View source for Lens

{{short description|Optical device which transmits and refracts light}}
{{Other uses|Lens (disambiguation)}}<!--No, this dablink is not superfluous. Some of the listed other uses of "lens" are ambiguous with "lens (optics)".-->

{{Use dmy dates|date=December 2013}}
[[File:Double-lens burning apparatus, Ehrenfried Walther von Tschirnhaus, Kieslingswalde (today Slawonice, Poland), c. 1690 - Mathematisch-Physikalischer Salon, Dresden - DSC08133.JPG|thumb|A [[Burning glass|burning apparatus]] consisting of two [[biconvex lens]]]]
A '''lens''' is a transmissive [[optics|optical]] device that focuses or disperses a [[light beam]] by means of [[refraction]]. A [[simple lens]] consists of a single piece of [[transparent material]], while a [[#Compound lenses|compound lens]] consists of several simple lenses (''elements''), usually arranged along a common [[Optical axis|axis]]. Lenses are made from materials such as [[glass]] or [[plastic]] and are [[Grinding (abrasive cutting)|ground]], [[Polishing|polished]], or [[Molding (process)|molded]] to the required shape. A lens can focus light to form an [[image]], unlike a [[Prism (optics)|prism]], which refracts light without focusing. Devices that similarly focus or disperse waves and radiation other than visible light are also called "lenses", such as [[microwave]] lenses, [[electron lens]]es, [[acoustic lens]]es, or [[explosive lens]]es.

Lenses are used in various imaging devices such as [[telescope]]s, [[binoculars]], and [[camera]]s. They are also used as visual aids in [[glasses]] to correct defects of vision such as [[Near-sightedness|myopia]] and [[Far-sightedness|hypermetropia]].

== History ==
{{See also|History of optics|Camera lens}}
{{expand section|history after 1823|date=January 2012}}
[[File:Optics from Roger Bacon's De multiplicatone specierum.jpg|thumb|right|Light being refracted by a spherical glass container full of water. [[Roger Bacon]], 13th century]]
[[File:LSST Telescope - L1 Lens of the camera.jpg|thumb|Lens for [[Large Synoptic Survey Telescope|LSST]], a planned sky surveying telescope{{update after|2024|08|reason= Mission planned for August 2024}}]]
The word ''[[:wikt:lens|lens]]'' comes from {{lang|la|[[Lens (genus)|lēns]]}}, the Latin name of the [[lentil]] (a seed of a lentil plant), because a double-convex lens is lentil-shaped. The lentil also gives its name to a [[Lens (geometry)|geometric figure]].{{Efn|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.
* {{cite book |last=Brians |first=Paul |year=2003 |title=Common Errors in English |publisher=Franklin, Beedle & Associates |isbn=978-1-887902-89-2 |page=[https://archive.org/details/commonerrorsinen0000bria/page/125 125] |url=https://archive.org/details/commonerrorsinen0000bria/page/125 |access-date=28 June 2009 }} Reports "lense" as listed in some dictionaries, but not generally considered acceptable.
* {{cite book |year=1995 |title=Merriam-Webster's Medical Dictionary |publisher=Merriam-Webster |isbn=978-0-87779-914-6 |page=[https://archive.org/details/isbn_9780877799146/page/368 368] |url-access=registration |url=https://archive.org/details/isbn_9780877799146/page/368 }} Lists "lense" as an acceptable alternate spelling.
* {{cite web |url=https://writingexplained.org/lens-or-lense |website=writingexplained.org |title=Lens or Lense – Which is Correct? |date=2017-04-30 |access-date=21 April 2018 |archive-date=21 April 2018 |archive-url=https://web.archive.org/web/20180421163426/https://writingexplained.org/lens-or-lense |url-status=live }} Analyses the almost negligible frequency of use and concludes that the misspelling is a result of a wrong singularisation of the plural (lenses).}}

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|s2cid=191384703 }}</ref> The so-called [[Nimrud lens]] is a rock crystal artifact dated to the 7th century BCE which may or may not have been used as a magnifying glass, or a burning glass.<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 |access-date=10 May 2008 |archive-date=1 February 2009 |archive-url=https://web.archive.org/web/20090201185740/http://news.bbc.co.uk/1/hi/sci/tech/380186.stm |url-status=live }}</ref><ref>{{cite web |title=The Nimrud lens/The Layard lens |publisher=The British Museum |website=Collection database |url=https://www.britishmuseum.org/research/search_the_collection_database/search_object_details.aspx?objectid=369215&partid=1 |access-date=25 November 2012 |archive-date=19 October 2012 |archive-url=https://web.archive.org/web/20121019022100/http://www.britishmuseum.org/research/search_the_collection_database/search_object_details.aspx?objectid=369215&partid=1 |url-status=live }}</ref><ref>{{Cite book | author=D. Brewster |year=1852 | chapter=On an account of a rock-crystal lens and decomposed glass found in Niniveh | title=Die Fortschritte der Physik | publisher=Deutsche Physikalische Gesellschaft |page=355 |language=de | chapter-url={{google books|plainurl=yes|id=bHwEAAAAYAAJ|page=355}}}}</ref> Others have suggested that certain [[Egyptian hieroglyphs]]  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}}

The oldest certain reference to the use of lenses is from [[Aristophanes]]' play ''[[The Clouds]]'' (424 BCE) mentioning a burning-glass.<ref name="The Clouds" /> 
[[Pliny the Elder]] (1st century) confirms that burning-glasses were known in the Roman period.<ref>[[Pliny the Elder]], ''The Natural History'' (trans. John Bostock) [https://www.perseus.tufts.edu/cgi-bin/ptext?lookup=Plin.+Nat.+37.10 Book XXXVII, Chap. 10] {{Webarchive|url=https://web.archive.org/web/20081004061452/https://www.perseus.tufts.edu/cgi-bin/ptext?lookup=Plin.+Nat.+37.10 |date=4 October 2008 }}.</ref>
Pliny also has the earliest known reference to the use of a [[corrective lens]] when he mentions that [[Nero]] was said to watch the [[gladiator]]ial 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) [https://www.perseus.tufts.edu/cgi-bin/ptext?lookup=Plin.+Nat.+37.16 Book XXXVII, Chap. 16] {{Webarchive|url=https://web.archive.org/web/20080928081650/https://www.perseus.tufts.edu/cgi-bin/ptext?lookup=Plin.+Nat.+37.16 |date=28 September 2008 }}</ref> Both Pliny and [[Seneca the Younger]] (3 BC–65 AD) described the magnifying effect of a glass globe filled with water.

