or [[transparency (optics)|transparent]] [[plastic]]. Elements which refract [[electromagnetic radiation]] outside the [[visual spectrum]] are also called lenses: for instance, a [[microwave]] lens can be made from [[paraffin wax]].
The archaic spelling '''''lense''''' is sometimes seen, but Merriam-Webster's medical dictionary is the only major dictionary that considers this to be correct.<ref>{{cite book | last=Brians | first=Paul | year=2003 | title=Common Errors in English | url=http://wsu.edu/~brians/errors/lense.html }}</ref>
==History==
{{seealso|History of optics}}
[[Image:GGB reflection in raindrops.jpg|thumb|right|The [[Golden Gate Bridge]] [[refraction|refracted]] in [[rain]] [[droplets]], which act as lenses]]
The oldest lens artefact is the [[Nimrud lens]], which is over three thousand years old, dating back to ancient [[Assyria]].<ref name="Nimrud lens">{{cite news |first=David |last=Whitehouse |title=World's oldest telescope? |url=http://news.bbc.co.uk/1/hi/sci/tech/380186.stm |date=1999-07-01 |work=BBC News |accessdate=2008-05-10}}</ref> [[David Brewster]] proposed that it may have been used as a [[magnifying glass]], or as a [[burning-glass]] to start fires by concentrating sunlight.<ref name="Nimrud lens"/><ref>{{cite journal | journal = Die Fortschritte der Physik | publisher = Deutsche Physikalische Gesellschaft |date=1852 | author = D. Brewster | title = On an account of a rock-crystal lens and decomposed glass found in Niniveh |language=German | url = http://books.google.com/books?id=bHwEAAAAYAAJ&pg=RA1-PA355&dq=niniveh+lens&as_brr=3&ei=ILaBR-mHEoGmswP6jqHDCw }}</ref> Assyrian craftsmen made intricate engravings, and could have used such a lens in their work. Another early reference to [[magnification]] dates back to [[ancient Egypt]]ian [[Egyptian hieroglyphs|hieroglyphs]] in the 8th century BC, which depict "simple glass meniscal lenses".<ref name=Kriss/>
The earliest written records of lenses date to [[History of Ancient Greece|Ancient Greece]], with [[Aristophanes]]' play ''[[The Clouds]]'' (424 BC) mentioning a burning-glass (a [[#Types of lenses|biconvex lens]] used to [[focus (optics)|focus]] the [[sun]]'s rays to produce fire). The writings of [[Pliny the Elder]] (23–79) also show that burning-glasses were known to the [[Roman Empire]],<ref>[[Pliny the Elder]], ''The Natural History'' (trans. John Bostock) [http://www.perseus.tufts.edu/cgi-bin/ptext?lookup=Plin.+Nat.+37.10 Book XXXVII, Chap. 10].</ref> and mentions what is arguably the earliest use of a [[corrective lens]]: [[Nero]] was said to watch the [[gladiator|gladiatorial games]] using an [[emerald]]<ref>Pliny the Elder, ''The Natural History'' (trans. John Bostock) [http://www.perseus.tufts.edu/cgi-bin/ptext?lookup=Plin.+Nat.+37.16 Book XXXVII, Chap. 16]</ref> (presumably concave to correct for [[myopia]], though the reference is vague). Both Pliny and [[Seneca the Younger]] (3 BC–65) described the magnifying effect of a glass globe filled with [[water]].
The word ''lens'' comes from the Latin name of the [[lentil]], because a double-convex lens is lentil-shaped. The genus of the lentil plant is ''[[Lens (genus)|Lens]]'', and the most commonly eaten species is ''Lens culinaris''. The lentil plant also gives its name to a [[Lens (geometry)|geometric figure]].
