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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, polished, or molded to the required 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.
Lenses are used in various imaging devices such as telescopes, binoculars, and cameras. They are also used as visual aids in glasses to correct defects of vision such as myopia and hypermetropia.
HistoryEdit
Template:See also Template:Expand section
The word lens comes from {{#invoke:Lang|lang}}, the Latin name of the lentil (a seed of a lentil plant), because a double-convex lens is lentil-shaped. The lentil also gives its name to a geometric figure.Template:Efn
Some scholars argue that the archeological evidence indicates that there was widespread use of lenses in antiquity, spanning several millennia.<ref>Template:Cite journal</ref> The so-called Nimrud lens is a rock crystal artifact dated to the 7th century BCE which may or may not have been used as a magnifying glass, or a burning glass.<ref name="Nimrud lens">Template:Cite news</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>Template:Cite book</ref> Others have suggested that certain Egyptian hieroglyphs depict "simple glass meniscal lenses".<ref name=Kriss>Template:Cite journal</ref>Template:Verify source
The oldest certain reference to the use of lenses is from Aristophanes' play The Clouds (424 BCE) mentioning a burning-glass.<ref name="The Clouds" /> Pliny the Elder (1st century) confirms that burning-glasses were known in the Roman period.<ref>Pliny the Elder, The Natural History (trans. John Bostock) Book XXXVII, Chap. 10 Template:Webarchive.</ref> Pliny also has the earliest known reference to the use of a corrective lens when he mentions that Nero was said to watch the gladiatorial games using an emerald (presumably concave to correct for nearsightedness, though the reference is vague).<ref>Pliny the Elder, The Natural History (trans. John Bostock) Book XXXVII, Chap. 16 Template:Webarchive</ref> Both Pliny and Seneca the Younger (3 BC–65 AD) described the magnifying effect of a glass globe filled with water.
Ptolemy (2nd century) wrote a book on Optics, 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 crystal Visby lenses may or may not have been intended for use as burning glasses.<ref>Template:Cite book</ref>
Spectacles were invented as an improvement of the "reading stones" of the high medieval period in Northern Italy in the second half of the 13th century.<ref>Template:Cite book</ref> This was the start of the optical industry of grinding and polishing lenses for spectacles, first in Venice and Florence in the late 13th century,<ref>Al Van Helden. The Galileo Project > Science > The Telescope Template:Webarchive. Galileo.rice.edu. Retrieved on 6 June 2012.</ref> and later in the spectacle-making centres in both the Netherlands and Germany.<ref>Template:Cite book</ref> Spectacle makers created improved types of lenses for the correction of vision based more on empirical knowledge gained from observing the effects of the lenses (probably without the knowledge of the rudimentary optical theory of the day).<ref>Template:Cite book</ref><ref>Template:Cite bookTemplate:Dead link</ref> The practical development and experimentation with lenses led to the invention of the compound optical microscope around 1595, and the refracting telescope in 1608, both of which appeared in the spectacle-making centres in the Netherlands.<ref>Microscopes: Time Line Template:Webarchive, Nobel Foundation. Retrieved 3 April 2009</ref><ref name="LZZginzib4C page 55">Template:Cite book</ref>
Template:Further With the invention of the telescope and microscope there was a great deal of experimentation with lens shapes in the 17th and early 18th centuries by those trying to correct chromatic errors seen in lenses. Opticians tried to construct lenses of varying forms of curvature, wrongly assuming errors arose from defects in the spherical figure of their surfaces.<ref>This paragraph is adapted from the 1888 edition of the Encyclopædia Britannica.</ref> Optical theory on refraction and experimentation was showing no single-element lens could bring all colours to a focus. This led to the invention of the compound achromatic lens by Chester Moore Hall in England in 1733, an invention also claimed by fellow Englishman John Dollond in a 1758 patent.
