Lens (optics)

A lens is a transmissive optical device that focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a compound lens consists of several simple lenses (elements), usually arranged along a common axis. Lenses are made from materials such as glass or plastic, and are ground and polished or molded to a desired shape. A lens can focus light to form an image, unlike a prism, which refracts light without focusing. Devices that similarly focus or disperse waves and radiation other than visible light are also called lenses, such as microwave lenses, electron lenses, acoustic lenses, or explosive lenses.

A biconvex lens
Lens and wavefronts
Lenses can be used to focus light


Optics from Roger Bacon's De multiplicatone specierum
Light being refracted by a spherical glass container full of water. Roger Bacon, 13th century

The word lens comes from lēns , the Latin name of the lentil, because a double-convex lens is lentil-shaped. The lentil plant also gives its name to a geometric figure.[1]

Some scholars argue that the archeological evidence indicates that there was widespread use of lenses in antiquity, spanning several millennia.[2] The so-called Nimrud lens is a rock crystal artifact dated to the 7th century BC which may or may not have been used as a magnifying glass, or a burning glass.[3][4][3][5] Others have suggested that certain Egyptian hieroglyphs depict "simple glass meniscal lenses".[6]

The oldest certain reference to the use of lenses is from Aristophanes' play The Clouds (424 BC) mentioning a burning-glass.[7] Pliny the Elder (1st century) confirms that burning-glasses were known in the Roman period.[8] Pliny also has the earliest known reference to the use of a corrective lens when he mentions that Nero was said to watch the gladiatorial games using an emerald (presumably concave to correct for nearsightedness, though the reference is vague).[9] Both Pliny and Seneca the Younger (3 BC–65 AD) described the magnifying effect of a glass globe filled with water.

Ptolemy (2nd century) wrote a book on Optics, which however survives only in the Latin translation of an incomplete and very poor Arabic translation. The book was, however, received, by medieval scholars in the Islamic world, and commented upon by Ibn Sahl (10th century), who was in turn improved upon by Alhazen (Book of Optics, 11th century). The Arabic translation of Ptolemy's Optics became available in Latin translation in the 12th century (Eugenius of Palermo 1154). Between the 11th and 13th century "reading stones" were invented. These were primitive plano-convex lenses initially made by cutting a glass sphere in half. The medieval (11th or 12th century) rock cystal Visby lenses may or may not have been intended for use as burning glasses.[10]

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.[11] This was the start of the optical industry of grinding and polishing lenses for spectacles, first in Venice and Florence in the late 13th century,[12] and later in the spectacle-making centres in both the Netherlands and Germany.[13] Spectacle makers created improved types of lenses for the correction of vision based more on empirical knowledge gained from observing the effects of the lenses (probably without the knowledge of the rudimentary optical theory of the day).[14][15] The practical development and experimentation with lenses led to the invention of the compound optical microscope around 1595, and the refracting telescope in 1608, both of which appeared in the spectacle-making centres in the Netherlands.[16][17]

With the invention of the telescope and microscope there was a great deal of experimentation with lens shapes in the 17th and early 18th centuries by those trying to correct chromatic errors seen in lenses. Opticians tried to construct lenses of varying forms of curvature, wrongly assuming errors arose from defects in the spherical figure of their surfaces.[18] Optical theory on refraction and experimentation was showing no single-element lens could bring all colours to a focus. This led to the invention of the compound achromatic lens by Chester Moore Hall in England in 1733, an invention also claimed by fellow Englishman John Dollond in a 1758 patent.

Construction of simple lenses

Most lenses are spherical lenses: their two surfaces are parts of the surfaces of spheres. Each surface can be convex (bulging outwards from the lens), concave (depressed into the lens), or planar (flat). The line joining the centres of the spheres making up the lens surfaces is called the axis of the lens. Typically the lens axis passes through the physical centre of the lens, because of the way they are manufactured. Lenses may be cut or ground after manufacturing to give them a different shape or size. The lens axis may then not pass through the physical centre of the lens.

Toric or sphero-cylindrical lenses have surfaces with two different radii of curvature in two orthogonal planes. They have a different focal power in different meridians. This forms an astigmatic lens. An example is eyeglass lenses that are used to correct astigmatism in someone's eye.

More complex are aspheric lenses. These are lenses where one or both surfaces have a shape that is neither spherical nor cylindrical. The more complicated shapes allow such lenses to form images with less aberration than standard simple lenses, but they are more difficult and expensive to produce.

Types of simple lenses

Types of lenses

Lenses are classified by the curvature of the two optical surfaces. A lens is biconvex (or double convex, or just convex) if both surfaces are convex. If both surfaces have the same radius of curvature, the lens is equiconvex. A lens with two concave surfaces is biconcave (or just concave). If one of the surfaces is flat, the lens is plano-convex or plano-concave depending on the curvature of the other surface. A lens with one convex and one concave side is convex-concave or meniscus. It is this type of lens that is most commonly used in corrective lenses.