The [[Islamic physics|Arabian physicist]] and [[Islamic mathematics|mathematician]], [[Ibn Sahl]] (c.940–c.1000), used what is now known as [[Snell's law]] to calculate the shape of lenses.<ref>Rashed, R. (1990). "A pioneer in anaclastics: Ibn Sahl on burning mirrors and lenses." Isis, 81, 464–491.</ref> [[Ibn al-Haytham]] (965–1038), known in the [[Western world|West]] as ''Alhazen'', wrote the first major [[Optics|optical]] treatise, the ''[[Book of Optics]]'', which described how the [[lens (anatomy)|lens]] in the human [[eye]] formed an image on the [[retina]]. The earliest "historical proof of a [[Magnifying glass|magnifying ''device'']], a convex lens forming a magnified image," also dates back to the ''Book of Optics''. Its [[Latin translations of the 12th century|translation into Latin]] in the 12th century was instrumental to the invention of [[Glasses|eyeglasses]] in 13th century [[Italy]].<ref name=Kriss>{{citation|last1=Kriss|first1=Timothy C.|last2=Kriss|first2=Vesna Martich|title=History of the Operating Microscope: From Magnifying Glass to Microneurosurgery|journal=Neurosurgery|volume=42|issue=4|pages=899–907|date=April 1998|doi=10.1097/00006123-199804000-00116}}</ref>
Excavations at the [[Viking]] harbour town of [[Fröjel]], [[Gotland]], [[Sweden]] discovered in 1999 the rock crystal [[Visby lenses]], produced by turning on pole-lathes at Fröjel in the 11th to 12th century, with an imaging quality comparable to that of 1950s aspheric lenses. The Viking lenses concentrate sunlight enough to ignite fires.
Widespread use of lenses did not occur until the use of [[reading stone]]s in the 11th century and the invention of [[spectacles]], probably in [[Italy]] in the 1280s. [[Nicholas of Cusa]] is believed to have been the first to discover the benefits of [[#Types of lenses|concave lenses]] for the treatment of [[myopia]] in 1451.
The [[Abbe sine condition]], due to [[Ernst Abbe]] (1860s), is a condition that must be fulfilled by a lens or other optical system in order for it to produce sharp images of off-axis as well as on-axis objects. It revolutionized the design of optical instruments such as [[microscopes]], and helped to establish the [[Carl Zeiss]] company as a leading supplier of optical instruments.
==Construction of simple lenses==
[[Image:Refraction through glasses 090306.jpg|thumb|200px|right|Image of the city of [[Seattle]] as seen through a lens.]]
Most lenses are ''spherical lenses'': their two surfaces are parts, with the same axis as each other, of the surfaces of spheres. Each surface can be [[wiktionary:convex|''convex'']] (bulging outwards from the lens), [[wiktionary:concave|''concave'']] (depressed into the lens), or ''planar'' (flat). The line joining the centres of the spheres making up the lens surfaces is called the ''axis'' of the lens. Typically the lens axis passes through the physical centre of the lens, because of the way they are manufactured. Lenses may be cut or ground after manufacturing to give them a different shape or size. The lens axis may then not pass through the physical centre of the lens.
[[Toric lens|Toric]] or sphero-cylindrical lenses have surfaces with two different radii of curvature in two orthogonal planes. They have a different [[focal power]] in different meridians. This is a form of deliberate [[astigmatism]].
More complex are [[aspheric lens]]es. These are lenses where one or both surfaces have a shape that is neither spherical nor cylindrical. Such lenses can produce images with much less aberration than standard simple lenses.
===Types of simple lenses=== <!--Many redirects point to this section title-->
[[Image:lens2a.png|right|500px]]
Lenses are classified by the curvature of the two optical surfaces. A lens is ''biconvex'' (or ''double convex'', or just ''convex'') if both surfaces are convex, A lens with two concave surfaces is ''biconcave'' (or just ''concave''). If one of the surfaces is flat, the lens is ''plano-convex'' or ''plano-concave'' depending on the curvature of the other surface. A lens with one convex and one concave side is ''convex-concave'' or ''meniscus''. It is this type of lens that is most commonly used in [[corrective lenses#Lens shapes|corrective lenses]].
If the lens is biconvex or plano-convex, a [[collimated]] or parallel beam of light travelling parallel to the lens axis and passing through the lens will be converged (or ''focused'') to a spot on the axis, at a certain distance behind the lens (known as the ''[[focal length]]''). In this case, the lens is called a ''positive'' or ''converging'' lens.
{|
|[[Image:lens1.svg|left|390px|Biconvex lens]]
|[[Image:Large convex lens.jpg|right|250px]]
|}
{{clr}}
If the lens is biconcave or plano-concave, a collimated beam of light passing through the lens is diverged (spread); the lens is thus called a ''negative'' or ''diverging'' lens. The beam after passing through the lens appears to be emanating from a particular point on the axis in front of the lens; the distance from this point to the lens is also known as the focal length, although it is negative with respect to the focal length of a converging lens.