Developments in transatlantic commerce were the impetus for the construction of modern lighthouses in the 18th century, which utilize a combination of elevated sightlines, lighting sources, and lenses to provide navigational aid overseas. With maximal distance of visibility needed in lighthouses, conventional convex lenses would need to be significantly sized which would negatively affect the development of lighthouses in terms of cost, design, and implementation. Fresnel lens were developed that considered these constraints by featuring less material through their concentric annular sectioning. They were first fully implemented into a lighthouse in 1823.<ref>Template:Cite journal</ref>
Construction of simple lensesEdit
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.
Types of simple lensesEdit
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. Convex-concave lenses are most commonly used in corrective lenses, since the shape minimizes some aberrations.
For a biconvex or plano-convex lens in a lower-index medium, 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. For a thin lens in air, the distance from the lens to the spot is the focal length of the lens, which is commonly represented by Template:Mvar in diagrams and equations. An extended hemispherical lens is a special type of plano-convex lens, in which the lens's curved surface is a full hemisphere and the lens is much thicker than the radius of curvature.
Another extreme case of a thick convex lens is a ball lens, whose shape is completely round. When used in novelty photography it is often called a "lensball". A ball-shaped lens has the advantage of being omnidirectional, but for most optical glass types, its focal point lies close to the ball's surface. Because of the ball's curvature extremes compared to the lens size, optical aberration is much worse than thin lenses, with the notable exception of chromatic aberration.
For a biconcave or plano-concave lens in a lower-index medium, a collimated beam of light passing through the lens is diverged (spread); the lens is thus called a negative or diverging lens. The beam, after passing through the lens, appears to emanate from a particular point on the axis in front of the lens. For a thin lens in air, the distance from this point to the lens is the focal length, though it is negative with respect to the focal length of a converging lens.
The behavior reverses when a lens is placed in a medium with higher refractive index than the material of the lens. In this case a biconvex or plano-convex lens diverges light, and a biconcave or plano-concave one converges it.
Convex-concave (meniscus) lenses can be either positive or negative, depending on the relative curvatures of the two surfaces. A negative meniscus lens has a steeper concave surface (with a shorter radius than the convex surface) and is thinner at the centre than at the periphery. Conversely, a positive meniscus lens has a steeper convex surface (with a shorter radius than the concave surface) and is thicker at the centre than at the periphery.
An ideal thin lens with two surfaces of equal curvature (also equal in the sign) would have zero optical power (as its focal length becomes infinity as shown in the lensmaker's equation), meaning that it would neither converge nor diverge light. All real lenses have a nonzero thickness, however, which makes a real lens with identical curved surfaces slightly positive. To obtain exactly zero optical power, a meniscus lens must have slightly unequal curvatures to account for the effect of the lens' thickness.
For a spherical surfaceEdit
For a single refraction for a circular boundary, the relation between object and its image in the paraxial approximation is given by<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
<math display="block">\frac {n_1}u + \frac {n_2}v = \frac {n_2-n_1}R</math>
where Template:Mvar is the radius of the spherical surface, Template:Math is the refractive index of the material of the surface, Template:Math is the refractive index of medium (the medium other than the spherical surface material), <math display="inline">u</math> is the on-axis (on the optical axis) object distance from the line perpendicular to the axis toward the refraction point on the surface (which height is h), and <math display="inline">v</math> is the on-axis image distance from the line. Due to paraxial approximation where the line of h is close to the vertex of the spherical surface meeting the optical axis on the left, <math display="inline">u</math> and <math display="inline">v</math> are also considered distances with respect to the vertex.
Moving <math display="inline">v</math> toward the right infinity leads to the first or object focal length <math display="inline">f_0</math> for the spherical surface. Similarly, <math display="inline">u</math> toward the left infinity leads to the second or image focal length <math>f_i</math>.<ref>Template:Cite book</ref>
<math display="block">\begin{align}
f_0 &= \frac{n_1}{n_2 - n_1} R,\\
f_i &= \frac{n_2}{n_2 - n_1} R
\end{align}</math>
Applying this equation on the two spherical surfaces of a lens and approximating the lens thickness to zero (so a thin lens) leads to the lensmaker's formula.