[[Ptolemy]] (2nd century) wrote a book on ''[[Optics (Ptolemy)|Optics]]'', 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 by [[Ibn Sahl (mathematician)|Ibn Sahl]] (10th century), who was in turn improved upon by [[Alhazen]] (''[[Book of Optics]]'', 11th century). The Arabic translation of Ptolemy's ''Optics'' became available in Latin translation in the 12th century ([[Eugenius of Palermo]] 1154). Between the 11th and 13th century "[[reading stone]]s" were invented. These were primitive plano-convex lenses initially made by cutting a glass sphere in half. The medieval (11th or 12th century) rock crystal [[Visby lens]]es may or may not have been intended for use as burning glasses.<ref>{{cite book |title=The Complete Book of Fire: Building Campfires for Warmth, Light, Cooking, and Survival |first1=Buck |last1=Tilton |publisher=Menasha Ridge Press |year=2005 |isbn=978-0-89732-633-9 |page=25 |url={{google books|plainurl=yes|id=Qgd4QB1Eje0C|p=PA25}}}}</ref>

[[Spectacles]] were invented as an improvement of the "reading stones" of the high medieval period in Northern Italy in the second half of the 13th century.<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=https://books.google.com/books?id=SaJlbWK_-FcC |author2=Steven John Livesey |author3=Faith Wallis |access-date=24 April 2011 |page=167 |archive-date=20 January 2023 |archive-url=https://web.archive.org/web/20230120115105/https://books.google.com/books?id=SaJlbWK_-FcC |url-status=live }}</ref> 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,<ref>Al Van Helden. [http://galileo.rice.edu/sci/instruments/telescope.html The Galileo Project > Science > The Telescope] {{Webarchive|url=https://web.archive.org/web/20040623033108/http://galileo.rice.edu/sci/instruments/telescope.html |date=23 June 2004 }}. 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=https://books.google.com/books?id=KAWwzHlDVksC |access-date=6 June 2012 |date=28 September 2003 |publisher=Courier Dover Publications |isbn=978-0-486-43265-6 |page=27 |archive-date=2 July 2023 |archive-url=https://web.archive.org/web/20230702172014/https://books.google.com/books?id=KAWwzHlDVksC |url-status=live }}</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={{google books|plainurl=yes|id=Gm4bqeBMR8cC|p=17}} |access-date=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={{google books|plainurl=yes|id=id=peIL7hVQUmwC|p=210}} |access-date=6 June 2012 |year=2007 |publisher=American Philosophical Society |isbn=978-0-87169-259-7 |page=210 }}{{Dead link|date=July 2023 |bot=InternetArchiveBot |fix-attempted=yes }}</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] {{Webarchive|url=https://web.archive.org/web/20100109122901/http://nobelprize.org/educational_games/physics/microscopes/timeline/index.html |date=9 January 2010 }}, 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={{google books|plainurl=yes|id=2LZZginzib4C|p=55}} |access-date=6 June 2012 |date=1 October 2007 |publisher=Allen & Unwin |isbn=978-1-74175-383-7 |page=55}}</ref>

{{further|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.<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.

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.<ref>{{Cite journal |last=Julia |first=Elton |date=July 18, 2013 |title=A Light to Lighten our Darkenss: Lighthouse Optics and the Later Development of Fresnel's Revolutionary Refracting Lens 1780-1900 |url=https://www.tandfonline.com/doi/abs/10.1179/175812109X449612 |journal=The International Journal for the History of Engineering & Technology |volume=79 |issue=2 |pages=72–76 |doi=10.1179/175812109X449612 |via=Taylor & Francis}}</ref>

== Construction of simple lenses<span class="anchor" id="simple_lens_anchor"></span>==
Most lenses are ''spherical lenses'': their two surfaces are parts of the surfaces of spheres. Each surface can be [[wikt:convex|''convex'']] (bulging outwards from the lens), [[wikt: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 forms an [[Astigmatism (optical systems)|astigmatic]] lens. An example is eyeglass lenses that are used to correct [[astigmatism]] in someone's eye.

===Types of simple lenses=== <!--Many redirects point to this section title-->

[[File:Lenses en.svg|Types of lenses|alt=Types of lenses|thumb]]
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''. Convex-concave lenses are most commonly used in [[corrective lenses#Lens shape|corrective lens]]es, since the shape minimizes some aberrations.

For a biconvex or plano-convex lens in a lower-index medium, a [[collimated light|collimated]] beam of light passing through the lens converges to a spot (a ''focus'') behind the lens. In this case, the lens is called a ''positive'' or ''converging'' lens. For a [[thin lens]] in air, the distance from the lens to the spot is the [[focal length]] of the lens, which is commonly represented by  {{mvar|f}}  in diagrams and equations. An [[extended hemispherical lens]] is a special type of plano-convex lens, in which the lens's curved surface is a full hemisphere and the lens is much thicker than the radius of curvature.

Another extreme case of a thick convex lens is a [[ball 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 most [[optical glass]] types, its focal point lies close to the ball's surface. Because of the ball's curvature extremes compared to the lens size, [[optical aberration]] is much worse than thin lenses, with the notable exception of [[chromatic aberration]].

{|
|-
|[[File:lens1.svg|left|390px|Biconvex lens]]
|[[File:Large convex lens.jpg|right|250px]]
|}
{{clear}}

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 a ''negative'' or ''diverging'' lens. The beam, after passing through the lens, appears to emanate from a particular point on the axis in front of the lens. For a thin lens in air, the distance from this point to the lens is the focal length, though it is negative with respect to the focal length of a converging lens.
{|
|-
|[[File:lens1b.svg|left|390px|Biconcave lens]]
|[[File:concave lens.jpg|right|250px]]
|}
{{clear}}

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.

[[File:Meniscus lenses.svg|thumb|right|upright=0.6|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. A ''negative meniscus'' lens has a steeper concave surface (with a shorter radius than the convex surface) and is thinner at the centre than at the periphery. Conversely, a ''positive meniscus'' lens has a steeper convex surface (with a shorter radius than the concave surface) and is thicker at the centre than at the periphery.

An ideal [[thin lens]] with two surfaces of equal curvature (also equal in the sign) would have zero [[optical power]] (as its focal length becomes infinity as shown in the [[#Lensmaker's equation|lensmaker'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.

{{clear}}

=== For a spherical surface ===
[[File:Refraction at spherical surface.svg|thumb|Simulation of refraction at spherical surface at [https://www.desmos.com/calculator/ax4rsqdot0 Desmos]]]
For a single refraction for a circular boundary, the relation between object and its image in the [[paraxial approximation]] is given by<ref>{{Cite web |date=2019-07-02 |title=4.4: Spherical Refractors |url=https://phys.libretexts.org/Courses/University_of_California_Davis/UCD%3A_Physics_9B__Waves_Sound_Optics_Thermodynamics_and_Fluids/04%3A_Geometrical_Optics/4.04%3A_Spherical_Refractors |access-date=2023-07-02 |website=Physics LibreTexts |language=en |archive-date=26 November 2022 |archive-url=https://web.archive.org/web/20221126132929/https://phys.libretexts.org/Courses/University_of_California_Davis/UCD%3A_Physics_9B__Waves_Sound_Optics_Thermodynamics_and_Fluids/04%3A_Geometrical_Optics/4.04%3A_Spherical_Refractors |url-status=live }}</ref><ref>{{Cite web |title=Refraction at Spherical Surfaces |url=https://personal.math.ubc.ca/~cass/courses/m309-01a/chu/MirrorsLenses/refraction-curved.htm |access-date=2023-07-02 |website=personal.math.ubc.ca |archive-date=26 October 2021 |archive-url=https://web.archive.org/web/20211026211612/https://personal.math.ubc.ca/~cass/courses/m309-01a/chu/MirrorsLenses/refraction-curved.htm |url-status=live }}</ref>

<math display="block">\frac {n_1}u + \frac {n_2}v = \frac {n_2-n_1}R</math>

where {{mvar|R}} is the radius of the spherical surface, {{math|''n''{{sub|2}}}} is the refractive index of the material of the surface, {{math|''n''{{sub|1}}}} is the refractive index of medium (the medium other than the spherical surface material), <math display="inline">u</math> 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 is ''h''), and <math display="inline">v</math> is the on-axis image distance from the line. Due to paraxial approximation where the line of ''h'' is close to the vertex of the spherical surface meeting the optical axis on the left, <math display="inline">u</math> and <math display="inline">v</math> are also considered distances with respect to the vertex.