{|
|[[Image:lens1b.svg|left|390px|Biconcave lens]]
|[[Image:concave lens.jpg|right|250px]]
|}
{{clr}}
Convex-concave (meniscus) lenses can be either positive or negative, depending on the relative curvatures of the two surfaces. A ''negative meniscus'' lens has a steeper concave surface and will be thinner at the centre than at the periphery. Conversely, a ''positive meniscus'' lens has a steeper convex surface and will be thicker at the centre than at the periphery. An ideal [[thin lens]] with two surfaces of equal curvature would have zero [[optical power]], meaning that it would neither converge nor diverge light. All real lenses have a nonzero thickness, however, which affects the optical power. To obtain exactly zero optical power, a meniscus lens must have slightly unequal curvatures to account for the effect of the lens' thickness.
===Lensmaker's equation===
The focal length of a lens ''in air'' can be calculated from the '''lensmaker's equation''':<ref>Greivenkamp, p.14; Hecht §6.1</ref>
:<math>\frac{1}{f} = (n-1) \left[ \frac{1}{R_1} - \frac{1}{R_2} + \frac{(n-1)d}{n R_1 R_2} \right],</math>
<!--
CAUTION TO EDITORS: This equation depends on an arbitrary sign convention
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where
:<math>f</math> is the focal length of the lens,
:<math>n</math> is the [[refractive index]] of the lens material,
:<math>R_1</math> is the radius of curvature of the lens surface closest to the light source,
:<math>R_2</math> is the radius of curvature of the lens surface farthest from the light source, and
:<math>d</math> is the thickness of the lens (the distance along the lens axis between the two [[surface vertex#Surface vertices|surface vertices]]).
====Sign convention of lens radii ''R''<sub>1</sub> and ''R''<sub>2</sub>====
{{main|Radius of curvature (optics)}}
The signs of the lens' radii of curvature indicate whether the corresponding surfaces are convex or concave. The [[sign convention]] used to represent this varies, but in this article if ''R''<sub>1</sub> is positive the first surface is convex, and if ''R''<sub>1</sub> is negative the surface is concave. The signs are reversed for the back surface of the lens: if ''R''<sub>2</sub> is positive the surface is concave, and if ''R''<sub>2</sub> is negative the surface is convex. If either radius is [[infinity|infinite]], the corresponding surface is flat.
====Thin lens equation====
If ''d'' is small compared to ''R''<sub>1</sub> and ''R''<sub>2</sub>, then the ''[[thin lens]]'' approximation can be made. For a lens in air, ''f'' is then given by
:<math>\frac{1}{f} \approx \left(n-1\right)\left[ \frac{1}{R_1} - \frac{1}{R_2} \right].</math><ref>Hecht, § 5.2.3</ref>
<!--
CAUTION TO EDITORS: This equation depends on an arbitrary sign convention
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The focal length ''f'' is positive for converging lenses, and negative for diverging lenses. The value 1/''f'' is known as the ''[[optical power]]'' of the lens, measured in ''[[dioptre]]s'', which are units equal to inverse meters (m<sup>−1</sup>).
Lenses have the same focal length when light travels from the back to the front as when light goes from the front to the back, although other properties of the lens, such as the [[Aberration in optical systems|aberrations]] are not necessarily the same in both directions.
==Imaging properties==
As mentioned above, a positive or converging lens in air will focus a collimated beam travelling along the lens axis to a spot (known as the [[Focus (optics)|focal point]]) at a distance ''f'' from the lens. Conversely, a [[point source]] of light placed at the focal point will be converted into a collimated beam by the lens. These two cases are examples of [[image]] formation in lenses. In the former case, an object at an infinite distance (as represented by a collimated beam of waves) is focused to an image at the focal point of the lens. In the latter, an object at the focal length distance from the lens is imaged at infinity. The plane perpendicular to the lens axis situated at a distance ''f'' from the lens is called the ''focal plane''.
[[Image:lens3.svg|550px]]
If the distances from the object to the lens and from the lens to the image are ''S''<sub>1</sub> and ''S''<sub>2</sub> respectively, for a lens of negligible thickness, in air, the distances are related by the '''thin lens formula''':
:<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math> .
<!--
CAUTION TO EDITORS: This equation depends on an arbitrary sign convention
(explained on the page). If the signs don't match your textbook, your book
is probably using a different sign convention.
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What this means is that, if an object is placed at a distance ''S''<sub>1</sub> along the axis in front of a positive lens of focal length ''f'', a screen placed at a distance ''S''<sub>2</sub> behind the lens will have a sharp image of the object projected onto it, as long as ''S''<sub>1</sub> > ''f'' (if the lens-to-screen distance ''S''<sub>2</sub> is varied slightly, the image will become less sharp). This is the principle behind [[photography]]. The image in this case is known as a ''[[real image]]''.
