![Lens and wavefronts](https://upload.wikimedia.org/wikipedia/commons/c/c7/Lens_and_wavefronts.gif)
A lens is a transmissive
optical
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultravio ...
device which focuses or disperses a
light beam
A light beam or beam of light is a directional projection of light energy radiating from a light source. Sunlight forms a light beam (a sunbeam) when filtered through media such as clouds, foliage, or windows. To artificially produce a light ...
by means of
refraction
In physics, refraction is the redirection of a wave as it passes from one medium to another. The redirection can be caused by the wave's change in speed or by a change in the medium. Refraction of light is the most commonly observed phenome ...
. A
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 aberr ...
consists of a single piece of
transparent material, while a
compound lens consists of several simple lenses (''elements''), usually arranged along a common
axis
An axis (plural ''axes'') is an imaginary line around which an object rotates or is symmetrical. Axis may also refer to:
Mathematics
* Axis of rotation: see rotation around a fixed axis
* Axis (mathematics), a designator for a Cartesian-coordinat ...
. Lenses are made from materials such as
glass
Glass is a non-crystalline, often transparent, amorphous solid that has widespread practical, technological, and decorative use in, for example, window panes, tableware, and optics. Glass is most often formed by rapid cooling ( quenching ...
or
plastic
Plastics are a wide range of synthetic or semi-synthetic materials that use polymers as a main ingredient. Their plasticity makes it possible for plastics to be moulded, extruded or pressed into solid objects of various shapes. This adaptab ...
, and are
ground and
polished or
molded to a desired shape. A lens can focus light to form an
image, unlike a
prism
Prism usually refers to:
* Prism (optics), a transparent optical component with flat surfaces that refract light
* Prism (geometry), a kind of polyhedron
Prism may also refer to:
Science and mathematics
* Prism (geology), a type of sedimentary ...
, 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
Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequencies between 300 MHz and 300 GHz respectively. Different sources define different frequency ra ...
lenses,
electron lens
Transmission electron microscopy (TEM) is a microscopy technique in which a beam of electrons is transmitted through a specimen to form an image. The specimen is most often an ultrathin section less than 100 nm thick or a suspension on a g ...
es,
acoustic lens
Acoustic may refer to:
Music Albums
* ''Acoustic'' (Above & Beyond album), 2014
* ''Acoustic'' (Deine Lakaien album), 2007
* ''Acoustic'' (Everything but the Girl album), 1992
* ''Acoustic'' (John Lennon album), 2004
* ''Acoustic'' (Love Amo ...
es, or
explosive lens
An explosive lens—as used, for example, in nuclear weapons—is a highly specialized shaped charge. In general, it is a device composed of several explosive charges. These charges are arranged and formed with the intent to control the shape ...
es.
Lenses are used in various imaging devices like
telescope
A telescope is a device used to observe distant objects by their emission, absorption, or reflection of electromagnetic radiation. Originally meaning only an optical instrument using lenses, curved mirrors, or a combination of both to observ ...
s,
binoculars and
camera
A camera is an optical instrument that can capture an image. Most cameras can capture 2D images, with some more advanced models being able to capture 3D images. At a basic level, most cameras consist of sealed boxes (the camera body), with a ...
s. They are also used as visual aids in
glasses
Glasses, also known as eyeglasses or spectacles, are vision eyewear, with lenses (clear or tinted) mounted in a frame that holds them in front of a person's eyes, typically utilizing a bridge over the nose and hinged arms (known as temples ...
to correct defects of vision such as
myopia and
hypermetropia
Far-sightedness, also known as long-sightedness, hypermetropia, or hyperopia, is a condition of the eye where distant objects are seen clearly but near objects appear blurred. This blurred effect is due to incoming light being focused behind, i ...
.
History
![LSST Telescope - L1 Lens of the camera](https://upload.wikimedia.org/wikipedia/commons/f/f5/LSST_Telescope_-_L1_Lens_of_the_camera.jpg)
The word ''
lens
A lens is a transmissive optical device which 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 ...
'' comes from ''
lēns'', the Latin name of the
lentil
The lentil (''Lens culinaris'' or ''Lens esculenta'') is an edible legume. It is an annual plant known for its lens-shaped seeds. It is about tall, and the seeds grow in pods, usually with two seeds in each. As a food crop, the largest pro ...
(a seed of a lentil plant), because a double-convex lens is lentil-shaped. The lentil also gives its name to a
geometric figure
A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type.
A plane shape or plane figure is constrained to lie o ...
.
Some scholars argue that the archeological evidence indicates that there was widespread use of lenses in antiquity, spanning several millennia. The so-called
Nimrud lens is a rock crystal artifact dated to the 7th century BCE which may or may not have been used as a magnifying glass, or a burning glass.
