Lens - Knowledge

40: 2475: 4873: 215: 227: 6159: 3774: 1182: 701: 7457: 6026:, 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. 1518: 1189: 184: 4861: 7566: 5247: 662: 610: 7424: 479: 4649: 3786:(so its direction does not change), and the third tray toward the first focal point and refracted to the direction parallel to the optical axis. 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. 647: 637: 6164: 6092: 600: 4633: 4574: 6031: 4332: 419:

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.

4599:. 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, 373:. 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 692:), 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. 3387: 4327: 6139:) 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 6083:. 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 6255:

In a multiple-lens system, each lens treats the previous lens's image as an object, and produces a new image of it, so that the imaging is cascaded through the lenses. This is easy to understand when the image from one lens is before the front focus of the next lens, but it remains true even when the

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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

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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.,

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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

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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

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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

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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,

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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

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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.

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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.

5300:) 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 2967: 7702:

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

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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

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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.

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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.

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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.

5034: 1508: 39: 804: 5888: 2788: 8493: 6462: 5115:, 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 3592: 6777: 6596: 1020: 7053: 3781:

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

9389: 2465: 7136: 3769: 3447: 328:, 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 8049: 9124: 7472:

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.

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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

6131:, 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 1272: 1025: 3000: 9050:

Yao, Jie; Liu, Zhaowei; Liu, Yongmin; Wang, Yuan; Sun, Cheng; Bartal, Guy; Stacy, Angelica M.; Zhang, Xiang (15 August 2008). "Optical Negative Refraction in Bulk Metamaterials of Nanowires".

2006: 1932: 1735: 7371: 5790: 1260: 7307:), 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 6143:

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.

7170: 5622: 5384: 3382:{\displaystyle {\frac {n_{1}}{f}}=(n_{2}-n_{1})({\frac {1}{R_{1}}}-{\frac {1}{R_{2}}})\rightarrow {\frac {1}{f}}=({\frac {n_{2}}{n_{1}}}-1)({\frac {1}{R_{1}}}-{\frac {1}{R_{2}}}).} 3178: 3843: 8725: 4322:{\displaystyle {\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}}} 580:, 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 5576: 5430: 3941: 7317:. 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 2668: 6589: 5084:), 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 ( 2993: 2673: 3891: 6536: 4343:

The above equations also hold for thick lenses (including compound lenses made by multiple lens elements) in air or vacuum (which refractive index can be treated as 1) if

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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

8531: 6058:-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 5740: 2034:

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

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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

1013: 720: 7624:; 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 3482: 620:

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

6976: 6377: 6347: 6317: 6287: 9213: 6384: 4395: 4368: 4129: 4102: 4075: 4048: 4021: 3994: 3967: 3621: 3509: 2511: 1855: 1828: 966: 118:, which refracts light without focusing. Devices that similarly focus or disperse waves and radiation other than visible light are also called "lenses", such as 9410: 8041: 6484: 3149: 1543: 4891:) forming a real, inverted image (as the image formed by the objective lens of a telescope or binoculars) rather than the upright, virtual image as seen in a 263:

Some scholars argue that the archeological evidence indicates that there was widespread use of lenses in antiquity, spanning several millennia. The so-called

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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.

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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. (

3502: 986: 939: 916: 896: 872: 848: 354:" 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 8755: 5231:, 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 573:

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.

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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.

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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

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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.

1392:{\displaystyle {\begin{aligned}\tan(i-\theta )&={\frac {h}{u}}\\\tan(\theta -r)&={\frac {h}{v}}\\\sin \theta &={\frac {h}{R}}\end{aligned}}} 6542:

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.,

5111:—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 5042:

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

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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

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Using an inappropriate measurement of magnification can be formally correct but yield a meaningless number. For instance, using a magnifying glass of

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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

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is sometimes seen. While it is listed as an alternative spelling in some dictionaries, most mainstream dictionaries do not list it as acceptable.

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Analyses the almost negligible frequency of use and concludes that the misspelling is a result of a wrong singularisation of the plural (lenses).

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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.

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means that rays reaching the surface have already passed the center of curvature. Consequently, for external lens surfaces as diagrammed above,

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Schottner, G (May 2003). "Scratch and Abrasion Resistant Coatings on Plastic Lenses—State of the Art, Current Developments and Perspectives".

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in the right figure. The similarities give the following equations and combining these results gives the Newtonian form of the lens equation.

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have at least one surface that is neither spherical nor cylindrical. The more complicated shapes allow such lenses to form images with less

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indicates a surface's center of curvature is further along in the direction of the ray travel (right, in the accompanying diagrams), while

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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

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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

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Kriss, Timothy C.; Kriss, Vesna Martich (April 1998). "History of the Operating Microscope: From Magnifying Glass to Microneurosurgery".

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Different lens materials may also be used to minimise chromatic aberration, such as specialised coatings or lenses made from the crystal

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Real image of a lamp is projected onto a screen (inverted). Reflections of the lamp from both surfaces of the biconvex lens are visible.

288:(1st century) confirms that burning-glasses were known in the Roman period. Pliny also has the earliest known reference to the use of a 677:

lens has a steeper convex surface (with a shorter radius than the concave surface) and is thicker at the centre than at the periphery.

8717: 2962:{\displaystyle {\frac {n_{1}}{u}}+{\frac {n_{1}}{v}}=(n_{2}-n_{1})({\frac {1}{R_{1}}}-{\frac {1}{R_{2}}})+{\frac {n_{2}d}{(v'-d)v'}}.} 1936: 1862: 1554: 9237: 399:

and experimentation was showing no single-element lens could bring all colours to a focus. This led to the invention of the compound

8276: 8464:"A Light to Lighten our Darkenss: Lighthouse Optics and the Later Development of Fresnel's Revolutionary Refracting Lens 1780-1900" 8141: 7375:

which is the ratio of the output beam width to the input beam width. Note the sign convention: a telescope with two convex lenses (

9447: 7322: 6943:{\displaystyle {\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})}}.} 6762:{\displaystyle {\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})}}.} 9356: 7546:

for microwaves. Improved versions have been made by other researchers. As of 2014 the superlens has not yet been demonstrated at

4612:. The magnifying glass creates a (magnified) virtual image behind the magnifying glass, but those rays are then re-imaged by the 1160:{\displaystyle {\begin{aligned}f_{0}&={\frac {n_{1}}{n_{2}-n_{1}}}R,\\f_{i}&={\frac {n_{2}}{n_{2}-n_{1}}}R\end{aligned}}} 669:

Convex-concave (meniscus) lenses can be either positive or negative, depending on the relative curvatures of the two surfaces. A

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or sphero-cylindrical lenses have surfaces with two different radii of curvature in two orthogonal planes. They have a different

7068:, it can be seen that the powers of thin lenses in contact are additive. The general case of multiple thin lenses in contact is 6206:

Even if a lens is designed to minimize or eliminate the aberrations described above, the image quality is still limited by the

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occurs because spherical surfaces are not the ideal shape for a lens, but are by far the simplest shape to which glass can be

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Grbic, A.; Eleftheriades, G. V. (2004). "Overcoming the Diffraction Limit with a Planar Left-handed Transmission-line Lens".

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types, its focal point lies close to the ball's surface. Because of the ball's curvature extremes compared to the lens size,

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point. In this case, the 1st lens images the object at its focal point. So, the thin lens formula for the 2nd lens becomes

7313:, since it produces no net convergence or divergence of the beam. Two lenses at this separation form the simplest type of 3794: 2474: 9460: 7931: 7434:

have curvature along only one axis. They are used to focus light into a line, or to convert the elliptical light from a

7412: 7411:) produces a positive magnification and the image is upright. For further information on simple optical telescopes, see 4505:

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

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is 5, meaning that the object appears 5 times larger to the eye than without the lens. When taking a picture of the

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of light placed at the focal point is converted into a collimated beam by the lens. These two cases are examples of

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is derived here with respect to the right figure. The 1st spherical lens surface (which meets the optical axis at

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The signs of the lens' radii of curvature indicate whether the corresponding surfaces are convex or concave. The

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As an object moves to the lens, the virtual image also gets closer to the lens while the image size is reduced.

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is negative, as it is for real images, the image is upside-down with respect to the object. For virtual images

256:(a seed of a lentil plant), because a double-convex lens is lentil-shaped. The lentil also gives its name to a 8997:

Valentine, J.; et al. (2008). "Three-dimensional optical metamaterial with a negative refractive index".

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the location of its real image. In that case even a negative lens can project a real image, as is done by a

7835: 7535: 3154: 3122:{\displaystyle {\frac {n_{1}}{u}}+{\frac {n_{1}}{v}}=(n_{2}-n_{1})({\frac {1}{R_{1}}}-{\frac {1}{R_{2}}}).} 2064: 7636:

are designed to attenuate light; sunglass lenses that also correct visual impairments can be custom made.

214: 9498: 9473: 7525: 5967: 5100:), a converging lens would form an inverted image, whereas a diverging lens would form an upright image. 570: 99: 9402: 7708: 4872: 2297:

This convention seems to be mainly used for this article, although there is another convention such as

8188: 8164: 5539: 5393: 3896: 9367: 8777: 7734: 1400: 9072: 6545: 2972: 5108: 4743:. This happens when the lens is inserted into a converging beam (being focused by a previous lens) 3850: 2298: 389: 329: 7280:

given for thin lenses in contact. It can be shown that the same formula works for thick lenses if

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The position of the focus of a spherical lens depends on the radii of curvature of the two facets.

