{"history":[],"article":{"id":"a521cf64-3817-4b61-bfc6-b1c2508e75c6","title":"Lens","slug":"lens","content":":::summary\n:::\n\n# \nA **lens** is a transmissive [optical](https://bliptext.com/articles/optics) device that focuses or disperses a [light beam](https://bliptext.com/articles/light-beam) by means of [refraction](https://bliptext.com/articles/refraction).\nA [simple lens](https://bliptext.com/articles/simple-lens) consists of a single piece of [transparent material](https://bliptext.com/articles/transparent-material), while a [compound lens](https://bliptext.com/articles/) consists of several [simple lens](https://bliptext.com/articles/simple-lens)es (elements), usually arranged along a common [axis](https://bliptext.com/articles/optical-axis).\nLenses are made from materials such as [glass](https://bliptext.com/articles/glass) or [plastic](https://bliptext.com/articles/plastic) and are [ground](https://bliptext.com/articles/grinding-abrasive-cutting), [polished](https://bliptext.com/articles/polishing), or [molded](https://bliptext.com/articles/molding-process) to the required shape.\nA lens can focus light to form an [image](https://bliptext.com/articles/image), unlike a [prism](https://bliptext.com/articles/prism-optics), which refracts light without focusing.\nDevices that similarly focus or disperse waves and radiation other than visible light are also called \"lenses\", such as [microwave](https://bliptext.com/articles/microwave) lenses, [electron lenses](https://bliptext.com/articles/electron-lens), [acoustic lenses](https://bliptext.com/articles/acoustic-lens), or [explosive lenses](https://bliptext.com/articles/explosive-lens).\nLenses are used in various imaging devices such as [telescopes](https://bliptext.com/articles/telescope), [binoculars](https://bliptext.com/articles/binoculars), and [cameras](https://bliptext.com/articles/camera).\nThey are also used as visual aids in [glasses](https://bliptext.com/articles/glasses) to correct defects of vision such as [myopia](https://bliptext.com/articles/near-sightedness) and [hypermetropia](https://bliptext.com/articles/far-sightedness).\n# [A burning apparatus consisting of two biconvex lens | upload.wikimedia.org/wikipedia/commons/c/c8/Double-lens///burning///apparatus%2C///Ehrenfried///Walther///von///Tschirnhaus%2C///Kieslingswalde///%28today///Slawonice%2C///Poland%29%2C///c.///1690///-///Mathematisch-Physikalischer///Salon%2C///Dresden///-///DSC08133.JPG]\n\n# History\nThe word [lens](https://bliptext.com/articles/wikt-lens) comes from lēns, the Latin name of the [lentil](https://bliptext.com/articles/lentil) (a seed of a [lentil](https://bliptext.com/articles/lentil) plant), because a double-convex [lens](https://bliptext.com/articles/wikt-lens) is [lentil](https://bliptext.com/articles/lentil)-shaped.\nThe lentil also gives its name to a [geometric figure](https://bliptext.com/articles/lens-geometry).\nSome scholars argue that the archeological evidence indicates that there was widespread use of lenses in antiquity, spanning several millennia.\nThe so-called [Nimrud lens](https://bliptext.com/articles/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.\nOthers have suggested that certain [Egyptian hieroglyphs](https://bliptext.com/articles/egyptian-hieroglyphs) depict \"simple glass meniscal lenses\".\nThe oldest certain reference to the use of lenses is from [Aristophanes](https://bliptext.com/articles/aristophanes)' play [The Clouds](https://bliptext.com/articles/the-clouds) (424 BCE) mentioning a burning-glass.\n[Pliny the Elder](https://bliptext.com/articles/pliny-the-elder) (1st century) confirms that burning-glasses were known in the Roman period.\nPliny also has the earliest known reference to the use of a [corrective lens](https://bliptext.com/articles/corrective-lens) when he mentions that [Nero](https://bliptext.com/articles/nero) was said to watch the [gladiatorial](https://bliptext.com/articles/gladiator) games using an [emerald](https://bliptext.com/articles/emerald) (presumably [concave](https://bliptext.com/articles/concave) to correct for [nearsightedness](https://bliptext.com/articles/myopia), though the reference is vague).\nBoth Pliny and [Seneca the Younger](https://bliptext.com/articles/seneca-the-younger) (3 BC–65 AD) described the magnifying effect of a glass globe filled with water.\n[Ptolemy](https://bliptext.com/articles/ptolemy) (2nd century) wrote a book on [Optics](https://bliptext.com/articles/optics-ptolemy), which however survives only in the Latin translation of an incomplete and very poor Arabic translation.\nThe book was, however, received by medieval scholars in the Islamic world, and commented upon by [Ibn Sahl](https://bliptext.com/articles/ibn-sahl-mathematician) (10th century), who was in turn improved upon by [Alhazen](https://bliptext.com/articles/alhazen) ([Book of Optics](https://bliptext.com/articles/book-of-optics), 11th century).\nThe Arabic translation of Ptolemy's Optics became available in Latin translation in the 12th century ([Eugenius of Palermo](https://bliptext.com/articles/eugenius-of-palermo) 1154).\nBetween the 11th and 13th century \"[reading stones](https://bliptext.com/articles/reading-stone)\" were invented.\nThese were primitive plano-convex lenses initially made by cutting a glass sphere in half.\nThe medieval (11th or 12th century) rock crystal [Visby lenses](https://bliptext.com/articles/visby-lens) may or may not have been intended for use as burning glasses.\n[Spectacles](https://bliptext.com/articles/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.\nThis was the start of the optical industry of grinding and polishing lenses for spectacles, first in Venice and Florence in the late 13th century, and later in the spectacle-making centres in both the [Netherlands](https://bliptext.com/articles/netherlands) and [Germany](https://bliptext.com/articles/germany).\nSpectacle 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).\nThe practical development and experimentation with lenses led to the invention of the compound [optical microscope](https://bliptext.com/articles/optical-microscope) around 1595, and the [refracting telescope](https://bliptext.com/articles/refracting-telescope) in 1608, both of which appeared in the spectacle-making centres in the [Netherlands](https://bliptext.com/articles/netherlands).\nWith 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.\nOpticians tried to construct lenses of varying forms of curvature, wrongly assuming errors arose from defects in the spherical figure of their surfaces.\nOptical theory on [refraction](https://bliptext.com/articles/refraction) and experimentation was showing no single-element lens could bring all colours to a focus.\nThis led to the invention of the compound [achromatic lens](https://bliptext.com/articles/achromatic-lens) by [Chester Moore Hall](https://bliptext.com/articles/chester-moore-hall) in [England](https://bliptext.com/articles/england) in 1733, an invention also claimed by fellow Englishman [John Dollond](https://bliptext.com/articles/john-dollond) in a 1758 patent.\nDevelopments 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.\nWith 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.\nFresnel lens were developed that considered these constraints by featuring less material through their concentric annular sectioning.\nThey were first fully implemented into a lighthouse in 1823.\n# [Light being refracted by a spherical glass container full of water. Roger Bacon, 13th century | upload.wikimedia.org/wikipedia/commons/8/86/Optics///from///Roger///Bacon%27s///De///multiplicatone///specierum.jpg]\n# [Lens for LSST, a planned sky surveying telescope | upload.wikimedia.org/wikipedia/commons/f/f5/LSST///Telescope///-///L1///Lens///of///the///camera.jpg]\n\n# Construction of simple lenses\nMost lenses are spherical lenses: their two surfaces are parts of the surfaces of spheres.\nEach surface can be convex (bulging outwards from the lens), concave (depressed into the lens), or planar (flat).\nThe line joining the centres of the spheres making up the lens surfaces is called the axis of the lens.\nTypically the lens axis passes through the physical centre of the lens, because of the way they are manufactured.\nLenses may be cut or ground after manufacturing to give them a different shape or size.\nThe lens axis may then not pass through the physical centre of the lens.\n[Toric](https://bliptext.com/articles/toric-lens) or sphero-cylindrical lenses have surfaces with two different radii of curvature in two orthogonal planes.\nThey have a different [focal power](https://bliptext.com/articles/focal-power) in different meridians.\nThis forms an [astigmatic](https://bliptext.com/articles/astigmatism-optical-systems) lens.\nAn example is eyeglass lenses that are used to correct [astigmatism](https://bliptext.com/articles/astigmatism) in someone's eye.\n\n## Types of simple lenses\nLenses are classified by the curvature of the two optical surfaces.\nA lens is bi[convex](https://bliptext.com/articles/convex) (or double [convex](https://bliptext.com/articles/convex), or just [convex](https://bliptext.com/articles/convex)) if both surfaces are [convex](https://bliptext.com/articles/convex).\nIf both surfaces have the same radius of curvature, the lens is equiconvex.\nA lens with two [concave](https://bliptext.com/articles/concave) surfaces is bi[concave](https://bliptext.com/articles/concave) (or just [concave](https://bliptext.com/articles/concave)).\nIf one of the surfaces is flat, the lens is plano-convex or plano-concave depending on the curvature of the other surface.\nA lens with one convex and one concave side is convex-concave or meniscus.\nConvex-concave lenses are most commonly used in [corrective lenses](https://bliptext.com/articles/corrective-lenses), since the shape minimizes some aberrations.\nFor a biconvex or plano-convex lens in a lower-index medium, a [collimated](https://bliptext.com/articles/collimated-light) beam of light passing through the lens converges to a spot (a focus) behind the lens.\nIn this case, the lens is called a positive or converging lens.\nFor a [thin lens](https://bliptext.com/articles/thin-lens) in air, the distance from the lens to the spot is the [focal length](https://bliptext.com/articles/focal-length) of the lens, which is commonly represented by f in diagrams and equations.\nAn [extended hemispherical lens](https://bliptext.com/articles/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.