Others have suggested that certain
Egyptian hieroglyphs
Egyptian hieroglyphs (, ) were the formal writing system used in Ancient Egypt, used for writing the Egyptian language. Hieroglyphs combined logographic, syllabic and alphabetic elements, with some 1,000 distinct characters.There were about 1, ...
depict "simple glass meniscal lenses".
The oldest certain reference to the use of lenses is from
Aristophanes
Aristophanes (; grc, Ἀριστοφάνης, ; c. 446 – c. 386 BC), son of Philippus, of the deme Kydathenaion ( la, Cydathenaeum), was a comic playwright or comedy-writer of ancient Athens and a poet of Old Attic Comedy. Eleven of his for ...
' play ''
The Clouds
''The Clouds'' ( grc, Νεφέλαι ''Nephelai'') is a Greek comedy play written by the playwright Aristophanes. A lampooning of intellectual fashions in classical Athens, it was originally produced at the City Dionysia in 423BC and was not ...
'' (424 BCE) mentioning a burning-glass.
Pliny the Elder
Gaius Plinius Secundus (AD 23/2479), called Pliny the Elder (), was a Roman author, naturalist and natural philosopher, and naval and army commander of the early Roman Empire, and a friend of the emperor Vespasian. He wrote the encyclopedic ' ...
(1st century) confirms that burning-glasses were known in the Roman period.
Pliny also has the earliest known reference to the use of a
corrective lens
A corrective lens is a lens (i.e. a transmissive optical device) that is typically worn in front of the eye to improve daily vision. The most common use is to treat refractive errors: myopia, hypermetropia, astigmatism, and presbyopia. G ...
when he mentions that
Nero
Nero Claudius Caesar Augustus Germanicus ( ; born Lucius Domitius Ahenobarbus; 15 December AD 37 – 9 June AD 68), was the fifth Roman emperor and final emperor of the Julio-Claudian dynasty, reigning from AD 54 unti ...
was said to watch the
gladiatorial games using an
emerald (presumably
concave
Concave or concavity may refer to:
Science and technology
* Concave lens
* Concave mirror
Mathematics
* Concave function, the negative of a convex function
* Concave polygon, a polygon which is not convex
* Concave set
* The concavity of a ...
to correct for
nearsightedness
Near-sightedness, also known as myopia and short-sightedness, is an eye disease where light focuses in front of, instead of on, the retina. As a result, distant objects appear blurry while close objects appear normal. Other symptoms may include ...
, though the reference is vague). Both Pliny and
Seneca the Younger
Lucius Annaeus Seneca the Younger (; 65 AD), usually known mononymously as Seneca, was a Stoic philosopher of Ancient Rome, a statesman, dramatist, and, in one work, satirist, from the post-Augustan age of Latin literature.
Seneca was born in ...
(3 BC–65 AD) described the magnifying effect of a glass globe filled with water.
Ptolemy
Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importance ...
(2nd century) wrote a book on ''
Optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultrav ...
'', 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 Eugenius of Palermo (also Eugene) ( la, Eugenius Siculus, el, Εὐγενἠς Εὐγένιος ὁ τῆς Πανόρμου, it, Eugenio da Palermo; 1130 – 1202) was an '' amiratus'' (admiral) of the Kingdom of Sicily in the late twelfth cen ...
1154). Between the 11th and 13th century "
reading stone
A reading stone is an approximately hemispherical lens that can be placed on top of text to magnify the letters so that people with presbyopia can read it more easily. Reading stones were among the earliest common uses of lenses.
The invention o ...
s" were invented. These were primitive plano-convex lenses initially made by cutting a glass sphere in half. The medieval (11th or 12th century) rock crystal
Visby lenses may or may not have been intended for use as burning glasses.
Spectacles
Glasses, also known as eyeglasses or spectacles, are vision eyewear, with lenses (clear or tinted) mounted in a frame that holds them in front of a person's eyes, typically utilizing a bridge over the nose and hinged arms (known as temples o ...
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. 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, and later in the spectacle-making centres in both the
Netherlands
)
, anthem = ( en, "William of Nassau")
, image_map =
, map_caption =
, subdivision_type = Sovereign state
, subdivision_name = Kingdom of the Netherlands
, established_title = Before independence
, established_date = Spanish Netherl ...
and
Germany
Germany,, officially the Federal Republic of Germany, is a country in Central Europe. It is the second most populous country in Europe after Russia, and the most populous member state of the European Union. Germany is situated betwe ...
.
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). The practical development and experimentation with lenses led to the invention of the compound
optical microscope
The optical microscope, also referred to as a light microscope, is a type of microscope that commonly uses visible light and a system of lenses to generate magnified images of small objects. Optical microscopes are the oldest design of microsc ...
around 1595, and the
refracting telescope
A refracting telescope (also called a refractor) is a type of optical telescope that uses a lens as its objective to form an image (also referred to a dioptric telescope). The refracting telescope design was originally used in spyglasses and a ...
in 1608, both of which appeared in the spectacle-making centres in the
Netherlands
)
, anthem = ( en, "William of Nassau")
, image_map =
, map_caption =
, subdivision_type = Sovereign state
, subdivision_name = Kingdom of the Netherlands
, established_title = Before independence
, established_date = Spanish Netherl ...