If the lens is biconvex or plano-convex, a collimated beam of light passing through the lens converges to a spot (a focus) behind the lens. In this case, the lens is called a positive or converging lens. The distance from the lens to the spot is the focal length of the lens, which is commonly abbreviated f in diagrams and equations. An extended hemispherical lens is a special type of plano-convex lens, in which the lens's curved surface is a full hemisphere and the lens is much thicker than the radius of curvature.

Biconvex lens
Large convex lens

If the lens is biconcave or plano-concave, a collimated beam of light passing through the lens is diverged (spread); the lens is thus called a negative or diverging lens. The beam, after passing through the lens, appears to emanate from a particular point on the axis in front of the lens. The distance from this point to the lens is also known as the focal length, though it is negative with respect to the focal length of a converging lens.

Biconcave lens
Concave lens

Convex-concave (meniscus) lenses can be either positive or negative, depending on the relative curvatures of the two surfaces. A negative meniscus lens has a steeper concave surface and is thinner at the centre than at the periphery. Conversely, a positive meniscus lens has a steeper convex surface and is thicker at the centre than at the periphery. An ideal thin lens with two surfaces of equal curvature would have zero optical power, meaning that it would neither converge nor diverge light. All real lenses have nonzero thickness, however, which makes a real lens with identical curved surfaces slightly positive. To obtain exactly zero optical power, a meniscus lens must have slightly unequal curvatures to account for the effect of the lens' thickness.

Lensmaker's equation

The focal length of a lens in air can be calculated from the lensmaker's equation:[19]


is the focal length of the lens,
is the refractive index of the lens material,
is the radius of curvature (with sign, see below) of the lens surface closer to the light source,
is the radius of curvature of the lens surface farther from the light source, and
is the thickness of the lens (the distance along the lens axis between the two surface vertices).

The focal length f is positive for converging lenses, and negative for diverging lenses. The reciprocal of the focal length, 1/f, is the optical power of the lens. If the focal length is in metres, this gives the optical power in dioptres (inverse metres).

Lenses have the same focal length when light travels from the back to the front as when light goes from the front to the back. Other properties of the lens, such as the aberrations are not the same in both directions.

Sign convention for radii of curvature R1 and R2

The signs of the lens' radii of curvature indicate whether the corresponding surfaces are convex or concave. The sign convention used to represent this varies, but in this article 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. Consequently, for external lens surfaces as diagrammed above, R1 > 0 and R2 < 0 indicate convex surfaces (used to converge light in a positive lens), while R1 < 0 and R2 > 0 indicate concave surfaces. The reciprocal of the radius of curvature is called the curvature. A flat surface has zero curvature, and its radius of curvature is infinity.

Thin lens approximation

If d is small compared to R1 and R2, then the thin lens approximation can be made. For a lens in air, f is then given by


Imaging properties

As mentioned above, a positive or converging lens in air focuses a collimated beam travelling along the lens axis to a spot (known as the focal point) at a distance f from the lens. Conversely, a point source of light placed at the focal point is converted into a collimated beam by the lens. These two cases are examples of image formation in lenses. In the former case, an object at an infinite distance (as represented by a collimated beam of waves) is focused to an image at the focal point of the lens. In the latter, an object at the focal length distance from the lens is imaged at infinity. The plane perpendicular to the lens axis situated at a distance f from the lens is called the focal plane.

If the distances from the object to the lens and from the lens to the image are S1 and S2 respectively, for a lens of negligible thickness, in air, the distances are related by the thin lens formula:[21][22][23]


This can also be put into the "Newtonian" form:


where and .

A camera lens forms a real image of a distant object.

Therefore, if an object is placed at a distance S1 > f from a positive lens of focal length f, we will find an image distance S2 according to this formula. If a screen is placed at a distance S2 on the opposite side of the lens, an image is formed on it. This sort of image, which can be projected onto a screen or image sensor, is known as a real image.

Virtual image formation using a positive lens as a magnifying glass.[25]

This is the principle of the camera, and of the human eye. The focusing adjustment of a camera adjusts S2, as using an image distance different from that required by this formula produces a defocused (fuzzy) image for an object at a distance of S1 from the camera. Put another way, modifying S2 causes objects at a different S1 to come into perfect focus.

In some cases S2 is negative, indicating that the image is formed on the opposite side of the lens from where those rays are being considered. Since the diverging light rays emanating from the lens never come into focus, and those rays are not physically present at the point where they appear to form an image, this is called a virtual image. Unlike real images, a virtual image cannot be projected on a screen, but appears to an observer looking through the lens as if it were a real object at the location of that virtual image. Likewise, it appears to a subsequent lens as if it were an object at that location, so that second lens could again focus that light into a real image, S1 then being measured from the virtual image location behind the first lens to the second lens. This is exactly what the eye does when looking through a magnifying glass. The magnifying glass creates a (magnified) virtual image behind the magnifying glass, but those rays are then re-imaged by the lens of the eye to create a real image on the retina.