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For a single refraction for a circular boundary, the relation between object and its image in the

8954: 7770: 7688: 7395:) produces a negative magnification, indicating an inverted image. A convex plus a concave lens ( 6489: 5950: 5677: 1800: 8924: 8686: 7870: 7864: 7486:

to make images that have an illusion of depth or that change when viewed from different angles.

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This Newtonian form of the lens equation can be derived by using a similarity between triangles

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depending on the curvature of the other surface. A lens with one convex and one concave side is

9067: 8463: 7263:{\displaystyle {\frac {1}{f}}={\frac {1}{f_{1}}}+{\frac {1}{f_{2}}}-{\frac {d}{f_{1}f_{2}}}\,.} 4442: 3669: 2483: 2014: 712: 17: 9339: 8072: 7687:

where the lens's spherical aberration corrects the opposite aberration in the mirror (such as

5489: 5449: 447:(flat). The line joining the centres of the spheres making up the lens surfaces is called the 7830: 7741: 6188: 5641: 5236: 874:

is the on-axis image distance from the line. Due to paraxial approximation where the line of

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image on the film corresponds to an angular size of the moon seen from earth of about 0.5°.

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FDTD Animation of Electromagnetic Propagation through Convex Lens (on- and off-axis) Video

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If the separation distance between two lenses is equal to the sum of their focal lengths (

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intermediate image is located within the focal length of the next lens or even beyond it.

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is the refractive index of medium (the medium other than the spherical surface material),

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has no limit in increasing so its extremum is only the minimum, at which the derivate of

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Reports "lense" as listed in some dictionaries, but not generally considered acceptable.

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for at least 2400 years. A modern application is the use of relatively large lenses to

7692: 7449: 6469: 5995: 5920: 5911: 4540: 4373: 4346: 4107: 4080: 4053: 4026: 3999: 3972: 3945: 3599: 3134: 2778:{\displaystyle {\frac {n_{2}}{-v'+d}}+{\frac {n_{1}}{v}}={\frac {n_{1}-n_{2}}{R_{2}}}.} 2489: 1833: 1806: 1528: 944: 585: 404: 374: 313: 9351: 8210:

The Complete Book of Fire: Building Campfires for Warmth, Light, Cooking, and Survival

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is close to the vertex of the spherical surface meeting the optical axis on the left,

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Nielsen, R.B.; Thoreson, M.D.; Chen, W.; Kristensen, A.; Hvam, J.M.; Shalaev, V. M.;

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Lenses do not form perfect images, and always introduce some degree of distortion or

5240: 4539:, as using an image distance different from that required by this formula produces a 170: 107: 9422: 9159: 9105: 8367: 6158: 5029:{\displaystyle M=-{\frac {S_{2}}{S_{1}}}={\frac {f}{f-S_{1}}}\ =-{\frac {f}{x_{1}}}} 2618:{\displaystyle {\frac {n_{1}}{u}}+{\frac {n_{2}}{v'}}={\frac {n_{2}-n_{1}}{R_{1}}}.} 9242: 9197: 9147: 9077: 9036: 9016: 8999: 8971: 8861: 8853: 8475: 8099: 7976: 7775: 7760: 7617: 7585: 7574: 7456: 7431: 7143: 6116: 6071: 4892: 4609: 4424: 4400: 3487: 1745: 971: 924: 901: 881: 857: 833: 542: 324: 154: 8975: 7715:, harvesting more energy without the need to use larger and more expensive cells. 3773: 1545:

of a spherical lens in air or vacuum for paraxial rays can be calculated from the

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is the radius of curvature of the lens surface farther from the light source; and

1503:{\displaystyle {\frac {n_{2}}{v}}+{\frac {n_{1}}{u}}={\frac {n_{2}-n_{1}}{R}}\,.} 400: 316:(3 BC–65 AD) described the magnifying effect of a glass globe filled with water. 289: 285: 257: 158: 636: 395:

arose from defects in the spherical figure of their surfaces. Optical theory on

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This paragraph is adapted from the 1888 edition of the Encyclopædia Britannica.

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Convex lenses produce an image of an object at infinity at their focus; if the

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from the object, produces a virtual image at infinity of infinite linear size:

4730:), according to the above formula. It is also possible for the object distance 4613: 3632: 2027:

of the lens. If the focal length is in metres, this gives the optical power in

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is the thickness of the lens (the distance along the lens axis between the two

1197: 705: 431:: their two surfaces are parts of the surfaces of spheres. Each surface can be 338: 131: 115: 9398: 9246: 9151: 799:{\displaystyle {\frac {n_{1}}{u}}+{\frac {n_{2}}{v}}={\frac {n_{2}-n_{1}}{R}}} 684:

with two surfaces of equal curvature (also equal in the sign) would have zero

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Lenses are classified by the curvature of the two optical surfaces. A lens is

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Thinking about Life: The History and Philosophy of Biology and Other Sciences

7923: 7704: 7680: 7664: 7633: 7625: 7593: 7445: 7309: 6042: 5940: 5925: 5883:{\displaystyle \vert S_{2}\vert <\vert f\vert ,S_{1}>\vert S_{2}\vert } 5304:

determines the image size while other rays help to locate the image location.

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This magnification formula provides two easy ways to distinguish converging (

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For the imaging by second lens surface, by taking the above sign convention,

2024: 1188: 685: 581: 351: 127: 123: 48: 44: 9081: 8073:"On an account of a rock-crystal lens and decomposed glass found in Niniveh" 9264: 9209: 9097: 9028: 8983: 8128: 7730: 7609: 7543: 7511: 7489: 7469: 7461: 6457:{\displaystyle {\frac {1}{f_{j}}}={\frac {1}{s_{oj}}}+{\frac {1}{s_{ij}}},} 6235: 4506: 3640: 3587:{\displaystyle {\frac {1}{f}}=(n-1)({\frac {1}{R_{1}}}-{\frac {1}{R_{2}}})} 562: 412: 275: 247: 87: 8111: 6259:

For a two-lens system the object distances of each lens can be denoted as

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is positive for converging lenses, and negative for diverging lenses. The

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The International Journal for the History of Engineering & Technology

7435: 6207: 6151: 5232: 5112: 4748: 4662:(focus of red ray path) into a magnified real image (green rays at focus) 4655: 467: 459: 382: 366: 264: 219: 174: 71: 9020: 499: 432: 183: 8824:

Physics 1C, Summer Session I, 2011 - University of California San Diego

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

7519: 7048:{\displaystyle {\frac {1}{f}}={\frac {1}{f_{1}}}+{\frac {1}{f_{2}}}\,.} 6140: 4909: 4860: 4511: 455: 396: 381:

in 1608, both of which appeared in the spectacle-making centres in the

361: 355: 280: 142: 67: 63: 9125:"Toward superlensing with metal–dielectric composites and multilayers" 8137:. Translated by Hickie, William James. Project Gutenberg. EBook #2562. 7988: 7967:

Sines, George; Sakellarakis, Yannis A. (1987). "Lenses in antiquity".

7640: 7565: 7531: 7492:

has two or more, or a graduated, focal lengths ground into the lens.

7479: 4520: 3673: 2333: 2136: 1760: 681: 577: 558: 297: 138: 119: 103: 8776:

There are always 3 "easy rays". For the third ray in this case, see

8233: 7573:

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

7276:

tends to zero, the focal length of the system tends to the value of

4577:

Virtual image formation using a positive lens as a magnifying glass.

3676:(a lens of negligible thickness) and from the lens to the image are 2139:. A flat surface has zero curvature, and its radius of curvature is 941:

toward the right infinity leads to the first or object focal length

609: 240: 218:

Light being refracted by a spherical glass container full of water.

8602:

Hecht, Eugene (2017). "Chapter 6.1 Thick Lenses and Lens Systems".

7980: 7785: 7780: 7754: 7551: 6211: 6147: 5286: 5246: 2140: 2041: 988:

toward the left infinity leads to the second or image focal length

5197:

In the extreme case where an object is an infinite distance away,

4467:

other than air or vacuum a more complicated analysis is required.

2460:{\displaystyle {\frac {1}{\ f\ }}\approx \left(n-1\right)\left\,.} 2135:

surfaces. The reciprocal of the radius of curvature is called the

661: 502:. If both surfaces have the same radius of curvature, the lens is 8865: 7538:

and claim to produce images at spatial resolutions exceeding the

7423: 4686:, the lens thus being used as a magnifying glass (rather than if 2028: 1517: 408: 370: 333: 319: 301: 150: 95: 8996: 8553:

Hecht, Eugene (2017). "5.2.2 Refraction at Spherical Surfaces".