\nAnother extreme case of a thick convex lens is a [ball lens](https://bliptext.com/articles/ball-lens), whose shape is completely round.\nWhen used in novelty photography it is often called a \"lensball\".\nA ball-shaped lens has the advantage of being omnidirectional, but for most [optical glass](https://bliptext.com/articles/optical-glass) types, its focal point lies close to the ball's surface.\nBecause of the ball's curvature extremes compared to the lens size, [optical aberration](https://bliptext.com/articles/optical-aberration) is much worse than thin lenses, with the notable exception of [chromatic aberration](https://bliptext.com/articles/chromatic-aberration).\nFor 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.\nThe beam, after passing through the lens, appears to emanate from a particular point on the axis in front of the lens.\nFor 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.\nThe behavior reverses when a lens is placed in a medium with higher refractive index than the material of the lens.\nIn this case a biconvex or plano-convex lens diverges light, and a biconcave or plano-concave one converges it.\nConvex-concave (meniscus) lenses can be either positive or negative, depending on the relative curvatures of the two surfaces.\nA 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.\nConversely, 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.\nAn ideal [thin lens](https://bliptext.com/articles/thin-lens) with two surfaces of equal curvature (also equal in the sign) would have zero [optical power](https://bliptext.com/articles/optical-power) (as its focal length becomes infinity as shown in the [lensmaker's equation](https://bliptext.com/articles/)), meaning that it would neither converge nor diverge light.\nAll real lenses have a nonzero thickness, however, which makes a real lens with identical curved surfaces slightly positive.\nTo obtain exactly zero optical power, a meniscus lens must have slightly unequal curvatures to account for the effect of the lens' thickness.\n# [450x450px | upload.wikimedia.org/wikipedia/commons/a/a7/Lenses///en.svg]\n# [Meniscus lenses: negative (top) and positive (bottom) | upload.wikimedia.org/wikipedia/commons/e/e8/Meniscus///lenses.svg]\n\n## For a spherical surface\nFor a single refraction for a circular boundary, the relation between object and its image in the [paraxial approximation](https://bliptext.com/articles/paraxial-approximation) is given by\nwhere R is the radius of the spherical surface,\nn2\nis the refractive index of the material of the surface,\nn1\nis the refractive index of medium (the medium other than the spherical surface material), u is the on-axis (on the optical axis) object distance from the line perpendicular to the axis toward the refraction point on the surface (which height is h), and v is the on-axis image distance from the line.\nDue to paraxial approximation where the line of h is close to the vertex of the spherical surface meeting the optical axis on the left, u and v are also considered distances with respect to the vertex.\nMoving v toward the right infinity leads to the first or object focal length f_0 for the spherical surface.\nSimilarly, u toward the left infinity leads to the second or image focal length f_i.\nApplying 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 formula](https://bliptext.com/articles/).\n# [Simulation of refraction at spherical surface at Desmos | upload.wikimedia.org/wikipedia/commons/4/4f/Refraction///at///spherical///surface.svg]\n\n### Derivation\nApplying [Snell's law](https://bliptext.com/articles/snell-s-law) on the spherical surface,\nAlso in the diagram,, and using [small angle approximation](https://bliptext.com/articles/small-angle-approximation) (paraxial approximation) and eliminating i, r, and θ,\n# [lens | upload.wikimedia.org/wikipedia/commons/1/1b/Refraction///in///spherical///surface.svg]\n# [The four cases of spherical refraction | upload.wikimedia.org/wikipedia/commons/8/8e/Four///spherical///refractions.png]\n\n## Lensmaker's equation\nThe (effective) focal length f of a spherical lens in air or vacuum for paraxial rays can be calculated from the **lensmaker's equation**:\nwhere\nThe focal length \\ f\\ is with respect to the [principal planes](https://bliptext.com/articles/cardinal-point-optics) of the lens, and the locations of the [principal planes](https://bliptext.com/articles/cardinal-point-optics) \\ h_1\\ and \\ h_2\\ with respect to the respective lens vertices are given by the following formulas, where it is a positive value if it is right to the respective vertex.\nThe focal length \\ f\\ is positive for converging lenses, and negative for diverging lenses.\nThe [reciprocal](https://bliptext.com/articles/multiplicative-inverse) of the focal length, is the [optical power](https://bliptext.com/articles/optical-power) of the lens.\nIf the focal length is in metres, this gives the optical power in [dioptres](https://bliptext.com/articles/dioptre) (reciprocal metres).\nLenses have the same focal length when light travels from the back to the front as when light goes from the front to the back.\nOther properties of the lens, such as the [aberrations](https://bliptext.com/articles/aberration-in-optical-systems) are not the same in both directions.\n\n### Sign convention for radii of curvature \n\nR{{sub|1}}\n\n and \n\nR{{sub|2}}\nThe signs of the lens' radii of curvature indicate whether the corresponding surfaces are convex or concave.\nThe [sign convention](https://bliptext.com/articles/sign-convention) used to represent this varies, but in this article a positive 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 R means that rays reaching the surface have already passed the center of curvature.\nConsequently, for external lens surfaces as diagrammed above,\nR1 > 0\nand\nR2 < 0\nindicate convex surfaces (used to converge light in a positive lens), while\nR1 < 0\nand\nR2 > 0\nindicate concave surfaces.\nThe reciprocal of the radius of [curvature](https://bliptext.com/articles/curvature) is called the [curvature](https://bliptext.com/articles/curvature).\nA flat surface has zero curvature, and its radius of curvature is [infinite](https://bliptext.com/articles/infinity).\n\n### Sign convention for other parameters\nThis convention seems to be mainly used for this article, although there is another convention such as [Cartesian sign convention](https://bliptext.com/articles/undefined) requiring different lens equation forms.\n\n### Thin lens approximation\nIf d is small compared to\nR1\nand\nR2\nthen the approximation can be made.\nFor a lens in air, f is then given by\n\n### Derivation\nThe spherical thin lens equation in [paraxial approximation](https://bliptext.com/articles/paraxial-approximation) is derived here with respect to the right figure.\nThe 1st spherical lens surface (which meets the optical axis at \\ V_1\\ as its vertex) images an on-axis object point O to the virtual image I, which can be described by the following equation, For the imaging by second lens surface, by taking the above sign convention, and Adding these two equations yields For the thin lens approximation where the 2nd term of the RHS (Right Hand Side) is gone, so\nThe focal length \\ f\\ of the thin lens is found by limiting\nSo, the Gaussian thin lens equation is\nFor the thin lens in air or vacuum where \\ n_1 = 1\\ can be assumed, \\ f\\ becomes\nwhere the subscript of 2 in \\ n_2\\ is dropped.\n# [A Diagram for a Spherical Lens Equation with Paraxial Rays. | upload.wikimedia.org/wikipedia/commons/8/86/A///Diagram///for///a///Spherical///Lens///Equation///with///Paraxial///Rays%2C///2024-08-27.png]\n\n# Imaging properties\nAs mentioned above, a positive or converging lens in air focuses a collimated beam travelling along the lens axis to a spot (known as the [focal point](https://bliptext.com/articles/focus-optics)) at a distance f from the lens.\nConversely, a [point source](https://bliptext.com/articles/point-source) of light placed at the focal point is converted into a collimated beam by the lens.\nThese two cases are examples of [image](https://bliptext.com/articles/image) formation in lenses.\nIn 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.\nIn the latter, an object at the focal length distance from the lens is imaged at infinity.\nThe plane perpendicular to the lens axis situated at a distance f from the lens is called the [Cardinal point (optics)](https://bliptext.com/articles/cardinal-point-optics).\n\n## Lens equation\nFor [paraxial rays](https://bliptext.com/articles/paraxial-approximation), if the distances from an object to a spherical [thin lens](https://bliptext.com/articles/thin-lens) (a lens of negligible thickness) and from the lens to the image are\nS1\nand\nS2\nrespectively, the distances are related by the (Gaussian) **thin lens formula**:\nThe 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 [the optical center of the lens](https://bliptext.com/articles/cardinal-point-optics) (so its direction does not change), and the third tray toward the first focal point and refracted to the direction parallel to the optical axis.\nThis is a simple ray tracing method easily used.\nTwo rays among the three are sufficient to locate the image point.\nBy 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.\nThe lens equation can also be put into the \"Newtonian\" form:\nwhere x_1 = S_1-f and x_1 is positive if it is left to the front focal point F_1, and x_2 is positive if it is right to the rear focal point F_2.\nBecause f^2 is positive, an object point and the corresponding imaging point made by a lens are always in opposite sides with respect to their respective focal points.\n(x_1 and x_2 are either positive or negative.)\nThis Newtonian form of the lens equation can be derived by using a similarity between triangles P1PO1F1 and L3L2F1 and another similarity between triangles L1L2F2 and P2P02F2 in the right figure.\nThe similarities give the following equations and combining these results gives the Newtonian form of the lens equation.\nThe 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 S_1, S_2, and f are with respect to the [principal planes](https://bliptext.com/articles/principal-plane) of the lens (f is the [effective focal length](https://bliptext.com/articles/effective-focal-length) in this case).