Moving <math display="inline">v</math> toward the right infinity leads to the first or object focal length <math display="inline">f_0</math> for the spherical surface. Similarly, <math display="inline">u</math> toward the left infinity leads to the second or image focal length <math>f_i</math>.<ref>{{Cite book |last=Hecht |first=Eugene |title=Optics |publisher=Pearson |year=2017 |isbn=978-1-292-09693-3 |edition=5th |pages=164 |language=English |chapter=5.2.2 Refraction at Spherical Surfaces}}</ref>

<math display="block">\begin{align}
 f_0 &= \frac{n_1}{n_2 - n_1} R,\\
 f_i &= \frac{n_2}{n_2 - n_1} R
\end{align}</math>

Applying this equation on the two spherical surfaces of a lens and approximating the lens thickness to zero (so a thin lens) leads to the [[#Lensmaker's equation|lensmaker's formula]].

==== Derivation ====
[[File:Refraction in spherical surface.svg|thumb]]
[[File:Four spherical refractions.png|thumb|The four cases of spherical refraction]]
Applying [[Snell's law]] on the spherical surface, <math>n_1 \sin i = n_2 \sin r\,.</math>

Also in the diagram,<math display="block">\begin{align}
\tan (i - \theta) &= \frac hu \\
\tan (\theta - r) &= \frac hv \\
\sin \theta &= \frac hR
\end{align}</math>, and using [[Small-angle approximation|small angle approximation]] (paraxial approximation) and eliminating {{mvar|i}}, {{mvar|r}}, and {{mvar|θ}},

<math display="block">\frac {n_2}v + \frac {n_1}u = \frac {n_2-n_1}R\,.</math>

===Lensmaker's equation===<!--Lensmaker's equation redirects here-->
[[File:Spherical Lens.gif|thumb|alt=Simulation of the effect of lenses with different curvatures of the two facets on a collimated Gaussian beam.|The position of the focus of a spherical lens depends on the radii of curvature of the two facets.]]
The (effective) focal length <math>f</math> of a spherical lens in air or vacuum for paraxial rays can be calculated from the '''lensmaker's equation''':<ref>{{harvnb|Greivenkamp|2004|p=14}}<br/>{{harvnb|Hecht|1987|loc=§&nbsp;6.1}}</ref><ref name="Hecht-2017" />

<math display="block"> \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]\ ,</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

* <math display="inline">\ n\ </math> is the [[refractive index]] of the lens material;
* <math display="inline">\ R_1\ </math> is the (signed, see [[#Sign convention for radii of curvature R1 and R2|below]]) [[radius of curvature]] of the lens surface closer to the light source;
* <math display="inline">\ R_2\ </math> is the radius of curvature of the lens surface farther from the light source; and
* <math display="inline">\ d\ </math> is the thickness of the lens (the distance along the lens axis between the two [[surface vertex#Surface vertices|surface vertices]]).



The focal length <math display="inline">\ f\ </math> is with respect to the [[Cardinal point (optics)|principal planes]] of the lens, and the locations of the principal planes <math display="inline">\ h_1\ </math> and <math display="inline">\ h_2\ </math> 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.<ref name="Hecht-2017">{{Cite book |last=Hecht |first=Eugene |title=Optics |publisher=Pearson |year=2017 |isbn=978-1-292-09693-3 |edition=5th |pages= |chapter=Chapter 6.1 Thick Lenses and Lens Systems}}</ref>

<math display="block">\ h_1 = -\ \frac{\ \left( n - 1 \right) f\ d ~}{\ n\ R_2\ }\ </math><math display="block">\ h_2 = -\ \frac{\ \left( n - 1 \right) f\ d ~}{\ n\ R_1\ }\ </math>

The focal length <math>\ f\ </math> is positive for converging lenses, and negative for diverging lenses. The [[multiplicative inverse|reciprocal]] of the focal length, <math display="inline">\ \tfrac{ 1 }{\ f\ }\ ,</math> is the [[optical power]] of the lens. If the focal length is in metres, this gives the optical power in [[dioptre]]s (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 the [[Aberration in optical systems|aberrations]] are not the same in both directions.

==== Sign convention for radii of curvature {{math|''R''{{sub|1}}}} and {{math|''R''{{sub|2}}}} ====
{{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,<ref>{{Cite web |title=Rule sign for concave and convex lens? |url=https://physics.stackexchange.com/questions/211345/rule-sign-for-concave-and-convex-lens |access-date=2024-10-27 |website=Physics Stack Exchange |language=en}}</ref> but in this article a ''positive'' {{mvar|R}} indicates a surface's center of curvature is further along in the direction of the ray travel (right, in the accompanying diagrams), while ''negative'' {{mvar|R}} means that rays reaching the surface have already passed the center of curvature. Consequently, for external lens surfaces as diagrammed above, {{math|''R''{{sub|1}} > 0}} and {{math|''R''{{sub|2}} < 0}} indicate ''convex'' surfaces (used to converge light in a positive lens), while {{math|''R''{{sub|1}} < 0}} and {{math|''R''{{sub|2}} > 0}} indicate ''concave'' surfaces. The reciprocal of the radius of curvature is called the [[curvature]]. A flat surface has zero curvature, and its radius of curvature is [[infinity|infinite]].

==== Sign convention for other parameters ====
{| class="wikitable sortable mw-collapsible"
|+ Sign convention for Gaussian lens equation<ref name="Hecht-2017a" />
! Parameter
! Meaning
! + Sign
! − Sign
|-
|align=center| {{mvar|s}}<sub>o</sub>
| The distance between an object and a lens.
| Real object
| Virtual object
|-
|align=center| {{mvar|s}}{{sub|i}}
| The distance between an image and a lens.
| Real image
| Virtual image
|-
|align=center| {{mvar|f}}
| The focal length of a lens.
| Converging lens
| Diverging lens
|-
|align=center| {{mvar|y}}{{sub|o}}
| The height of an object from the optical axis.
| Erect object
| Inverted object
|-
|align=center| {{mvar|y}}{{sub|i}}
| The height of an image from the optical axis
| Erect image
| Inverted image
|-
|align=center| {{mvar|M}}{{sub|T}}
| The transverse magnification in imaging (&nbsp;{{math|{{=}}}} the ratio of {{mvar|y}}{{sub|i}} to {{mvar|y}}{{sub|o}}&nbsp;).
| Erect image
| Inverted image
|}
This convention seems to be mainly used for this article, although there is another convention such as [http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/lenseq.html#c2 Cartesian sign convention] requiring different lens equation forms.

