A lens is a transmissive optics 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 Optical axis. Lenses are made from materials such as glass or plastic and are ground, Polishing, 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, , , or .
Lenses are used in various imaging devices such as , binoculars, and . They are also used as visual aids in glasses to correct defects of vision such as Near-sightedness and Far-sightedness.
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 depict "simple glass meniscal lenses".
The oldest certain reference to the use of lenses is from Aristophanes' play The Clouds (424 BCE) mentioning a burning-glass. Pliny the Elder (1st century) confirms that burning-glasses were known in the Roman period.Pliny the Elder, The Natural History (trans. John Bostock) Book XXXVII, Chap. 10 . Pliny also has the earliest known reference to the use of a corrective lens when he mentions that Nero was said to watch the games using an emerald (presumably to correct for myopia, though the reference is vague).Pliny the Elder, The Natural History (trans. John Bostock) Book XXXVII, Chap. 16 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 "" 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 may or may not have been intended for use as burning glasses.
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.
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,Al Van Helden. The Galileo Project > Science > The Telescope . Galileo.rice.edu. Retrieved on 6 June 2012. and later in the spectacle-making centres in both the Netherlands and Germany. 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).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.This paragraph is adapted from the 1888 edition of the Encyclopædia Britannica. 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.
Toric lens 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.
For a biconvex or plano-convex lens in a lower-index medium, a collimated light 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 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.
where is the radius of the spherical surface, is the refractive index of the material of the surface, is the refractive index of medium (the medium other than the spherical surface material), 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 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, and are also considered distances with respect to the vertex.
Moving toward the right infinity leads to the first or object focal length for the spherical surface. Similarly, toward the left infinity leads to the second or image focal length .
f_0 &= \frac{n_1}{n_2 - n_1} R,\\ f_i &= \frac{n_2}{n_2 - n_1} R\end{align}
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.
Also in the diagram,, and using small angle approximation (paraxial approximation) and eliminating , , and ,
where
The focal length is with respect to the principal planes of the lens, and the locations of the principal planes and 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.
The focal length is positive for converging lenses, and negative for diverging lenses. The reciprocal of the focal length, is the optical power of the lens. If the focal length is in metres, this gives the optical power in (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 Gaussian lens equation ! Parameter ! Meaning ! + Sign ! − Sign | |||
o | The distance between an object and a lens. | Real object | Virtual object |
The distance between an image and a lens. | Real image | Virtual image | |
The focal length of a lens. | Converging lens | Diverging lens | |
The height of an object from the optical axis. | Erect object | Inverted object | |
The height of an image from the optical axis | Erect image | Inverted image | |
The transverse magnification in imaging ( the ratio of to ). | Erect image | Inverted image |
The focal length of the thin lens is found by limiting
So, the Gaussian thin lens equation is
For the thin lens in air or vacuum where can be assumed, becomes
where the subscript of 2 in is dropped.
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:
where and is positive if it is left to the front focal point , and is positive if it is right to the rear focal point . Because 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. ( and are either positive or negative.)
This Newtonian form of the lens equation can be derived by using a similarity between triangles P1 PO1 F1 and L3 L2 F1 and another similarity between triangles L1 L2 F2 and P2 P02 F2 in the right figure. The similarities give the following equations and combining these results gives the Newtonian form of the lens equation.
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 , , and are with respect to the of the lens ( is the effective focal length in this case). This is because of triangle similarities like the thin lens case above; similarity between triangles P1 PO1 F1 and L3 H1 F1 and another similarity between triangles L1 'H2 F2 and P2 P02 F2 in the right figure. If distances or pass through a medium other than air or vacuum, then a more complicated analysis is required.
If an object is placed at a distance from a positive lens of focal length , we will find an image at a distance according to this formula. If a screen is placed at a distance 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 , as using an image distance different from that required by this formula produces a defocused (fuzzy) image for an object at a distance of from the camera. Put another way, modifying causes objects at a different to come into perfect focus.
In some cases, 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 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, 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.
Using a positive lens of focal length , 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 () can only produce a virtual image (), according to the above formula. It is also possible for the object distance 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) the location of its real image. In that case even a negative lens can project a real image, as is done by a Barlow lens.
For a given lens with the focal length f, the minimum distance between an object and the real image is 4 f ( S1 = S2 = 2 f). 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.)
where 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 is negative, as it is for real images, the image is upside-down with respect to the object. For virtual images is positive, so the image is upright.
This magnification formula provides two easy ways to distinguish converging () and diverging () 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 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 or 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 focal length, held from the eye and from the object, produces a virtual image at infinity of infinite linear size: . But the 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 lens, one is not concerned with the linear magnification Rather, the plate scale of the camera is about , from which one can conclude that the 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.
; and K (at ) has a double-size, virtual and upright image.
Note that the images of letters H, I, J, and i are located far away from the lens such that they are not shown here. What is also shown here that the ray that is parallelly incident on the lens and refracted toward the second focal point f determines the image size while other rays help to locate the image location.]]
+Images of Real Objects Formed by Thin Lenses !Lens Type !Object Location !Image Type !Image Location !Lateral Image Orientation !Image Magnification !Remark | ||||||
Converging lens (or positive lens) | Real (rays converging to each image point) | Inverted (opposite to the object orientation) | Diminished | |||
Converging lens | Real | Inverted | Same size | |||
Converging lens | Real | Inverted | Magnified | |||
Converging lens | ||||||
Converging lens | Virtual (rays apparently diverging from each image point) | 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 | Erect | Diminished |
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.
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.
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 and , and the image distances as and and . If the lenses are thin, each satisfies the thin lens formula
If the distance between the two lenses is , then . (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. 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., . So, the thin lens formula for the 1st lens becomes
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
A simplest case is where thin lenses are placed in contact (). Then the combined focal length of the lenses is given by
Since is the power of a lens with focal length , it can be seen that the powers of thin lenses in contact are additive. The general case of multiple thin lenses in contact is
where is the number of lenses.
If two thin lenses are separated in air by some distance , then the focal length for the combined system is given by
As tends to zero, the focal length of the system tends to the value of given for thin lenses in contact. It can be shown that the same formula works for thick lenses if is taken as the distance between their principal planes.
If the separation distance between two lenses is equal to the sum of their focal lengths (), 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
which is the ratio of the output beam width to the input beam width. Note the sign convention: a telescope with two convex lenses (, ) produces a negative magnification, indicating an inverted image. A convex plus a concave lens () produces a positive magnification and the image is upright. For further information on simple optical telescopes, see Refracting telescope § Refracting telescope designs.
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.
are arrays of 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 the optic axis, or transforms a laser beam into a ring.
Diffractive optical elements can function as lenses.
are made from negative index metamaterials and claim to produce images at spatial resolutions exceeding the diffraction limit. The first superlenses were made in 2004 using such a metamaterial for microwaves. Improved versions have been made by other researchers. the superlens has not yet been demonstrated at visible or near-infrared wavelengths.
A prototype flat ultrathin lens, with no curvature has been developed.
Lenses are used as for the correction of 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 human eyeball; their curvature may not be axially symmetric to correct for astigmatism. sunglass lens are designed to attenuate light; sunglass lenses that also correct visual impairments can be custom made.
Other uses are in imaging systems such as , binoculars, telescopes, , and Movie projector. 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 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 for at least 2400 years.[6] 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 , 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.
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