\nThis is because of triangle similarities like the thin lens case above; similarity between triangles P1PO1F1 and L3H1F1 and another similarity between triangles L1'H2F2 and P2P02F2 in the right figure.\nIf distances\nS1\nor\nS2\npass through a [medium](https://bliptext.com/articles/medium-optics) other than air or vacuum, then a more complicated analysis is required.\nIf an object is placed at a distance\nS1 > f\nfrom a positive lens of focal length f, we will find an image at a distance\nS2\naccording to this formula.\nIf a screen is placed at a distance\nS2\non the opposite side of the lens, an image is formed on it.\nThis sort of image, which can be projected onto a screen or [image sensor](https://bliptext.com/articles/image-sensor), is known as a [real image](https://bliptext.com/articles/real-image).\nThis is the principle of the [camera](https://bliptext.com/articles/camera), and also of the [human eye](https://bliptext.com/articles/human-eye), in which the [retina](https://bliptext.com/articles/retina) serves as the image sensor.\nThe focusing adjustment of a camera adjusts\nS2\n, as using an image distance different from that required by this formula produces a [defocused](https://bliptext.com/articles/defocus-aberration) (fuzzy) image for an object at a distance of\nS1\nfrom the camera.\nPut another way, modifying\nS2\ncauses objects at a different\nS1\nto come into perfect focus.\nIn some cases,\nS2\nis negative, indicating that the image is formed on the opposite side of the lens from where those rays are being considered.\nSince the diverging light rays emanating from the lens never come into focus, and those rays are not physically present at the point where they to form an image, this is called a [virtual image](https://bliptext.com/articles/virtual-image).\nUnlike 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.\nLikewise, 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,\nS1\nthen being measured from the virtual image location behind the first lens to the second lens.\nThis is exactly what the eye does when looking through a [magnifying glass](https://bliptext.com/articles/magnifying-glass).\nThe magnifying glass creates a (magnified) virtual image behind the magnifying glass, but those rays are then re-imaged by the [lens of the eye](https://bliptext.com/articles/lens-anatomy) to create a real image on the [retina](https://bliptext.com/articles/retina).\nUsing a positive lens of focal length f, a virtual image results when\nS1 < f\n, the lens thus being used as a magnifying glass (rather than if\nS1 ≫ f\nas for a camera). Using a negative lens (\nf < 0\n) with a (\nS1 > 0\n) can only produce a virtual image (\nS2 < 0\n), according to the above formula. It is also possible for the object distance\nS1\nto be negative, in which case the lens sees a so-called virtual object.\nThis happens when the lens is inserted into a converging beam (being focused by a previous lens) the location of its real image.\nIn that case even a negative lens can project a real image, as is done by a [Barlow lens](https://bliptext.com/articles/barlow-lens).\nFor a given lens with the focal length f, the minimum distance between an object and the real image is 4f (S1 = S2 = 2f).\nThis is derived by letting L = S1 + S2, expressing S2 in terms of S1 by the lens equation (or expressing S1 in terms of S2), and equating the derivative of L with respect to S1 (or S2) to zero.\n(Note that L has no limit in increasing so its extremum is only the minimum, at which the derivate of L is zero.)\n# [Single thin lens imaging with chief rays | upload.wikimedia.org/wikipedia/commons/7/76/Single///Lens///Imaging%2C///2024-05-30.png]\n# [A diagram of imaging with a single thick lens imaging. H1 and H2 are principal points where 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 nodal points. | upload.wikimedia.org/wikipedia/commons/3/3d/Single///Thick///Lens///Imaging%2C///2024-10-07.png]\n# [A camera lens forms a real image of a distant object. | upload.wikimedia.org/wikipedia/commons/8/86/lens3.svg]\n# [Virtual image formation using a positive lens as a magnifying glass. | upload.wikimedia.org/wikipedia/commons/e/e7/lens3b.svg]\n\n## Magnification\nThe linear [magnification](https://bliptext.com/articles/magnification) of an imaging system using a single lens is given by\nwhere M is the magnification factor defined as the ratio of the size of an image compared to the size of the object.\nThe sign convention here dictates that if M is negative, as it is for real images, the image is upside-down with respect to the object.\nFor virtual images M is positive, so the image is upright.\nThis magnification formula provides two easy ways to distinguish converging (\nf > 0\n) and diverging (\nf < 0\n) lenses: For an object very close to the lens (\n0 < S1 <\n), 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 (\n> 0\n), a converging lens would form an inverted image, whereas a diverging lens would form an upright image.\nLinear magnification M is not always the most useful measure of magnifying power.\nFor instance, when characterizing a visual telescope or binoculars that produce only a virtual image, one would be more concerned with the [angular magnification](https://bliptext.com/articles/magnification)—which expresses how much larger a distant object appears through the telescope compared to the naked eye.\nIn the case of a camera one would quote the [plate scale](https://bliptext.com/articles/plate-scale), which compares the apparent (angular) size of a distant object to the size of the real image produced at the focus.\nThe plate scale is the reciprocal of the focal length of the camera lens; lenses are categorized as [long-focus lenses](https://bliptext.com/articles/long-focus-lens) or [wide-angle lenses](https://bliptext.com/articles/wide-angle-lens) according to their focal lengths.\nUsing an inappropriate measurement of magnification can be formally correct but yield a meaningless number.\nFor instance, using a magnifying glass of 5 cm focal length, held 20 cm from the eye and 5 cm from the object, produces a virtual image at infinity of infinite linear size:\nM = ∞\n. But the is 5, meaning that the object appears 5 times larger to the eye than without the lens.\nWhen taking a picture of the [moon](https://bliptext.com/articles/moon) using a camera with a 50 mm lens, one is not concerned with the linear magnification\nM ≈ -50 mm / 380,000 km = -1.3.\nRather, the plate scale of the camera is about 1 °, from which one can conclude that the 0.5 mm image on the film corresponds to an angular size of the moon seen from earth of about 0.5°.\nIn the extreme case where an object is an infinite distance away,\nS1 = ∞\n,\nS2 = f\nand\nM = −f/∞ = 0\n, indicating that the object would be imaged to a single point in the focal plane.\nIn fact, the diameter of the projected spot is not actually zero, since [diffraction](https://bliptext.com/articles/diffraction) places a lower limit on the size of the [point spread function](https://bliptext.com/articles/point-spread-function).\nThis is called the [diffraction limit](https://bliptext.com/articles/diffraction-limit).\n[[File:Thin lens images.svg|thumb|none|500px|**I**mages 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. ****E**** (at\n2f\n) has an equal-size, real and inverted image; **I** (at f) has its image at [infinity](https://bliptext.com/articles/infinity); and **K** (at\nf/2\n) has a double-size, virtual and upright image.\nNote that the images of letters H, I, J, and i are located far away from the lens such that they are not shown here.\nWhat 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.]]\n\n## Table for thin lens imaging properties\n\n# Aberrations\nLenses do not form perfect images, and always introduce some degree of distortion or aberration that makes the image an imperfect replica of the object.\nCareful design of the lens system for a particular application minimizes the aberration.\nSeveral types of aberration affect image quality, including spherical aberration, coma, and chromatic aberration.\n\n## Spherical aberration\nSpherical 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 [ground and polished](https://bliptext.com/articles/fabrication-and-testing-of-optical-components), and so are often used.\nSpherical 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.\nThis manifests itself as a blurring of the image.\nSpherical aberration can be minimised with normal lens shapes by carefully choosing the surface curvatures for a particular application.\nFor 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.\n# [400px | upload.wikimedia.org/wikipedia/commons/0/0b/lens5.svg]\n\n## Coma\nComa, or comatic aberration, derives its name from the [comet](https://bliptext.com/articles/comet)-like appearance of the aberrated image.\nComa 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.\nRays that pass through the centre of a lens of focal length f are focused at a point with distance\nf tan θ\nfrom the axis.\nRays 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).\nIn 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).\nThe sum of all these circles results in a V-shaped or comet-like flare.\nAs 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.\nLenses in which both spherical aberration and coma are minimised are called bestform lenses.\n# [400px | upload.wikimedia.org/wikipedia/commons/0/0f/lens-coma.svg]\n\n## Chromatic aberration\nChromatic aberration is caused by the [dispersion](https://bliptext.com/articles/dispersion-optics) of the lens material—the variation of its [refractive index](https://bliptext.com/articles/refractive-index), n, with the wavelength of light.\nSince, from [the formulae above](https://bliptext.com/articles/), f is dependent upon n, it follows that light of different wavelengths is focused to different positions.\nChromatic aberration of a lens is seen as fringes of colour around the image.\nIt can be minimised by using an [achromatic doublet](https://bliptext.com/articles/achromatic-lens) (or achromat) in which two materials with differing dispersion are bonded together to form a single lens.\nThis reduces the amount of chromatic aberration over a certain range of wavelengths, though it does not produce perfect correction.\nThe use of achromats was an important step in the development of the optical microscope.\nAn [apochromat](https://bliptext.com/articles/apochromat) is a lens or lens system with even better chromatic aberration correction, combined with improved spherical aberration correction.