.
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. Optical theory on
refraction
In physics, refraction is the redirection of a wave as it passes from one medium to another. The redirection can be caused by the wave's change in speed or by a change in the medium. Refraction of light is the most commonly observed phenome ...
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
An achromatic lens or achromat is a lens that is designed to limit the effects of chromatic and spherical aberration. Achromatic lenses are corrected to bring two wavelengths (typically red and blue) into focus on the same plane.
The most comm ...
by
Chester Moore Hall in
England
England is a country that is part of the United Kingdom. It shares land borders with Wales to its west and Scotland to its north. The Irish Sea lies northwest and the Celtic Sea to the southwest. It is separated from continental Europe b ...
in 1733, an invention also claimed by fellow Englishman
John Dollond
John Dollond FRS (10 June O.S. (21 June N.S.) 170630 November 1761) was an English optician, known for his successful optics business and his patenting and commercialization of achromatic doublets.
Biography
Dollond was the son of a Hugue ...
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
Astigmatism is a type of refractive error due to rotational asymmetry in the eye's refractive power. This results in distorted or blurred vision at any distance. Other symptoms can include eyestrain, headaches, and trouble driving at n ...
in someone's eye.
Types of simple lenses
![Lenses en](https://upload.wikimedia.org/wikipedia/commons/a/a7/Lenses_en.svg)
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
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
. If both surfaces have the same radius of curvature, the lens is ''equiconvex''. A lens with two
concave
Concave or concavity may refer to:
Science and technology
* Concave lens
* Concave mirror
Mathematics
* Concave function, the negative of a convex function
* Concave polygon, a polygon which is not convex
* Concave set
* The concavity of a ...
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 lens
A corrective lens is a lens (i.e. a transmissive optical device) that is typically worn in front of the eye to improve daily vision. The most common use is to treat refractive errors: myopia, hypermetropia, astigmatism, and presbyopia. G ...
es.
If the lens is biconvex or plano-convex, a
collimated
A collimated beam of light or other electromagnetic radiation has parallel rays, and therefore will spread minimally as it propagates. A perfectly collimated light beam, with no divergence, would not disperse with distance. However, diffraction p ...
beam of light passing through the lens converges to a spot (a ''focus'') behind the lens. In this case, the lens is called a ''positive'' or ''converging'' lens. For a
thin lens
In optics, a thin lens is a lens with a thickness (distance along the optical axis between the two surfaces of the lens) that is negligible compared to the radii of curvature of the lens surfaces. Lenses whose thickness is not negligible are so ...
in air, the distance from the lens to the spot is the
focal length of the lens, which is commonly represented by ''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.
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. For a thin lens in air, the distance from this point to the lens is the focal length, though it is negative with respect to the focal length of a converging lens.
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
In optics, a thin lens is a lens with a thickness (distance along the optical axis between the two surfaces of the lens) that is negligible compared to the radii of curvature of the lens surfaces. Lenses whose thickness is not negligible are so ...
with two surfaces of equal curvature would have zero
optical power
In optics, optical power (also referred to as dioptric power, refractive power, focusing power, or convergence power) is the degree to which a lens, mirror, or other optical system converges or diverges light. It is equal to the reciprocal of the ...
, 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
![Spherical Lens](https://upload.wikimedia.org/wikipedia/commons/8/81/Spherical_Lens.gif)
The focal length of a lens ''in air'' can be calculated from the lensmaker's equation:
:
where
:
is the focal length of the lens,
:
is the
refractive index
In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium.
The refractive index determines how much the path of light is bent, or ...
of the lens material,
:
is the
radius of curvature
In differential geometry, the radius of curvature, , is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius o ...
(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
Reciprocal may refer to:
In mathematics
* Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal''
* Reciprocal polynomial, a polynomial obtained from another pol ...
of the focal length, 1/''f'', is the
optical power
In optics, optical power (also referred to as dioptric power, refractive power, focusing power, or convergence power) is the degree to which a lens, mirror, or other optical system converges or diverges light. It is equal to the reciprocal of the ...
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 ''R''1 and ''R''2
The signs of the lens' radii of curvature indicate whether the corresponding surfaces are convex or concave. The
sign convention
In physics, a sign convention is a choice of the physical significance of signs (plus or minus) for a set of quantities, in a case where the choice of sign is arbitrary. "Arbitrary" here means that the same physical system can be correctly describ ...
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, and indicate ''convex'' surfaces (used to converge light in a positive lens), while and 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
infinite
Infinite may refer to:
Mathematics
* Infinite set, a set that is not a finite set
*Infinity, an abstract concept describing something without any limit
Music
*Infinite (group), a South Korean boy band
*''Infinite'' (EP), debut EP of American m ...