8331: 7131:{\displaystyle {\frac {1}{f}}=\sum _{k=1}^{N}{\frac {1}{f_{k}}}} 6123:, with the wavelength of light. Since, from the formulae above, 4758:, the minimum distance between an object and the real image is 4 7656: 7589: 7503: 6379:. If the lenses are thin, each satisfies the thin lens formula 4621: 4524: 4516: 4496:

according to this formula. If a screen is placed at a distance

646: 541:

For a biconvex or plano-convex lens in a lower-index medium, a

309: 253: 146: 59: 9470: 8079:(in German). Deutsche Physikalische Gesellschaft. p. 355. 7628:; their curvature may not be axially symmetric to correct for 6079:

focused to a ring-shaped image in the focal plane, known as a

6062:. Rays that pass through the centre of a lens of focal length 5250:

Images of black letters in a thin convex lens of focal length

4648: 3764:{\displaystyle {1 \over f}={1 \over S_{1}}+{1 \over S_{2}}\,.} 588:

is much worse than thin lenses, with the notable exception of

545:

beam of light passing through the lens converges to a spot (a

358:

may or may not have been intended for use as burning glasses.

9118: 7765: 7722: 6055: 3789:

The lens equation can also be put into the "Newtonian" form:

3644: 3442:{\displaystyle {\frac {1}{u}}+{\frac {1}{v}}={\frac {1}{f}}.} 466:

lens. An example is eyeglass lenses that are used to correct

111: 91: 8234:

Glick, Thomas F.; Steven John Livesey; Faith Wallis (2005).

7167:, then the focal length for the combined system is given by 6952:

A simplest case is where thin lenses are placed in contact (

6163: 6091: 2111:

surfaces (used to converge light in a positive lens), while

478: 9421: 8237:

Medieval science, technology, and medicine: an encyclopedia

6150:. This naturally occurring substance has the highest known 5152: 2478:

A Diagram for a Spherical Lens Equation with Paraxial Rays.

346:

became available in Latin translation in the 12th century (

293: 9364: 9340:

A chapter from an online textbook on refraction and lenses

5744:

Virtual (rays apparently diverging from each image point)

3694:

respectively, the distances are related by the (Gaussian)

918:

are also considered distances with respect to the vertex.

274:

The oldest certain reference to the use of lenses is from

7699: 7284:

is taken as the distance between their principal planes.

7163:

If two thin lenses are separated in air by some distance

5308: 5159:

lens, one is not concerned with the linear magnification

4573: 110:

to the required shape. A lens can focus light to form an

7744:-resistant coatings are available to help control this. 7569:

A watch with a plano-convex lens over the date indicator

7438:

into a round beam. They are also used in motion picture

6030: 4739:

to be negative, in which case the lens sees a so-called

4632: 4331: 2995:, the 2nd term of the RHS (Right Hand Side) is gone, so 2271:

The transverse magnification in imaging (= the ratio of

821:

is the refractive index of the material of the surface,

411:

in 1733, an invention also claimed by fellow Englishman

8332:

Paul S. Agutter; Denys N. Wheatley (12 December 2008).

7542:. The first superlenses were made in 2004 using such a 2146: 1803:

of the lens, and the locations of the principal planes

8891:

Hecht, Eugene (2017). "Back and Front Focal Lengths".

7146: 6492: 6355: 6325: 6295: 6265: 5254:

are shown in red. Selected rays are shown for letters

4427: 4403: 4376: 4349: 4110: 4083: 4056: 4029: 4002: 3975: 3948: 3602: 3490: 3457: 2631: 2492: 1836: 1809: 974: 947: 927: 904: 884: 860: 836: 688:(as its focal length becomes infinity as shown in the 561:

in air, the distance from the lens to the spot is the

549:) behind the lens. In this case, the lens is called a 7325: 7173: 7076: 6990: 6958: 6780: 6599: 6591:. So, the thin lens formula for the 1st lens becomes 6548: 6538:. (The 2nd lens images the image of the first lens.) 6472: 6387: 5819: 5752: 5715: 5680: 5644: 5588: 5542: 5492: 5452: 5396: 5350: 4938: 4930:

of an imaging system using a single lens is given by

4202: 3899: 3853: 3797: 3706: 3512: 3399: 3188: 3157: 3137: 3003: 2975: 2791: 2676: 2527: 2349: 2073:

used to represent this varies, but in this article a

1939: 1865: 1557: 1531: 1423: 1270: 1206: 1023: 994: 723: 665:

Meniscus lenses: negative (top) and positive (bottom)

342:, 11th century). The Arabic translation of Ptolemy's 9408:

The Use of Magnifying Lenses in the Classical World

8412: 8274:

The Galileo Project > Science > The Telescope

7620:.) Most lenses used for other purposes have strict 7413:

Refracting telescope § Refracting telescope designs

6154:, indicating that the material has low dispersion. 4023:is positive if it is right to the rear focal point 3969:is positive if it is left to the front focal point 2521:, which can be described by the following equation, 137:Lenses are used in various imaging devices such as 9268: 9181: 8951: 8817:"Lecture 9 Notes: 07 / 13 - Multiple-lens systems" 7919:Lists "lense" as an acceptable alternate spelling. 7896: 7365: 7262: 7152: 7130: 7047: 6970: 6942: 6761: 6583: 6530: 6478: 6456: 6371: 6341: 6311: 6281: 5882: 5784: 5734: 5689: 5663: 5616: 5570: 5514: 5474: 5424: 5378: 5028: 4433: 4409: 4389: 4362: 4321: 4123: 4096: 4069: 4042: 4015: 3988: 3961: 3935: 3885: 3837: 3763: 3615: 3586: 3496: 3476: 3441: 3381: 3172: 3143: 3121: 2987: 2961: 2777: 2662: 2617: 2505: 2459: 2000: 1926: 1849: 1822: 1729: 1537: 1502: 1391: 1254: 1159: 1007: 980: 960: 933: 910: 890: 866: 842: 798: 534:. Convex-concave lenses are most commonly used in 9432:. Vol. 16 (11th ed.). pp. 421–427. 7966: 3672:, if the distances from an object to a spherical 704:Simulation of refraction at spherical surface at 576:Another extreme case of a thick convex lens is a 332:(10th century), who was in turn improved upon by 312:, though the reference is vague). Both Pliny and 27:Optical device which transmits and refracts light 9485: 9467:– Open source lens simulator (downloadable java) 8790:Hecht, Eugene (2017). "Thin-Lens Combinations". 8369:Renaissance Vision from Spectacles to Telescopes 7667:when applied to the human eye; others produce a 5067:) lenses: For an object very close to the lens ( 2001:{\displaystyle h_{2}=-{\frac {f(n-1)d}{R_{1}n}}} 1927:{\displaystyle h_{1}=-{\frac {f(n-1)d}{R_{2}n}}} 1730:{\displaystyle {\frac {1}{\ f\ }}=(n-1)\left\ ,} 422: 8372:. American Philosophical Society. p. 210. 5186:Rather, the plate scale of the camera is about 2233:The height of an object from the optical axis. 1767:of the lens surface closer to the light source; 565:of the lens, which is commonly represented by 518:). If one of the surfaces is flat, the lens is 194: with: history after 1823. You can help by 8915: 8655: 8653: 8627: 8625: 8623: 8422:Stargazer: The Life and Times of the Telescope 8365: 5434:Inverted (opposite to the object orientation) 5277:) has an equal-size, real and inverted image; 2513:as its vertex) images an on-axis object point 2337:approximation can be made. For a lens in air, 538:, since the shape minimizes some aberrations. 9049: 8660:Hecht, Eugene (2017). "Thin-Lens Equations". 8597: 8595: 8593: 8591: 8292: 8283:. Galileo.rice.edu. Retrieved on 6 June 2012. 7869:. Franklin, Beedle & Associates. p. 5975: 5314:Images of Real Objects Formed by Thin Lenses 4543:(fuzzy) image for an object at a distance of 2252:The height of an image from the optical axis 1170: 689: 8990: 8127: 8123: 8121: 7366:{\displaystyle M=-{\frac {f_{2}}{f_{1}}}\,,} 6169: 5877: 5864: 5845: 5839: 5833: 5820: 5766: 5753: 4552:from the camera. Put another way, modifying 4530:The focusing adjustment of a camera adjusts 9309: 9183:"Good-Bye to Curved Lens: New Lens Is Flat" 8650: 8620: 8577: 8418: 8070: 5388:Real (rays converging to each image point) 2179:The distance between an object and a lens. 2152:Sign convention for Gaussian lens equation 245: 8588: 8409:, Nobel Foundation. Retrieved 3 April 2009 8299:. Courier Dover Publications. p. 27. 8089: 8009: 8003: 7639:Other uses are in imaging systems such as 7522:, or transforms a laser beam into a ring. 6466:If the distance between the two lenses is 6210:of light passing through the lens' finite 6115:of the lens material—the variation of its 5982: 5968: 5785:{\displaystyle \vert S_{2}\vert >S_{1}} 4642:lens produces a demagnified virtual image. 2304: 2198:The distance between an image and a lens. 1255:{\displaystyle n_{1}\sin i=n_{2}\sin r\,.} 695: 350:1154). Between the 11th and 13th century " 9238:Journal of Sol-Gel Science and Technology 9234: 9112: 9071: 8118: 7740:Lenses can become scratched and abraded. 7359: 7256: 7064:is the power of a lens with focal length 7041: 5266:in blue, green and orange, respectively. 4699:as for a camera). Using a negative lens ( 4164:and another similarity between triangles 3929: 3831: 3757: 3451:For the thin lens in air or vacuum where 2453: 2301:requiring different lens equation forms. 1496: 1403:(paraxial approximation) and eliminating 1248: 473: 365:the spectacle-making centres in both the 90:. Lenses are made from materials such as 8947: 8945: 8814: 8632:Hecht, Eugene (2017). "Finite Imagery". 7564: 7455: 7422: 5245: 5235:places a lower limit on the size of the 4912:may also be viewed when put on a screen. 4572: 4330: 3777:Single thin lens imaging with chief rays 3772: 2473: 1516: 1187: 1180: 812:is the radius of the spherical surface, 699: 660: 477: 225: 213: 38: 8909: 6201: 6097: 6008: 5794:Erect (same to the object orientation) 5190:, from which one can conclude that the 4754:For a given lens with the focal length 3391:So, the Gaussian thin lens equation is 2042:Sign convention for radii of curvature 1512: 284:(424 BCE) mentioning a burning-glass. 271:depict "simple glass meniscal lenses". 149:. They are also used as visual aids in 14: 9486: 8895:(5th ed.). Pearson. p. 181. 8794:(5th ed.). Pearson. p. 178. 8557:(5th ed.). Pearson. p. 164. 8206: 7862: 7663:. Some of these instruments produce a 7418: 5617:{\displaystyle \infty >S_{2}>2f} 5379:{\displaystyle \infty >S_{1}>2f} 5309:Table for thin lens imaging properties 5050:is positive, so the image is upright. 4669:Using a positive lens of focal length 4487:, we will find an image at a distance 3151:of the thin lens is found by limiting 2969:For the thin lens approximation where 1192:The four cases of spherical refraction 968:for the spherical surface. Similarly, 462:in different meridians. This forms an 9290: 9263: 9179: 8942: 8890: 8851: 8789: 8718:"Resource Lesson: Thin Lens Equation" 8659: 8631: 8601: 8582: 8552: 8504:from the original on 26 November 2022 8461: 6066:are focused at a point with distance 4483:from a positive lens of focal length 4470:If an object is placed at a distance 4131:are either positive or negative.) 3626: 3173:{\displaystyle u\rightarrow -\infty } 2038:are not the same in both directions. 153:to correct defects of vision such as 8697:from the original on 12 October 2000 8534:from the original on 26 October 2021 8254:from the original on 20 January 2023 8083: 8052:from the original on 19 October 2012 8022:from the original on 1 February 2009 7899:Merriam-Webster's Medical Dictionary 4812:by the lens equation (or expressing 4720:) can only produce a virtual image ( 2482:The spherical thin lens equation in 2147:Sign convention for other parameters 178: 79: 8715: 8293:Henry C. King (28 September 2003). 8213:. Menasha Ridge Press. p. 25. 7924:"Lens or Lense – Which is Correct?" 5342:Converging lens (or positive lens) 4595:to form an image, this is called a 3838:{\displaystyle f^{2}=x_{1}x_{2}\,,} 234:, a planned sky surveying telescope 86:), usually arranged along a common 82:consists of several simple lenses ( 62:device that focuses or disperses a 24: 9202:10.1038/scientificamerican0515-22b 8872:from the original on 29 April 2024 8833:from the original on 18 April 2024 8758:from the original on 10 March 2015 8524:"Refraction at Spherical Surfaces" 7934:from the original on 21 April 2018 6978:). Then the combined focal length 6221: 6174:Other kinds of aberration include 5805:Diverging lens (or negative lens) 5684: 5589: 5351: 5123:according to their focal lengths. 