A negative lens produces a demagnified virtual image.
Barlow lens
A Barlow lens (B) reimages a virtual object (focus of red ray path) into a magnified real image (green rays at focus)

Using a positive lens of focal length f, a virtual image results when S1 < f, the lens thus being used as a magnifying glass (rather than if S1 >> f as for a camera). Using a negative lens (f < 0) with a real object (S1 > 0) can only produce a virtual image (S2 < 0), according to the above formula. It is also possible for the object distance S1 to be negative, in which case the lens sees a so-called virtual object. This happens when the lens is inserted into a converging beam (being focused by a previous lens) before the location of its real image. In that case even a negative lens can project a real image, as is done by a Barlow lens.

Real image of a lamp is projected onto a screen (inverted). Reflections of the lamp from both surfaces of the biconvex lens are visible.
Convex lens (magnifying glass) and upside-down image
A convex lens (fS1) forming a real, inverted image rather than the upright, virtual image as seen in a magnifying glass

For a thin lens, the distances S1 and S2 are measured from the object and image to the position of the lens, as described above. When the thickness of the lens is not much smaller than S1 and S2 or there are multiple lens elements (a compound lens), one must instead measure from the object and image to the principal planes of the lens. If distances S1 or S2 pass through a medium other than air or vacuum a more complicated analysis is required.


The linear magnification of an imaging system using a single lens is given by


where M is the magnification factor defined as the ratio of the size of an image compared to the size of the object. The sign convention here dictates that if M is negative, as it is for real images, the image is upside-down with respect to the object. For virtual images M is positive, so the image is upright.

Linear magnification M is not always the most useful measure of magnifying power. For instance, when characterizing a visual telescope or binoculars that produce only a virtual image, one would be more concerned with the angular magnification—which expresses how much larger a distant object appears through the telescope compared to the naked eye. In the case of a camera one would quote the plate scale, which compares the apparent (angular) size of a distant object to the size of the real image produced at the focus. The plate scale is the reciprocal of the focal length of the camera lens; lenses are categorized as long-focus lenses or wide-angle lenses according to their focal lengths.

Using an inappropriate measurement of magnification can be formally correct but yield a meaningless number. For instance, using a magnifying glass of 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: M = ∞. But the angular magnification is 5, meaning that the object appears 5 times larger to the eye than without the lens. When taking a picture of the moon using a camera with a 50 mm lens, one is not concerned with the linear magnification M−50 mm / 380000 km = −1.3×10−10. Rather, the plate scale of the camera is about 1°/mm, 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°.

In the extreme case where an object is an infinite distance away, S1 = ∞, S2 = f and M = −f/∞= 0, indicating that the object would be imaged to a single point in the focal plane. In fact, the diameter of the projected spot is not actually zero, since diffraction places a lower limit on the size of the point spread function. This is called the diffraction limit.

Thin lens images
Images of black letters in a thin convex lens of focal length f are shown in red. Selected rays are shown for letters E, I and K in blue, green and orange, respectively. Note that E (at 2f) has an equal-size, real and inverted image; I (at f) has its image at infinity; and K (at f/2) has a double-size, virtual and upright image.


Optical aberration
Out-of-focus image of a spoke target..svg Defocus

HartmannShack 1lenslet.svg Tilt
Spherical aberration 3.svg Spherical aberration
Astigmatism.svg Astigmatism
Lens coma.svg Coma
Barrel distortion.svg Distortion
Field curvature.svg Petzval field curvature
Chromatic aberration lens diagram.svg Chromatic aberration

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

Spherical aberration

Spherical aberration occurs because spherical surfaces are not the ideal shape for a lens, but are by far the simplest shape to which glass can be ground and polished, and so are often used. Spherical aberration causes beams parallel to, but distant from, the lens axis to be focused in a slightly different place than beams close to the axis. This manifests itself as a blurring of the image. Lenses in which closer-to-ideal, non-spherical surfaces are used are called aspheric lenses. These were formerly complex to make and often extremely expensive, but advances in technology have greatly reduced the manufacturing cost for such lenses. Spherical aberration can be minimised by carefully choosing the surface curvatures for a particular application. For instance, a plano-convex lens, which is used to focus a collimated beam, produces a sharper focal spot when used with the convex side towards the beam source.




Coma, or comatic aberration, derives its name from the comet-like appearance of the aberrated image. Coma occurs when an object off the optical axis of the lens is imaged, where rays pass through the lens at an angle to the axis θ. Rays that pass through the centre of a lens of focal length f are focused at a point with distance f tan θ from the axis. Rays passing through the outer margins of the lens are focused at different points, either further from the axis (positive coma) or closer to the axis (negative coma). In general, a bundle of parallel rays passing through the lens at a fixed distance from the centre of the lens are focused to a ring-shaped image in the focal plane, known as a comatic circle. The sum of all these circles results in a V-shaped or comet-like flare. As with spherical aberration, coma can be minimised (and in some cases eliminated) by choosing the curvature of the two lens surfaces to match the application. Lenses in which both spherical aberration and coma are minimised are called bestform lenses.