[[Image:lens3b.svg|360]]
Note that if ''S''<sub>1</sub> < ''f'', ''S''<sub>2</sub> becomes negative, the image is apparently positioned on the same side of the lens as the object. Although this kind of image, known as a ''[[virtual image]]'', cannot be projected on a screen, an observer looking through the lens will see the image in its apparent calculated position. A [[magnifying glass]] creates this kind of image.
The ''[[magnification]]'' of the lens is given by:
:<math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math> ,
<!--
CAUTION TO EDITORS: This equation depends on an arbitrary sign convention
(explained on the page). If the signs don't match your textbook, your book
is probably using a different sign convention.
-->
where ''M'' is the magnification factor; if |''M''|>1, the image is larger than the object.
Notice the sign convention here shows that, if ''M'' is negative, as it is for real images, the image is upside-down with respect to the object. For virtual images, ''M'' is positive and the image is upright.
In the special case that ''S''<sub>1</sub> = ∞, then ''S''<sub>2</sub> = ''f'' and ''M'' = −''f'' / ∞ = 0. This corresponds to a collimated beam being focused to a single spot at the focal point. The size of the image in this case is not actually zero, since [[diffraction]] effects place a lower limit on the size of the image (see [[Rayleigh criterion]]).
[[Image:lens4.svg|470px]]
The formulas above may also be used for negative (diverging) lens by using a negative focal length (''f''), but for these lenses only virtual images can be formed.
For the case of lenses that are not thin, or for more complicated multi-lens optical systems, the same formulas can be used, but ''S''<sub>1</sub> and ''S''<sub>2</sub> are interpreted differently. If the system is in air or [[vacuum]], ''S''<sub>1</sub> and ''S''<sub>2</sub> are measured from the front and rear [[principal plane]]s of the system, respectively. Imaging in media with an index of refraction greater than 1 is more complicated, and is beyond the scope of this article.
==Aberrations==
{{main|Aberration in optical systems}}
Lenses do not form perfect images, and there is always some degree of distortion or ''aberration'' introduced by the lens which causes the image to be an imperfect replica of the object. Careful design of the lens system for a particular application ensures that the aberration is minimized. There are several different types of aberration which can affect image quality.
===Spherical aberration===
''[[Spherical aberration]]'' occurs because spherical surfaces are not the ideal shape with which to make a lens, but they are by far the simplest shape to which glass can be [[Fabrication and testing of optical components|ground and polished]] and so are often used. Spherical aberration causes beams parallel to, but distant from, the lens axis to be focused in a slightly different place than beams close to the axis. This manifests itself as a blurring of the image. Lenses in which closer-to-ideal, non-spherical surfaces are used are called [[aspheric lens|''aspheric'' lenses]]. These were formerly complex to make and often extremely expensive, but advances in technology have greatly reduced the manufacturing cost for such lenses. Spherical aberration can be minimised by careful choice of the curvature of the surfaces for a particular application: for instance, a plano-convex lens which is used to focus a collimated beam produces a sharper focal spot when used with the convex side towards the beam.
[[Image:lens5.svg|400px]]
===Coma===
Another type of aberration is ''[[coma (optics)|coma]]'', which derives its name from the [[comet]]-like appearance of the aberrated image. Coma occurs when an object off the optical axis of the lens is imaged, where rays pass through the lens at an angle to the axis θ. Rays which pass through the centre of the lens of focal length ''f'' are focused at a point with distance ''f'' tan θ from the axis. Rays passing through the outer margins of the lens are focused at different points, either further from the axis (positive coma) or closer to the axis (negative coma). In general, a bundle of parallel rays passing through the lens at a fixed distance from the centre of the lens are focused to a ring-shaped image in the focal plane, known as a ''comatic circle''. The sum of all these circles results in a V-shaped or comet-like flare. As with spherical aberration, coma can be minimised (and in some cases eliminated) by choosing the curvature of the two lens surfaces to match the application. Lenses in which both spherical aberration and coma are minimised are called ''bestform'' lenses.