==== Thin lens approximation ====
If {{mvar|d}} is small compared to {{math|''R''{{sub|1}}}} and {{math|''R''{{sub|2}}}} then the {{dfn|[[thin lens]]}} approximation can be made. For a lens in air, {{mvar|f}}&thinsp; is then given by<ref name="Hecht-2017b">{{Cite book |last=Hecht |first=Eugene |title=Optics |publisher=Pearson |year=2017 |isbn=978-1-292-09693-3 |edition=5th |language=English |chapter=Thin-Lens Equations}}</ref>

<math display="block">\ \frac{ 1 }{\ f\ } \approx \left( n - 1 \right) \left[\ \frac{ 1 }{\ R_1\ } - \frac{ 1 }{\ R_2\ }\ \right] ~.</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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==== Derivation ====
[[File:A Diagram for a Spherical Lens Equation with Paraxial Rays, 2024-08-27.png|thumb|A Diagram for a Spherical Lens Equation with Paraxial Rays.]]
The spherical thin lens equation in [[paraxial approximation]] is derived here with respect to the right figure.<ref name="Hecht-2017b" /> The 1st spherical lens surface (which meets the optical axis at <math display="inline">\ V_1\ </math> as its vertex) images an on-axis object point ''O'' to the virtual image ''I'', which can be described by the following equation,<math display="block">\ \frac{\ n_1\ }{\ u\ } + \frac{\ n_2\ }{\ v'\ } = \frac{\ n_2 - n_1\ }{\ R_1\ } ~.</math> For the imaging by second lens surface, by taking the above sign convention, <math display="inline">\ u' = - v' + d\ </math> and <math display="block">\ \frac{ n_2 }{\ -v' + d\ } + \frac{\ n_1\ }{\ v\ } = \frac{\ n_1 - n_2\ }{\ R_2\ } ~.</math> Adding these two equations yields <math display="block">\ \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'\ } ~.</math> For the thin lens approximation where <math>\ d \rightarrow 0\ ,</math> the 2nd term of the RHS (Right Hand Side) is gone, so

<math display="block">\ \frac{\ n_1\ }{ u } + \frac{\ n_1\ }{ v } = \left( n_2 - n_1 \right) \left( \frac{ 1 }{\ R_1\ } - \frac{ 1 }{\ R_2\ } \right) ~.</math>

The focal length <math>\ f\ </math> of the thin lens is found by limiting <math>\ u \rightarrow - \infty\ ,</math>

<math display="block">\ \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) ~.</math>

So, the Gaussian thin lens equation is

<math display="block">\ \frac{ 1 }{\ u\ } + \frac{ 1 }{\ v\ } = \frac{ 1 }{\ f\ } ~.</math>

For the thin lens in air or vacuum where <math display="inline">\ n_1 = 1\ </math> can be assumed, <math display="inline">\ f\ </math> becomes

<math display="block">\ \frac{ 1 }{\ f\ } = \left( n - 1 \right)\left(\frac{ 1 }{\ R_1\ } - \frac{ 1 }{\ R_2\ } \right)\ </math>

where the subscript of 2 in <math display="inline">\ n_2\ </math> 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 the [[Focus (optics)|focal point]]) at a distance {{mvar|f}} from the lens. Conversely, a [[point source]] of light placed at the focal point is 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 {{mvar|f}} from the lens is called the ''[[Cardinal point (optics)#Focal planes|{{dfn|focal plane}}]]''.

=== Lens equation ===
For [[Paraxial approximation|paraxial rays]], if the distances from an object to a spherical [[thin lens]] (a lens of negligible thickness) and from the lens to the image are {{math|''S''{{sub|1}}}} and {{math|''S''{{sub|2}}}} respectively, the distances are related by the (Gaussian) '''thin lens formula''':<ref>{{cite web |url=http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/lenseq.html |title=Thin Lens Equation |website=Hyperphysics |first=Carl R. |last=Nave |publisher=Georgia State University |access-date=March 17, 2015 |archive-date=12 October 2000 |archive-url=https://web.archive.org/web/20001012073640/http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/lenseq.html |url-status=live }}</ref><ref>{{cite web |url=http://dev.physicslab.org/Document.aspx?doctype=3&filename=GeometricOptics_ThinLensEquation.xml |title=Resource Lesson: Thin Lens Equation |website=PhysicsLab.org |first=Catharine H. |last=Colwell |access-date=March 17, 2015 |archive-date=2 April 2015 |archive-url=https://web.archive.org/web/20150402160324/http://dev.physicslab.org/Document.aspx?doctype=3&filename=GeometricOptics_ThinLensEquation.xml |url-status=live }}</ref><ref>{{cite web |url=http://www.physicsclassroom.com/class/refrn/Lesson-5/The-Mathematics-of-Lenses |title=The Mathematics of Lenses |website=The Physics Classroom |access-date=March 17, 2015 |archive-date=10 March 2015 |archive-url=https://web.archive.org/web/20150310061631/http://www.physicsclassroom.com/class/refrn/Lesson-5/The-Mathematics-of-Lenses |url-status=live }}</ref>

<math display="block">{1\over f} = {1\over S_1} + {1\over S_2}\,.</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.
-->[[File:Single Lens Imaging, 2024-05-30.png|thumb|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 crossing [[Cardinal point (optics)#Optical center|the 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:<ref name="Hecht-2017a">{{Cite book |last=Hecht |first=Eugene |title=Optics |publisher=Pearson |year=2017 |isbn=978-1-292-09693-3 |edition=5th |pages= |language=English |chapter=Finite Imagery}}</ref>

<math display="block">f^2 = x_1 x_2\,,</math>

where <math>x_1 = S_1-f</math> and <math>x_2 = S_2-f\,.</math> <math display="inline">x_1</math> is positive if it is left to the front focal point <math display="inline">F_1</math>, and <math display="inline">x_2</math> is positive if it is right to the rear focal point <math display="inline">F_2</math>. Because <math display="inline">f^2</math> 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. (<math display="inline">x_1</math> and <math display="inline">x_2</math> are either positive or negative.)   

This Newtonian form of the lens equation can be derived by using a similarity between triangles ''P''<sub>1</sub>''P''<sub>O1</sub>''F''<sub>1</sub> and ''L''<sub>3</sub>''L''<sub>2</sub>''F''<sub>1</sub> and another similarity between triangles ''L''<sub>1</sub>''L''<sub>2</sub>''F''<sub>2</sub> and ''P''<sub>2</sub>''P''<sub>02</sub>''F''<sub>2</sub> in the right figure. The similarities give the following equations and combining these results gives the Newtonian form of the lens equation.