.
Thin lens approximation
If ''d'' is small compared to ''R''
1 and ''R''
2, then the ''
thin lens
In optics, a thin lens is a lens with a thickness (distance along the optical axis between the two surfaces of the lens) that is negligible compared to the radii of curvature of the lens surfaces. Lenses whose thickness is not negligible are so ...
'' 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
A point source is a single identifiable ''localised'' source of something. A point source has negligible extent, distinguishing it from other source geometries. Sources are called point sources because in mathematical modeling, these sources ca ...
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 ''S''
1 and ''S''
2 respectively, for a lens of negligible thickness (
thin lens
In optics, a thin lens is a lens with a thickness (distance along the optical axis between the two surfaces of the lens) that is negligible compared to the radii of curvature of the lens surfaces. Lenses whose thickness is not negligible are so ...
), in air, the distances are related by the thin lens formula:
:
This can also be put into the "Newtonian" form:
:
where
and
.
![lens3](https://upload.wikimedia.org/wikipedia/commons/7/71/Lens3.svg)
Therefore, if an object is placed at a distance from a positive lens of focal length ''f'', we will find an image distance ''S''
2 according to this formula. If a screen is placed at a distance ''S''
2 on the opposite side of the lens, an image is formed on it. This sort of image, which can be projected onto a screen or
image sensor
An image sensor or imager is a sensor that detects and conveys information used to make an image. It does so by converting the variable attenuation of light waves (as they pass through or reflect off objects) into signals, small bursts of c ...
, is known as a ''
real image''. This is the principle of the
camera
A camera is an optical instrument that can capture an image. Most cameras can capture 2D images, with some more advanced models being able to capture 3D images. At a basic level, most cameras consist of sealed boxes (the camera body), with a ...
, and also of the
human eye, in which the
retina
The retina (from la, rete "net") is the innermost, light-sensitive layer of tissue of the eye of most vertebrates and some molluscs. The optics of the eye create a focused two-dimensional image of the visual world on the retina, which then ...
serves as the image sensor.
The focusing adjustment of a camera adjusts ''S''
2, as using an image distance different from that required by this formula produces a
defocused (fuzzy) image for an object at a distance of ''S''
1 from the camera. Put another way, modifying ''S''
2 causes objects at a different ''S''
1 to come into perfect focus.
![lens3b](https://upload.wikimedia.org/wikipedia/commons/9/97/Lens3b.svg)
In some cases ''S''
2 is negative, indicating that the image is formed on the opposite side of the lens from where those rays are being considered. Since the diverging light rays emanating from the lens never come into focus, and those rays are not physically present at the point where they ''appear'' to form an image, this is called a
virtual image
In optics, an ''image'' is defined as the collection of focus points of light rays coming from an object. A '' real image'' is the collection of focus points made by converging rays, while a virtual image is the collection of focus points ma ...
. 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, ''S''
1 then being measured from the virtual image location behind the first lens to the second lens. This is exactly what the eye does when looking through a
magnifying glass. The magnifying glass creates a (magnified) virtual image behind the magnifying glass, but those rays are then re-imaged by the
lens of the eye
The lens, or crystalline lens, is a transparent biconvex structure in the eye that, along with the cornea, helps to refract light to be focused on the retina. By changing shape, it functions to change the focal length of the eye so that it ca ...
to create a ''real image'' on the
retina
The retina (from la, rete "net") is the innermost, light-sensitive layer of tissue of the eye of most vertebrates and some molluscs. The optics of the eye create a focused two-dimensional image of the visual world on the retina, which then ...
.
Using a positive lens of focal length ''f'', a virtual image results when , the lens thus being used as a magnifying glass (rather than if as for a camera). Using a negative lens () with a ''real object'' () can only produce a virtual image (), according to the above formula. It is also possible for the object distance ''S''
1 to be negative, in which case the lens sees a so-called ''virtual object''. This happens when the lens is inserted into a converging beam (being focused by a previous lens) ''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
The Barlow lens, named after Peter Barlow, is a diverging lens which, used in series with other optics in an optical system, increases the effective focal length of an optical system as perceived by all components that are after it in the system. ...
.
For a
thin lens
In optics, a thin lens is a lens with a thickness (distance along the optical axis between the two surfaces of the lens) that is negligible compared to the radii of curvature of the lens surfaces. Lenses whose thickness is not negligible are so ...
, the distances ''S''
1 and ''S''
2 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 ''S''
1 and ''S''
2 or there are multiple lens elements (a
compound lens), one must instead measure from the object and image to the
principal plane
In Gaussian optics, the cardinal points consist of three pairs of points located on the optical axis of a rotationally symmetric, focal, optical system. These are the '' focal points'', the principal points, and the nodal points. For ''ideal'' s ...
s of the lens. If distances ''S''
1 or ''S''
2 pass through a
medium
Medium may refer to:
Science and technology
Aviation
*Medium bomber, a class of war plane
*Tecma Medium, a French hang glider design
Communication
* Media (communication), tools used to store and deliver information or data
* Medium of ...
other than air or vacuum a more complicated analysis is required.