[[Image:lens-coma.svg|400px]]
===Chromatic aberration===
''[[Chromatic aberration]]'' is caused by the [[dispersion (optics)|dispersion]] of the lens material—the variation of its [[refractive index]] ''n'' with the wavelength of light. Since, from the formulae above, ''f'' is dependent upon ''n'', it follows that different wavelengths of light will be focused to different positions. Chromatic aberration of a lens is seen as fringes of colour around the image. It can be minimised by using an [[Achromatic lens|achromatic doublet]] (or ''achromat'') in which two materials with differing dispersion are bonded together to form a single lens. This reduces the amount of chromatic aberration over a certain range of wavelengths, though it does not produce perfect correction. The use of achromats was an important step in the development of the optical microscope. An [[apochromatic|''apochromat'']] is a lens or lens system which has even better correction of chromatic aberration, combined with improved correction of spherical aberration. Apochromats are much more expensive than achromats.
Different lens materials may also be used to minimize chromatic aberration, such as specialized coatings or lenses made from the crystal [[fluorite]]. This naturally occurring substance has the highest known [[Abbe number]], indicating that the material has low dispersion.
[[Image:lens6a.svg|400px]]
[[Image:lens6b.svg|400px]]
===Other types of aberration===
Other kinds of aberration include ''[[field curvature]]'', ''[[barrel distortion|barrel]] '' and ''[[pincushion distortion]]'', and ''[[astigmatism]]''.
===Aperture diffraction===
Even if a lens is designed to minimize or eliminate the aberrations described above, the image quality is still limited by the [[diffraction]] of light passing through the lens' finite [[aperture]]. A [[diffraction-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==
{{seealso|Photographic lens|Doublet (lens)|Achromat}}
Simple lenses are subject to the [[#Aberrations|optical aberration]]s discussed above. In many cases these aberrations can be compensated for to a great extent by using a combination of simple lenses with complementary aberrations. A ''compound lens'' is a collection of simple lenses of different shapes and made of materials of different refractive indices, arranged one after the other with a common axis.
The simplest case is where lenses are placed in contact: if the lenses of focal lengths ''f''<sub>1</sub> and ''f''<sub>2</sub> are "[[thin lens|thin]]", the combined focal length ''f'' of the lenses is given by
:<math>\frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2}</math> .
Since 1/''f'' is the power of a lens, it can be seen that the powers of thin lenses in contact are additive.
If two thin lenses are separated in air by some distance ''d'', the focal length for the combined system is given by
:<math>\frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2}-\frac{d}{f_1 f_2}</math> .
The distance from the second lens to the focal point of the combined lenses is called the ''back focal length'' (BFL).
:<math> \mbox{BFL} = \frac{f_2 (d - f_1) } { d - (f_1 +f_2) }</math> .
As ''d'' tends to zero, the value of the BFL tends to the value of ''f'' given for thin lenses in contact.
If the separation distance is equal to the sum of the focal lengths (''d'' = ''f''<sub>1</sub>+''f''<sub>2</sub>), the combined focal length and BFL are infinite. This corresponds to a pair of lenses that transform a parallel (collimated) beam into another collimated beam. This type of system is called ''afocal'', since it produces no net convergence or divergence of the beam. Two lenses at this separation form the simplest type of [[optical telescope]]. Although the system does not alter the divergence of a collimated beam, it does alter the width of the beam. The magnification of such a telescope is given by
:<math>M = \frac{-f_2}{f_1}</math> ,
which is the ratio of the input beam width to the output beam width. Note the sign convention: a telescope with two convex lenses (''f''<sub>1</sub> > 0, ''f''<sub>2</sub> > 0) produces a negative magnification, indicating an inverted image. A convex plus a concave lens (''f''<sub>1</sub> > 0 > ''f''<sub>2</sub>) produces a positive magnification and the image is upright.
== Uses of lenses ==<!-- This section is linked from [[Philately]] -->
A single convex lens mounted in a frame with a handle or stand is a [[magnifying glass]].
Lenses are used as [[prosthetic]]s for the correction of [[visual impairment]]s such as [[myopia]], [[hyperopia]], [[presbyopia]], and [[astigmatism]]. (See [[corrective lens]], [[contact lens]], [[eyeglasses]].) Most lenses used for other purposes have strict [[axial symmetry]]; eyeglass lenses are only approximately symmetric. They are usually shaped to fit in a roughly oval, not circular, frame; the optical centers are placed over the [[eyeball]]s; their curvature may not be axially symmetric to correct for [[astigmatism]]. [[Sunglasses]] lenses may be designed to attenuate light without refraction.