DerivationEdit
Applying Snell's law on the spherical surface, <math>n_1 \sin i = n_2 \sin r\,.</math>
Also in the diagram,<math display="block">\begin{align} \tan (i - \theta) &= \frac hu \\ \tan (\theta - r) &= \frac hv \\ \sin \theta &= \frac hR \end{align}</math>, and using small angle approximation (paraxial approximation) and eliminating Template:Mvar, Template:Mvar, and Template:Mvar,
<math display="block">\frac {n_2}v + \frac {n_1}u = \frac {n_2-n_1}R\,.</math>
Lensmaker's equationEdit
The (effective) focal length <math>f</math> of a spherical lens in air or vacuum for paraxial rays can be calculated from the lensmaker's equation:<ref>Template:Harvnb
Template:Harvnb</ref><ref name="Hecht-2017" />
<math display="block"> \frac{ 1 }{\ f\ } = \left( n - 1 \right) \left[\ \frac{ 1 }{\ R_1\ } - \frac{ 1 }{\ R_2\ } + \frac{\ \left( n - 1 \right)\ d ~}{\ n\ R_1\ R_2\ }\ \right]\ ,</math> where
- <math display="inline">\ n\ </math> is the refractive index of the lens material;
- <math display="inline">\ R_1\ </math> is the (signed, see below) radius of curvature of the lens surface closer to the light source;
- <math display="inline">\ R_2\ </math> is the radius of curvature of the lens surface farther from the light source; and
- <math display="inline">\ d\ </math> is the thickness of the lens (the distance along the lens axis between the two surface vertices).
The focal length <math display="inline">\ f\ </math> is with respect to the principal planes of the lens, and the locations of the principal planes <math display="inline">\ h_1\ </math> and <math display="inline">\ h_2\ </math> with respect to the respective lens vertices are given by the following formulas, where it is a positive value if it is right to the respective vertex.<ref name="Hecht-2017">Template:Cite book</ref>
<math display="block">\ h_1 = -\ \frac{\ \left( n - 1 \right) f\ d ~}{\ n\ R_2\ }\ </math><math display="block">\ h_2 = -\ \frac{\ \left( n - 1 \right) f\ d ~}{\ n\ R_1\ }\ </math>
The focal length <math>\ f\ </math> is positive for converging lenses, and negative for diverging lenses. The reciprocal of the focal length, <math display="inline">\ \tfrac{ 1 }{\ f\ }\ ,</math> is the optical power of the lens. If the focal length is in metres, this gives the optical power in dioptres (reciprocal metres).
Lenses have the same focal length when light travels from the back to the front as when light goes from the front to the back. Other properties of the lens, such as the aberrations are not the same in both directions.
Sign convention for radii of curvature Template:Math and Template:Math Edit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The signs of the lens' radii of curvature indicate whether the corresponding surfaces are convex or concave. The sign convention used to represent this varies,<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> but in this article a positive Template:Mvar indicates a surface's center of curvature is further along in the direction of the ray travel (right, in the accompanying diagrams), while negative Template:Mvar means that rays reaching the surface have already passed the center of curvature. Consequently, for external lens surfaces as diagrammed above, Template:Math and Template:Math indicate convex surfaces (used to converge light in a positive lens), while Template:Math and Template:Math 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.
Sign convention for other parametersEdit
| Parameter | Meaning | + Sign | − Sign |
|---|---|---|---|
| Template:Mvaro | The distance between an object and a lens. | Real object | Virtual object |
| Template:Mvari | The distance between an image and a lens. | Real image | Virtual image |
| Template:Mvar | The focal length of a lens. | Converging lens | Diverging lens |
| Template:Mvaro | The height of an object from the optical axis. | Erect object | Inverted object |
| Template:Mvari | The height of an image from the optical axis | Erect image | Inverted image |
| Template:MvarT | The transverse magnification in imaging ( Template:Math the ratio of Template:Mvari to Template:Mvaro ). | Erect image | Inverted image |
This convention is used in this article. Other conventions such as the Cartesian sign convention change the form of the equations.