Magnification
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.
This magnification formula provides two easy ways to distinguish converging (''f > 0'') and diverging (''f < 0'') lenses: For an object very close to the lens (), 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 (), a converging lens would form an inverted image, whereas a diverging lens would form an upright image.
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
The plate scale of a telescope connects the angular separation of an object with the linear separation of its image at the focal plane
If focal length f is measured in mm, the plate scale in radians per mm is given by angular separation ''θ'' and ...
, which compares the apparent (angular) size of a distant object to the size of the real image produced at the focus. The plate scale is the reciprocal of the focal length of the camera lens; lenses are categorized as
long-focus lens
In photography, a long-focus lens is a camera lens which has a focal length that is longer than the diagonal measure of the film or sensor that receives its image.
It is used to make distant objects appear magnified with magnification increasin ...
es or
wide-angle lens
In photography and cinematography, a wide-angle lens refers to a lens whose focal length is substantially smaller than the focal length of a normal lens for a given film plane. This type of lens allows more of the scene to be included in the ...
es according to their focal lengths.
Using an inappropriate measurement of magnification can be formally correct but yield a meaningless number. For instance, using a magnifying glass of 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: . 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
The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...
using a camera with a 50 mm lens, one is not concerned with the linear magnification 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, , and , 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
The resolution of an optical imaging system a microscope, telescope, or camera can be limited by factors such as imperfections in the lenses or misalignment. However, there is a principal limit to the resolution of any optical system, due to t ...
.
Aberrations
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. Spherical aberration can be minimised with normal lens shapes by carefully choosing the surface curvatures for a particular application. For instance, a plano-convex lens, which is used to focus a collimated beam, produces a sharper focal spot when used with the convex side towards the beam source.
Coma
''Coma'', or ''comatic aberration'', derives its name from the
comet
A comet is an icy, small Solar System body that, when passing close to the Sun, warms and begins to release gases, a process that is called outgassing. This produces a visible atmosphere or coma, and sometimes also a tail. These phenomena ...
-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 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
Dispersion may refer to:
Economics and finance
* Dispersion (finance), a measure for the statistical distribution of portfolio returns
* Price dispersion, a variation in prices across sellers of the same item
*Wage dispersion, the amount of variat ...
of the lens material—the variation of its
refractive index
In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium.
The refractive index determines how much the path of light is bent, or ...
, ''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
An achromatic lens or achromat is a lens that is designed to limit the effects of chromatic and spherical aberration. Achromatic lenses are corrected to bring two wavelengths (typically red and blue) into focus on the same plane.
The most comm ...
(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
An apochromat, or apochromatic lens (apo), is a photographic or other lens that has better correction of chromatic and spherical aberration than the much more common achromat lenses.
Explanation
Chromatic aberration is the phenomenon of differen ...
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
Fluorite (also called fluorspar) is the mineral form of calcium fluoride, CaF2. It belongs to the halide minerals. It crystallizes in isometric cubic habit, although octahedral and more complex isometric forms are not uncommon.
The Mohs sca ...
. This naturally occurring substance has the highest known
Abbe number
In optics and lens design, the Abbe number, also known as the V-number or constringence of a transparent material, is an approximate measure of the material's dispersion (change of refractive index versus wavelength), with high values of ''V'' in ...
, indicating that the material has low dispersion.
Other types of aberration
Other kinds of aberration include ''
field curvature
Petzval field curvature, named for Joseph Petzval, describes the optical aberration in which a flat object normal to the optical axis (or a non-flat object past the hyperfocal distance) cannot be brought properly into focus on a flat image pl ...
'', ''
barrel '' and ''
pincushion distortion
In geometric optics, distortion is a deviation from rectilinear projection; a projection in which straight lines in a scene remain straight in an image. It is a form of optical aberration.
Radial distortion
Although distortion can be irre ...
'', and ''
astigmatism
Astigmatism is a type of refractive error due to rotational asymmetry in the eye's refractive power. This results in distorted or blurred vision at any distance. Other symptoms can include eyestrain, headaches, and trouble driving at n ...
''.
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
In optics, an aperture is a hole or an opening through which light travels. More specifically, the aperture and focal length of an optical system determine the cone angle of a bundle of rays that come to a focus in the image plane.
An ...
. A
diffraction-limited
The resolution of an optical imaging system a microscope, telescope, or camera can be limited by factors such as imperfections in the lenses or misalignment. However, there is a principal limit to the resolution of any optical system, due to t ...
