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| | <br><br>Though many moms are gainfully employed, yet a lot of them prefer to stay at home in order to appear after their youngsters. The internet technology has opened new ways through which such moms can make enough money from the comfort of their homes. If you genuinely mom and that suits you staying at home in order to after your children, you can still make enough money at the convenience your home through any of these five ways.<br><br>Long, well-written copy vends. Believe it or not, if that say speaks to your target audience in language that's about them, their problems, their world, and their needs, they'll read much more you'd are lead to believe. The key, though, is that an individual say has to be about them, not your firm - at least not formerly.<br><br>To offer good place to start, 100 % possible look into any for the following marketers. The following are niches that have proven to be profitable for literally many years and they will continue for you to become profitable for centuries to you should come.<br><br>If I took Spike Lee seriously I could envision a president slinking down on his chair along with a devious watch in his eye. Around him are his collaborators and staff. He asks them if within the armed forces has any weapon or [http://www.ractiv.com/ technology] which to direct the hurricane toward New Orleans. They assure him they have not any such knowledge. Undaunted the president asks if the demolition experts could possibly blow within the levees thus wiping the actual poor individuals New Orleans. Far fetched? Not towards the mind of Spike Lee. It is a case of public record that Mr. Lee made just such remarks on CNN within evening edition of CNN, august 26, 2006.<br><br>Biodiesels could be found at gas stations nationwide, but sometimes can be hard uncover. It is also becoming since expensive as regular diesel engine. It is better for nature, but the actual same for our wallet. Water is free and easily found. There are companies that make it simple drive your on water systems. Running your car on water is remarkable the cheapest ways to boost gas mileage.<br><br>Having very best kind of tools just helps to be able to retain customers because of one's good work but additionally enable in which improve your technical know-how. A typical set of tools for cutting hair would consist of at least eight items. These are adjustable blade clipper, detachable blade clipper, straight razor, hair styling razor, shears, blending shears, clipper combs and corded trimmers. These eight products are considered always be the bare essentials.<br><br>Car found this item you ought to look onto a phone through answering podium. This may not be necessary a lot of people have a cellular phone with voicemail to perform same function as a landline or home phone line. Ensuring that you have a method of contact for business emergencies is necessary. If there is an trouble with a shipment or product you can be reached quickly to immediate resolution to individuals.<br><br>In property never keep clutter and your house clean, clean with salty water. Store water aqaurium regarding north east, keep the centre open, have a ghee candle in the south distance. Place heavy objects in the southwest of home. Sleep with your head pointing east or south, to prevent the east. Always sleep with a window open so fresh prana can enters your house. |
| {{Other uses|Lens (disambiguation){{!}}Lens}}<!--No, this dablink is not superfluous. Some of the listed other uses of "lens" are ambiguous with "lens (optics)".-->
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| [[File:BiconvexLens.jpg|thumb|right|250px|A biconvex lens.]]
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| [[File:Lens and wavefronts.gif|right|frame|Lenses can be used to focus light.]]
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| A '''lens''' is an [[optics|optical]] device which [[transmittance|transmits]] and [[refraction|refracts]] [[light]], converging or diverging the [[light beam|beam]].{{Citation needed|date=September 2010}} A [[simple lens]] consists 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. 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]].
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| The variant spelling '''''lense''''' is sometimes seen. While it is listed as an alternative spelling in some dictionaries, most mainstream dictionaries do not list it as acceptable.<ref>{{Cite book| last=Brians | first=Paul | year=2003 | title=Common Errors in English |publisher=Franklin, Beedle & Associates |isbn=1-887902-89-9| page=125| url=http://wsu.edu/~brians/errors/lense.html |accessdate=28 June 2009}} Reports "lense" as listed in some dictionaries, but not generally considered acceptable.</ref><ref>{{Cite book|title=Merriam-Webster's Medical Dictionary |publisher=Merriam-Webster |year=1995 |isbn=0-87779-914-8 |page=368}} Lists "lense" as an acceptable alternate spelling.</ref>
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| ==History==
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| {{expand section|history after 1758|date=January 2012}}
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| {{See also|History of optics|Camera lens}}
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| [[File:Nimrud lens British Museum.jpg|thumb|right|The [[Nimrud lens]]]]
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| 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]].