Thin lens approximationEdit
If Template:Mvar is small compared to Template:Math and Template:Math then the Template:Dfn approximation can be made. For a lens in air, Template:Mvar is then given by<ref name="Hecht-2017b">Template:Cite book</ref>
<math display="block">\ \frac{ 1 }{\ f\ } \approx \left( n - 1 \right) \left[\ \frac{ 1 }{\ R_1\ } - \frac{ 1 }{\ R_2\ }\ \right] ~.</math>
DerivationEdit
The spherical thin lens equation in paraxial approximation is derived here with respect to the right figure.<ref name="Hecht-2017b" /> The 1st spherical lens surface (which meets the optical axis at <math display="inline">\ V_1\ </math> as its vertex) images an on-axis object point O to the virtual image I, which can be described by the following equation,<math display="block">\ \frac{\ n_1\ }{\ u\ } + \frac{\ n_2\ }{\ v'\ } = \frac{\ n_2 - n_1\ }{\ R_1\ } ~.</math> For the imaging by second lens surface, by taking the above sign convention, <math display="inline">\ u' = - v' + d\ </math> and <math display="block">\ \frac{ n_2 }{\ -v' + d\ } + \frac{\ n_1\ }{\ v\ } = \frac{\ n_1 - n_2\ }{\ R_2\ } ~.</math> Adding these two equations yields <math display="block">\ \frac{\ n_1\ }{ u } + \frac{\ n_1\ }{ v } = \left( n_2 - n_1 \right) \left( \frac{ 1 }{\ R_1\ } - \frac{ 1 }{\ R_2\ } \right) + \frac{\ n_2\ d\ }{\ \left(\ v' - d\ \right)\ v'\ } ~.</math> For the thin lens approximation where <math>\ d \rightarrow 0\ ,</math> the 2nd term of the RHS (Right Hand Side) is gone, so
<math display="block">\ \frac{\ n_1\ }{ u } + \frac{\ n_1\ }{ v } = \left( n_2 - n_1 \right) \left( \frac{ 1 }{\ R_1\ } - \frac{ 1 }{\ R_2\ } \right) ~.</math>
The focal length <math>\ f\ </math> of the thin lens is found by limiting <math>\ u \rightarrow - \infty\ ,</math>
<math display="block">\ \frac{\ n_1\ }{\ f\ } = \left( n_2 - n_1 \right) \left( \frac{ 1 }{\ R_1\ } - \frac{ 1 }{\ R_2\ } \right) \rightarrow \frac{ 1 }{\ f\ } = \left( \frac{\ n_2\ }{\ n_1\ } - 1 \right) \left( \frac{ 1 }{\ R_1\ } - \frac{ 1 }{\ R_2\ } \right) ~.</math>
So, the Gaussian thin lens equation is
<math display="block">\ \frac{ 1 }{\ u\ } + \frac{ 1 }{\ v\ } = \frac{ 1 }{\ f\ } ~.</math>
For the thin lens in air or vacuum where <math display="inline">\ n_1 = 1\ </math> can be assumed, <math display="inline">\ f\ </math> becomes
<math display="block">\ \frac{ 1 }{\ f\ } = \left( n - 1 \right)\left(\frac{ 1 }{\ R_1\ } - \frac{ 1 }{\ R_2\ } \right)\ </math>
where the subscript of 2 in <math display="inline">\ n_2\ </math> is dropped.
Imaging propertiesEdit
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 Template:Mvar 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 Template:Mvar from the lens is called the [[Cardinal point (optics)#Focal planes|Template:Dfn]].