Another use is in imaging systems such as a [[monocular]], [[binoculars]], [[telescope]], [[spotting scope]], [[telescopic sight|telescopic gun sight]], [[theodolite]], [[microscope]], [[camera]] ([[photographic lens]]) and [[projector]]. Some of these instruments produce a [[virtual image]] when applied to the human eye; others produce a [[real image]] which can be captured on [[photographic film]] or an [[optical sensor]].
Convex lenses produce an image of an object at infinity at their focus; if the [[sun]] is imaged, all the infrared energy incident on the lens is concentrated on the small image. A large lens will concentrate enough energy to heat an inflammable object on which the image falls to burning point. Such lenses, which do not need to be even approximately optically accurate, have been used as [[burning-glass]]es for hundreds of years. A modern application is the use of relatively large lenses to concentrate solar energy on relatively small [[photovoltaic cell]]s, harvesting more energy without the need to use larger, more expensive, cells.
[[Radio astronomy]] and [[radar]] systems often use [[dielectric lens]]es, commonly called a [[lens antenna]] to refract [[electromagnetic radiation]] into a collector antenna. The [[Square Kilometre Array]] [[radio telescope]], scheduled to be operational by 2020[http://www.skatelescope.org/pages/page_genpub.htm], will employ such lenses to get a collection area nearly 30 times greater than any previous antenna.
== See also ==
* [[Aberration in optical systems]]
* [[Anti-fog]]ging treatment of optical surfaces
* [[Axicon]]
* [[Back focal plane]]
* [[Bokeh]]
* [[Cardinal point (optics)]]
* [[Corrective lens]]
* [[Eyepiece]]
* [[F-number]]
* [[Fresnel lens]]
* [[Gradient index lens]]
* [[Gravitational lens]]
* [[History of lensmaking]]
* [[Lens (anatomy)]]
* [[List of lens designs]]
* [[Microscope]]
* [[Microlens]]
* [[Numerical aperture]]
* [[Optical coating]]s
* [[Optical lens design]]
* [[Optical lenticular]]
* [[Photochromic lens]]
* [[Photographic lens]]
* [[Prime lens]]
* [[Prism (optics)]]
* [[Ray tracing (physics)|Ray tracing]]
* [[Superlens]]
* [[Telescope]]
* [[Zoom lens]]
==References==
===General===
*{{cite book | first=Eugene|last=Hecht|year=1987|title=Optics|edition=2nd ed.|publisher=Addison Wesley|id=ISBN 0-201-11609-X}} Chapters 5 & 6.
*{{cite book | first=John E. | last=Greivenkamp | year=2004 | title=Field Guide to Geometrical Optics | publisher=SPIE | others=SPIE Field Guides vol. '''FG01''' | id=ISBN 0-8194-5294-7 }}
===Footnotes===
<references />
== External links ==
{{Commons|Lens}}
*[http://www.bbc.co.uk/radio4/history/inourtime/inourtime.shtml History of Optics (audio mp3)] by Simon Schaffer, Professor in History and Philosophy of Science at the [[University of Cambridge]], Jim Bennett, Director of the Museum of the History of Science at the [[University of Oxford]] and Emily Winterburn, Curator of Astronomy at the [[National Maritime Museum]] (recorded by the [[BBC]]).
* [http://www.lightandmatter.com/html_books/5op/ch04/ch04.html a chapter from an online textbook on refraction and lenses]
* [http://www.physnet.org/modules/pdfmodules/m223.pdf ''Thin Spherical Lenses ''] on [http://www.physnet.org Project PHYSNET].
* [http://www.digitalartform.com/lenses.htm Lens article at ''digitalartform.com'']
* [http://www.phy.ntnu.edu.tw/java/Lens/lens_e.html Thin Lens Java applet]
* [http://home.comcast.net/~hebsed/enoch.htm Article on Ancient Egyptian lenses]
* [http://www3.usal.es/%7Ehistologia/aplicacion/english/museum/microsco/micros01/micros01.htm picture of the Ninive rock crystal lens]
* [http://www.vias.org/simulations/simusoft_lenses.html Learning by Simulations] - Concave and Convex Lenses
* [http://luminous-landscape.com/tutorials/resolution.shtml Do Sensors “Outresolve” Lenses?]; on lens and sensor resolution interaction.
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[[Category:Optical devices]]
[[Category:Geometrical optics]]
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[[sr:Сочиво (оптика)]]
[[fi:Linssi (optiikka)]]
[[sv:Lins]]
[[te:కటకము]]
[[th:เลนส์เว้า]]
[[vi:Thấu kính]]
[[tr:Mercek]]
[[uk:Лінза]]
[[zh:透镜]]