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| The oldest lens artifact is the [[Nimrud lens]], dating back 2700 years 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=1 July 1999 |work=BBC News |accessdate=10 May 2008}}</ref><ref>{{cite web |title=The Nimrud lens/The Layard lens |publisher=The British Museum |work=Collection database |url=http://www.britishmuseum.org/research/search_the_collection_database/search_object_details.aspx?objectid=369215&partid=1|accessdate=25 November 2012}}</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 |year=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/?id=bHwEAAAAYAAJ&pg=RA1-PA355}}</ref> 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>{{Cite journal|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|pmid=9574655}}</ref>{{Verify source|date=July 2013}}
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| 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). Some scholars argue that the archeological evidence indicates that there was widespread use of lenses in antiquity, spanning several millennia.<ref>{{cite journal |title=Lenses in antiquity |first1=George |last1=Sines |first2=Yannis A. |last2=Sakellarakis |journal=American Journal of Archaeology |volume=91 |issue=2 |year=1987 |pages=191–196 |jstor=505216 |doi=10.2307/505216 }}</ref> Such lenses were used by artisans for fine work, and for authenticating [[Seal (emblem)|seal]] impressions. The writings of [[Pliny the Elder]] (23–79) 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 written reference to a [[corrective lens]]: [[Nero]] was said to watch the [[gladiator|gladiatorial games]] using an [[emerald]] (presumably [[wikt:concave|concave]] to correct for [[myopia|nearsightedness]], though the reference is vague).<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> Both Pliny and [[Seneca the Younger]] (3 BC–65) described the magnifying effect of a glass globe filled with [[water]].
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| Excavations at the [[Viking]] harbour town of [[Fröjel]], [[Gotland]], [[Sweden]] discovered in 1999 the rock crystal [[Visby lenses]], produced by turning on [[pole lathe]]s at Fröjel in the 11th to 12th century, with an imaging quality comparable to that of 1950s [[aspheric lens]]es. The Viking lenses were capable of concentrating enough sunlight to ignite fires.<ref>{{cite book
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| |title=The Complete Book of Fire: Building Campfires for Warmth, Light, Cooking, and Survival
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| |first1=Buck
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| |last1=Tilton
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| |publisher=Menasha Ridge Press
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| |year=2005
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| |isbn=0-89732-633-4
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| |page=25
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| |url=http://books.google.com/books?id=Qgd4QB1Eje0C&pg=PA25}}
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| </ref>
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| Between the 11th and 13th century "[[reading stone]]s" were invented. Often used by [[monk]]s to assist in [[Illuminated manuscript|illuminating]] manuscripts, these were primitive [[plano-convex lens]]es initially made by cutting a glass sphere in half. As the stones were experimented with, it was slowly understood that shallower lenses [[magnification|magnified]] more effectively.
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| Lenses came into widespread use in Europe with the invention of [[spectacles]], probably in [[Italy]] in the 1280s.<ref>{{cite book|last=Glick|first=Thomas F.|title=Medieval science, technology, and medicine: an encyclopedia|year=2005|publisher=Routledge|isbn=978-0-415-96930-7|url=http://books.google.com/?id=SaJlbWK_-FcC&pg=PA167|coauthors=Steven John Livesey, Faith Wallis|accessdate=24 April 2011|page=167}}</ref> This was the start of the optical industry of grinding and polishing lenses for spectacles, first in Venice and Florence in the thirteenth century,<ref>Al Van Helden. [http://galileo.rice.edu/sci/instruments/telescope.html '''The Galileo Project > Science > The Telescope]. Galileo.rice.edu. Retrieved on 6 June 2012.</ref> and later in the spectacle-making centres in both the [[Netherlands]] and Germany.<ref>{{cite book|author=Henry C. King |title=The History of the Telescope |url=http://books.google.com/books?id=KAWwzHlDVksC&pg=PR1 |accessdate=6 June 2012 |date=28 September 2003 |publisher=Courier Dover Publications |isbn=978-0-486-43265-6 |page=27}}</ref> 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).<ref>{{cite book|author1=Paul S. Agutter |author2=Denys N. Wheatley |title=Thinking about Life: The History and Philosophy of Biology and Other Sciences |url=http://books.google.com/books?id=Gm4bqeBMR8cC&pg=PA17 |accessdate=6 June 2012 |date=12 December 2008 |publisher=Springer |isbn=978-1-4020-8865-0 |page=17}}</ref><ref>{{cite book|author=Vincent Ilardi|title=Renaissance Vision from Spectacles to Telescopes|url=http://books.google.com/books?id=peIL7hVQUmwC&pg=PA210|accessdate=6 June 2012|year=2007|publisher=American Philosophical Society|isbn=978-0-87169-259-7|page=210}}</ref> The practical development and experimentation with lenses led to the invention of the compound [[optical microscope]] around 1595, and the [[refracting telescope]] in 1608, both of which appeared in the spectacle-making centres in the [[Netherlands]].<ref>[http://nobelprize.org/educational_games/physics/microscopes/timeline/index.html Microscopes: Time Line], Nobel Foundation. Retrieved 3 April 2009</ref><ref name="LZZginzib4C page 55">{{cite book|author=Fred Watson |title=Stargazer: The Life and Times of the Telescope |url=http://books.google.com/books?id=2LZZginzib4C&pg=PA55 |accessdate=6 June 2012 |date=1 October 2007 |publisher=Allen & Unwin |isbn=978-1-74175-383-7 |page=55}}</ref>
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| 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 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.<ref>This paragraph is adapted from the 1888 edition of the Encyclopædia Britannica.</ref> Optical theory on [[refraction]] and experimentation was showing no single-element lens could bring all colours to a focus. This led to the invention of the compound [[achromatic lens]] by [[Chester Moore Hall]] in [[England]] in 1733, an invention also claimed by fellow Englishman [[John Dollond]] in a 1758 patent.