\nApochromats are much more expensive than achromats.\nDifferent lens materials may also be used to minimise chromatic aberration, such as specialised coatings or lenses made from the crystal [fluorite](https://bliptext.com/articles/fluorite).\nThis naturally occurring substance has the highest known [Abbe number](https://bliptext.com/articles/abbe-number), indicating that the material has low dispersion.\n# [400px | upload.wikimedia.org/wikipedia/commons/a/aa/Chromatic///aberration///lens///diagram.svg]\n# [400px | upload.wikimedia.org/wikipedia/commons/1/15/Lens6b-en.svg]\n\n## Other types of aberration\nOther kinds of aberration include [field curvature](https://bliptext.com/articles/field-curvature), barrel and pincushion distortion, and [astigmatism](https://bliptext.com/articles/astigmatism-optical-systems).\n\n## Aperture diffraction\nEven if a lens is designed to minimize or eliminate the aberrations described above, the image quality is still limited by the [diffraction](https://bliptext.com/articles/diffraction) of light passing through the lens' finite [aperture](https://bliptext.com/articles/aperture).\nA [diffraction-limited](https://bliptext.com/articles/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.\n\n# Compound lenses \nSimple lenses are subject to the [optical aberrations](https://bliptext.com/articles/) discussed above.\nIn many cases these aberrations can be compensated for to a great extent by using a combination of simple lenses with complementary aberrations.\nA 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.\nIn 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.\nAs shown [above](https://bliptext.com/articles/), 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.\nFor 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.\nThis imaging cascade by each lens surface justifies the imaging cascade by each lens.\nFor a two-lens system the object distances of each lens can be denoted as s_{o1} and s_{o2}, and the image distances as and s_{i1} and s_{i2}.\nIf the lenses are thin, each satisfies the thin lens formula\nIf the distance between the two lenses is d, then.\n(The 2nd lens images the image of the first lens.)\nFFD (Front Focal Distance) is defined as the distance between the front (left) focal point of an optical system and its nearest optical surface vertex.\nIf 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).\nTo do this, the image of the 1st lens is located at the focal point of the 2nd lens, i.e.,.\nSo, the thin lens formula for the 1st lens becomes\nBFD (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.\nIf 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.\nIn this case, the 1st lens images the object at its focal point.\nSo, the thin lens formula for the 2nd lens becomes\nA simplest case is where thin lenses are placed in contact (d = 0).\nThen the combined focal length f of the lenses is given by\nSince\n1/f\nis the power of a lens with focal length f, it can be seen that the powers of thin lenses in contact are additive.\nThe general case of multiple thin lenses in contact is\nwhere N is the number of lenses.\nIf two thin lenses are separated in air by some distance d, then the focal length for the combined system is given by\nAs d tends to zero, the focal length of the system tends to the value of f given for thin lenses in contact.\nIt can be shown that the same formula works for thick lenses if d is taken as the distance between their principal planes.\nIf the separation distance between two lenses is equal to the sum of their focal lengths (\nd = f1 + f2\n), 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](https://bliptext.com/articles/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](https://bliptext.com/articles/refracting-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\nwhich is the ratio of the output beam width to the input beam width.\nNote the sign convention: a telescope with two convex lenses (\nf1 > 0\n,\nf2 > 0\n) produces a negative magnification, indicating an inverted image. A convex plus a concave lens (\nf1 > 0 > f2\n) produces a positive magnification and the image is upright. For further information on simple optical telescopes, see [Refracting telescope § Refracting telescope designs](https://bliptext.com/articles/refracting-telescope).\n\n# Non spherical types\n[Cylindrical lenses](https://bliptext.com/articles/cylindrical-lens) have curvature along only one axis.\nThey are used to focus light into a line, or to convert the elliptical light from a [laser diode](https://bliptext.com/articles/laser-diode) into a round beam.\nThey are also used in motion picture [anamorphic lenses](https://bliptext.com/articles/anamorphic-lens).\n[Aspheric lenses](https://bliptext.com/articles/aspheric-lens) have at least one surface that is neither spherical nor cylindrical.\nThe more complicated shapes allow such lenses to form images with less [aberration](https://bliptext.com/articles/optical-aberration) than standard simple lenses, but they are more difficult and expensive to produce.\nThese were formerly complex to make and often extremely expensive, but advances in technology have greatly reduced the manufacturing cost for such lenses.\nA [Fresnel lens](https://bliptext.com/articles/fresnel-lens) has its optical surface broken up into narrow rings, allowing the lens to be much thinner and lighter than conventional lenses.\nDurable Fresnel lenses can be molded from plastic and are inexpensive.\n[Lenticular lenses](https://bliptext.com/articles/lenticular-lens) are arrays of [microlenses](https://bliptext.com/articles/microlens) that are used in [lenticular printing](https://bliptext.com/articles/lenticular-printing) to make images that have an illusion of depth or that change when viewed from different angles.\n[Bifocal lens](https://bliptext.com/articles/bifocal-lens) has two or more, or a graduated, focal lengths ground into the lens.\nA [gradient index lens](https://bliptext.com/articles/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.\nAn [axicon](https://bliptext.com/articles/axicon) has a [conical](https://bliptext.com/articles/cone-geometry) optical surface.\nIt images a [point source](https://bliptext.com/articles/point-source) into a line the [optic axis](https://bliptext.com/articles/optic-axis), or transforms a laser beam into a ring.\n[Diffractive optical elements](https://bliptext.com/articles/diffractive-optical-element) can function as lenses.\n[Superlenses](https://bliptext.com/articles/superlens) are made from [negative index metamaterials](https://bliptext.com/articles/negative-index-metamaterials) and claim to produce images at spatial resolutions exceeding the [diffraction limit](https://bliptext.com/articles/diffraction-limit).\nThe first superlenses were made in 2004 using such a [metamaterial](https://bliptext.com/articles/metamaterial) for microwaves.\nImproved versions have been made by other researchers.\nthe superlens has not yet been demonstrated at [visible](https://bliptext.com/articles/visible-frequency) or near-[infrared](https://bliptext.com/articles/infrared) wavelengths.\nA prototype flat ultrathin lens, with no curvature has been developed.\n# [An aspheric biconvex lens. | upload.wikimedia.org/wikipedia/commons/4/43/Pfeilh%C3%B6he.svg]\n# [Close-up view of a flat Fresnel lens. | upload.wikimedia.org/wikipedia/commons/c/cd/Flat///flexible///plastic///sheet///lens.JPG]\n\n# Uses\nA single convex lens mounted in a frame with a handle or stand is a [magnifying glass](https://bliptext.com/articles/magnifying-glass).\nLenses are used as [prosthetics](https://bliptext.com/articles/prosthetic) for the correction of [refractive errors](https://bliptext.com/articles/refractive-error) such as [myopia](https://bliptext.com/articles/myopia), [hypermetropia](https://bliptext.com/articles/hypermetropia), [presbyopia](https://bliptext.com/articles/presbyopia), and [astigmatism](https://bliptext.com/articles/astigmatism-optical-systems).\n(See [corrective lens](https://bliptext.com/articles/corrective-lens), [contact lens](https://bliptext.com/articles/contact-lens), [eyeglasses](https://bliptext.com/articles/eyeglasses), [intraocular lens](https://bliptext.com/articles/intraocular-lens).) Most lenses used for other purposes have strict [axial symmetry](https://bliptext.com/articles/axial-symmetry); eyeglass lenses are only approximately symmetric.\nThey are usually shaped to fit in a roughly oval, not circular, frame; the optical centres are placed over the [eyeballs](https://bliptext.com/articles/human-eyeball); their curvature may not be axially symmetric to correct for [astigmatism](https://bliptext.com/articles/astigmatism-optical-systems).\n[Sunglasses' lenses](https://bliptext.com/articles/sunglass-lens) are designed to attenuate light; sunglass lenses that also correct visual impairments can be custom made.\nOther uses are in imaging systems such as [monoculars](https://bliptext.com/articles/monocular), [binoculars](https://bliptext.com/articles/binoculars), [telescopes](https://bliptext.com/articles/optical-telescope), [microscopes](https://bliptext.com/articles/microscope), [cameras](https://bliptext.com/articles/camera) and [projectors](https://bliptext.com/articles/movie-projector).\nSome of these instruments produce a [virtual image](https://bliptext.com/articles/virtual-image) when applied to the human eye; others produce a [real image](https://bliptext.com/articles/real-image) that can be captured on [photographic film](https://bliptext.com/articles/photographic-film) or an [optical sensor](https://bliptext.com/articles/optical-sensor), or can be viewed on a screen.\nIn these devices lenses are sometimes paired up with [curved mirrors](https://bliptext.com/articles/curved-mirror) to make a [catadioptric system](https://bliptext.com/articles/catadioptric-system) where the lens's spherical aberration corrects the opposite aberration in the mirror (such as [Schmidt](https://bliptext.com/articles/schmidt-corrector-plate) and [meniscus](https://bliptext.com/articles/meniscus-corrector) correctors).\nConvex lenses produce an image of an object at infinity at their focus; if the [sun](https://bliptext.com/articles/sun) is imaged, much of the visible and infrared light incident on the lens is concentrated into the small image.\nA large lens creates enough intensity to burn a flammable object at the focal point.