Lens equationEdit
For paraxial rays, if the distances from an object to a spherical thin lens (a lens of negligible thickness) and from the lens to the image are Template:Math and Template:Math respectively, the distances are related by the (Gaussian) thin lens formula:<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
<math display="block">{1\over f} = {1\over S_1} + {1\over S_2}\,.</math>
The right figure shows how the image of an object point can be found by using three rays; the first ray parallelly incident on the lens and refracted toward the second focal point of it, the second ray crossing the optical center of the lens (so its direction does not change), and the third ray toward the first focal point and refracted to the direction parallel to the optical axis. This is a simple ray tracing method easily used. Two rays among the three are sufficient to locate the image point. By moving the object along the optical axis, it is shown that the second ray determines the image size while other rays help to locate the image location.
The lens equation can also be put into the "Newtonian" form:<ref name="Hecht-2017a">Template:Cite book</ref>
<math display="block">f^2 = x_1 x_2\,,</math>
where <math>x_1 = S_1-f</math> and <math>x_2 = S_2-f\,.</math> <math display="inline">x_1</math> is positive if it is left to the front focal point <math display="inline">F_1</math>, and <math display="inline">x_2</math> is positive if it is right to the rear focal point <math display="inline">F_2</math>. Because <math display="inline">f^2</math> is positive, an object point and the corresponding imaging point made by a lens are always in opposite sides with respect to their respective focal points. (<math display="inline">x_1</math> and <math display="inline">x_2</math> are either positive or negative.)
This Newtonian form of the lens equation can be derived by using a similarity between triangles P1PO1F1 and L3L2F1 and another similarity between triangles L1L2F2 and P2P02F2 in the right figure. The similarities give the following equations and combining these results gives the Newtonian form of the lens equation.
<math display="block">\begin{array}{lcr} \frac{y_1}{x_1} = \frac{\left\vert y_2 \right\vert}{f} \\ \frac{y_1}{f} = \frac{\left\vert y_2 \right\vert}{x_2} \end{array} </math>
The above equations also hold for thick lenses (including a compound lens made by multiple lenses, that can be treated as a thick lens) in air or vacuum (which refractive index can be treated as 1) if <math display="inline">S_1</math>, <math display="inline">S_2</math>, and <math display="inline">f</math> are with respect to the principal planes of the lens (<math display="inline">f</math> is the effective focal length in this case).<ref name="Hecht-2017" /> This is because of triangle similarities like the thin lens case above; similarity between triangles P1PO1F1 and L3H1F1 and another similarity between triangles L1'H2F2 and P2P02F2 in the right figure. If distances Template:Math or Template:Math pass through a medium other than air or vacuum, then a more complicated analysis is required.
If an object is placed at a distance Template:Math from a positive lens of focal length Template:Mvar, we will find an image at a distance Template:Math according to this formula. If a screen is placed at a distance Template:Math on the opposite side of the lens, an image is formed on it. This sort of image, which can be projected onto a screen or image sensor, is known as a real image. This is the principle of the camera, and also of the human eye, in which the retina serves as the image sensor.
The focusing adjustment of a camera adjusts Template:Math, as using an image distance different from that required by this formula produces a defocused (fuzzy) image for an object at a distance of Template:Math from the camera. Put another way, modifying Template:Math causes objects at a different Template:Math to come into perfect focus.
In some cases, Template:Math 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 Template:Em 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, Template:Math 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.
Template:Multiple image Using a positive lens of focal length Template:Mvar, a virtual image results when Template:Math, the lens thus being used as a magnifying glass (rather than if Template:Math as for a camera). Using a negative lens (Template:Math) with a Template:Em (Template:Math) can only produce a virtual image (Template:Math), according to the above formula. It is also possible for the object distance Template:Math 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) Template:Em the location of its real image. In that case even a negative lens can project a real image, as is done by a Barlow lens.