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| ==Construction of simple lenses==
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| Most lenses are ''spherical lenses'': their two surfaces are parts of the surfaces of spheres, with the lens axis ideally perpendicular to both surfaces. Each surface can be [[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.
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| [[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]].
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|
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| 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 [[Optical aberration|aberration]] than standard simple lenses. These in turn evolved into freeform (digital/adaptive/corrected curve) spectacle lenses, where up to 20,000 ray paths are calculated from the eye to the image taking into account the position of the eye and the differing back vertex distance of the lens surface and its pantoscopic tilt and face form angle. The lens surface(s) are digitally adapted at nanometre levels (normally by a diamond stylus) to eliminate spherical aberration, coma and oblique astigmatism. This type of lens design almost completely fulfills the sagittal and tangential image shell requirements first described by Tscherning in 1925 and further described by Wollaston and Ostwalt.{{Citation needed|date=April 2012}} These advanced designs of spectacle lens can improve the visual performance by up to 70% particularly in the periphery.{{Citation needed|date=April 2012}}
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| ===Types of simple lenses=== <!--Many redirects point to this section title-->
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| [[File:Lenses en.svg|450px|right|Types of lenses]]
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| 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 [[wikt:convex|convex]]. If both surfaces have the same radius of curvature, the lens is ''equiconvex''. A lens with two [[wikt:concave|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 shape|corrective lenses]].
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| If the lens is biconvex or plano-convex, a [[collimated light|collimated]] beam of light passing through the lens will be converged (or ''focused'') to a spot behind the lens. In this case, the lens is called a ''positive'' or ''converging'' lens. The distance from the lens to the spot is the [[focal length]] of the lens, which is commonly abbreviated ''f'' in diagrams and equations.
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| {|
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| |[[File:lens1.svg|left|390px|Biconvex lens]]
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| |[[File:Large convex lens.jpg|right|250px]]
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| |}
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| {{clr}}
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| 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.
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| {|
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| |[[File:lens1b.svg|left|390px|Biconcave lens]]
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| |[[File:concave lens.jpg|right|250px]]
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| |}
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| {{clr}}
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| 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 nonzero thickness, however, which causes a real lens with identical curved surfaces to be 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.
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| ===Lensmaker's equation===<!--Lensmaker's equation redirects here-->
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| The focal length of a lens ''in air'' can be calculated from the '''lensmaker's equation''':<ref>Greivenkamp, p.14; Hecht §6.1</ref>
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| :<math> {P} = \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>
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| <!--
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| CAUTION TO EDITORS: This equation depends on an arbitrary sign convention
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| (explained on the page). If the signs don't match your textbook, your book
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| is probably using a different sign convention.
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| -->
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| where
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| :<math>P</math> is the power of the lens,
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| :<math>f</math> is the focal length of the lens,
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| :<math>n</math> is the [[refractive index]] of the lens material,
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| :<math>R_1</math> is the radius of curvature (with sign, see below) of the lens surface closest to the light source,
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| :<math>R_2</math> is the radius of curvature of the lens surface farthest from the light source, and
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| :<math>d</math> is the thickness of the lens (the distance along the lens axis between the two [[surface vertex#Surface vertices|surface vertices]]).