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
In optics, aberration is a property of optical systems, such as lenses, that causes light to be spread out over some region of space rather than focused to a point. Aberrations cause the image formed by a lens to be blurred or distorted, with th ...
discussed above. In many cases these aberrations can be compensated for to a great extent by using a combination of simple lenses with complementary aberrations. A ''compound lens'' is a collection of simple lenses of different shapes and made of materials of different refractive indices, arranged one after the other with a common axis.
The simplest case is where lenses are placed in contact: if the lenses of focal lengths ''f''
1 and ''f''
2 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'' = ''f''
1 + ''f''
2), 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
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 wh ...
'', since it produces no net convergence or divergence of the beam. Two lenses at this separation form the simplest type of
optical telescope
An optical telescope is a telescope that gathers and focuses light mainly from the visible part of the electromagnetic spectrum, to create a magnified image for direct visual inspection, to make a photograph, or to collect data through elect ...
. 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 (''f''
1 > 0, ''f''
2 > 0) produces a negative magnification, indicating an inverted image. A convex plus a concave lens (''f''
1 > 0 > ''f''
2) produces a positive magnification and the image is upright. For further information on simple optical telescopes, see
Refracting telescope § Refracting telescope designs.
Non spherical types
Cylindrical lens
A cylindrical lens is a lens which focuses light into a line instead of a point, as a spherical lens would. The curved face or faces of a cylindrical lens are sections of a cylinder, and focus the image passing through it into a line parallel to ...
es have curvature along only one axis. They are used to focus light into a line, or to convert the elliptical light from a
laser diode
The laser diode chip removed and placed on the eye of a needle for scale
A laser diode (LD, also injection laser diode or ILD, or diode laser) is a semiconductor device similar to a light-emitting diode in which a diode pumped directly with e ...
into a round beam. They are also used in motion picture
anamorphic lens
Anamorphic format is the cinematography technique of shooting a widescreen picture on standard 35 mm film or other visual recording media with a non-widescreen native aspect ratio. It also refers to the projection format in which a distorte ...
es.
Aspheric lens
An aspheric lens or asphere (often labeled ''ASPH'' on eye pieces) is a lens (optics), lens whose surface profiles are not portions of a sphere or Cylinder (geometry), cylinder. In photography, a camera lens, lens assembly that includes an aspheri ...
es have at least one surface 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. These were formerly complex to make and often extremely expensive, but advances in technology have greatly reduced the manufacturing cost for such lenses.
![Flat flexible plastic sheet lens](https://upload.wikimedia.org/wikipedia/commons/c/cd/Flat_flexible_plastic_sheet_lens.JPG)
A
Fresnel lens
A Fresnel lens ( ; ; or ) is a type of composite compact lens developed by the French physicist Augustin-Jean Fresnel (1788–1827) for use in lighthouses. It has been called "the invention that saved a million ships."
The design allows the c ...
has its optical surface broken up into narrow rings, allowing the lens to be much thinner and lighter than conventional lenses. Durable Fresnel lenses can be molded from plastic and are inexpensive.
Lenticular lens
A lenticular lens is an array of lenses, designed so that when viewed from slightly different angles, different parts of the image underneath are shown. The most common example is the lenses used in lenticular printing, where the technology is used ...
es are arrays of
microlens
A microlens is a small lens, generally with a diameter less than a millimetre (mm) and often as small as 10 micrometres (µm). The small sizes of the lenses means that a simple design can give good optical quality but sometimes unwanted effects ...
es that are used in
lenticular printing
Lenticular printing is a technology in which lenticular lenses (a technology also used for 3D displays) are used to produce printed images with an illusion of depth, or the ability to change or move as they are viewed from different angles.
Ex ...
to make images that have an illusion of depth or that change when viewed from different angles.
Bifocal lens
Bifocals are glasses, eyeglasses with two distinct optical powers. Bifocals are commonly prescribed to people with presbyopia who also require a correction for myopia, hyperopia, and/or astigmatism (eye), astigmatism.
History
Benjamin Franklin ...
has two or more, or a graduated, focal lengths ground into the lens.
A
gradient index lens has flat optical surfaces, but has a radial or axial variation in index of refraction that causes light passing through the lens to be focused.
An
axicon
An axicon is a specialized type of lens (optics), lens which has a Cone (geometry), conical surface. An axicon transforms a laser beam into a ring shaped distribution. They can be convex or concave and be made of any optical material. The combinat ...
has a
conical optical surface. It images a
point source
A point source is a single identifiable ''localised'' source of something. A point source has negligible extent, distinguishing it from other source geometries. Sources are called point sources because in mathematical modeling, these sources ca ...
into a line ''along'' the
optic axis
An optical axis is a line along which there is some degree of rotational symmetry in an optical system such as a camera lens, microscope or telescopic sight.
The optical axis is an imaginary line that defines the path along which light propagat ...
, or transforms a laser beam into a ring.
Diffractive optical element
Diffraction is defined as the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a s ...
s can function as lenses.