For a given lens with the focal length f, the minimum distance between an object and the real image is 4f (S1 = S2 = 2f). This is derived by letting L = S1 + S2, expressing S2 in terms of S1 by the lens equation (or expressing S1 in terms of S2), and equating the derivative of L with respect to S1 (or S2) to zero. (Note that L has no limit in increasing so its extremum is only the minimum, at which the derivate of L is zero.)
MagnificationEdit
The linear magnification of an imaging system using a single lens is given by
<math display="block"> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1}\ = - \frac{f}{x_1}</math>
where Template:Mvar 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 Template:Mvar is negative, as it is for real images, the image is upside-down with respect to the object. For virtual images Template:Mvar is positive, so the image is upright.
This magnification formula provides two easy ways to distinguish converging (Template:Math) and diverging (Template:Math) lenses: For an object very close to the lens (Template:Math), 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 (Template:Math), a converging lens would form an inverted image, whereas a diverging lens would form an upright image.
Linear magnification Template:Mvar 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 Template:Val focal length, held Template:Val from the eye and Template:Val from the object, produces a virtual image at infinity of infinite linear size: Template:Math. But the Template:Dfn 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 Template:Val lens, one is not concerned with the linear magnification Template:Math Rather, the plate scale of the camera is about Template:Val, from which one can conclude that the Template:Val 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, Template:Math, Template:Math and Template:Math, 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.
Table for thin lens imaging propertiesEdit
| Lens Type | Object Location | Image Type | Image Location | Lateral Image Orientation | Image Magnification | Remark |
|---|---|---|---|---|---|---|
| Converging lens (or positive lens) | <math>\infty > S_1 > 2f</math> | Real (rays converging to each image point) | <math>f < S_2 < 2f</math> | Inverted (opposite to the object orientation) | Diminished | |
| Converging lens | <math>S_1 = 2f</math> | Real | <math>S_2 = 2f</math> | Inverted | Same size | |
| Converging lens | <math>f < S_1 < 2f</math> | Real | <math>\infty > S_2 > 2f</math> | Inverted | Magnified | |
| Converging lens | <math>S_1 = f</math> | <math>\plusmn \infty</math> | ||||
| Converging lens | <math>S_1 < f</math> | Virtual (rays apparently diverging from each image point) | <math>\vert S_2 \vert > S_1 </math> | Erect (same to the object orientation) | Magnified | As an object moves to the lens, the virtual image also gets closer to the lens while the image size is reduced. |
| Diverging lens (or negative lens) | Anywhere | Virtual | <math>\vert S_2 \vert < \vert f \vert, S_1 > \vert S_2 \vert</math> | Erect | Diminished |
AberrationsEdit
Template:Optical aberration {{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}
Lenses do not form perfect images, and always introduce some degree of distortion or aberration that makes the image an imperfect replica of the object. Careful design of the lens system for a particular application minimizes the aberration. Several types of aberration affect image quality, including spherical aberration, coma, and chromatic aberration.
Spherical aberrationEdit
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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 laterally distant from, the lens axis to be focused in a slightly different place than beams close to the axis. This manifests itself as a blurring of the image. Spherical aberration can be minimised with normal lens shapes by carefully choosing the surface curvatures for a particular application. For instance, a plano-convex lens, which is used to focus a collimated beam, produces a sharper focal spot when used with the convex side towards the beam source.
ComaEdit
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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 Template:Mvar. Rays that pass through the centre of a lens of focal length Template:Mvar are focused at a point with distance Template:Math from the axis. Rays passing through the outer margins of the lens are focused at different points, either further from the axis (positive coma) or closer to the axis (negative coma). In general, a bundle of parallel rays passing through the lens at a fixed distance from the centre of the lens are focused to a ring-shaped image in the focal plane, known as a comatic circle (see each circle of the image in the below figure). The sum of all these circles results in a V-shaped or comet-like flare. As with spherical aberration, coma can be minimised (and in some cases eliminated) by choosing the curvature of the two lens surfaces to match the application. Lenses in which both spherical aberration and coma are minimised are called bestform lenses.