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| ====Sign convention of lens radii ''R''<sub>1</sub> and ''R''<sub>2</sub>====
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| {{Main|Radius of curvature (optics)}}
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| 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. With this convention the signs are determined by the shapes of the lens surfaces, and are independent of the direction in which light travels through the lens.
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| ====Thin lens equation====
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| 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
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| :<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>
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| <!--
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| CAUTION TO EDITORS: This equation depends on an arbitrary sign convention
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| (explained on the page). If the signs don't match your textbook, your book
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| is probably using a different sign convention.
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| -->
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| The focal length ''f'' is positive for converging lenses, and negative for diverging lenses. The [[Multiplicative inverse|reciprocal]] of the focal length, 1/''f'', is the [[optical power]] of the lens. If the focal length is in metres, this gives the optical power in [[dioptre]]s (inverse metres).
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| 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.
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| ==Imaging properties==
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| [[File:Reflectionprojection.jpg|right|thumb|This image has three visible reflections and one visible projection of the same lamp; two reflections are on a biconvex lens.]]
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| 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''.
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| [[File:lens3.svg|550px]]
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| [[File:GGB reflection in raindrops.jpg|thumb|right|The [[Golden Gate Bridge]] [[refraction|refracted]] in [[rain]] [[droplets]], which act as lenses]]
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| [[File:Convex lens flipped image.JPG|thumb|right|Image of a plant as seen through a biconvex lens.]]
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| 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'''
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| :<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f} </math> .
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| <!--
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| CAUTION TO EDITORS: This equation depends on an arbitrary sign convention
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| (explained on the page). If the signs don't match your textbook, your book
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| -->
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| This can also be put into the "Newtonian" form:
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| :<math>x_1 x_2 = f^2,\!</math> <ref>Hecht (2002), p. 120.</ref>
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| where <math>x_1 = S_1-f</math> and <math>x_2 = S_2-f</math>.
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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]] and the [[human eye]]. The image in this case is known as a ''[[real image]]''.
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| [[File:lens3b.svg|360]]
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| 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.
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| The ''[[magnification]]'' of the lens is given by:
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| :<math> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1} </math> ,
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| <!--
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| CAUTION TO EDITORS: This equation depends on an arbitrary sign convention
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| (explained on the page). If the signs don't match your textbook, your book
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| -->
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| where ''M'' is the magnification factor; if |''M''|>1, the image is larger than the object.
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| 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.
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| 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]]).
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| [[File:lens4.svg|470px]]
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| 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.
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| 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.
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| [[Image:Thin lens images.svg|thumb|none|500px|Images of black letters in a thin convex lens of focal length ''f'' are shown in red. Selected rays are shown for letters '''E''', '''I''' and '''K''' in blue, green and orange, respectively. Note that '''E''' (at 2''f'') has an equal-size, real and inverted image; '''I''' (at ''f'') has its image at [[infinity]]; and '''K''' (at ''f''/2) has a double-size, virtual and upright image.]]
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| ==Aberrations==
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| {{Optical aberration}}
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| {{Main|Optical aberration}}
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| 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.
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| ===Spherical aberration===
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| {{Main|Spherical aberration}}
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| ''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'' lenses. These were formerly complex to make and often extremely expensive, but advances in technology have greatly reduced the manufacturing cost for such lenses. Spherical aberration can be minimised by careful choice of the curvature of the surfaces for a particular application: for instance, a plano-convex lens which is used to focus a collimated beam produces a sharper focal spot when used with the convex side towards the beam source.
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| [[File:lens5.svg|400px]]
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| ===Coma===
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| {{Main|Coma (optics)}}
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| Another type of aberration is ''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'' [[Tangent function|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.
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| [[File:lens-coma.svg|400px]]
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| ===Chromatic aberration===
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| {{Main|Chromatic aberration}}
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| ''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 [[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.
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| Different lens materials may also be used to minimise chromatic aberration, such as specialised 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.
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| [[File:Chromatic abberation lens diagram.svg|400px]]
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| [[File:Lens6b-en.svg|400px]]
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| ===Other types of aberration===
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| Other kinds of aberration include ''[[field curvature]]'', ''[[barrel distortion|barrel]] '' and ''[[pincushion distortion]]'', and ''[[astigmatism]]''.
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| ===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 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.
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| ==Compound lenses==
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| {{See also|Photographic lens|Doublet (lens)|Achromat}}
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| 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.