Chromatic aberration

Chromatic aberration is caused by the dispersion of the lens material—the variation of its refractive index, n, with the wavelength of light. Since, from the formulae above, f is dependent upon n, it follows that light of different wavelengths is focused to different positions. Chromatic aberration of a lens is seen as fringes of colour around the image. It can be minimised by using an achromatic doublet (or achromat) in which two materials with differing dispersion are bonded together to form a single lens. This reduces the amount of chromatic aberration over a certain range of wavelengths, though it does not produce perfect correction. The use of achromats was an important step in the development of the optical microscope. An apochromat is a lens or lens system with even better chromatic aberration correction, combined with improved spherical aberration correction. Apochromats are much more expensive than achromats.

Different lens materials may also be used to minimise chromatic aberration, such as specialised coatings or lenses made from the crystal fluorite. This naturally occurring substance has the highest known Abbe number, indicating that the material has low dispersion.

Chromatic aberration lens diagram

Chromatic aberration lens diagram

Other types of aberration

Other kinds of aberration include field curvature, barrel and pincushion distortion, and astigmatism.

Aperture diffraction

Even if a lens is designed to minimize or eliminate the aberrations described above, the image quality is still limited by the diffraction of light passing through the lens' finite aperture. A diffraction-limited lens is one in which aberrations have been reduced to the point where the image quality is primarily limited by diffraction under the design conditions.

Compound lenses

Simple lenses are subject to the optical aberrations discussed above. In many cases these aberrations can be compensated for to a great extent by using a combination of simple lenses with complementary aberrations. A compound lens is a collection of simple lenses of different shapes and made of materials of different refractive indices, arranged one after the other with a common axis.

The simplest case is where lenses are placed in contact: if the lenses of focal lengths f1 and f2 are "thin", the combined focal length f of the lenses is given by

Since 1/f is the power of a lens, it can be seen that the powers of thin lenses in contact are additive.

If two thin lenses are separated in air by some distance d, the focal length for the combined system is given by

The distance from the front focal point of the combined lenses to the first lens is called the front focal length (FFL):


Similarly, the distance from the second lens to the rear focal point of the combined system is the back focal length (BFL):

As d tends to zero, the focal lengths tend to the value of f given for thin lenses in contact.

If the separation distance is equal to the sum of the focal lengths (d = f1+f2), the FFL and BFL are infinite. This corresponds to a pair of lenses that transform a parallel (collimated) beam into another collimated beam. This type of system is called an afocal system, since it produces no net convergence or divergence of the beam. Two lenses at this separation form the simplest type of optical telescope. Although the system does not alter the divergence of a collimated beam, it does alter the width of the beam. The magnification of such a telescope is given by

which is the ratio of the output beam width to the input beam width. Note the sign convention: a telescope with two convex lenses (f1 > 0, f2 > 0) produces a negative magnification, indicating an inverted image. A convex plus a concave lens (f1 > 0 > f2) produces a positive magnification and the image is upright. For further information on simple optical telescopes, see Refracting telescope § Refracting telescope designs.

Other types

Cylindrical lenses have curvature in only one direction. They are used to focus light into a line, or to convert the elliptical light from a laser diode into a round beam. They are also used in motion picture anamorphic lenses.

Flat flexible plastic sheet lens
Close-up view of a flat Fresnel lens.

A Fresnel lens has its optical surface broken up into narrow rings, allowing the lens to be much thinner and lighter than conventional lenses. Durable Fresnel lenses can be molded from plastic and are inexpensive.

Lenticular lenses are arrays of microlenses that are used in lenticular printing to make images that have an illusion of depth or that change when viewed from different angles.

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

An axicon has a conical optical surface. It images a point source into a line along the optic axis, or transforms a laser beam into a ring.[27]

Diffractive optical elements can function as lenses.

Superlenses are made from negative index metamaterials and claim to produce images at spatial resolutions exceeding the diffraction limit.[28] The first superlenses were made in 2004 using such a metamaterial for microwaves.[28] Improved versions have been made by other researchers.[29][30] As of 2014 the superlens has not yet been demonstrated at visible or near-infrared wavelengths.[31]

A prototype flat ultrathin lens, with no curvature has been developed.[32]


A single convex lens mounted in a frame with a handle or stand is a magnifying glass.

Lenses are used as prosthetics for the correction of visual impairments such as myopia, hypermetropia, presbyopia, and astigmatism. (See corrective lens, contact lens, eyeglasses.) Most lenses used for other purposes have strict axial symmetry; eyeglass lenses are only approximately symmetric. They are usually shaped to fit in a roughly oval, not circular, frame; the optical centres are placed over the eyeballs; their curvature may not be axially symmetric to correct for astigmatism. Sunglasses' lenses are designed to attenuate light; sunglass lenses that also correct visual impairments can be custom made.

Other uses are in imaging systems such as monoculars, binoculars, telescopes, microscopes, cameras and projectors. Some of these instruments produce a virtual image when applied to the human eye; others produce a real image that can be captured on photographic film or an optical sensor, or can be viewed on a screen. In these devices lenses are sometimes paired up with curved mirrors to make a catadioptric system where the lens's spherical aberration corrects the opposite aberration in the mirror (such as Schmidt and meniscus correctors).