Chromatic aberrationEdit
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Chromatic aberration is caused by the dispersion of the lens material—the variation of its refractive index, Template:Mvar, with the wavelength of light. Since, from the formulae above, Template:Mvar is dependent upon Template:Mvar, 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.
File:Chromatic aberration lens diagram.svg File:Lens6b-en.svg
Other types of aberrationEdit
Other kinds of aberration include field curvature, barrel and pincushion distortion, and astigmatism.
Aperture diffractionEdit
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 Edit
Template:See also 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.
In a multiple-lens system, if the purpose of the system is to image an object, then the system design can be such that each lens treats the image made by the previous lens as an object, and produces the new image of it, so the imaging is cascaded through the lenses.<ref>Template:Cite book</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> As shown above, the Gaussian lens equation for a spherical lens is derived such that the 2nd surface of the lens images the image made by the 1st lens surface. For multi-lens imaging, 3rd lens surface (the front surface of the 2nd lens) can image the image made by the 2nd surface, and 4th surface (the back surface of the 2nd lens) can also image the image made by the 3rd surface. This imaging cascade by each lens surface justifies the imaging cascade by each lens.
For a two-lens system the object distances of each lens can be denoted as <math display="inline">s_{o1}</math> and <math display="inline">s_{o2}</math>, and the image distances as and <math display="inline">s_{i1}</math> and <math display="inline">s_{i2}</math>. If the lenses are thin, each satisfies the thin lens formula
<math display="block">\frac{1}{f_j} = \frac{1}{s_{oj}} + \frac{1}{s_{ij}},</math>
If the distance between the two lenses is <math>d</math>, then <math display="inline">s_{o2} = d - s_{i1}</math>. (The 2nd lens images the image of the first lens.)
FFD (Front Focal Distance) is defined as the distance between the front (left) focal point of an optical system and its nearest optical surface vertex.<ref>Template:Cite journal</ref> If an object is located at the front focal point of the system, then its image made by the system is located infinitely far way to the right (i.e., light rays from the object is collimated after the system). To do this, the image of the 1st lens is located at the focal point of the 2nd lens, i.e., <math>s_{i1} = d - f_2 </math>. So, the thin lens formula for the 1st lens becomes<ref>Template:Cite book</ref>
<math display="block">\frac{1}{f_1} = \frac{1}{FFD} + \frac{1}{d - f_2} \rightarrow FFD = \frac{f_1(d - f_2)}{d - (f_1 + f_2)}. </math>
BFD (Back Focal Distance) is similarly defined as the distance between the back (right) focal point of an optical system and its nearest optical surface vertex. If an object is located infinitely far away from the system (to the left), then its image made by the system is located at the back focal point. In this case, the 1st lens images the object at its focal point. So, the thin lens formula for the 2nd lens becomes
<math display="block">\frac{1}{f_2} = \frac{1}{BFD} + \frac{1}{d - f_1} \rightarrow BFD = \frac{f_2(d - f_1)}{d - (f_1 + f_2)}.</math>
A simplest case is where thin lenses are placed in contact (<math>d = 0</math>). Then the combined focal length Template:Mvar of the lenses is given by
<math display="block">\frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2}\,.</math>
Since Template:Math is the power of a lens with focal length Template:Mvar, it can be seen that the powers of thin lenses in contact are additive. The general case of multiple thin lenses in contact is
<math display="block">\frac{1}{f} =\sum_{k = 1}^N \frac{1}{f_k}</math>
where <math display="inline">N</math> is the number of lenses.