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| 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
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| :<math>\frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2}.</math>
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| Since 1/''f'' is the power of a lens, it can be seen that the powers of thin lenses in contact are additive.
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| If two thin lenses are separated in air by some distance ''d'', the focal length for the combined system is given by
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| :<math>\frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2}-\frac{d}{f_1 f_2}.</math>
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| The distance from the front focal point of the combined lenses to the first lens is called the ''front focal length'' (FFL):
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| :<math>{\mbox{FFL}} = \frac{f_{1}(f_{2} - d)}{(f_1 + f_2) - d} .</math><ref>Hecht (2002), p. 168.</ref>
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|
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| Similarly, the distance from the second lens to the rear focal point of the combined system is the ''back focal length'' (BFL):
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| :<math>\mbox{BFL} = \frac{f_2 (d - f_1) } { d - (f_1 +f_2) }.</math>
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| As ''d'' tends to zero, the focal lengths tend to the value of ''f'' given for thin lenses in contact.
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| If the separation distance is equal to the sum of the focal lengths (''d'' = ''f''<sub>1</sub>+''f''<sub>2</sub>), the FFL 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 an ''[[afocal system]]'', 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
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| :<math>M = -\frac{f_2}{f_1},</math>
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| which is the ratio of the output beam width to the input 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.
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| ==Other types==
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| [[Cylindrical lens]]es have curvature in only one direction. They are used to focus light into a line, or to convert the elliptical light from a [[laser diode]] into a round beam.
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| [[File:Flat flexible plastic sheet lens.JPG|thumb|240px|Close-up view of a flat [[Fresnel lens]].]]
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| A [[Fresnel 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.
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| [[Lenticular lens]]es are arrays of [[microlens]]es that are used in [[lenticular printing]] to make images that have an illusion of depth or that change when viewed from different angles.
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| A [[gradient 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.
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| An [[axicon]] has a [[Cone (geometry)|conical]] optical surface. It images a [[point source]] into a line ''along'' the [[optic axis]], or transforms a laser beam into a ring.<ref>{{cite web|url=http://www.optics.arizona.edu/OPTI696/2005/axicon_Proteep.pdf| author=Proteep Mallik| title=The Axicon| year=2005| accessdate=22 November 2007}}</ref>
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| [[Superlens]]es are made from [[metamaterial]]s with negative index of refraction. They can achieve higher resolution than is allowed by conventional optics.
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| ==Uses==
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| A single convex lens mounted in a frame with a handle or stand is a [[magnifying glass]].
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| 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 centres are placed over the [[human eyeball|eyeball]]s; their curvature may not be axially symmetric to correct for [[astigmatism]]. [[sunglass lens|Sunglasses' lens]]es are designed to attenuate light; sunglass lenses that also correct visual impairments can be custom made.
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| Other uses are in imaging systems such as [[monocular]]s, [[binoculars]], [[optical telescope|telescope]]s, [[microscope]]s, [[camera]]s and [[Movie projector|projector]]s. 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]], or can be viewed on a screen. In these devices lenses are sometimes paired up with [[curved mirror]]s to make a [[catadioptric system]] where the lens's spherical aberration corrects the opposite aberration in the mirror (such as [[Schmidt corrector plate|Schmidt]] and [[Meniscus corrector|meniscus]] correctors).
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| Convex lenses produce an image of an object at infinity at their focus; if the [[sun]] is imaged, much of the visible and infrared light incident on the lens is concentrated into the small image. A large lens will create enough intensity to burn a flammable object at the focal point. Since ignition can be achieved even with a poorly made lens, lenses have been used as [[burning-glass]]es for at least 2400 years.<ref>{{Cite journal|last=Aristophanes |title=[[The Clouds]] |year=424 BC |authorlink=Aristophanes}}</ref> 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 and more expensive cells.
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| [[Radio astronomy]] and [[radar]] systems often use [[dielectric lens]]es, commonly called a [[lens antenna]] to refract [[electromagnetic radiation]] into a collector antenna.