\nSince ignition can be achieved even with a poorly made lens, lenses have been used as [burning-glasses](https://bliptext.com/articles/burning-glass) for at least 2400 years.\nA modern application is the use of relatively large lenses to [concentrate solar energy](https://bliptext.com/articles/concentrator-photovoltaics) on relatively small [photovoltaic cells](https://bliptext.com/articles/photovoltaic-cell), harvesting more energy without the need to use larger and more expensive cells.\n[Radio astronomy](https://bliptext.com/articles/radio-astronomy) and [radar](https://bliptext.com/articles/radar) systems often use [dielectric lenses](https://bliptext.com/articles/dielectric-lens), commonly called a [lens antenna](https://bliptext.com/articles/lens-antenna) to refract [electromagnetic radiation](https://bliptext.com/articles/electromagnetic-radiation) into a collector antenna.\nLenses can become scratched and abraded.\n[Abrasion](https://bliptext.com/articles/abrasion-mechanical)-resistant coatings are available to help control this.\n# [A watch with a plano-convex lens over the date indicator | upload.wikimedia.org/wikipedia/commons/1/11/Steeldive///SD8205-2.jpg]\n\n## Simulations","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T17:21:26.764Z","updated_at":"2025-01-25T17:21:26.764Z","search_vector":"'len':1","search_content":" \n \n\n# \nA **lens** is a transmissive [optical](https://bliptext.com/articles/optics) device that focuses or disperses a [light beam](https://bliptext.com/articles/light-beam) by means of [refraction](https://bliptext.com/articles/refraction).\nA [simple lens](https://bliptext.com/articles/simple-lens) consists of a single piece of [transparent material](https://bliptext.com/articles/transparent-material), while a [compound lens](https://bliptext.com/articles/) consists of several [simple lens](https://bliptext.com/articles/simple-lens)es (elements), usually arranged along a common [axis](https://bliptext.com/articles/optical-axis).\nLenses are made from materials such as [glass](https://bliptext.com/articles/glass) or [plastic](https://bliptext.com/articles/plastic) and are [ground](https://bliptext.com/articles/grinding-abrasive-cutting), [polished](https://bliptext.com/articles/polishing), or [molded](https://bliptext.com/articles/molding-process) to the required shape.\nA lens can focus light to form an [image](https://bliptext.com/articles/image), unlike a [prism](https://bliptext.com/articles/prism-optics), which refracts light without focusing.\nDevices that similarly focus or disperse waves and radiation other than visible light are also called \"lenses\", such as [microwave](https://bliptext.com/articles/microwave) lenses, [electron lenses](https://bliptext.com/articles/electron-lens), [acoustic lenses](https://bliptext.com/articles/acoustic-lens), or [explosive lenses](https://bliptext.com/articles/explosive-lens).\nLenses are used in various imaging devices such as [telescopes](https://bliptext.com/articles/telescope), [binoculars](https://bliptext.com/articles/binoculars), and [cameras](https://bliptext.com/articles/camera).\nThey are also used as visual aids in [glasses](https://bliptext.com/articles/glasses) to correct defects of vision such as [myopia](https://bliptext.com/articles/near-sightedness) and [hypermetropia](https://bliptext.com/articles/far-sightedness).\n# [A burning apparatus consisting of two biconvex lens | upload.wikimedia.org/wikipedia/commons/c/c8/Double-lens///burning///apparatus%2C///Ehrenfried///Walther///von///Tschirnhaus%2C///Kieslingswalde///%28today///Slawonice%2C///Poland%29%2C///c.///1690///- ///Mathematisch-Physikalischer///Salon%2C///Dresden///-///DSC08133.JPG]\n\n# History\nThe word [lens](https://bliptext.com/articles/wikt-lens) comes from lēns, the Latin name of the [lentil](https://bliptext.com/articles/lentil) (a seed of a [lentil](https://bliptext.com/articles/lentil) plant), because a double-convex [lens](https://bliptext.com/articles/wikt-lens) is [lentil](https://bliptext.com/articles/lentil)-shaped.\nThe lentil also gives its name to a [geometric figure](https://bliptext.com/articles/lens-geometry).\nSome scholars argue that the archeological evidence indicates that there was widespread use of lenses in antiquity, spanning several millennia.\nThe so-called [Nimrud lens](https://bliptext.com/articles/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.\nOthers have suggested that certain [Egyptian hieroglyphs](https://bliptext.com/articles/egyptian-hieroglyphs) depict \"simple glass meniscal lenses\".\nThe oldest certain reference to the use of lenses is from [Aristophanes](https://bliptext.com/articles/aristophanes)' play [The Clouds](https://bliptext.com/articles/the-clouds) (424 BCE) mentioning a burning-glass.\n[Pliny the Elder](https://bliptext.com/articles/pliny-the-elder) (1st century) confirms that burning-glasses were known in the Roman period.\nPliny also has the earliest known reference to the use of a [corrective lens](https://bliptext.com/articles/corrective-lens) when he mentions that [Nero](https://bliptext.com/articles/nero) was said to watch the [gladiatorial](https://bliptext.com/articles/gladiator) games using an [emerald](https://bliptext.com/articles/emerald) (presumably [concave](https://bliptext.com/articles/concave) to correct for [nearsightedness](https://bliptext.com/articles/myopia), though the reference is vague).\nBoth Pliny and [Seneca the Younger](https://bliptext.com/articles/seneca-the-younger) (3 BC–65 AD) described the magnifying effect of a glass globe filled with water.\n[Ptolemy](https://bliptext.com/articles/ptolemy) (2nd century) wrote a book on [Optics](https://bliptext.com/articles/optics-ptolemy), which however survives only in the Latin translation of an incomplete and very poor Arabic translation.\nThe book was, however, received by medieval scholars in the Islamic world, and commented upon by [Ibn Sahl](https://bliptext.com/articles/ibn-sahl-mathematician) (10th century), who was in turn improved upon by [Alhazen](https://bliptext.com/articles/alhazen) ([Book of Optics](https://bliptext.com/articles/book-of-optics), 11th century).\nThe Arabic translation of Ptolemy's Optics became available in Latin translation in the 12th century ([Eugenius of Palermo](https://bliptext.com/articles/eugenius-of-palermo) 1154).\nBetween the 11th and 13th century \"[reading stones](https://bliptext.com/articles/reading-stone)\" were invented.\nThese were primitive plano-convex lenses initially made by cutting a glass sphere in half.\nThe medieval (11th or 12th century) rock crystal [Visby lenses](https://bliptext.com/articles/visby-lens) may or may not have been intended for use as burning glasses.\n[Spectacles](https://bliptext.com/articles/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.\nThis was the start of the optical industry of grinding and polishing lenses for spectacles, first in Venice and Florence in the late 13th century, and later in the spectacle-making centres in both the [Netherlands](https://bliptext.com/articles/netherlands) and [Germany](https://bliptext.com/articles/germany).\nSpectacle 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).\nThe practical development and experimentation with lenses led to the invention of the compound [optical microscope](https://bliptext.com/articles/optical-microscope) around 1595, and the [refracting telescope](https://bliptext.com/articles/refracting-telescope) in 1608, both of which appeared in the spectacle-making centres in the [Netherlands](https://bliptext.com/articles/netherlands).\nWith 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.\nOpticians tried to construct lenses of varying forms of curvature, wrongly assuming errors arose from defects in the spherical figure of their surfaces.\nOptical theory on [refraction](https://bliptext.com/articles/refraction) and experimentation was showing no single-element lens could bring all colours to a focus.\nThis led to the invention of the compound [achromatic lens](https://bliptext.com/articles/achromatic-lens) by [Chester Moore Hall](https://bliptext.com/articles/chester-moore-hall) in [England](https://bliptext.com/articles/england) in 1733, an invention also claimed by fellow Englishman [John Dollond](https://bliptext.com/articles/john-dollond) in a 1758 patent.\nDevelopments 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.\nWith 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.\nFresnel lens were developed that considered these constraints by featuring less material through their concentric annular sectioning.\nThey were first fully implemented into a lighthouse in 1823.\n# [Light being refracted by a spherical glass container full of water. Roger Bacon, 13th century | upload.wikimedia.org/wikipedia/commons/8/86/Optics///from///Roger///Bacon%27s///De///multiplicatone///specierum.jpg]\n# [Lens for LSST, a planned sky surveying telescope | upload.wikimedia.org/wikipedia/commons/f/f5/LSST///Telescope///-///L1///Lens///of///the///camera.jpg]\n\n# Construction of simple lenses \nMost lenses are spherical lenses: their two surfaces are parts of the surfaces of spheres.\nEach surface can be convex (bulging outwards from the lens), concave (depressed into the lens), or planar (flat).\nThe line joining the centres of the spheres making up the lens surfaces is called the axis of the lens.\nTypically the lens axis passes through the physical centre of the lens, because of the way they are manufactured.\nLenses may be cut or ground after manufacturing to give them a different shape or size.\nThe lens axis may then not pass through the physical centre of the lens.\n[Toric](https://bliptext.com/articles/toric-lens) or sphero-cylindrical lenses have surfaces with two different radii of curvature in two orthogonal planes.\nThey have a different [focal power](https://bliptext.com/articles/focal-power) in different meridians.\nThis forms an [astigmatic](https://bliptext.com/articles/astigmatism-optical-systems) lens.\nAn example is eyeglass lenses that are used to correct [astigmatism](https://bliptext.com/articles/astigmatism) in someone's eye.\n\n## Types of simple lenses\nLenses are classified by the curvature of the two optical surfaces.\nA lens is bi[convex](https://bliptext.com/articles/convex) (or double [convex](https://bliptext.com/articles/convex), or just [convex](https://bliptext.com/articles/convex)) if both surfaces are [convex](https://bliptext.com/articles/convex).