If two thin lenses are separated in air by some distance Template:Mvar, then the focal length for the combined system is given by <math display="block">\frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2}-\frac{d}{f_1 f_2}\,.</math>
As Template:Mvar tends to zero, the focal length of the system tends to the value of Template:Mvar given for thin lenses in contact. It can be shown that the same formula works for thick lenses if Template:Mvar is taken as the distance between their principal planes.<ref name="Hecht-2017" />
If the separation distance between two lenses is equal to the sum of their focal lengths (Template:Math), then the FFD and BFD are infinite. This corresponds to a pair of lenses that transforms a parallel (collimated) beam into another collimated beam. This type of system is called an afocal system, since it produces no net convergence or divergence of the beam. Two lenses at this separation form the simplest type of optical telescope. Although the system does not alter the divergence of a collimated beam, it does alter the (transverse) width of the beam. The magnification of such a telescope is given by
<math display="block">M = -\frac{f_2}{f_1}\,,</math>
which is the ratio of the output beam width to the input beam width. Note the sign convention: a telescope with two convex lenses (Template:Math, Template:Math) produces a negative magnification, indicating an inverted image. A convex plus a concave lens (Template:Math) produces a positive magnification and the image is upright. For further information on simple optical telescopes, see Refracting telescope § Refracting telescope designs.
Non spherical typesEdit
Cylindrical lenses have curvature along only one axis. They are used to focus light into a line, or to convert the elliptical light from a laser diode into a round beam. They are also used in motion picture anamorphic lenses.
Aspheric lenses 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.
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.
Bifocal lens has two or more, or a graduated, focal lengths ground into the lens.
A gradient index lens has flat optical surfaces, but has a radial or axial variation in index of refraction that causes light passing through the lens to be focused.
An axicon has a conical optical surface. It images a point source into a line Template:Em the optic axis, or transforms a laser beam into a ring.<ref name=Proteep>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
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.<ref name=Grbic>Template:Cite journal</ref> The first superlenses were made in 2004 using such a metamaterial for microwaves.<ref name=Grbic /> Improved versions have been made by other researchers.<ref name=Valenitne-J.>Template:Cite journal</ref><ref>Template:Cite journal</ref> Template:As of the superlens has not yet been demonstrated at visible or near-infrared wavelengths.<ref name=mielsen10>Template:Cite journal</ref>
A prototype flat ultrathin lens, with no curvature has been developed.<ref>Template:Cite journal</ref>
UsesEdit
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 refractive errors such as myopia, hypermetropia, presbyopia, and astigmatism. (See corrective lens, contact lens, eyeglasses, intraocular lens.) Most lenses used for other purposes have strict axial symmetry; eyeglass lenses are only approximately symmetric. They are usually shaped to fit in a roughly oval, not circular, frame; the optical centres are placed over the 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.<ref name="The Clouds">Template:Cite book[1] Template:Webarchive</ref> 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.<ref>Template:Cite news</ref>
See alsoEdit
- Anti-fogging treatment of optical surfaces
- Back focal plane
- Bokeh
- Cardinal point (optics)
- Caustic (optics)
- Eyepiece
- F-number
- Gravitational lens
- Lens (anatomy)
- List of lens designs
- Numerical aperture
- Optical coatings
- Optical lens design
- Photochromic lens
- Prism (optics)
- Ray tracing
- Ray transfer matrix analysis
NotesEdit
ReferencesEdit
BibliographyEdit
- Template:Cite book Chapters 5 & 6.
- Template:Cite book
- Template:Cite book
External linksEdit
- A chapter from an online textbook on refraction and lenses Template:Webarchive
- Thin Spherical Lenses Template:Webarchive (.pdf) on Project PHYSNET Template:Webarchive.
- Lens article at digitalartform.com
- Article on Ancient Egyptian lenses Template:Webarchive
- Template:YouTube
- The Use of Magnifying Lenses in the Classical World Template:Webarchive
- Template:Cite EB1911 (with 21 diagrams)
SimulationsEdit
- Learning by Simulations Template:Webarchive – Concave and Convex Lenses
- OpticalRayTracer Template:Webarchive – Open source lens simulator (downloadable java)
- Animations demonstrating lens Template:Webarchive by QED