Superlens A superlens, or super lens, is a lens which uses metamaterials to go beyond the diffraction limit. For example, in 1995, Guerra combined a transparent grating having 50nm lines and spaces (the "metamaterial") with a conventional microscope immersio ...
es are made from
negative index metamaterials
Negative-index metamaterial or negative-index material (NIM) is a metamaterial whose refractive index for an electromagnetic wave has a negative value over some frequency range.
NIMs are constructed of periodic basic parts called unit cells, wh ...
and claim to produce images at spatial resolutions exceeding the
diffraction limit
The resolution of an optical imaging system a microscope, telescope, or camera can be limited by factors such as imperfections in the lenses or misalignment. However, there is a principal limit to the resolution of any optical system, due to t ...
.
The first superlenses were made in 2004 using such a
metamaterial
A metamaterial (from the Greek word μετά ''meta'', meaning "beyond" or "after", and the Latin word ''materia'', meaning "matter" or "material") is any material engineered to have a property that is not found in naturally occurring materials. ...
for microwaves.
Improved versions have been made by other researchers.
the superlens has not yet been demonstrated at
visible or near-
infrared
Infrared (IR), sometimes called infrared light, is electromagnetic radiation (EMR) with wavelengths longer than those of visible light. It is therefore invisible to the human eye. IR is generally understood to encompass wavelengths from around ...
wavelengths.
A prototype flat ultrathin lens, with no curvature has been developed.
Uses
A single convex lens mounted in a frame with a handle or stand is a
magnifying glass.
Lenses are used as
prosthetic
In medicine, a prosthesis (plural: prostheses; from grc, πρόσθεσις, prósthesis, addition, application, attachment), or a prosthetic implant, is an artificial device that replaces a missing body part, which may be lost through trau ...
s for the correction of
refractive error
Refractive error, also known as refraction error, is a problem with focusing light accurately on the retina due to the shape of the eye and or cornea. The most common types of refractive error are near-sightedness, far-sightedness, astigmatism ...
s such as
myopia,
hypermetropia
Far-sightedness, also known as long-sightedness, hypermetropia, or hyperopia, is a condition of the eye where distant objects are seen clearly but near objects appear blurred. This blurred effect is due to incoming light being focused behind, i ...
,
presbyopia
Presbyopia is physiological insufficiency of accommodation associated with the aging of the eye that results in progressively worsening ability to focus clearly on close objects. Also known as age-related farsightedness (or age-related long si ...
, and
astigmatism
Astigmatism is a type of refractive error due to rotational asymmetry in the eye's refractive power. This results in distorted or blurred vision at any distance. Other symptoms can include eyestrain, headaches, and trouble driving at n ...
. (See
corrective lens
A corrective lens is a lens (i.e. a transmissive optical device) that is typically worn in front of the eye to improve daily vision. The most common use is to treat refractive errors: myopia, hypermetropia, astigmatism, and presbyopia. G ...
,
contact lens,
eyeglasses
Glasses, also known as eyeglasses or spectacles, are vision eyewear, with lenses (clear or tinted) mounted in a frame that holds them in front of a person's eyes, typically utilizing a bridge over the nose and hinged arms (known as temples or ...
.) Most lenses used for other purposes have strict
axial symmetry
Axial symmetry is symmetry around an axis; an object is axially symmetric if its appearance is unchanged if rotated around an axis. ; 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
Astigmatism is a type of refractive error due to rotational asymmetry in the eye's refractive power. This results in distorted or blurred vision at any distance. Other symptoms can include eyestrain, headaches, and trouble driving at n ...
.
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
monocular
A monocular is a compact refracting telescope used to magnify images of distant objects, typically using an optical prism to ensure an erect image, instead of using relay lenses like most telescopic sights. The volume and weight of a mono ...
s,
binoculars,
telescopes
A telescope is a device used to observe distant objects by their emission, absorption, or reflection of electromagnetic radiation. Originally meaning only an optical instrument using lenses, curved mirrors, or a combination of both to observe ...
,
microscope
A microscope () is a laboratory instrument used to examine objects that are too small to be seen by the naked eye. Microscopy is the science of investigating small objects and structures using a microscope. Microscopic means being invisi ...
s,
camera
A camera is an optical instrument that can capture an image. Most cameras can capture 2D images, with some more advanced models being able to capture 3D images. At a basic level, most cameras consist of sealed boxes (the camera body), with a ...
s and
projectors. Some of these instruments produce a
virtual image
In optics, an ''image'' is defined as the collection of focus points of light rays coming from an object. A '' real image'' is the collection of focus points made by converging rays, while a virtual image is the collection of focus points ma ...
when applied to the human eye; others produce a
real image that can be captured on
photographic film or an
optical sensor, or can be viewed on a screen. In these devices lenses are sometimes paired up with
curved mirror
A curved mirror is a mirror with a curved reflecting surface. The surface may be either ''convex'' (bulging outward) or ''concave'' (recessed inward). Most curved mirrors have surfaces that are shaped like part of a sphere, but other shapes are ...
s 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
The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
is imaged, much of the visible and infrared light incident on the lens is concentrated into the small image. A large lens creates enough intensity to burn a flammable object at the focal point. Since ignition can be achieved even with a poorly made lens, lenses have been used as
burning-glass
A burning glass or burning lens is a large convex lens that can concentrate the sun's rays onto a small area, heating up the area and thus resulting in ignition of the exposed surface. Burning mirrors achieve a similar effect by using reflecting ...
es for at least 2400 years.