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| Lenses can become scratched and abraded. [[abrasion (mechanical)|Abrasion]]-resistant coatings are available to help control this.<ref>{{Cite news
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| | last = Schottner
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| | first = G
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| | title = Scratch and Abrasion Resistant Coatings on Plastic Lenses—State of the Art, Current Developments and Perspectives
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| | newspaper = [[Journal of Sol-Gel Science and Technology]]
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| | pages = 71–79
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| | date = May 2003
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| | url = http://www.springerlink.com/content/wu963135883p31r8/
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| | accessdate =28 December 2009
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| }}</ref>
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| ==See also==
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| {{colbegin|3}}
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| * [[Anti-fog]]ging treatment of optical surfaces
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| * [[Back focal plane]]
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| * [[Bokeh]]
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| * [[Cardinal point (optics)]]
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| * [[Eyepiece]]
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| * [[F-number]]
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| * [[Gravitational lens]]
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| * [[Lens (anatomy)]]
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| * [[List of lens designs]]
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| * [[Numerical aperture]]
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| * [[Optical coating]]s
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| * [[Optical lens design]]
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| * [[Photochromic lens]]
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| * [[Prism (optics)]]
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| * [[Ray tracing (physics)|Ray tracing]]
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| * [[Ray transfer matrix analysis]]
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| {{colend}}
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| ==References==
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| {{Reflist|35em}}
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| ==Bibliography==
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| *{{Cite book| first=Eugene|last=Hecht|year=1987|title=Optics|edition=2nd|publisher=Addison Wesley|isbn=0-201-11609-X}} Chapters 5 & 6.
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| *{{Cite book| first=Eugene|last=Hecht|year=2002|title=Optics|edition=4th|publisher=Addison Wesley|isbn=0-321-18878-0}}
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| *{{Cite book| first=John E. | last=Greivenkamp | year=2004 | title=Field Guide to Geometrical Optics | publisher=SPIE | others=SPIE Field Guides vol. '''FG01''' | isbn=0-8194-5294-7 }}
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| ==External links==
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| {{Commons|Lens}}
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| [[File:ThinLens.gif|thumb|right|Thin lens simulation]]
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| *[http://books.google.com/books?id=cuzYl4hx-B8C&pg=PA58 Applied photographic optics Book]
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| *[http://books.google.com/books?id=J0RX1mbhzAEC&printsec=toc&dq=bk7+optical+glass+construction&source=gbs_summary_s&cad=0#PRA1-PA58 The properties of optical glass]
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| *[http://books.google.com/books?id=_T9dX14rz64C&pg=PT415 Handbook of Ceramics, Glasses, and Diamonds]
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| * [http://books.google.com/books?id=KdYclkhSfTAC&pg=PT49 Optical glass construction]
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| *[http://www.bbc.co.uk/radio4/history/inourtime/inourtime_20070301.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]]).
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| * [http://www.lightandmatter.com/html_books/5op/ch04/ch04.html a chapter from an online textbook on refraction and lenses]
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| * [http://www.physnet.org/modules/pdf_modules/m223.pdf ''Thin Spherical Lenses ''] on [http://www.physnet.org Project PHYSNET].
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| * [http://www.digitalartform.com/lenses.htm Lens article at ''digitalartform.com'']
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| * [http://home.comcast.net/~hebsed/enoch.htm Article on Ancient Egyptian lenses]
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| * [http://www3.usal.es/%7Ehistologia/aplicacion/english/museum/microsco/micros01/micros01.htm picture of the Ninive rock crystal lens]
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| * [http://luminous-landscape.com/tutorials/resolution.shtml Do Sensors “Outresolve” Lenses?]; on lens and sensor resolution interaction.
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| * [http://www.cvimellesgriot.com/products/Documents/TechnicalGuide/fundamental-Optics.pdf Fundamental optics]
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| * [http://www.youtube.com/watch?v=4COYF4by8Sc FDTD Animation of Electromagnetic Propagation through Convex Lens (on- and off-axis) ] on YouTube
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| * [http://www.academia.edu/467038/The_Use_of_Magnifying_Lenses_in_the_Classical_World The Use of Magnifying Lenses in the Classical World]
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| ===Simulations===
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| * [http://www.vias.org/simulations/simusoft_lenses.html Learning by Simulations] – Concave and Convex Lenses
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| * [http://www.arachnoid.com/OpticalRayTracer/ OpticalRayTracer – [[GPL|Open source]] lens simulator (downloadable java)]
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| * [http://www.youtube.com/watch?v=jvdfJXUTHHw Video with a simulation of light while it passes a convex lens]
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| * [http://qed.wikina.org/lens/ Animations demonstrating lens] by QED
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| {{DEFAULTSORT:Lens (Optics)}}
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| [[Category:Lenses| ]]
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| [[Category:Optical devices]]
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| [[Category:Geometrical optics]]
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