\nIf both surfaces have the same radius of curvature, the lens is equiconvex.\nA lens with two [concave](https://bliptext.com/articles/concave) surfaces is bi[concave](https://bliptext.com/articles/concave) (or just [concave](https://bliptext.com/articles/concave)).\nIf one of the surfaces is flat, the lens is plano-convex or plano-concave depending on the curvature of the other surface.\nA lens with one convex and one concave side is convex-concave or meniscus.\nConvex-concave lenses are most commonly used in [corrective lenses](https://bliptext.com/articles/corrective-lenses), since the shape minimizes some aberrations.\nFor a biconvex or plano-convex lens in a lower-index medium, a [collimated](https://bliptext.com/articles/collimated-light) beam of light passing through the lens converges to a spot (a focus) behind the lens.\nIn this case, the lens is called a positive or converging lens.\nFor a [thin lens](https://bliptext.com/articles/thin-lens) in air, the distance from the lens to the spot is the [focal length](https://bliptext.com/articles/focal-length) of the lens, which is commonly represented by f in diagrams and equations.\nAn [extended hemispherical lens](https://bliptext.com/articles/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.\nAnother extreme case of a thick convex lens is a [ball lens](https://bliptext.com/articles/ball-lens), whose shape is completely round.\nWhen used in novelty photography it is often called a \"lensball\".\nA ball-shaped lens has the advantage of being omnidirectional, but for most [optical glass](https://bliptext.com/articles/optical-glass) types, its focal point lies close to the ball's surface.\nBecause of the ball's curvature extremes compared to the lens size, [optical aberration](https://bliptext.com/articles/optical-aberration) is much worse than thin lenses, with the notable exception of [chromatic aberration](https://bliptext.com/articles/chromatic-aberration).\nFor 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.\nThe beam, after passing through the lens, appears to emanate from a particular point on the axis in front of the lens.\nFor 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.\nThe behavior reverses when a lens is placed in a medium with higher refractive index than the material of the lens.\nIn this case a biconvex or plano-convex lens diverges light, and a biconcave or plano-concave one converges it.\nConvex-concave (meniscus) lenses can be either positive or negative, depending on the relative curvatures of the two surfaces.\nA 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.\nConversely, 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.\nAn ideal [thin lens](https://bliptext.com/articles/thin-lens) with two surfaces of equal curvature (also equal in the sign) would have zero [optical power](https://bliptext.com/articles/optical-power) (as its focal length becomes infinity as shown in the [lensmaker's equation](https://bliptext.com/articles/)), meaning that it would neither converge nor diverge light.\nAll real lenses have a nonzero thickness, however, which makes a real lens with identical curved surfaces slightly positive.\nTo obtain exactly zero optical power, a meniscus lens must have slightly unequal curvatures to account for the effect of the lens' thickness.\n# [450x450px | upload.wikimedia.org/wikipedia/commons/a/a7/Lenses///en.svg]\n# [Meniscus lenses: negative (top) and positive (bottom) | upload.wikimedia.org/wikipedia/commons/e/e8/Meniscus///lenses.svg]\n\n## For a spherical surface\nFor a single refraction for a circular boundary, the relation between object and its image in the [paraxial approximation](https://bliptext.com/articles/paraxial-approximation) is given by\nwhere R is the radius of the spherical surface,\nn2\nis the refractive index of the material of the surface,\nn1\nis the refractive index of medium (the medium other than the spherical surface material), u is the on-axis (on the optical axis) object distance from the line perpendicular to the axis toward the refraction point on the surface (which height is h), and v is the on-axis image distance from the line.\nDue to paraxial approximation where the line of h is close to the vertex of the spherical surface meeting the optical axis on the left, u and v are also considered distances with respect to the vertex.\nMoving v toward the right infinity leads to the first or object focal length f_0 for the spherical surface.\nSimilarly, u toward the left infinity leads to the second or image focal length f_i.\nApplying 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 formula](https://bliptext.com/articles/).\n# [Simulation of refraction at spherical surface at Desmos | upload.wikimedia.org/wikipedia/commons/4/4f/Refraction///at///spherical///surface.svg]\n\n### Derivation\nApplying [Snell's law](https://bliptext.com/articles/snell-s-law) on the spherical surface,\nAlso in the diagram,, and using [small angle approximation](https://bliptext.com/articles/small-angle-approximation) (paraxial approximation) and eliminating i, r, and θ,\n# [lens | upload.wikimedia.org/wikipedia/commons/1/1b/Refraction///in///spherical///surface.svg]\n# [The four cases of spherical refraction | upload.wikimedia.org/wikipedia/commons/8/8e/Four///spherical///refractions.png]\n\n## Lensmaker's equation\nThe (effective) focal length f of a spherical lens in air or vacuum for paraxial rays can be calculated from the **lensmaker's equation**:\nwhere\nThe focal length \\ f\\ is with respect to the [principal planes](https://bliptext.com/articles/cardinal-point-optics) of the lens, and the locations of the [principal planes](https://bliptext.com/articles/cardinal-point-optics) \\ h_1\\ and \\ h_2\\ with respect to the respective lens vertices are given by the following formulas, where it is a positive value if it is right to the respective vertex.\nThe focal length \\ f\\ is positive for converging lenses, and negative for diverging lenses.\nThe [reciprocal](https://bliptext.com/articles/multiplicative-inverse) of the focal length, is the [optical power](https://bliptext.com/articles/optical-power) of the lens.\nIf the focal length is in metres, this gives the optical power in [dioptres](https://bliptext.com/articles/dioptre) (reciprocal metres).\nLenses have the same focal length when light travels from the back to the front as when light goes from the front to the back.\nOther properties of the lens, such as the [aberrations](https://bliptext.com/articles/aberration-in-optical-systems) are not the same in both directions.\n\n### Sign convention for radii of curvature \n\nR{{sub|1}}\n\n and \n\nR{{sub|2}}\nThe signs of the lens' radii of curvature indicate whether the corresponding surfaces are convex or concave.\nThe [sign convention](https://bliptext.com/articles/sign-convention) used to represent this varies, but in this article a positive 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 R means that rays reaching the surface have already passed the center of curvature.\nConsequently, for external lens surfaces as diagrammed above,\nR1 > 0\nand\nR2 0\nindicate concave surfaces.\nThe reciprocal of the radius of [curvature](https://bliptext.com/articles/curvature) is called the [curvature](https://bliptext.com/articles/curvature).\nA flat surface has zero curvature, and its radius of curvature is [infinite](https://bliptext.com/articles/infinity).\n\n### Sign convention for other parameters\nThis convention seems to be mainly used for this article, although there is another convention such as [Cartesian sign convention](https://bliptext.com/articles/undefined) requiring different lens equation forms.\n\n### Thin lens approximation\nIf d is small compared to\nR1\nand\nR2\nthen the approximation can be made.\nFor a lens in air, f is then given by\n\n### Derivation\nThe spherical thin lens equation in [paraxial approximation](https://bliptext.com/articles/paraxial-approximation) is derived here with respect to the right figure.\nThe 1st spherical lens surface (which meets the optical axis at \\ V_1\\ as its vertex) images an on-axis object point O to the virtual image I, which can be described by the following equation, For the imaging by second lens surface, by taking the above sign convention, and Adding these two equations yields For the thin lens approximation where the 2nd term of the RHS (Right Hand Side) is gone, so\nThe focal length \\ f\\ of the thin lens is found by limiting\nSo, the Gaussian thin lens equation is\nFor the thin lens in air or vacuum where \\ n_1 = 1\\ can be assumed, \\ f\\ becomes\nwhere the subscript of 2 in \\ n_2\\ is dropped.\n# [A Diagram for a Spherical Lens Equation with Paraxial Rays. | upload.wikimedia.org/wikipedia/commons/8/86/A///Diagram///for///a///Spherical///Lens///Equation///with///Paraxial///Rays%2C///2024-08-27.png]\n\n# Imaging properties\nAs mentioned above, a positive or converging lens in air focuses a collimated beam travelling along the lens axis to a spot (known as the [focal point](https://bliptext.com/articles/focus-optics)) at a distance f from the lens.\nConversely, a [point source](https://bliptext.com/articles/point-source) of light placed at the focal point is converted into a collimated beam by the lens.\nThese two cases are examples of [image](https://bliptext.com/articles/image) formation in lenses.\nIn 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.\nIn the latter, an object at the focal length distance from the lens is imaged at infinity.\nThe plane perpendicular to the lens axis situated at a distance f from the lens is called the [Cardinal point (optics)](https://bliptext.com/articles/cardinal-point-optics).\n\n## Lens equation\nFor [paraxial rays](https://bliptext.com/articles/paraxial-approximation), if the distances from an object to a spherical [thin lens](https://bliptext.com/articles/thin-lens) (a lens of negligible thickness) and from the lens to the image are\nS1\nand\nS2\nrespectively, the distances are related by the (Gaussian) **thin lens formula**:\nThe 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 [the optical center of the lens](https://bliptext.com/articles/cardinal-point-optics) (so its direction does not change), and the third tray toward the first focal point and refracted to the direction parallel to the optical axis.\nThis is a simple ray tracing method easily used.\nTwo rays among the three are sufficient to locate the image point.\nBy 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.