4826:), and equating the derivative of 3167: 2785:Adding these two equations yields, 645: 635: 608: 598: 437:(bulging outwards from the lens), 25: 9510: 9333: 9312:Field Guide to Geometrical Optics 8728:from the original on 2 April 2015 8482:– via Taylor & Francis. 8425:. Allen & Unwin. p. 55. 8042:"The Nimrud lens/The Layard lens" 8010:Whitehouse, David (1 July 1999). 7903:. Merriam-Webster. 1995. p. 6319:, and the image distances as and 6244:Simple lenses are subject to the 3648:lens axis situated at a distance 9295:(4th ed.). Addison Wesley. 9275:(2nd ed.). Addison Wesley. 9241:. Vol. 27. pp. 71–79. 9216:from the original on 19 May 2015 8684: 8313:from the original on 2 July 2023 8104:10.1097/00006123-199804000-00116 6245: 6162: 6157: 6090: 6029: 5571:{\displaystyle f<S_{1}<2f} 5425:{\displaystyle f<S_{2}<2f} 4919: 4871: 4859: 4647: 4631: 3936:{\displaystyle x_{2}=S_{2}-f\,.} 3663: 182: 9257: 9228: 9173: 9043: 8884: 8845: 8808: 8783: 8770: 8740: 8709: 8678: 8571: 8546: 8516: 8486: 8455: 8446: 8393: 8359: 8325: 8286: 8266: 8227: 8200: 7969:American Journal of Archaeology 4673:, a virtual image results when 4515:. This is the principle of the 2469: 9438: 8419:Fred Watson (1 October 2007). 8177: 8150: 8064: 8034: 7960: 7849: 6931: 6905: 6894: 6875: 6847: 6750: 6724: 6713: 6694: 6666: 6584:{\displaystyle s_{i1}=d-f_{2}} 5903: 4780:). This is derived by letting 4561:causes objects at a different 3784:the optical center of the lens 3581: 3541: 3538: 3526: 3373: 3333: 3330: 3297: 3281: 3278: 3238: 3235: 3209: 3161: 3113: 3073: 3070: 3044: 2988:{\displaystyle d\rightarrow 0} 2979: 2942: 2925: 2901: 2861: 2858: 2832: 1974: 1962: 1900: 1888: 1673: 1661: 1591: 1579: 1335: 1323: 1293: 1281: 569:in diagrams and equations. An 443:(depressed into the lens), or 322:(2nd century) wrote a book on 74:consists of a single piece of 13: 1: 9471:Animations demonstrating lens 9310:Greivenkamp, John E. (2004). 8976:10.1103/PhysRevLett.92.117403 8462:Julia, Elton (18 July 2013). 7953: 7757:treatment of optical surfaces 7510:optical surface. It images a 3886:{\displaystyle x_{1}=S_{1}-f} 3639:from the lens. Conversely, a 1525:The (effective) focal length 1176: 423:Construction of simple lenses 8693:. Georgia State University. 8296:The History of the Telescope 7836:Ray transfer matrix analysis 7536:negative index metamaterials 7526:Diffractive optical elements 6531:{\textstyle s_{o2}=d-s_{i1}} 6054:, derives its name from the 4570:to come into perfect focus. 4527:serves as the image sensor. 4445:in this case). If distances 3652:from the lens is called the 3596:where the subscript of 2 in 2214:The focal length of a lens. 2065:Radius of curvature (optics) 7: 9454:– Concave and Convex Lenses 8748:"The Mathematics of Lenses" 8494:"4.4: Spherical Refractors" 8077:Die Fortschritte der Physik 8012:"World's oldest telescope?" 7747: 5690:{\displaystyle \pm \infty } 571:extended hemispherical lens 10: 9515: 8240:. Routledge. p. 167. 7737:into a collector antenna. 7427:An aspheric biconvex lens. 6982:of the lenses is given by 6225: 6101: 6040: 6012: 5993: 5735:{\displaystyle S_{1}<f} 5331:Lateral Image Orientation 2062: 1200:on the spherical surface, 387: 168: 164: 29: 9314:. SPIE Field Guides vol. 9152:10.1007/s00340-010-4065-z 8815:Vlasenko, Alexey (2011). 8778:File:Lens3b third ray.svg 8664:(5th ed.). Pearson. 8636:(5th ed.). Pearson. 8606:(5th ed.). Pearson. 8194:28 September 2008 at the 7735:electromagnetic radiation 7160:is the number of lenses. 6170:Other types of aberration 2341: is then given by 2299:Cartesian sign convention 1401:small angle approximation 9413:13 November 2017 at the 9345:17 December 2009 at the 8480:10.1179/175812109X449612 8338:. Springer. p. 17. 7866:Common Errors in English 7842: 7709:concentrate solar energy 7671:that can be captured on 7528:can function as lenses. 7460:Close-up view of a flat 5515:{\displaystyle S_{2}=2f} 5475:{\displaystyle S_{1}=2f} 4417:are with respect to the 390:History of the telescope 252:, the Latin name of the 9450:21 January 2010 at the 9445:Learning by Simulations 9429:Encyclopædia Britannica 9387:Ancient Egyptian lenses 9247:10.1023/A:1022684011222 9082:10.1126/science.1157566 8955:Physical Review Letters 8916:Proteep Mallik (2005). 8852:Paschotta, Dr Rüdiger. 8366:Vincent Ilardi (2007). 7771:Cardinal point (optics) 7560: 6036: 5951:Petzval field curvature 5664:{\displaystyle S_{1}=f} 2305:Thin lens approximation 1799:is with respect to the 696:For a spherical surface 498:) if both surfaces are 9463:6 October 2010 at the 9353:Thin Spherical Lenses 9291:Hecht, Eugene (2002). 9180:Patel, Prachi (2015). 8716:Colwell, Catharine H. 8405:9 January 2010 at the 8400:Microscopes: Time Line 8187:(trans. John Bostock) 8170:4 October 2008 at the 8163:(trans. John Bostock) 8048:. The British Museum. 7584:for the correction of 7570: 7465: 7428: 7367: 7264: 7154: 7132: 7110: 7049: 6972: 6944: 6763: 6585: 6532: 6480: 6458: 6373: 6343: 6313: 6283: 5884: 5786: 5736: 5691: 5665: 5618: 5572: 5516: 5476: 5426: 5380: 5305: 5155:using a camera with a 5030: 4844:) to zero. (Note that 4578: 4443:effective focal length 4435: 4411: 4391: 4364: 4340: 4335:A camera lens forms a 4323: 4125: 4098: 4071: 4044: 4017: 3990: 3963: 3937: 3887: 3839: 3778: 3765: 3617: 3588: 3498: 3478: 3443: 3383: 3174: 3145: 3123: 2989: 2963: 2779: 2664: 2619: 2507: 2484:paraxial approximation 2479: 2461: 2002: 1928: 1851: 1824: 1731: 1539: 1522: 1504: 1393: 1256: 1193: 1185: 1161: 1009: 982: 962: 935: 912: 892: 868: 844: 800: 713:paraxial approximation 708: 666: 650: 641: 613: 604: 483: 474:Types of simple lenses 296:was said to watch the 292:when he mentions that 246: 235: 223: 51: 9420:Henker, Otto (1911). 9359:13 March 2020 at the 8752:The Physics Classroom 8207:Tilton, Buck (2005). 8189:Book XXXVII, Chap. 16 8165:Book XXXVII, Chap. 10 7863:Brians, Paul (2003). 7855:The variant spelling 7568: 7459: 7426: 7368: 7265: 7155: 7133: 7090: 7050: 6973: 6945: 6764: 6586: 6533: 6481: 6459: 6374: 6344: 6314: 6284: 6189:pincushion distortion 5885: 5787: 5737: 5692: 5666: 5619: 5573: 5517: 5477: 5427: 5381: 5249: 5239:. This is called the 5237:point spread function 5148:angular magnification 5109:angular magnification 5103:Linear magnification 5031: 4576: 4436: 4412: 4392: 4365: 4334: 4324: 4126: 4099: 4072: 4045: 4018: 3991: 3964: 3938: 3888: 3840: 3776: 3766: 3618: 3589: 3499: 3479: 3444: 3384: 3175: 3146: 3124: 2990: 2964: 2780: 2665: 2663:{\textstyle u'=-v'+d} 2620: 2517:to the virtual image 2508: 2477: 2462: 2313:is small compared to 2017:of the focal length, 2003: 1929: 1852: 1825: 1748:of the lens material; 1732: 1540: 1520: 1505: 1394: 1257: 1191: 1184: 1162: 1010: 1008:{\displaystyle f_{i}} 983: 963: 936: 913: 893: 869: 845: 801: 703: 664: 649: 639: 612: 602: 481: 388:Further information: 377:around 1595, and the 229: 217: 42: 32:Lens (disambiguation) 9476:4 April 2012 at the 8858:www.rp-photonics.com 8687:"Thin Lens Equation" 8528:personal.math.ubc.ca 8279:23 June 2004 at the 8144:28 June 2017 at the 8131:(22 January 2013) . 8071:D. Brewster (1852). 