Convex lenses produce an image of an object at infinity at their focus; if the sun is imaged, much of the visible and infrared light incident on the lens is concentrated into the small image. A large lens creates enough intensity to burn a flammable object at the focal point. Since ignition can be achieved even with a poorly made lens, lenses have been used as burning-glasses for at least 2400 years.[7] A modern application is the use of relatively large lenses to concentrate solar energy on relatively small photovoltaic cells, harvesting more energy without the need to use larger and more expensive cells.

Radio astronomy and radar systems often use dielectric lenses, commonly called a lens antenna to refract electromagnetic radiation into a collector antenna.

Lenses can become scratched and abraded. Abrasion-resistant coatings are available to help control this.[33]

See also


  1. ^ The variant spelling lense is sometimes seen. While it is listed as an alternative spelling in some dictionaries, most mainstream dictionaries do not list it as acceptable.
    • Brians, Paul (2003). Common Errors in English. Franklin, Beedle & Associates. p. 125. ISBN 978-1-887902-89-2. Retrieved 28 June 2009. Reports "lense" as listed in some dictionaries, but not generally considered acceptable.
    • Merriam-Webster's Medical Dictionary. Merriam-Webster. 1995. p. 368. ISBN 978-0-87779-914-6. Lists "lense" as an acceptable alternate spelling.
    • "Lens or Lense – Which is Correct?". writingexplained.org. 2017-04-30. Analyses the almost negligible frequency of use and concludes that the misspelling is a result of a wrong singularisation of the plural (lenses).
  2. ^ Sines, George; Sakellarakis, Yannis A. (1987). "Lenses in antiquity". American Journal of Archaeology. 91 (2): 191–196. doi:10.2307/505216. JSTOR 505216.
  3. ^ a b Whitehouse, David (1 July 1999). "World's oldest telescope?". BBC News. Retrieved 10 May 2008.
  4. ^ "The Nimrud lens/The Layard lens". Collection database. The British Museum. Retrieved 25 November 2012.
  5. ^ D. Brewster (1852). "On an account of a rock-crystal lens and decomposed glass found in Niniveh". Die Fortschritte der Physik (in German). Deutsche Physikalische Gesellschaft. p. 355.
  6. ^ Kriss, Timothy C.; Kriss, Vesna Martich (April 1998). "History of the Operating Microscope: From Magnifying Glass to Microneurosurgery". Neurosurgery. 42 (4): 899–907. doi:10.1097/00006123-199804000-00116. PMID 9574655.
  7. ^ a b Aristophanes (22 Jan 2013) [First performed in 423 BC]. The Clouds. Translated by Hickie, William James. Project Gutenberg. EBook #2562.[1]
  8. ^ Pliny the Elder, The Natural History (trans. John Bostock) Book XXXVII, Chap. 10.
  9. ^ Pliny the Elder, The Natural History (trans. John Bostock) Book XXXVII, Chap. 16
  10. ^ Tilton, Buck (2005). The Complete Book of Fire: Building Campfires for Warmth, Light, Cooking, and Survival. Menasha Ridge Press. p. 25. ISBN 978-0-89732-633-9.
  11. ^ Glick, Thomas F.; Steven John Livesey; Faith Wallis (2005). Medieval science, technology, and medicine: an encyclopedia. Routledge. p. 167. ISBN 978-0-415-96930-7. Retrieved 24 April 2011.
  12. ^ Al Van Helden. The Galileo Project > Science > The Telescope. Galileo.rice.edu. Retrieved on 6 June 2012.
  13. ^ Henry C. King (28 September 2003). The History of the Telescope. Courier Dover Publications. p. 27. ISBN 978-0-486-43265-6. Retrieved 6 June 2012.
  14. ^ Paul S. Agutter; Denys N. Wheatley (12 December 2008). Thinking about Life: The History and Philosophy of Biology and Other Sciences. Springer. p. 17. ISBN 978-1-4020-8865-0. Retrieved 6 June 2012.
  15. ^ Vincent Ilardi (2007). Renaissance Vision from Spectacles to Telescopes. American Philosophical Society. p. 210. ISBN 978-0-87169-259-7. Retrieved 6 June 2012.
  16. ^ Microscopes: Time Line, Nobel Foundation. Retrieved 3 April 2009
  17. ^ Fred Watson (1 October 2007). Stargazer: The Life and Times of the Telescope. Allen & Unwin. p. 55. ISBN 978-1-74175-383-7. Retrieved 6 June 2012.
  18. ^ This paragraph is adapted from the 1888 edition of the Encyclopædia Britannica.
  19. ^ Greivenkamp 2004, p. 14
    Hecht 1987, § 6.1
  20. ^ Hecht 1987, § 5.2.3.
  21. ^ Nave, Carl R. "Thin Lens Equation". Hyperphysics. Georgia State University. Retrieved March 17, 2015.
  22. ^ Colwell, Catharine H. "Resource Lesson: Thin Lens Equation". PhysicsLab.org. Retrieved March 17, 2015.
  23. ^ "The Mathematics of Lenses". The Physics Classroom. Retrieved March 17, 2015.
  24. ^ Hecht 2002, p. 120.
  25. ^ There are always 3 "easy rays". For the third ray in this case, see File:Lens3b third ray.svg.
  26. ^ Hecht 2002, p. 168.
  27. ^ Proteep Mallik (2005). "The Axicon" (PDF). Archived from the original (PDF) on 23 November 2009. Retrieved 22 November 2007.
  28. ^ a b Grbic, A.; Eleftheriades, G. V. (2004). "Overcoming the Diffraction Limit with a Planar Left-handed Transmission-line Lens". Physical Review Letters. 92 (11): 117403. Bibcode:2004PhRvL..92k7403G. doi:10.1103/PhysRevLett.92.117403. PMID 15089166.
  29. ^ Valentine, J.; et al. (2008). "Three-dimensional optical metamaterial with a negative refractive index". Nature. 455 (7211): 376–9. Bibcode:2008Natur.455..376V. doi:10.1038/nature07247. PMID 18690249.
  30. ^ Yao, Jie; Liu, Zhaowei; Liu, Yongmin; Wang, Yuan; Sun, Cheng; Bartal, Guy; Stacy, Angelica M.; Zhang, Xiang (2008-08-15). "Optical Negative Refraction in Bulk Metamaterials of Nanowires". Science. 321 (5891): 930. Bibcode:2008Sci...321..930Y. CiteSeerX doi:10.1126/science.1157566. ISSN 0036-8075. PMID 18703734.
  31. ^ Nielsen, R.B.; Thoreson, M.D.; Chen, W.; Kristensen, A.; Hvam, J.M.; Shalaev, V. M.; Boltasseva, A. (2010). "Toward superlensing with metal–dielectric composites and multilayers" (PDF). Applied Physics B. 100 (1): 93. Bibcode:2010ApPhB.100...93N. doi:10.1007/s00340-010-4065-z. Archived from the original (PDF) on 9 March 2013.
  32. ^ Patel, Prachi. "Good-Bye to Curved Lens: New Lens Is Flat". Retrieved 2015-05-16.
  33. ^ Schottner, G (May 2003). "Scratch and Abrasion Resistant Coatings on Plastic Lenses—State of the Art, Current Developments and Perspectives". Journal of Sol-Gel Science and Technology. pp. 71–79. Retrieved 28 December 2009.