<math display="block">\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}
</math>
[[File:Single Thick Lens Imaging, 2024-10-07.png|thumb|A diagram of imaging with a single thick lens imaging. ''H''<sub>1</sub> and ''H''<sub>2</sub> are principal points where [[Cardinal point (optics)#Principal planes and points|principal planes]] of the thick lens cross the optical axis. If the object and image spaces are the same medium, then these points are also [[Cardinal point (optics)#Nodal points|nodal points]].]]
[[File:lens3.svg|thumb|A camera lens forms a ''real image'' of a distant object.]]
The above equations also hold for thick lenses (including a compound lens made by multiple lenses, that can be treated as a thick lens) in air or vacuum (which refractive index can be treated as 1) if <math display="inline">S_1</math>, <math display="inline">S_2</math>, and <math display="inline">f</math> are with respect to the [[principal plane]]s of the lens (<math display="inline">f</math> is the [[effective focal length]] in this case).<ref name="Hecht-2017" /> This is because of triangle similarities like the thin lens case above; similarity between triangles ''P''<sub>1</sub>''P''<sub>O1</sub>''F''<sub>1</sub> and ''L''<sub>3</sub>''H''<sub>1</sub>''F''<sub>1</sub> and another similarity between triangles ''L''<sub>1</sub>'''H''<sub>2</sub>''F''<sub>2</sub> and ''P''<sub>2</sub>''P''<sub>02</sub>''F''<sub>2</sub> in the right figure. If distances {{math|''S''{{sub|1}}}} or {{math|''S''{{sub|2}}}} pass through a [[Medium (optics)|medium]] other than air or vacuum, then a more complicated analysis is required.

If an object is placed at a distance {{math|''S''{{sub|1}} > ''f''}} from a positive lens of focal length {{mvar|f}}, we will find an image at a distance {{math|''S''{{sub|2}}}} according to this formula. If a screen is placed at a distance {{math|''S''{{sub|2}}}} on the opposite side of the lens, an image is formed on it. This sort of image, which can be projected onto a screen or [[image sensor]], is known as a ''[[real image]]''. This is the principle of the [[camera]], and also of the [[human eye]], in which the [[retina]] serves as the image sensor.

The focusing adjustment of a camera adjusts {{math|''S''{{sub|2}}}}, as using an image distance different from that required by this formula produces a [[Defocus aberration|defocused]] (fuzzy) image for an object at a distance of {{math|''S''{{sub|1}}}} from the camera. Put another way, modifying {{math|''S''{{sub|2}}}} causes objects at a different {{math|''S''{{sub|1}}}} to come into perfect focus.

[[File:lens3b.svg|thumb|Virtual image formation using a positive lens as a magnifying glass.<ref>There are always 3 "easy rays". For the third ray in this case, see [[:File:Lens3b third ray.svg]].</ref>]]
In some cases, {{math|''S''{{sub|2}}}} is negative, indicating that the image is formed on the opposite side of the lens from where those rays are being considered. Since the diverging light rays emanating from the lens never come into focus, and those rays are not physically present at the point where they {{em|appear}} to form an image, this is called a [[virtual 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, {{math|''S''{{sub|1}}}} then being measured from the virtual image location behind the first lens to the second lens. This is exactly what the eye does when looking through a [[magnifying glass]]. The magnifying glass creates a (magnified) virtual image behind the magnifying glass, but those rays are then re-imaged by the [[Lens (anatomy)|lens of the eye]] to create a ''real image'' on the [[retina]].

{{multiple image
| width = 180
| image1 = lens4.svg
| caption1 = A ''negative'' lens produces a demagnified virtual image.
| image2 = Barlow lens.svg
| caption2 = A [[Barlow lens]] (B) reimages a ''virtual object'' (focus of red ray path) into a magnified real image (green rays at focus)
| align = 
| direction = 
| total_width = 
| alt1 = 
}}
Using a positive lens of focal length {{mvar|f}}, a virtual image results when {{math|''S''{{sub|1}} < ''f''}}, the lens thus being used as a magnifying glass (rather than if {{math|''S''{{sub|1}} ≫ ''f''}} as for a camera). Using a negative lens ({{math|''f'' < 0}}) with a {{em|real object}} ({{math|''S''{{sub|1}} > 0}}) can only produce a virtual image ({{math|''S''{{sub|2}} < 0}}), according to the above formula. It is also possible for the object distance {{math|''S''{{sub|1}}}} to be negative, in which case the lens sees a so-called ''virtual object''. This happens when the lens is inserted into a converging beam (being focused by a previous lens) {{em|before}} the location of its real image. In that case even a negative lens can project a real image, as is done by a [[Barlow lens]].

For a given lens with the focal length ''f'', the minimum distance between an object and the real image is 4''f'' (''S''<sub>1</sub> = ''S''<sub>2</sub> = 2''f''). This is derived by letting ''L'' = ''S''<sub>1</sub> + ''S''<sub>2</sub>, expressing ''S''<sub>2</sub> in terms of ''S''<sub>1</sub> by the lens equation (or expressing ''S''<sub>1</sub> in terms of ''S''<sub>2</sub>), and equating the derivative of ''L'' with respect to ''S''<sub>1</sub> (or ''S''<sub>2</sub>) to zero. (Note that ''L'' has no limit in increasing so its extremum is only the minimum, at which the derivate of ''L'' is zero.) 

{{multiple image
| width2            = 250
| image1            = Reflectionprojection.jpg
| caption1          = Real image of a lamp is projected onto a screen (inverted). Reflections of the lamp from both surfaces of the biconvex lens are visible.
| image2            = Convex lens (magnifying glass) and upside-down image.jpg
| caption2          = A convex lens ({{math|''f'' ≪ ''S''{{sub|1}}}}) 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 a [[magnifying glass]] ({{math|''f'' > ''S''{{sub|1}}}}). This [[real image]] may also be viewed when put on a screen.
| align             = 
| direction         = 
| total_width       = 
| alt1              = 
}}

=== Magnification ===
The linear ''[[magnification]]'' of an imaging system using a single lens is given by

<math display="block"> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1}\ = - \frac{f}{x_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 {{mvar|M}} is the magnification factor defined as the ratio of the size of an image compared to the size of the object. The sign convention here dictates that if {{mvar|M}} is negative, as it is for real images, the image is upside-down with respect to the object. For virtual images {{mvar|M}} is positive, so the image is upright.

This magnification formula provides two easy ways to distinguish converging ({{math|''f'' > 0}}) and diverging ({{math|''f'' < 0}}) lenses: For an object very close to the lens ({{math|1=0 < ''S''{{sub|1}} < {{abs|''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 ({{math|1=''S''{{sub|1}} > {{abs|''f''}} > 0}}), a converging lens would form an inverted image, whereas a diverging lens would form an upright image.

Linear magnification {{mvar|M}} is not always the most useful measure of magnifying power. For instance, when characterizing a visual telescope or binoculars that produce only a virtual image, one would be more concerned with the [[Magnification#Angular magnification|angular 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 the [[plate 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 as [[long-focus lens]]es or [[wide-angle lens]]es according to their focal lengths.