7928:writingexplained.org 7801:List of lens designs 7729:, commonly called a 7711:on relatively small 7323: 7171: 7144: 7074: 6988: 6956: 6778: 6597: 6546: 6490: 6470: 6385: 6353: 6323: 6293: 6263: 6202:Aperture diffraction 6109:Chromatic aberration 6104:Chromatic aberration 6098:Chromatic aberration 6020:Spherical aberration 6015:Spherical aberration 6009:Spherical aberration 5956:Chromatic aberration 5931:Spherical aberration 5817: 5750: 5713: 5678: 5642: 5586: 5540: 5490: 5450: 5394: 5348: 5334:Image Magnification 4936: 4425: 4401: 4374: 4347: 4339:of a distant object. 4200: 4108: 4081: 4054: 4027: 4000: 3973: 3946: 3897: 3851: 3795: 3704: 3600: 3510: 3488: 3477:{\textstyle n_{1}=1} 3455: 3397: 3186: 3155: 3135: 3001: 2973: 2789: 2674: 2629: 2525: 2490: 2347: 1937: 1863: 1834: 1807: 1759:is the (signed, see 1555: 1547:lensmaker's equation 1529: 1513:Lensmaker's equation 1421: 1268: 1264:Also in the diagram, 1204: 1021: 992: 972: 945: 925: 902: 882: 858: 834: 721: 690:lensmaker's equation 590:chromatic aberration 379:refracting telescope 269:Egyptian hieroglyphs 76:transparent material 30:For other uses, see 9392:25 May 2022 at the 9370:14 May 2017 at the 9287:Chapters 5 & 6. 9189:Scientific American 9144:2010ApPhB.100...93N 9064:2008Sci...321..930Y 9021:10.1038/nature07247 9013:2008Natur.455..376V 8968:2004PhRvL..92k7403G 8930:on 23 November 2009 8185:The Natural History 8161:The Natural History 8046:Collection database 7816:Optical lens design 7685:catadioptric system 7580:Lenses are used as 7497:gradient index lens 7484:lenticular printing 7419:Non spherical types 6971:{\displaystyle d=0} 6372:{\textstyle s_{i2}} 6342:{\textstyle s_{i1}} 6312:{\textstyle s_{o2}} 6282:{\textstyle s_{o1}} 6246:optical aberrations 6216:diffraction-limited 6024:ground and polished 5315: 5285:) has its image at 5130:focal length, held 2153: 1765:radius of curvature 1171:lensmaker's formula 348:Eugenius of Palermo 9499:Optical components 9434:(with 21 diagrams) 9380:digitalartform.com 8498:Physics LibreTexts 7806:Numerical aperture 7791:Gravitational lens 7725:systems often use 7713:photovoltaic cells 7634:Sunglasses' lenses 7571: 7466: 7432:Cylindrical lenses 7429: 7363: 7260: 7150: 7128: 7045: 6968: 6940: 6759: 6581: 6528: 6476: 6454: 6369: 6339: 6309: 6279: 6133:achromatic doublet 6127:is dependent upon 6052:comatic aberration 5996:Optical aberration 5912:Optical aberration 5880: 5782: 5732: 5687: 5661: 5614: 5568: 5512: 5472: 5422: 5376: 5313: 5306: 5026: 4579: 4519:, and also of the 4431: 4407: 4390:{\textstyle S_{2}} 4387: 4363:{\textstyle S_{1}} 4360: 4341: 4319: 4317: 4124:{\textstyle x_{2}} 4121: 4097:{\textstyle x_{1}} 4094: 4070:{\textstyle f^{2}} 4067: 4043:{\textstyle F_{2}} 4040: 4016:{\textstyle x_{2}} 4013: 3989:{\textstyle F_{1}} 3986: 3962:{\textstyle x_{1}} 3959: 3933: 3883: 3835: 3779: 3761: 3627:Imaging properties 3616:{\textstyle n_{2}} 3613: 3584: 3494: 3474: 3439: 3379: 3170: 3141: 3119: 2985: 2959: 2775: 2660: 2615: 2506:{\textstyle V_{1}} 2503: 2480: 2457: 2151: 2031:(inverse metres). 1998: 1924: 1850:{\textstyle h_{2}} 1847: 1823:{\textstyle h_{1}} 1820: 1727: 1535: 1523: 1500: 1389: 1387: 1252: 1194: 1186: 1157: 1155: 1005: 978: 961:{\textstyle f_{0}} 958: 931: 908: 888: 864: 840: 796: 709: 667: 651: 642: 614: 605: 586:optical aberration 506:. A lens with two 484: 470:in someone's eye. 415:in a 1758 patent. 405:Chester Moore Hall 375:optical microscope 314:Seneca the Younger 236: 224: 58:is a transmissive 52: 47:consisting of two 9325:978-0-8194-5294-8 9302:978-0-321-18878-6 9282:978-0-201-11609-0 9132:Applied Physics B 8902:978-1-292-09693-3 8801:978-1-292-09693-3 8671:978-1-292-09693-3 8643:978-1-292-09693-3 8613:978-1-292-09693-3 8564:978-1-292-09693-3 8432:978-1-74175-383-7 8379:978-0-87169-259-7 8345:978-1-4020-8865-0 8306:978-0-486-43265-6 8247:978-0-415-96930-7 8220:978-0-89732-633-9 8183:Pliny the Elder, 7930:. 30 April 2017. 7914:978-0-87779-914-6 7880:978-1-887902-89-2 7821:Photochromic lens 7727:dielectric lenses 7673:photographic film 7586:refractive errors 7540:diffraction limit 7482:that are used in 7476:Lenticular lenses 7440:anamorphic lenses 7357: 7315:optical telescope 7254: 7222: 7202: 7182: 7126: 7085: 7039: 7019: 6999: 6935: 6845: 6817: 6796: 6754: 6664: 6636: 6615: 6479:{\displaystyle d} 6449: 6426: 6403: 6228:Photographic lens 6111:is caused by the 5992: 5991: 5901: 5900: 5241:diffraction limit 5134:from the eye and 5121:wide-angle lenses 5117:long-focus lenses 5060:) and diverging ( 5024: 5002: 4998: 4970: 4313: 4278: 4257: 4229: 3755: 3735: 3715: 3696:thin lens formula 3579: 3559: 3521: 3434: 3421: 3408: 3371: 3351: 3322: 3292: 3276: 3256: 3204: 3144:{\displaystyle f} 3131:The focal length 3111: 3091: 3039: 3019: 2954: 2899: 2879: 2827: 2807: 2770: 2728: 2708: 2610: 2568: 2543: 2446: 2444: 2431: 2418: 2416: 2403: 2366: 2364: 2358: 2295: 2294: 1996: 1922: 1795:The focal length 1723: 1714: 1692: 1684: 1678: 1660: 1650: 1648: 1635: 1622: 1620: 1607: 1574: 1572: 1566: 1538:{\displaystyle f} 1494: 1459: 1439: 1383: 1353: 1311: 1148: 1082: 794: 759: 739: 675:positive meniscus 671:negative meniscus 655: 654: 618: 617: 536:corrective lenses 212: 211: 171:History of optics 45:burning apparatus 16:(Redirected from 9506: 9458:OpticalRayTracer 9433: 9425: 9378:Lens article at 9329: 9306: 9286: 9274: 9251: 9250: 9232: 9226: 9225: 9223: 9221: 9185: 9177: 9171: 9170: 9169:on 9 March 2013. 9168: 9162:. Archived from 9129: 9116: 9110: 9109: 9075: 9047: 9041: 9040: 8994: 8988: 8987: 8949: 8940: 8939: 8937: 8935: 8929: 8923:. Archived from 8922: 8913: 8907: 8906: 8888: 8882: 8881: 8879: 8877: 8854:"focal distance" 8849: 8843: 8842: 8840: 8838: 8832: 8821: 8812: 8806: 8805: 8787: 8781: 8774: 8768: 8767: 8765: 8763: 8744: 8738: 8737: 8735: 8733: 8713: 8707: 8706: 8704: 8702: 8682: 8676: 8675: 8657: 8648: 8647: 8629: 8618: 8617: 8599: 8586: 8578:Greivenkamp 2004 8575: 8569: 8568: 8550: 8544: 8543: 8541: 8539: 8520: 8514: 8513: 8511: 8509: 8490: 8484: 8483: 8459: 8453: 8450: 8444: 8443: 8441: 8439: 8416: 8410: 8397: 8391: 8390: 8388: 8386: 8363: 8357: 8356: 8354: 8352: 8329: 8323: 8322: 8320: 8318: 8290: 8284: 8270: 8264: 8263: 8261: 8259: 8231: 8225: 8224: 8204: 8198: 8181: 8175: 8154: 8148: 8138: 8125: 8116: 8115: 8087: 8081: 8080: 8068: 8062: 8061: 8059: 8057: 8038: 8032: 8031: 8029: 8027: 8007: 8001: 8000: 7964: 7947: 7943: 7941: 7939: 7918: 7902: 7891: 7889: 7887: 7853: 7811:Optical coatings 7776:Caustic (optics) 7761:Back focal plane 7618:intraocular lens 7575:magnifying glass 7410: 7394: 7384: 7372: 7370: 7369: 7364: 7358: 7356: 7355: 7346: 7345: 7336: 7306: 7283: 7279: 7275: 7269: 7267: 7266: 7261: 7255: 7253: 7252: 7251: 7242: 7241: 7228: 7223: 7221: 7220: 7208: 7203: 7201: 7200: 7188: 7183: 7175: 7166: 7159: 7157: 7156: 7151: 7137: 7135: 7134: 7129: 7127: 7125: 7124: 7112: 7109: 7104: 7086: 7078: 7067: 7063: 7054: 7052: 7051: 7046: 7040: 7038: 7037: 7025: 7020: 7018: 7017: 7005: 7000: 6992: 6981: 6977: 6975: 6974: 6969: 6949: 6947: 6946: 6941: 6936: 6934: 6930: 6929: 6917: 6916: 6897: 6893: 6892: 6874: 6873: 6863: 6846: 6844: 6843: 6842: 6823: 6818: 6816: 6802: 6797: 6795: 6794: 6782: 6768: 6766: 6765: 6760: 6755: 6753: 6749: 6748: 6736: 6735: 6716: 6712: 6711: 6693: 6692: 6682: 6665: 6663: 6662: 6661: 6642: 6637: 6635: 6621: 6616: 6614: 6613: 6601: 6590: 6588: 6587: 6582: 6580: 6579: 6561: 6560: 6537: 6535: 6534: 6529: 6527: 6526: 6505: 6504: 6485: 6483: 6482: 6477: 6463: 6461: 6460: 6455: 6450: 6448: 6447: 6432: 6427: 6425: 6424: 6409: 6404: 6402: 6401: 6389: 6378: 6376: 6375: 