\nThe lens equation can also be put into the \"Newtonian\" form:\nwhere x_1 = S_1-f and x_1 is positive if it is left to the front focal point F_1, and x_2 is positive if it is right to the rear focal point F_2.\nBecause f^2 is positive, an object point and the corresponding imaging point made by a lens are always in opposite sides with respect to their respective focal points.\n(x_1 and x_2 are either positive or negative.)\nThis Newtonian form of the lens equation can be derived by using a similarity between triangles P1PO1F1 and L3L2F1 and another similarity between triangles L1L2F2 and P2P02F2 in the right figure.\nThe similarities give the following equations and combining these results gives the Newtonian form of the lens equation.\nThe 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 S_1, S_2, and f are with respect to the [principal planes](https://bliptext.com/articles/principal-plane) of the lens (f is the [effective focal length](https://bliptext.com/articles/effective-focal-length) in this case).\nThis is because of triangle similarities like the thin lens case above; similarity between triangles P1PO1F1 and L3H1F1 and another similarity between triangles L1'H2F2 and P2P02F2 in the right figure.\nIf distances\nS1\nor\nS2\npass through a [medium](https://bliptext.com/articles/medium-optics) other than air or vacuum, then a more complicated analysis is required.\nIf an object is placed at a distance\nS1 > f\nfrom a positive lens of focal length f, we will find an image at a distance\nS2\naccording to this formula.\nIf a screen is placed at a distance\nS2\non the opposite side of the lens, an image is formed on it.\nThis sort of image, which can be projected onto a screen or [image sensor](https://bliptext.com/articles/image-sensor), is known as a [real image](https://bliptext.com/articles/real-image).\nThis is the principle of the [camera](https://bliptext.com/articles/camera), and also of the [human eye](https://bliptext.com/articles/human-eye), in which the [retina](https://bliptext.com/articles/retina) serves as the image sensor.\nThe focusing adjustment of a camera adjusts\nS2\n, as using an image distance different from that required by this formula produces a [defocused](https://bliptext.com/articles/defocus-aberration) (fuzzy) image for an object at a distance of\nS1\nfrom the camera.\nPut another way, modifying\nS2\ncauses objects at a different\nS1\nto come into perfect focus.\nIn some cases,\nS2\nis negative, indicating that the image is formed on the opposite side of the lens from where those rays are being considered.\nSince the diverging light rays emanating from the lens never come into focus, and those rays are not physically present at the point where they to form an image, this is called a [virtual image](https://bliptext.com/articles/virtual-image).\nUnlike 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.\nLikewise, 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,\nS1\nthen being measured from the virtual image location behind the first lens to the second lens.\nThis is exactly what the eye does when looking through a [magnifying glass](https://bliptext.com/articles/magnifying-glass).\nThe magnifying glass creates a (magnified) virtual image behind the magnifying glass, but those rays are then re-imaged by the [lens of the eye](https://bliptext.com/articles/lens-anatomy) to create a real image on the [retina](https://bliptext.com/articles/retina).\nUsing a positive lens of focal length f, a virtual image results when\nS1 0\n) can only produce a virtual image (\nS2 0\n) and diverging (\nf 0\n), a converging lens would form an inverted image, whereas a diverging lens would form an upright image.\nLinear magnification M is not always the most useful measure of magnifying power.\nFor instance, when characterizing a visual telescope or binoculars that produce only a virtual image, one would be more concerned with the [angular magnification](https://bliptext.com/articles/magnification)—which expresses how much larger a distant object appears through the telescope compared to the naked eye.\nIn the case of a camera one would quote the [plate scale](https://bliptext.com/articles/plate-scale), which compares the apparent (angular) size of a distant object to the size of the real image produced at the focus.\nThe plate scale is the reciprocal of the focal length of the camera lens; lenses are categorized as [long-focus lenses](https://bliptext.com/articles/long-focus-lens) or [wide-angle lenses](https://bliptext.com/articles/wide-angle-lens) according to their focal lengths.\nUsing an inappropriate measurement of magnification can be formally correct but yield a meaningless number.\nFor instance, using a magnifying glass of 5 cm focal length, held 20 cm from the eye and 5 cm from the object, produces a virtual image at infinity of infinite linear size:\nM = ∞\n. But the is 5, meaning that the object appears 5 times larger to the eye than without the lens.\nWhen taking a picture of the [moon](https://bliptext.com/articles/moon) using a camera with a 50 mm lens, one is not concerned with the linear magnification\nM ≈ -50 mm / 380,000 km = -1.3.\nRather, the plate scale of the camera is about 1 °, from which one can conclude that the 0.5 mm image on the film corresponds to an angular size of the moon seen from earth of about 0.5°.\nIn the extreme case where an object is an infinite distance away,\nS1 = ∞\n,\nS2 = f\nand\nM = −f/∞ = 0\n, indicating that the object would be imaged to a single point in the focal plane.\nIn fact, the diameter of the projected spot is not actually zero, since [diffraction](https://bliptext.com/articles/diffraction) places a lower limit on the size of the [point spread function](https://bliptext.com/articles/point-spread-function).\nThis is called the [diffraction limit](https://bliptext.com/articles/diffraction-limit).\n[[File:Thin lens images.svg|thumb|none|500px|**I**mages 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. ****E**** (at\n2f\n) has an equal-size, real and inverted image; **I** (at f) has its image at [infinity](https://bliptext.com/articles/infinity); and **K** (at\nf/2\n) has a double-size, virtual and upright image.\nNote that the images of letters H, I, J, and i are located far away from the lens such that they are not shown here.\nWhat 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.]]\n\n## Table for thin lens imaging properties\n\n# Aberrations\nLenses do not form perfect images, and always introduce some degree of distortion or aberration that makes the image an imperfect replica of the object.\nCareful design of the lens system for a particular application minimizes the aberration.\nSeveral types of aberration affect image quality, including spherical aberration, coma, and chromatic aberration.\n\n## Spherical aberration\nSpherical 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 [ground and polished](https://bliptext.com/articles/fabrication-and-testing-of-optical-components), and so are often used.\nSpherical 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.\nThis manifests itself as a blurring of the image.\nSpherical aberration can be minimised with normal lens shapes by carefully choosing the surface curvatures for a particular application.\nFor 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.\n# [400px | upload.wikimedia.org/wikipedia/commons/0/0b/lens5.svg]\n\n## Coma\nComa, or comatic aberration, derives its name from the [comet](https://bliptext.com/articles/comet)-like appearance of the aberrated image.\nComa 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.\nRays that pass through the centre of a lens of focal length f are focused at a point with distance\nf tan θ\nfrom the axis.\nRays 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).\nIn 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).\nThe sum of all these circles results in a V-shaped or comet-like flare.\nAs 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.\nLenses in which both spherical aberration and coma are minimised are called bestform lenses.\n# [400px | upload.wikimedia.org/wikipedia/commons/0/0f/lens-coma.svg]\n\n## Chromatic aberration\nChromatic aberration is caused by the [dispersion](https://bliptext.com/articles/dispersion-optics) of the lens material—the variation of its [refractive index](https://bliptext.com/articles/refractive-index), n, with the wavelength of light.\nSince, from [the formulae above](https://bliptext.com/articles/), f is dependent upon n, it follows that light of different wavelengths is focused to different positions.\nChromatic aberration of a lens is seen as fringes of colour around the image.\nIt can be minimised by using an [achromatic doublet](https://bliptext.com/articles/achromatic-lens) (or achromat) in which two materials with differing dispersion are bonded together to form a single lens.\nThis reduces the amount of chromatic aberration over a certain range of wavelengths, though it does not produce perfect correction.\nThe use of achromats was an important step in the development of the optical microscope.\nAn [apochromat](https://bliptext.com/articles/apochromat) is a lens or lens system with even better chromatic aberration correction, combined with improved spherical aberration correction.\nApochromats are much more expensive than achromats.\nDifferent lens materials may also be used to minimise chromatic aberration, such as specialised coatings or lenses made from the crystal [fluorite](https://bliptext.com/articles/fluorite).\nThis naturally occurring substance has the highest known [Abbe number](https://bliptext.com/articles/abbe-number), indicating that the material has low dispersion.\n# [400px | upload.wikimedia.org/wikipedia/commons/a/aa/Chromatic///aberration///lens///diagram.svg]\n# [400px | upload.wikimedia.org/wikipedia/commons/1/15/Lens6b-en.svg]\n\n## Other types of aberration\nOther kinds of aberration include [field curvature](https://bliptext.com/articles/field-curvature), barrel and pincushion distortion, and [astigmatism](https://bliptext.com/articles/astigmatism-optical-systems).\n\n## Aperture diffraction\nEven if a lens is designed to minimize or eliminate the aberrations described above, the image quality is still limited by the [diffraction](https://bliptext.com/articles/diffraction) of light passing through the lens' finite [aperture](https://bliptext.com/articles/aperture).\nA [diffraction-limited](https://bliptext.com/articles/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.