  • Hecht, Eugene (1987). Optics (2nd ed.). Addison Wesley. ISBN 978-0-201-11609-0. Chapters 5 & 6.
  • Hecht, Eugene (2002). Optics (4th ed.). Addison Wesley. ISBN 978-0-321-18878-6.
  • Greivenkamp, John E. (2004). Field Guide to Geometrical Optics. SPIE Field Guides vol. FG01. SPIE. ISBN 978-0-8194-5294-8.

External links


Afocal system

In optics an afocal system (a system without focus) is an optical system that produces no net convergence or divergence of the beam, i.e. has an infinite effective focal length. This type of system can be created with a pair of optical elements where the distance between the elements is equal to the sum of each element's focal length (d = f1+f2). A simple example of an afocal optical system is an optical telescope imaging a star, the light entering the system is at infinity and the image it forms is at infinity (the light is collimated). Although the system does not alter the divergence of a collimated beam, it does alter the width of the beam, increasing magnification. The magnification of such a telescope is given by

Afocal systems are used in laser optics, for instance as beam expanders, Infrared and forward looking infrared systems, camera zoom lenses and telescopic lens attachments such as teleside converters, and photography setups combining cameras and telescopes (Afocal photography).


Bronica also Zenza Bronica (in Japanese: ゼンザブロニカ) was a Japanese manufacturer of classic medium-format roll film cameras and photographic equipment based in Tokyo, Japan. Their single-lens reflex (SLR) system-cameras competed with Pentax, Hasselblad, Mamiya and others in the medium-format camera market.


The Cineon System was one of the first computer based digital film system created by Kodak in the early 1990s. It was an integrated suite of components consisting a Motion picture film scanner, a film recorder and workstation hardware with software (the Cineon Digital Film Workstation) for compositing, visual effects, image restoration and color management.The system was first released in September 1992 to Cinesite Hollywood. The workstations were initially built on Sun-Transputer based hardware. In July 1993 version 2.1.3 of the software was released for Silicon Graphics Inc, SGI Onyx hardware. The software was withdrawn from sale by 1997, although a number of customers continued to use it beyond that date.

As an end-to-end solution for 4K resolution, 10 bit digital film production and Digital intermediate the system was one of the first. The three major components of the system (scanner, workstation software, and recorder) have all received separate AMPAS Scientific and Technical Awards.The Cineon project was also responsible for the creation of the Cineon ( .cin) 10 bit log file format, designed to handle digital film frames. Although the product is no longer for sale, Cineon file format that Kodak defined was for a long time commonly used in the film visual effects world, and formed the basis for the newer SMPTE-standardised Digital Picture Exchange (DPX) format.