6370: 6368: 6367: 6348: 6346: 6345: 6340: 6338: 6337: 6318: 6316: 6315: 6310: 6308: 6307: 6288: 6286: 6285: 6280: 6278: 6277: 6222:Compound lenses 6166: 6161: 6130: 6126: 6122: 6117:refractive index 6094: 6077: 6065: 6061: 6033: 5984: 5977: 5970: 5908: 5907: 5889: 5887: 5886: 5881: 5876: 5875: 5860: 5859: 5832: 5831: 5791: 5789: 5788: 5783: 5781: 5780: 5765: 5764: 5741: 5739: 5738: 5733: 5725: 5724: 5707:Converging lens 5696: 5694: 5693: 5688: 5670: 5668: 5667: 5662: 5654: 5653: 5636:Converging lens 5623: 5621: 5620: 5615: 5604: 5603: 5577: 5575: 5574: 5569: 5558: 5557: 5534:Converging lens 5521: 5519: 5518: 5513: 5502: 5501: 5481: 5479: 5478: 5473: 5462: 5461: 5444:Converging lens 5431: 5429: 5428: 5423: 5412: 5411: 5385: 5383: 5382: 5377: 5366: 5365: 5322:Object Location 5316: 5312: 5299: 5284: 5276: 5253: 5230: 5219: 5206: 5193: 5189: 5185: 5183: 5181: 5175: 5173: 5167: 5158: 5144: 5137: 5133: 5129: 5106: 5099: 5097: 5083: 5081: 5066: 5059: 5049: 5045: 5041: 5035: 5033: 5032: 5027: 5025: 5023: 5022: 5010: 5000: 4999: 4997: 4996: 4995: 4976: 4971: 4969: 4968: 4959: 4958: 4949: 4907: 4893:magnifying glass 4890: 4875: 4863: 4830:with respect to 4738: 4729: 4719: 4705: 4698: 4685: 4672: 4651: 4635: 4610:magnifying glass 4607: 4589: 4569: 4560: 4551: 4538: 4509:, is known as a 4504: 4495: 4486: 4482: 4462: 4453: 4440: 4438: 4437: 4432: 4419:principal planes 4416: 4414: 4413: 4408: 4396: 4394: 4393: 4388: 4386: 4385: 4369: 4367: 4366: 4361: 4359: 4358: 4328: 4326: 4325: 4320: 4318: 4314: 4312: 4311: 4302: 4298: 4297: 4284: 4279: 4274: 4273: 4264: 4258: 4253: 4249: 4248: 4235: 4230: 4228: 4227: 4218: 4217: 4208: 4130: 4128: 4127: 4122: 4120: 4119: 4103: 4101: 4100: 4095: 4093: 4092: 4076: 4074: 4073: 4068: 4066: 4065: 4049: 4047: 4046: 4041: 4039: 4038: 4022: 4020: 4019: 4014: 4012: 4011: 3995: 3993: 3992: 3987: 3985: 3984: 3968: 3966: 3965: 3960: 3958: 3957: 3942: 3940: 3939: 3934: 3922: 3921: 3909: 3908: 3892: 3890: 3889: 3884: 3876: 3875: 3863: 3862: 3844: 3842: 3841: 3836: 3830: 3829: 3820: 3819: 3807: 3806: 3770: 3768: 3767: 3762: 3756: 3754: 3753: 3741: 3736: 3734: 3733: 3721: 3716: 3708: 3693: 3684: 3651: 3638: 3635:) at a distance 3622: 3620: 3619: 3614: 3612: 3611: 3593: 3591: 3590: 3585: 3580: 3578: 3577: 3565: 3560: 3558: 3557: 3545: 3522: 3514: 3503: 3501: 3500: 3495: 3484:can be assumed, 3483: 3481: 3480: 3475: 3467: 3466: 3448: 3446: 3445: 3440: 3435: 3427: 3422: 3414: 3409: 3401: 3388: 3386: 3385: 3380: 3372: 3370: 3369: 3357: 3352: 3350: 3349: 3337: 3323: 3321: 3320: 3311: 3310: 3301: 3293: 3285: 3277: 3275: 3274: 3262: 3257: 3255: 3254: 3242: 3234: 3233: 3221: 3220: 3205: 3200: 3199: 3190: 3179: 3177: 3176: 3171: 3150: 3148: 3147: 3142: 3128: 3126: 3125: 3120: 3112: 3110: 3109: 3097: 3092: 3090: 3089: 3077: 3069: 3068: 3056: 3055: 3040: 3035: 3034: 3025: 3020: 3015: 3014: 3005: 2994: 2992: 2991: 2986: 2968: 2966: 2965: 2960: 2955: 2953: 2952: 2935: 2923: 2919: 2918: 2908: 2900: 2898: 2897: 2885: 2880: 2878: 2877: 2865: 2857: 2856: 2844: 2843: 2828: 2823: 2822: 2813: 2808: 2803: 2802: 2793: 2784: 2782: 2781: 2776: 2771: 2769: 2768: 2759: 2758: 2757: 2745: 2744: 2734: 2729: 2724: 2723: 2714: 2709: 2707: 2700: 2688: 2687: 2678: 2669: 2667: 2666: 2661: 2653: 2639: 2624: 2622: 2621: 2616: 2611: 2609: 2608: 2599: 2598: 2597: 2585: 2584: 2574: 2569: 2567: 2559: 2558: 2549: 2544: 2539: 2538: 2529: 2512: 2510: 2509: 2504: 2502: 2501: 2466: 2464: 2463: 2458: 2452: 2448: 2447: 2445: 2442: 2441: 2440: 2429: 2424: 2419: 2417: 2414: 2413: 2412: 2401: 2396: 2389: 2385: 2367: 2365: 2362: 2356: 2351: 2340: 2330: 2321: 2312: 2239:Inverted object 2217:Conversing lens 2154: 2150: 2130: 2120: 2106: 2096: 2086: 2079: 2059: 2050: 2022: 2012: 2007: 2005: 2004: 1999: 1997: 1995: 1991: 1990: 1980: 1957: 1949: 1948: 1933: 1931: 1930: 1925: 1923: 1921: 1917: 1916: 1906: 1883: 1875: 1874: 1856: 1854: 1853: 1848: 1846: 1845: 1829: 1827: 1826: 1821: 1819: 1818: 1801:principal planes 1798: 1786:surface vertices 1783: 1777: 1758: 1746:refractive index 1743: 1736: 1734: 1733: 1728: 1721: 1720: 1716: 1715: 1713: 1712: 1711: 1702: 1701: 1690: 1685: 1682: 1676: 1658: 1656: 1651: 1649: 1646: 1645: 1644: 1633: 1628: 1623: 1621: 1618: 1617: 1616: 1605: 1600: 1575: 1573: 1570: 1564: 1559: 1544: 1542: 1541: 1536: 1509: 1507: 1506: 1501: 1495: 1490: 1489: 1488: 1476: 1475: 1465: 1460: 1455: 1454: 1445: 1440: 1435: 1434: 1425: 1414: 1410: 1406: 1398: 1396: 1395: 1390: 1388: 1384: 1376: 1354: 1346: 1312: 1304: 1261: 1259: 1258: 1253: 1238: 1237: 1216: 1215: 1166: 1164: 1163: 1158: 1156: 1149: 1147: 1146: 1145: 1133: 1132: 1122: 1121: 1112: 1103: 1102: 1083: 1081: 1080: 1079: 1067: 1066: 1056: 1055: 1046: 1037: 1036: 1014: 1012: 1011: 1006: 1004: 1003: 987: 985: 984: 979: 967: 965: 964: 959: 957: 956: 940: 938: 937: 932: 917: 915: 914: 909: 897: 895: 894: 889: 873: 871: 870: 865: 849: 847: 846: 841: 829: 820: 811: 805: 803: 802: 797: 795: 790: 789: 788: 776: 775: 765: 760: 755: 754: 745: 740: 735: 734: 725: 632: 631: 595: 594: 568: 429:spherical lenses 427:Most lenses are 258:geometric figure 251: 207: 204: 186: 179: 132:explosive lenses 21: 9514: 9513: 9509: 9508: 9507: 9505: 9504: 9503: 9484: 9483: 9478:Wayback Machine 9465:Wayback Machine 9452:Wayback Machine 9441: 9419: 9415:Wayback Machine 9394:Wayback Machine 9372:Wayback Machine 9365:Project PHYSNET 9361:Wayback Machine 9347:Wayback Machine 9336: 9326: 9303: 9283: 9260: 9255: 9254: 9233: 9229: 9219: 9217: 9178: 9174: 9166: 9127: 9117: 9113: 9073:10.1.1.716.4426 9048: 9044: 9007:(7211): 376–9. 8995: 8991: 8950: 8943: 8933: 8931: 8927: 8920: 8914: 8910: 8903: 8889: 8885: 8875: 8873: 8850: 8846: 8836: 8834: 8830: 8819: 8813: 8809: 8802: 8788: 8784: 8775: 8771: 8761: 8759: 8746: 8745: 8741: 8731: 8729: 8714: 8710: 8700: 8698: 8683: 8679: 8672: 8658: 8651: 8644: 8630: 8621: 8614: 8600: 8589: 8581: 8576: 8572: 8565: 8551: 8547: 8537: 8535: 8522: 8521: 8517: 8507: 8505: 8500:. 2 July 2019. 8492: 8491: 8487: 8460: 8456: 8451: 8447: 8437: 8435: 8433: 8417: 8413: 8407:Wayback Machine 8398: 8394: 8384: 8382: 8380: 8364: 8360: 8350: 8348: 8346: 8330: 8326: 8316: 8314: 8307: 8291: 8287: 8281:Wayback Machine 8272:Al Van Helden. 8271: 8267: 8257: 8255: 8248: 8232: 8228: 8221: 8205: 8201: 8196:Wayback Machine 8182: 8178: 8172:Wayback Machine 8157:Pliny the Elder 8155: 8151: 8146:Wayback Machine 8126: 8119: 8088: 8084: 8069: 8065: 8055: 8053: 8040: 8039: 8035: 8025: 8023: 8008: 8004: 7965: 7961: 7956: 7951: 7950: 7937: 7935: 7922: 7915: 7895: 7885: 7883: 7881: 7854: 7850: 7845: 7840: 7750: 7719:Radio astronomy 7705:burning-glasses 7606:corrective lens 7563: 7446:Aspheric lenses 7421: 7409: 7402: 7396: 7392: 7386: 7382: 7376: 7351: 7347: 7341: 7337: 7335: 7324: 7321: 7320: 7305: 7298: 7288: 7281: 7277: 7273: 7247: 7243: 7237: 7233: 7232: 7227: 7216: 7212: 7207: 7196: 7192: 7187: 7174: 7172: 7169: 7168: 7164: 7145: 7142: 7141: 7120: 7116: 7111: 7105: 7094: 7077: 7075: 7072: 7071: 7065: 7058: 7033: 7029: 7024: 7013: 7009: 7004: 6991: 6989: 6986: 6985: 6979: 6957: 6954: 6953: 6925: 6921: 6912: 6908: 6898: 6888: 6884: 6869: 6865: 6864: 6862: 6838: 6834: 6827: 6822: 6806: 6801: 6790: 6786: 6781: 6779: 6776: 6775: 6744: 6740: 6731: 6727: 6717: 6707: 6703: 6688: 6684: 6683: 6681: 6657: 6653: 6646: 6641: 6625: 6620: 6609: 6605: 6600: 6598: 6595: 6594: 6575: 6571: 6553: 6549: 6547: 6544: 6543: 6519: 6515: 6497: 6493: 6491: 6488: 6487: 6471: 6468: 