\n\n# Compound lenses \nSimple lenses are subject to the [optical aberrations](https://bliptext.com/articles/) discussed above.\nIn many cases these aberrations can be compensated for to a great extent by using a combination of simple lenses with complementary aberrations.\nA 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.\nIn 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.\nAs shown [above](https://bliptext.com/articles/), 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.\nFor 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.\nThis imaging cascade by each lens surface justifies the imaging cascade by each lens.\nFor a two-lens system the object distances of each lens can be denoted as s_{o1} and s_{o2}, and the image distances as and s_{i1} and s_{i2}.\nIf the lenses are thin, each satisfies the thin lens formula\nIf the distance between the two lenses is d, then.\n(The 2nd lens images the image of the first lens.)\nFFD (Front Focal Distance) is defined as the distance between the front (left) focal point of an optical system and its nearest optical surface vertex.\nIf 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).\nTo do this, the image of the 1st lens is located at the focal point of the 2nd lens, i.e.,.\nSo, the thin lens formula for the 1st lens becomes\nBFD (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.\nIf 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.\nIn this case, the 1st lens images the object at its focal point.\nSo, the thin lens formula for the 2nd lens becomes\nA simplest case is where thin lenses are placed in contact (d = 0).\nThen the combined focal length f of the lenses is given by\nSince\n1/f\nis the power of a lens with focal length f, it can be seen that the powers of thin lenses in contact are additive.\nThe general case of multiple thin lenses in contact is\nwhere N is the number of lenses.\nIf two thin lenses are separated in air by some distance d, then the focal length for the combined system is given by\nAs d tends to zero, the focal length of the system tends to the value of f given for thin lenses in contact.\nIt can be shown that the same formula works for thick lenses if d is taken as the distance between their principal planes.\nIf the separation distance between two lenses is equal to the sum of their focal lengths (\nd = f1 + f2\n), 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](https://bliptext.com/articles/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](https://bliptext.com/articles/refracting-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\nwhich is the ratio of the output beam width to the input beam width.\nNote the sign convention: a telescope with two convex lenses (\nf1 > 0\n,\nf2 > 0\n) produces a negative magnification, indicating an inverted image. A convex plus a concave lens (\nf1 > 0 > f2\n) produces a positive magnification and the image is upright. For further information on simple optical telescopes, see [Refracting telescope § Refracting telescope designs](https://bliptext.com/articles/refracting-telescope).\n\n# Non spherical types\n[Cylindrical lenses](https://bliptext.com/articles/cylindrical-lens) have curvature along only one axis.\nThey are used to focus light into a line, or to convert the elliptical light from a [laser diode](https://bliptext.com/articles/laser-diode) into a round beam.\nThey are also used in motion picture [anamorphic lenses](https://bliptext.com/articles/anamorphic-lens).\n[Aspheric lenses](https://bliptext.com/articles/aspheric-lens) have at least one surface that is neither spherical nor cylindrical.\nThe more complicated shapes allow such lenses to form images with less [aberration](https://bliptext.com/articles/optical-aberration) than standard simple lenses, but they are more difficult and expensive to produce.\nThese were formerly complex to make and often extremely expensive, but advances in technology have greatly reduced the manufacturing cost for such lenses.\nA [Fresnel lens](https://bliptext.com/articles/fresnel-lens) has its optical surface broken up into narrow rings, allowing the lens to be much thinner and lighter than conventional lenses.\nDurable Fresnel lenses can be molded from plastic and are inexpensive.\n[Lenticular lenses](https://bliptext.com/articles/lenticular-lens) are arrays of [microlenses](https://bliptext.com/articles/microlens) that are used in [lenticular printing](https://bliptext.com/articles/lenticular-printing) to make images that have an illusion of depth or that change when viewed from different angles.\n[Bifocal lens](https://bliptext.com/articles/bifocal-lens) has two or more, or a graduated, focal lengths ground into the lens.\nA [gradient index lens](https://bliptext.com/articles/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.\nAn [axicon](https://bliptext.com/articles/axicon) has a [conical](https://bliptext.com/articles/cone-geometry) optical surface.\nIt images a [point source](https://bliptext.com/articles/point-source) into a line the [optic axis](https://bliptext.com/articles/optic-axis), or transforms a laser beam into a ring.\n[Diffractive optical elements](https://bliptext.com/articles/diffractive-optical-element) can function as lenses.\n[Superlenses](https://bliptext.com/articles/superlens) are made from [negative index metamaterials](https://bliptext.com/articles/negative-index-metamaterials) and claim to produce images at spatial resolutions exceeding the [diffraction limit](https://bliptext.com/articles/diffraction-limit).\nThe first superlenses were made in 2004 using such a [metamaterial](https://bliptext.com/articles/metamaterial) for microwaves.\nImproved versions have been made by other researchers.\nthe superlens has not yet been demonstrated at [visible](https://bliptext.com/articles/visible-frequency) or near-[infrared](https://bliptext.com/articles/infrared) wavelengths.\nA prototype flat ultrathin lens, with no curvature has been developed.\n# [An aspheric biconvex lens. | upload.wikimedia.org/wikipedia/commons/4/43/Pfeilh%C3%B6he.svg]\n# [Close-up view of a flat Fresnel lens. | upload.wikimedia.org/wikipedia/commons/c/cd/Flat///flexible///plastic///sheet///lens.JPG]\n\n# Uses\nA single convex lens mounted in a frame with a handle or stand is a [magnifying glass](https://bliptext.com/articles/magnifying-glass).\nLenses are used as [prosthetics](https://bliptext.com/articles/prosthetic) for the correction of [refractive errors](https://bliptext.com/articles/refractive-error) such as [myopia](https://bliptext.com/articles/myopia), [hypermetropia](https://bliptext.com/articles/hypermetropia), [presbyopia](https://bliptext.com/articles/presbyopia), and [astigmatism](https://bliptext.com/articles/astigmatism-optical-systems).\n(See [corrective lens](https://bliptext.com/articles/corrective-lens), [contact lens](https://bliptext.com/articles/contact-lens), [eyeglasses](https://bliptext.com/articles/eyeglasses), [intraocular lens](https://bliptext.com/articles/intraocular-lens).) Most lenses used for other purposes have strict [axial symmetry](https://bliptext.com/articles/axial-symmetry); eyeglass lenses are only approximately symmetric.\nThey are usually shaped to fit in a roughly oval, not circular, frame; the optical centres are placed over the [eyeballs](https://bliptext.com/articles/human-eyeball); their curvature may not be axially symmetric to correct for [astigmatism](https://bliptext.com/articles/astigmatism-optical-systems).\n[Sunglasses' lenses](https://bliptext.com/articles/sunglass-lens) are designed to attenuate light; sunglass lenses that also correct visual impairments can be custom made.\nOther uses are in imaging systems such as [monoculars](https://bliptext.com/articles/monocular), [binoculars](https://bliptext.com/articles/binoculars), [telescopes](https://bliptext.com/articles/optical-telescope), [microscopes](https://bliptext.com/articles/microscope), [cameras](https://bliptext.com/articles/camera) and [projectors](https://bliptext.com/articles/movie-projector).\nSome of these instruments produce a [virtual image](https://bliptext.com/articles/virtual-image) when applied to the human eye; others produce a [real image](https://bliptext.com/articles/real-image) that can be captured on [photographic film](https://bliptext.com/articles/photographic-film) or an [optical sensor](https://bliptext.com/articles/optical-sensor), or can be viewed on a screen.\nIn these devices lenses are sometimes paired up with [curved mirrors](https://bliptext.com/articles/curved-mirror) to make a [catadioptric system](https://bliptext.com/articles/catadioptric-system) where the lens's spherical aberration corrects the opposite aberration in the mirror (such as [Schmidt](https://bliptext.com/articles/schmidt-corrector-plate) and [meniscus](https://bliptext.com/articles/meniscus-corrector) correctors).\nConvex lenses produce an image of an object at infinity at their focus; if the [sun](https://bliptext.com/articles/sun) is imaged, much of the visible and infrared light incident on the lens is concentrated into the small image.\nA large lens creates enough intensity to burn a flammable object at the focal point.\nSince ignition can be achieved even with a poorly made lens, lenses have been used as [burning-glasses](https://bliptext.com/articles/burning-glass) for at least 2400 years.\nA modern application is the use of relatively large lenses to [concentrate solar energy](https://bliptext.com/articles/concentrator-photovoltaics) on relatively small [photovoltaic cells](https://bliptext.com/articles/photovoltaic-cell), harvesting more energy without the need to use larger and more expensive cells.\n[Radio astronomy](https://bliptext.com/articles/radio-astronomy) and [radar](https://bliptext.com/articles/radar) systems often use [dielectric lenses](https://bliptext.com/articles/dielectric-lens), commonly called a [lens antenna](https://bliptext.com/articles/lens-antenna) to refract [electromagnetic radiation](https://bliptext.com/articles/electromagnetic-radiation) into a collector antenna.\nLenses can become scratched and abraded.\n[Abrasion](https://bliptext.com/articles/abrasion-mechanical)-resistant coatings are available to help control this.\n# [A watch with a plano-convex lens over the date indicator | upload.wikimedia.org/wikipedia/commons/1/11/Steeldive///SD8205-2.jpg]\n\n## Simulations","revisionCount":0},"hasMore":false,"nextCursor":null}