/ref> A modern application is the use of relatively large lenses to concentrator photovoltaics, concentrate solar energy on relatively small photovoltaic cell
A solar cell, or photovoltaic cell, is an electronic device that converts the energy of light directly into electricity by the photovoltaic effect, which is a physical and chemical phenomenon. s, harvesting more energy without the need to use larger and more expensive cells.
Radio astronomy
Radio astronomy is a subfield of astronomy that studies celestial objects at radio frequencies. The first detection of radio waves from an astronomical object was in 1933, when Karl Jansky at Bell Telephone Laboratories reported radiation comin ...
and radar
Radar is a detection system that uses radio waves to determine the distance ('' ranging''), angle, and radial velocity of objects relative to the site. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, we ...
systems often use dielectric lens
A Luneburg lens (original German ''Lüneburg lens'', sometimes incorrectly spelled ''Luneberg lens'') is a spherically symmetric gradient-index lens. A typical Luneburg lens's refractive index ''n'' decreases radially from the center to the outer ...
es, commonly called a lens antenna
A lens antenna is a microwave antenna that uses a shaped piece of microwave-transparent material to bend and focus the radio waves by refraction, as an optical lens does for light. Typically it consists of a small feed antenna such as a patch a ...
to refract electromagnetic radiation
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) li ...
into a collector antenna.
Lenses can become scratched and abraded. Abrasion-resistant coatings are available to help control this.
See also
* Anti-fog
Anti-fog agents, also known as anti-fogging agents and treatments, are chemicals that prevent the condensation of water in the form of small droplets on a surface which resemble fog. Anti-fog treatments were first developed by NASA during Project ...
ging treatment of optical surfaces
* Back focal plane
* Bokeh
In photography, bokeh ( or ; ) is the aesthetic quality of the blur produced in out-of-focus parts of an image. Bokeh has also been defined as "the way the lens renders out-of-focus points of light". Differences in lens aberrations and ...
* Cardinal point (optics)
In Gaussian optics, the cardinal points consist of three pairs of points located on the optical axis of a rotationally symmetric, focal, optical system. These are the '' focal points'', the principal points, and the nodal points. For ''ideal'' s ...
* Caustic (optics)
In optics, a caustic or caustic network is the envelope of light rays which have been reflected or refracted by a curved surface or object, or the projection of that envelope of rays on another surface. The caustic is a curve or surface to ...
* Eyepiece
An eyepiece, or ocular lens, is a type of lens that is attached to a variety of optical devices such as telescopes and microscopes. It is named because it is usually the lens that is closest to the eye when someone looks through the device. The ...
* F-number
In optics, the f-number of an optical system such as a camera lens is the ratio of the system's focal length to the diameter of the entrance pupil ("clear aperture").Smith, Warren ''Modern Optical Engineering'', 4th Ed., 2007 McGraw-Hill Pro ...
* Gravitational lens
A gravitational lens is a distribution of matter (such as a cluster of galaxies) between a distant light source and an observer that is capable of bending the light from the source as the light travels toward the observer. This effect is known ...
* Lens (anatomy)
* List of lens designs
* Numerical aperture
* Optical coatings
* Optical lens design
Optical lens design is the process of designing a lens to meet a set of performance requirements and constraints, including cost and manufacturing limitations. Parameters include surface profile types (spherical, aspheric, holographic, diffractive ...
* Photochromic lens
A photochromic lens is an optical lens that darkens on exposure to light of sufficiently high frequency, most commonly ultraviolet (UV) radiation. In the absence of activating light, the lenses return to their clear state. Photochromic lenses may ...
* Prism (optics)
* Ray tracing
* Ray transfer matrix analysis
Ray transfer matrix analysis (also known as ABCD matrix analysis) is a mathematical form for performing ray tracing calculations in sufficiently simple problems which can be solved considering only paraxial rays. Each optical element (surface, in ...
References
Bibliography
* Chapters 5 & 6.
*
*
External links
A chapter from an online textbook on refraction and lenses
''Thin Spherical Lenses ''
(.pdf) o
Project PHYSNET
* Article o
Ancient Egyptian lenses
*
The Use of Magnifying Lenses in the Classical World
* (with 21 diagrams)
Simulations
– Concave and Convex Lenses
OpticalRayTracer
– Open source lens simulator (downloadable java)
*
Animations demonstrating lens
by QED
{{Authority control
Optical components