Depth of focus

Depth of focus is a lens optics concept that measures the tolerance of placement of the image plane (the film plane in a camera) in relation to the lens. In a camera, depth of focus indicates the tolerance of the film's displacement within the camera and is therefore sometimes referred to as "lens-to-film tolerance".


DocuTech is the name given to a line of electronic production-publishing systems produced by Xerox Corporation. It allowed paper documents to be scanned, electronically edited, and then printed on demand. DocuTech systems were the last known to use the XNS protocol for networking.

The very first DocuTech system, known as the DocuTech Production Publisher was announced on October 2, 1990. Its 135 page-per minute, black and white, xerographic print engine and attached finisher module were largely based on ones previously developed for the Xerox 5090 Duplicator system (announced by Xerox in 1988). However, image generation in the DocuTech was performed using a digitally driven, dual-beam, Laser ROS (Raster Output Scanner) rather than by the light-lens optics and exposure lamps found in the "analog" 5090 system. The system's scanner module allowed document scanning in a number of modes including manually from the platen or automatically using a 23 page-per-minute recirculating document feeder. The scanner also had a semi-automatic side feeder which could be used to scan large originals and computer fan-fold (CFF) input. The entire system was controlled by an electronic sub-system (ESS) of a proprietary Xerox design. The ESS incorporated a large number of Xerox proprietary Mesa processors which were specifically designed for high-speed image processing, 32 MegaBytes of RAM, I/O control interfaces for communicating with the Printer and Scanner modules, as well as 3 disk drives which contained system software and space for storing images (including those for saved jobs).

With its ability to digitally scan, edit and store documents for later retrieval, and also its ability to output stitched or tape-bound books, the DocuTech Production Publisher was arguably the first fully integrated "print-on-demand" publishing system. In fact, the Xerox DocuTech line of publishing systems is largely credited with establishing the "print-on-demand" industry.

In late 1991, Xerox re-branded the original DocuTech Production Publisher as the DocuTech Production Publisher Model 135 (DT135). This was done to distinguish it from the DocuTech Production Publisher Model 90 which the company anticipated announcing in 1992. The model numbers were chosen to reflect the page-per-minute print speeds of the two models. The controller and scanner where common for both models, but the Model 90 used a different print engine based on one developed for the previously announced Xerox 4090 printer.

The original DocuTech Production Publisher was capable of scanning and then printing black-and-white pages at up to 135 pages per minutes (for letter or A4 sizes) with an output resolution of 600 x 600 dots per inch (dpi). Scanned documents could be saved to a special memory area on disk known as the "Save Queue" where they could be retained, edited if desired (using the built in editor), and later printed "on demand". The system was also capable of printing on sheet sizes up to 14x17 inches. Another important feature of this earliest DocuTech was its ability to perform signature imposition and generate "2-up signatures" (later 4-up was added) in the proper page imposition order to create signature booklets. (note: Folding, trimming, and stitching of booklets was done by an optional accessory known as a C.P. Bourg Signature Booklet Maker or SBM-1, which could be attached in-line to the system's output finisher.)

In June 1992 Xerox announced the DocuTech 135 Network Publisher which augmented the earlier DocuTech's capability by enabling it to receive and print documents transmitted over a network. Although this system's network connectivity was limited to Xerox's proprietary XNS network, a DocuTech Network Server was also offered which enabled the now growing family of DocuTech Publishing Systems to be utilized with a broader set of networks.

The DocuTech 6135 is an improved version of the DT135, with a Sun Microsystems workstation controller replacing the original controller and scanner. Additional improvements include an optional VLD laser assembly, which uses sub-pixel dot positioning, while not truly increasing the print resolution to 600 x 1200 dpi, improves the halftone quality.

The DocuTech system's main competitor in the field of print-on-demand production plant is IBM's InfoPrint system. In addition, there are a number of other competitors in the field, led by the Kodak Digimaster Production Printer, which is sold under a number of different brand names. Xerox retired the original DocuTech 1xx platform in favor of the DocuTech 61xx and Nuvera systems (originally introduced as the DocuTech 100/120 Copier/Printer).


A headlamp is a lamp attached to the front of a vehicle to light the road ahead. Headlamps are also often called headlights, but in the most precise usage, headlamp is the term for the device itself and headlight is the term for the beam of light produced and distributed by the device.

Headlamp performance has steadily improved throughout the automobile age, spurred by the great disparity between daytime and nighttime traffic fatalities: the US National Highway Traffic Safety Administration states that nearly half of all traffic-related fatalities occur in the dark, despite only 25% of traffic travelling during darkness.Other vehicles, such as trains and aircraft, are required to have headlamps. Bicycle headlamps are often used on bicycles, and are required in some jurisdictions. They can be powered by a battery or a small generator mechanically integrated into the workings of the bicycles.

Index of optics articles

Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light. Because light is an electromagnetic wave, other forms of electromagnetic radiation such as X-rays, microwaves, and radio waves exhibit similar properties.