Using an inappropriate measurement of magnification can be formally correct but yield a meaningless number. For instance, using a magnifying glass of {{val|5|u=cm}} focal length, held {{val|20|u=cm}} from the eye and {{val|5|u=cm}} from the object, produces a virtual image at infinity of infinite linear size: {{math|1=''M'' = ∞}}. But the ''{{dfn|angular magnification}}'' is 5, meaning that the object appears 5 times larger to the eye than without the lens. When taking a picture of the [[moon]] using a camera with a {{val|50|u=mm}} lens, one is not concerned with the linear magnification {{math|1=''M'' ≈ {{val|-50|u=mm}} / {{val|380000|u=km}} = {{val|-1.3|e=-10}}.}} Rather, the plate scale of the camera is about {{val|1|u=°|up=mm}}, from which one can conclude that the {{val|0.5|u=mm}} image on the film corresponds to an angular size of the moon seen from earth of about 0.5°.

In the extreme case where an object is an infinite distance away, {{math|1=''S''{{sub|1}} = ∞}}, {{math|1=''S''{{sub|2}} = ''f''}} and {{math|1=''M'' = −''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, since [[diffraction]] places a lower limit on the size of the [[point spread function]]. This is called the [[diffraction limit]].

[[File:Thin lens images.svg|thumb|Images of black letters in a thin convex lens of focal length {{mvar|f}} are shown in red. Selected rays are shown for letters '''E''', '''I''' and '''K''' in blue, green and orange, respectively. '''E''' (at {{math|2''f''}}) has an equal-size, real and inverted image; '''I''' (at {{mvar|f}}) has its image at [[infinity]]; and '''K''' (at {{math|''f''/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 point ''f'' determines the image size while other rays help to locate the image location.]]

=== Table for thin lens imaging properties ===
{| class="wikitable"
|+Images of Real Objects Formed by Thin Lenses<ref name="Hecht-2017a" /> 
!Lens Type
!Object Location
!Image Type
!Image Location
!Lateral Image Orientation
!Image Magnification
!Remark
|-
|Converging lens (or positive lens)
|<math>\infty > S_1 > 2f</math>
|Real (rays converging to each image point)
|<math>f < S_2 < 2f</math>
|Inverted (opposite to the object orientation)
|Diminished
|
|-
|Converging lens
|<math>S_1 = 2f</math>
|Real
|<math>S_2 = 2f</math>
|Inverted
|Same size
|
|-
|Converging lens
|<math>f < S_1 < 2f</math>
|Real
|<math>\infty > S_2 > 2f</math>
|Inverted
|Magnified
|
|-
|Converging lens
|<math>S_1 = f</math>
|
|<math>\plusmn \infty</math>
|
|
|
|-
|Converging lens
|<math>S_1 < f</math>
|Virtual (rays apparently diverging from each image point)
|<math>\vert S_2 \vert  > S_1 </math>
|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
|<math>\vert S_2 \vert < \vert f \vert, S_1 > \vert S_2 \vert</math>
|Erect
|Diminished
|
|}

== Aberrations ==
{{Optical aberration}}
{{Main|Optical aberration}}

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

=== Spherical aberration ===
{{Main|Spherical aberration}}

''Spherical aberration'' occurs because spherical surfaces are not the ideal shape for a lens, but are by far the simplest shape to which glass can be [[Fabrication and testing of optical components|ground 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.

[[File:lens5.svg|frameless]]

=== Coma ===
{{Main|Coma (optics)}}

''Coma'', or ''comatic aberration'', 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 {{mvar|θ}}. Rays that pass through the centre of a lens of focal length {{mvar|f}} are focused at a point with distance {{math|''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'' (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 called ''bestform'' lenses.

[[File:lens-coma.svg|frameless]]

=== Chromatic aberration ===
{{Main|Chromatic aberration}}

''Chromatic aberration'' is caused by the [[dispersion (optics)|dispersion]] of the lens material—the variation of its [[refractive index]], {{mvar|n}}, with the wavelength of light. Since, from [[#Lensmaker's equation|the formulae above]], {{mvar|f}} is dependent upon {{mvar|n}}, 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 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 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 crystal [[fluorite]]. This naturally occurring substance has the highest known [[Abbe number]], indicating that the material has low dispersion.

[[File:Chromatic aberration lens diagram.svg|frameless]]
[[File:Lens6b-en.svg|frameless]]

=== Other types of aberration ===
Other kinds of aberration include ''[[field curvature]]'', [[Distortion (optics)|''barrel ''and ''pincushion distortion'']], and ''[[Astigmatism (optical systems)|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 <span class="anchor" id="compound_lens_anchor"></span> ==
{{See also|Photographic lens|Doublet (lens)|Triplet lens|Achromatic lens}}
Simple lenses are subject to the [[#Aberrations|optical aberrations]] discussed above. In many cases these aberrations can be compensated for to a great extent by using a combination of simple lenses with complementary aberrations. A ''compound lens'' is a collection of simple lenses of different shapes and made of materials of different refractive indices, arranged one after the other with a common axis.

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.<ref>{{Cite book |last=Hecht |first=Eugene |title=Optics |publisher=Pearson |year=2017 |isbn=978-1-292-09693-3 |edition=5th |pages=178 |language=English |chapter=Thin-Lens Combinations}}</ref><ref>{{Cite web |last=Vlasenko |first=Alexey |date=2011 |title=Lecture 9 Notes: 07 / 13 - Multiple-lens systems |url=https://courses.physics.ucsd.edu/2011/Summer/session1/physics1c/lecture9.pdf |url-status=live |archive-url=https://web.archive.org/web/20240418224408/https://courses.physics.ucsd.edu/2011/Summer/session1/physics1c/lecture9.pdf |archive-date=18 April 2024 |access-date=2024-04-19 |website=Physics 1C, Summer Session I, 2011 - University of California San Diego}}</ref> As shown [[#Derivation 2|above]], 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 <math display="inline">s_{o1}</math> and <math display="inline">s_{o2}</math>, and the image distances as and <math display="inline">s_{i1}</math> and <math display="inline">s_{i2}</math>. If the lenses are thin, each satisfies the thin lens formula

<math display="block">\frac{1}{f_j} = \frac{1}{s_{oj}} + \frac{1}{s_{ij}},</math>

If the distance between the two lenses is <math>d</math>, then <math display="inline">s_{o2} = d - s_{i1}</math>. (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.<ref>{{Cite journal |last=Paschotta |first=Dr Rüdiger |title=focal distance |url=https://www.rp-photonics.com/focal_distance.html |access-date=2024-04-29 |website=www.rp-photonics.com |language=en |doi=10.61835/6as |archive-date=29 April 2024 |archive-url=https://web.archive.org/web/20240429224301/https://www.rp-photonics.com/focal_distance.html |url-status=live }}</ref> 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., <math>s_{i1} = d - f_2 </math>. So, the thin lens formula for the 1st lens becomes<ref>{{Cite book |last=Hecht |first=Eugene |title=Optics |publisher=Pearson |year=2017 |isbn=978-1-292-09693-3 |edition=5th |pages=181 |language=English |chapter=Back and Front Focal Lengths}}</ref>

<math display="block">\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)}. </math>

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

<math display="block">\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)}.</math>

A simplest case is where thin lenses are placed in contact (<math>d = 0</math>). Then the combined focal length {{mvar|f}} of the lenses is given by

<math display="block">\frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2}\,.</math>

Since {{math|1/''f''}} is the power of a lens with focal length {{mvar|f}}, it can be seen that the powers of thin lenses in contact are additive. The general case of multiple thin lenses in contact is

<math display="block">\frac{1}{f} =\sum_{k = 1}^N \frac{1}{f_k}</math>

where <math display="inline">N</math> is the number of lenses.