Laboratory school

A laboratory school or demonstration school is an elementary or secondary school operated in association with a university, college, or other teacher education institution and used for the training of future teachers, educational experimentation, educational research, and professional development.Many laboratory schools follow a model of experiential education based on the original Laboratory School run by John Dewey at the University of Chicago. Many laboratory schools are still in operation in the United States and around the globe. They are known by many names: laboratory schools, demonstration schools, campus schools, model schools, university affiliated schools, child development schools, etc., and most have a connection to a college or university. Each university affiliated school has a unique relationship with a college or university and a different grade configuration. Some lab schools are only for preschool or kindergarten children, some are preschool through fifth or sixth grade, and some continue through high school.

Khan Lab School in Silicon Valley is one of the few laboratory schools not affiliated with a college or university. It is affiliated with Khan Academy, a non-profit educational organization. The school's experimentation with abolishing grade levels was featured on Voice of America in 2016.


Lensing is a surname. Notable people with the surname include:

Kees Lensing (born 1978), Namibian rugby union player

Vicki Lensing (born 1957), American politician

Wilhelmina Elisabeth Lensing (1847–1925), Dutch feminist, politician and writer


A microscope (from the Ancient Greek: μικρός, mikrós, "small" and σκοπεῖν, skopeîn, "to look" or "see") is an instrument used to see objects that are too small to be seen by the naked eye. Microscopy is the science of investigating small objects and structures using such an instrument. Microscopic means invisible to the eye unless aided by a microscope.

There are many types of microscopes, and they may be grouped in different ways. One way is to describe the way the instruments interact with a sample to create images, either by sending a beam of light or electrons to a sample in its optical path, or by scanning across, and a short distance from the surface of a sample using a probe. The most common microscope (and the first to be invented) is the optical microscope, which uses light to pass through a sample to produce an image. Other major types of microscopes are the fluorescence microscope, the electron microscope (both the transmission electron microscope and the scanning electron microscope) and the various types of scanning probe microscopes.

Modular optoelectronic multispectral scanner

The modular optoelectronic multispectral scanner (MOMS) is a scanning system for spaceborne, geoscientific remote sensing applications used in satellite navigation systems for sensing atmospheric and oceanic systems. The scanner is combination of separate spectrometer blocks.


The pupil is a hole located in the center of the iris of the eye that allows light to strike the retina. It appears black because light rays entering the pupil are either absorbed by the tissues inside the eye directly, or absorbed after diffuse reflections within the eye that mostly miss exiting the narrow pupil.

In humans the pupil is round, but other species, such as some cats, have vertical slit pupils, goats have horizontally oriented pupils, and some catfish have annular types. In optical terms, the anatomical pupil is the eye's aperture and the iris is the aperture stop. The image of the pupil as seen from outside the eye is the entrance pupil, which does not exactly correspond to the location and size of the physical pupil because it is magnified by the cornea. On the inner edge lies a prominent structure, the collarette, marking the junction of the embryonic pupillary membrane covering the embryonic pupil.

Simple lens

In optics, a simple lens or singlet lens is a lens consisting of a single simple element. Typical examples include a magnifying glass or a lens in a pair of simple reading glasses.Simple lenses are prone to aberrations, especially chromatic aberration. They cannot be used for precise imaging and make poor camera lenses. They are commonly used for laser applications, however, where the beams are both monochromatic (minimizing chromatic aberration) and narrow (minimizing spherical aberration).

Some cameras with fixed lenses have been made using a simple lens, usually a meniscus lens with the convex face facing outward. In such examples the lens aperture is made small and in some cases (such as the Kodak Brownie 127 camera), the film plane is curved to reduce the impact of aberrations.


Singlet may refer to:

singlet state, in theoretical physics, a quantum state with zero spin

Singlet fission, in molecular photophysics

in spectroscopy, an entity appearing as a single peak; see NMR spectroscopy

in optics, a single lens element, the building blocks of lens systems; see lens (optics)

a one-piece collarless garment, also known as a sleeveless shirt or vest

wrestling singlet, a one-piece garment specific to wrestling

BID/60, a British encryption machine

Singlet oxygen, the common name used for an excited form of molecular oxygen

Soft focus

In photography, soft focus is a lens flaw, in which the lens forms images that are blurred due to spherical aberration. A soft focus lens deliberately introduces spherical aberration in order to give the appearance of blurring the image while retaining sharp edges; it is not the same as an out-of-focus image, and the effect cannot be achieved simply by defocusing a sharp lens. Soft focus is also the name of the style of photograph produced by such a lens.


A spherometer is an instrument for the precise measurement of the radius of curvature of a sphere or a curved surface. Originally, these instruments were primarily used by opticians to measure the curvature of the surface of a lens.


In physics, a wavefront is the locus of points characterized by propagation of positions of identical phase: propagation of a point in 1D, a curve in 2D or a surface in 3D. For an electromagnetic wave, the wavefront is represented as a surface of identical phase, and can be modified with conventional optics. For instance, a lens can change the shape of optical wavefronts from planar to spherical as the lens introduces a spatial phase variation across the beam shape.

This page is based on a Wikipedia article written by authors (here).
Text is available under the CC BY-SA 3.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.