If two thin lenses are separated in air by some distance {{mvar|d}}, then the focal length for the combined system is given by
<math display="block">\frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2}-\frac{d}{f_1 f_2}\,.</math>

As {{mvar|d}} tends to zero, the focal length of the system tends to the value of {{mvar|f}} given for thin lenses in contact. It can be shown that the same formula works for thick lenses if {{mvar|d}} is taken as the distance between their principal planes.<ref name="Hecht-2017" />

If the separation distance between two lenses is equal to the sum of their focal lengths ({{math|1=''d'' = ''f''{{sub|1}} + ''f''{{sub|2}}}}), 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 an ''[[afocal system]]'', since it produces no net convergence or divergence of the beam. Two lenses at this separation form the simplest type of [[Refracting telescope|optical 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

<math display="block">M = -\frac{f_2}{f_1}\,,</math>

which is the ratio of the output beam width to the input beam width. Note the sign convention: a telescope with two convex lenses ({{math|''f''{{sub|1}} > 0}}, {{math|''f''{{sub|2}} > 0}}) produces a negative magnification, indicating an inverted image. A convex plus a concave lens ({{math|''f''{{sub|1}} > 0 > ''f''{{sub|2}}}}) produces a positive magnification and the image is upright. For further information on simple optical telescopes, see [[Refracting telescope#Refracting telescope designs|Refracting telescope § Refracting telescope designs]].

== Non spherical types ==
[[File:Pfeilhöhe.svg|thumb|right|upright|An aspheric biconvex lens.]]
[[Cylindrical lens]]es have curvature along only one axis. They are used to focus light into a line, or to convert the elliptical light from a [[laser diode]] into a round beam. They are also used in motion picture [[anamorphic lens]]es.

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

[[File:Flat flexible plastic sheet lens.JPG|thumb|Close-up view of a flat [[Fresnel lens]].]]
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.

[[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.

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

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.

An [[axicon]] has a [[Cone (geometry)|conical]] optical surface. It images a [[point source]] into a line {{em|along}} the [[optic axis]], or transforms a laser beam into a ring.<ref name=Proteep>{{cite web|url=http://www.optics.arizona.edu/OPTI696/2005/axicon_Proteep.pdf |author=Proteep Mallik |title=The Axicon |year=2005 |access-date=22 November 2007 |url-status=dead |archive-url=https://web.archive.org/web/20091123101108/http://www.optics.arizona.edu/OPTI696/2005/axicon_Proteep.pdf |archive-date=23 November 2009 }}</ref>

[[Diffractive optical element]]s can function as lenses.

[[Superlens]]es are made from [[negative index metamaterials]] and claim to produce images at spatial resolutions exceeding the [[diffraction limit]].<ref name=Grbic>{{cite journal
 |last1=Grbic |first1=A.
 |last2=Eleftheriades |first2=G. V.
 |year=2004
 |title=Overcoming the Diffraction Limit with a Planar Left-handed Transmission-line Lens
 |journal= [[Physical Review Letters]]
 |volume=92 |issue=11 |page=117403
 |doi=10.1103/PhysRevLett.92.117403
 |pmid=15089166 |bibcode=2004PhRvL..92k7403G}}</ref> The first superlenses were made in 2004 using such a [[metamaterial]] for microwaves.<ref name=Grbic /> Improved versions have been made by other researchers.<ref name=Valenitne-J.>{{cite journal
 |last1=Valentine |first1=J.
 |year=2008
 |title=Three-dimensional optical metamaterial with a negative refractive index
 |journal=[[Nature (journal)|Nature]]
 |volume=455 |issue=7211 |doi=10.1038/nature07247
 |pmid=18690249
 |bibcode = 2008Natur.455..376V
 |display-authors=1
 |last2=Zhang
 |first2=Shuang
 |last3=Zentgraf
 |first3=Thomas
 |last4=Ulin-Avila
 |first4=Erick
 |last5=Genov
 |first5=Dentcho A.
 |last6=Bartal
 |first6=Guy
 |last7=Zhang
 |first7=Xiang
 |pages=376–9 |s2cid=4314138
 }}</ref><ref>{{Cite journal|last1=Yao|first1=Jie|last2=Liu|first2=Zhaowei|last3=Liu|first3=Yongmin|last4=Wang|first4=Yuan|last5=Sun|first5=Cheng|last6=Bartal|first6=Guy|last7=Stacy|first7=Angelica M.|last8=Zhang|first8=Xiang|date=2008-08-15|title=Optical Negative Refraction in Bulk Metamaterials of Nanowires|journal=Science|language=en|volume=321|issue=5891|pages=930|doi=10.1126/science.1157566|issn=0036-8075|pmid=18703734|bibcode=2008Sci...321..930Y|citeseerx=10.1.1.716.4426|s2cid=20978013}}</ref> {{As of|2014}} the superlens has not yet been demonstrated at [[visible frequency|visible]] or near-[[infrared]] wavelengths.<ref name=mielsen10>{{cite journal|doi=10.1007/s00340-010-4065-z |url=http://cmip.pratt.duke.edu/tomuri2009/sites/cmip.pratt.duke.edu.tomuri2009/files/pubs_purdue/2010_APB_MDC_Superlensing.pdf |title=Toward superlensing with metal–dielectric composites and multilayers |year=2010 |last1=Nielsen |first1=R.B. |last2=Thoreson |first2=M.D. |last3=Chen |first3=W. |last4=Kristensen |first4=A. |last5=Hvam |first5=J.M. |last6=Shalaev |first6=V. M. |last7=Boltasseva |first7=A. |author-link7=Alexandra Boltasseva |journal=Applied Physics B |volume=100 |issue=1 |page=93 |bibcode=2010ApPhB.100...93N |s2cid=39903291 |url-status=dead |archive-url=https://web.archive.org/web/20130309022433/http://cmip.pratt.duke.edu/tomuri2009/sites/cmip.pratt.duke.edu.tomuri2009/files/pubs_purdue/2010_APB_MDC_Superlensing.pdf |archive-date=9 March 2013 }}</ref>

A prototype flat ultrathin lens, with no curvature has been developed.<ref>{{cite journal | title=Good-Bye to Curved Lens: New Lens Is Flat | url=http://www.scientificamerican.com/article/good-bye-to-curved-lens-new-lens-is-flat | access-date=2015-05-16 | first=Prachi | last=Patel | journal=[[Scientific American]] | year=2015 | volume=312 | issue=5 | page=22 | doi=10.1038/scientificamerican0515-22b | pmid=26336702 | archive-date=19 May 2015 | archive-url=https://web.archive.org/web/20150519063304/http://www.scientificamerican.com/article/good-bye-to-curved-lens-new-lens-is-flat/ | url-status=live | url-access=subscription }}</ref>

== Uses ==
[[File:Steeldive SD8205-2.jpg|thumb|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 a [[magnifying glass]].

Lenses are used as [[prosthetic]]s for the correction of [[refractive error]]s such as [[myopia]], [[hypermetropia]], [[presbyopia]], and [[Astigmatism (optical systems)|astigmatism]]. (See [[corrective lens]], [[contact lens]], [[eyeglasses]], [[intraocular lens]].) 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|eyeballs]]; their curvature may not be axially symmetric to correct for [[Astigmatism (optical systems)|astigmatism]]. [[sunglass lens|Sunglasses' lens]]es are designed to attenuate light; sunglass lenses that also correct visual impairments can be custom made.

Other uses are in imaging systems such as [[monocular]]s, [[binoculars]], [[optical telescope|telescopes]], [[microscope]]s, [[camera]]s and [[Movie projector|projectors]]. Some of these instruments produce a [[virtual image]] when applied to the human eye; others produce a [[real image]] that 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).

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 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 as [[burning-glass]]es for at least 2400 years.<ref name="The Clouds">{{Cite book |last=Aristophanes |author-link=Aristophanes |title=The Clouds |title-link=The Clouds |date=22 Jan 2013 |publisher=Project Gutenberg |translator-last=Hickie |translator-first=William James |id=EBook #2562 |orig-year=First performed in 423 BC}}[http://www.gutenberg.org/files/2562/2562-h/2562-h.htm] {{Webarchive|url=https://web.archive.org/web/20170628204155/http://www.gutenberg.org/files/2562/2562-h/2562-h.htm|date=28 June 2017}}</ref> A modern application is the use of relatively large lenses to [[concentrator photovoltaics|concentrate solar energy]] on relatively small [[photovoltaic cell]]s, harvesting more energy without the need to use larger and 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.

Lenses can become scratched and abraded. [[abrasion (mechanical)|Abrasion]]-resistant coatings are available to help control this.<ref>{{Cite news
 | last = Schottner
 | first = G
 | title = Scratch and Abrasion Resistant Coatings on Plastic Lenses—State of the Art, Current Developments and Perspectives
 | newspaper = [[Journal of Sol-Gel Science and Technology]]
 | pages = 71–79
 | date = May 2003
 | volume = 27
 | doi = 10.1023/A:1022684011222
 }}</ref>

== See also ==
{{div col|colwidth=22em}}
* [[Anti-fog]]ging treatment 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 coating]]s
* [[Optical lens design]]
* [[Photochromic lens]]
* [[Prism (optics)]]
* [[Ray tracing (physics)|Ray tracing]]
* [[Ray transfer matrix analysis]]
{{div col end}}

== Notes ==
{{notelist}}

== References ==
{{Reflist}}

== Bibliography ==
* {{cite book |last=Hecht |first=Eugene |author-link=Eugene Hecht |url-access=registration |year=1987 |url=https://archive.org/details/optics0000hech |title=Optics |edition=2nd |publisher=Addison Wesley |isbn=978-0-201-11609-0}} Chapters 5 & 6.
* {{cite book |last=Hecht |first=Eugene |year=2002 |title=Optics |edition=4th |publisher=Addison Wesley |isbn=978-0-321-18878-6}}
* {{cite book |last=Greivenkamp |first=John E. |year=2004 |title=Field Guide to Geometrical Optics |publisher=SPIE |others=SPIE Field Guides vol. '''FG01''' |isbn=978-0-8194-5294-8 }}

== External links ==
{{Commons|Lenses}}
* [http://www.lightandmatter.com/html_books/5op/ch04/ch04.html A chapter from an online textbook on refraction and lenses] {{Webarchive|url=https://web.archive.org/web/20091217113846/http://www.lightandmatter.com/html_books/5op/ch04/ch04.html |date=17 December 2009 }}
* [http://www.physnet.org/modules/pdf_modules/m223.pdf ''Thin Spherical Lenses ''] {{Webarchive|url=https://web.archive.org/web/20200313051457/http://www.physnet.org/modules/pdf_modules/m223.pdf |date=13 March 2020 }} (.pdf) on [http://www.physnet.org/ Project PHYSNET] {{Webarchive|url=https://web.archive.org/web/20170514213748/http://www.physnet.org/ |date=14 May 2017 }}.
* [https://web.archive.org/web/20160304054022/http://www.digitalartform.com/lenses.htm Lens article at ''digitalartform.com'']
* Article on [https://pubmed.ncbi.nlm.nih.gov/11624467/#:~:text=The%20lenses%20were%20ground%20from,sense%2C%20these%20were%20multifocal%20lenses. Ancient Egyptian lenses] {{Webarchive|url=https://web.archive.org/web/20220525041434/https://pubmed.ncbi.nlm.nih.gov/11624467/#:~:text=The%20lenses%20were%20ground%20from,sense%2C%20these%20were%20multifocal%20lenses. |date=25 May 2022 }}
* {{YouTube|id=4COYF4by8Sc FDTD Animation of Electromagnetic Propagation through Convex Lens (on- and off-axis)}}
* [https://www.academia.edu/467038/The_Use_of_Magnifying_Lenses_in_the_Classical_World The Use of Magnifying Lenses in the Classical World] {{Webarchive|url=https://web.archive.org/web/20171113201612/http://www.academia.edu/467038/The_Use_of_Magnifying_Lenses_in_the_Classical_World |date=13 November 2017 }}
* {{cite EB1911 |wstitle=Lens |volume=16 |pages=421–427 |first=Otto |last=Henker |short=1 |ref=none}} (with 21 diagrams)

=== Simulations ===
* [http://www.vias.org/simulations/simusoft_lenses.html Learning by Simulations] {{Webarchive|url=https://web.archive.org/web/20100121063732/http://www.vias.org/simulations/simusoft_lenses.html |date=21 January 2010 }} – Concave and Convex Lenses
* [http://www.arachnoid.com/OpticalRayTracer/ OpticalRayTracer] {{Webarchive|url=https://web.archive.org/web/20101006040411/http://www.arachnoid.com/OpticalRayTracer/ |date=6 October 2010 }} – Open source lens simulator (downloadable java)
* [http://qed.wikina.org/lens/ Animations demonstrating lens] {{Webarchive|url=https://web.archive.org/web/20120404024718/http://qed.wikina.org/lens/ |date=4 April 2012 }} by QED

{{Authority control}}

[[Category:Lenses| ]]
[[Category:Optical components]]