6467: 6440: 6436: 6431: 6417: 6413: 6408: 6397: 6393: 6388: 6386: 6383: 6382: 6360: 6356: 6354: 6351: 6350: 6330: 6326: 6324: 6321: 6320: 6300: 6296: 6294: 6291: 6290: 6270: 6266: 6264: 6261: 6260: 6242: 6240:Achromatic lens 6224: 6204: 6177:field curvature 6172: 6128: 6124: 6120: 6106: 6100: 6067: 6063: 6059: 6045: 6039: 6017: 6011: 5998: 5988: 5906: 5871: 5867: 5855: 5851: 5827: 5823: 5818: 5815: 5814: 5776: 5772: 5760: 5756: 5751: 5748: 5747: 5720: 5716: 5714: 5711: 5710: 5679: 5676: 5675: 5649: 5645: 5643: 5640: 5639: 5599: 5595: 5587: 5584: 5583: 5553: 5549: 5541: 5538: 5537: 5497: 5493: 5491: 5488: 5487: 5457: 5453: 5451: 5448: 5447: 5407: 5403: 5395: 5392: 5391: 5361: 5357: 5349: 5346: 5345: 5328:Image Location 5311: 5294: 5282: 5271: 5251: 5221: 5214: 5208: 5204: 5198: 5191: 5187: 5179: 5177: 5171: 5169: 5165: 5160: 5156: 5149: 5139: 5135: 5131: 5127: 5104: 5093: 5091: 5085: 5077: 5075: 5068: 5061: 5054: 5047: 5043: 5039: 5018: 5014: 5009: 4991: 4987: 4980: 4975: 4964: 4960: 4954: 4950: 4948: 4937: 4934: 4933: 4922: 4917: 4916: 4915: 4914: 4913: 4906: 4896: 4889: 4879: 4878:A convex lens ( 4876: 4868: 4867: 4864: 4843: 4836: 4825: 4818: 4811: 4804: 4797: 4790: 4775: 4768: 4737: 4731: 4727: 4721: 4717: 4711: 4700: 4693: 4687: 4680: 4674: 4670: 4667: 4666: 4665: 4664: 4663: 4658:(B) reimages a 4652: 4644: 4643: 4636: 4614:lens of the eye 4606: 4600: 4588: 4582: 4581:In some cases, 4568: 4562: 4559: 4553: 4550: 4544: 4537: 4531: 4523:, in which the 4503: 4497: 4494: 4488: 4484: 4477: 4471: 4463:pass through a 4461: 4455: 4452: 4446: 4426: 4423: 4422: 4402: 4399: 4398: 4381: 4377: 4375: 4372: 4371: 4354: 4350: 4348: 4345: 4344: 4316: 4315: 4307: 4303: 4293: 4289: 4285: 4283: 4269: 4265: 4263: 4260: 4259: 4244: 4240: 4236: 4234: 4223: 4219: 4213: 4209: 4207: 4203: 4201: 4198: 4197: 4192: 4188: 4184: 4177: 4173: 4169: 4162: 4158: 4154: 4147: 4143: 4139: 4115: 4111: 4109: 4106: 4105: 4088: 4084: 4082: 4079: 4078: 4061: 4057: 4055: 4052: 4051: 4034: 4030: 4028: 4025: 4024: 4007: 4003: 4001: 3998: 3997: 3980: 3976: 3974: 3971: 3970: 3953: 3949: 3947: 3944: 3943: 3917: 3913: 3904: 3900: 3898: 3895: 3894: 3871: 3867: 3858: 3854: 3852: 3849: 3848: 3825: 3821: 3815: 3811: 3802: 3798: 3796: 3793: 3792: 3749: 3745: 3740: 3729: 3725: 3720: 3707: 3705: 3702: 3701: 3692: 3686: 3683: 3677: 3666: 3657: 3649: 3636: 3629: 3607: 3603: 3601: 3598: 3597: 3573: 3569: 3564: 3553: 3549: 3544: 3513: 3511: 3508: 3507: 3489: 3486: 3485: 3462: 3458: 3456: 3453: 3452: 3426: 3413: 3400: 3398: 3395: 3394: 3365: 3361: 3356: 3345: 3341: 3336: 3316: 3312: 3306: 3302: 3300: 3284: 3270: 3266: 3261: 3250: 3246: 3241: 3229: 3225: 3216: 3212: 3195: 3191: 3189: 3187: 3184: 3183: 3156: 3153: 3152: 3136: 3133: 3132: 3105: 3101: 3096: 3085: 3081: 3076: 3064: 3060: 3051: 3047: 3030: 3026: 3024: 3010: 3006: 3004: 3002: 2999: 2998: 2974: 2971: 2970: 2945: 2928: 2924: 2914: 2910: 2909: 2907: 2893: 2889: 2884: 2873: 2869: 2864: 2852: 2848: 2839: 2835: 2818: 2814: 2812: 2798: 2794: 2792: 2790: 2787: 2786: 2764: 2760: 2753: 2749: 2740: 2736: 2735: 2733: 2719: 2715: 2713: 2693: 2689: 2683: 2679: 2677: 2675: 2672: 2671: 2646: 2632: 2630: 2627: 2626: 2604: 2600: 2593: 2589: 2580: 2576: 2575: 2573: 2560: 2554: 2550: 2548: 2534: 2530: 2528: 2526: 2523: 2522: 2497: 2493: 2491: 2488: 2487: 2472: 2436: 2432: 2428: 2423: 2408: 2404: 2400: 2395: 2394: 2390: 2375: 2371: 2355: 2350: 2348: 2345: 2344: 2338: 2336: 2329: 2323: 2320: 2314: 2310: 2307: 2291:Inverted image 2283: 2276: 2267: 2258:Inverted image 2248: 2229: 2220:Diverging lens 2194: 2185:Virtual object 2175: 2149: 2128: 2122: 2118: 2112: 2104: 2098: 2094: 2088: 2084: 2077: 2071:sign convention 2067: 2061: 2058: 2052: 2049: 2043: 2018: 2010: 1986: 1982: 1981: 1958: 1956: 1944: 1940: 1938: 1935: 1934: 1912: 1908: 1907: 1884: 1882: 1870: 1866: 1864: 1861: 1860: 1841: 1837: 1835: 1832: 1831: 1814: 1810: 1808: 1805: 1804: 1796: 1792: 1781: 1776: 1770: 1757: 1751: 1741: 1707: 1703: 1697: 1693: 1686: 1657: 1655: 1640: 1636: 1632: 1627: 1612: 1608: 1604: 1599: 1598: 1594: 1563: 1558: 1556: 1553: 1552: 1530: 1527: 1526: 1515: 1484: 1480: 1471: 1467: 1466: 1464: 1450: 1446: 1444: 1430: 1426: 1424: 1422: 1419: 1418: 1412: 1408: 1404: 1386: 1385: 1375: 1368: 1356: 1355: 1345: 1338: 1314: 1313: 1303: 1296: 1271: 1269: 1266: 1265: 1233: 1229: 1211: 1207: 1205: 1202: 1201: 1179: 1154: 1153: 1141: 1137: 1128: 1124: 1123: 1117: 1113: 1111: 1104: 1098: 1094: 1091: 1090: 1075: 1071: 1062: 1058: 1057: 1051: 1047: 1045: 1038: 1032: 1028: 1024: 1022: 1019: 1018: 999: 995: 993: 990: 989: 973: 970: 969: 952: 948: 946: 943: 942: 926: 923: 922: 903: 900: 899: 883: 880: 879: 859: 856: 855: 835: 832: 831: 828: 822: 819: 813: 809: 784: 780: 771: 767: 766: 764: 750: 746: 744: 730: 726: 724: 722: 719: 718: 698: 566: 482:Types of lenses 476: 425: 401:achromatic lens 392: 310:nearsightedness 308:to correct for 300:games using an 290:corrective lens 286:Pliny the Elder 208: 202: 199: 192:needs expansion 177: 167: 128:acoustic lenses 124:electron lenses 35: 28: 23: 22: 15: 12: 11: 5: 9512: 9502: 9501: 9496: 9482: 9481: 9468: 9455: 9440: 9437: 9436: 9435: 9417: 9405: 9396: 9383: 9375: 9349: 9335: 9334:External links 9332: 9331: 9330: 9324: 9307: 9301: 9288: 9281: 9259: 9256: 9253: 9252: 9227: 9172: 9121:Boltasseva, A. 9111: 9042: 8989: 8962:(11): 117403. 8941: 8908: 8901: 8883: 8844: 8807: 8800: 8782: 8769: 8739: 8722:PhysicsLab.org 8708: 8685:Nave, Carl R. 8677: 8670: 8649: 8642: 8619: 8612: 8587: 8570: 8563: 8545: 8515: 8485: 8454: 8445: 8431: 8411: 8392: 8378: 8358: 8344: 8324: 8305: 8285: 8265: 8246: 8226: 8219: 8199: 8176: 8149: 8117: 8098:(4): 899–907. 8082: 8063: 8033: 8002: 7981:10.2307/505216 7975:(2): 191–196. 7958: 7957: 7955: 7952: 7949: 7948: 7946: 7945: 7920: 7913: 7893: 7879: 7847: 7846: 7844: 7841: 7839: 7838: 7833: 7828: 7826:Prism (optics) 7823: 7818: 7813: 7808: 7803: 7798: 7796:Lens (anatomy) 7793: 7788: 7783: 7778: 7773: 7768: 7763: 7758: 7751: 7749: 7746: 7681:curved mirrors 7677:optical sensor 7622:axial symmetry 7562: 7559: 7534:are made from 7517: 7478:are arrays of 7420: 7417: 7407: 7400: 7390: 7380: 7362: 7354: 7350: 7344: 7340: 7334: 7331: 7328: 7303: 7296: 7259: 7250: 7246: 7240: 7236: 7231: 7226: 7219: 7215: 7211: 7206: 7199: 7195: 7191: 7186: 7181: 7178: 7153:{\textstyle N} 7149: 7123: 7119: 7115: 7108: 7103: 7100: 7097: 7093: 7089: 7084: 7081: 7044: 7036: 7032: 7028: 7023: 7016: 7012: 7008: 7003: 6998: 6995: 6967: 6964: 6961: 6939: 6933: 6928: 6924: 6920: 6915: 6911: 6907: 6904: 6901: 6896: 6891: 6887: 6883: 6880: 6877: 6872: 6868: 6861: 6858: 6855: 6852: 6849: 6841: 6837: 6833: 6830: 6826: 6821: 6815: 6812: 6809: 6805: 6800: 6793: 6789: 6785: 6758: 6752: 6747: 6743: 6739: 6734: 6730: 6726: 6723: 6720: 6715: 6710: 6706: 6702: 6699: 6696: 6691: 6687: 6680: 6677: 6674: 6671: 6668: 6660: 6656: 6652: 6649: 6645: 6640: 6634: 6631: 6628: 6624: 6619: 6612: 6608: 6604: 6578: 6574: 6570: 6567: 6564: 6559: 6556: 6552: 6525: 6522: 6518: 6514: 6511: 6508: 6503: 6500: 6496: 6475: 6453: 6446: 6443: 6439: 6435: 6430: 6423: 6420: 6416: 6412: 6407: 6400: 6396: 6392: 6366: 6363: 6359: 6336: 6333: 6329: 6306: 6303: 6299: 6276: 6273: 6269: 6232:Doublet (lens) 6223: 6220: 6203: 6200: 6171: 6168: 6102:Main article: 6099: 6096: 6081:comatic circle 6041:Main article: 6038: 6035: 6013:Main article: 6010: 6007: 5994:Main article: 5990: 5989: 5987: 5986: 5979: 5972: 5964: 5961: 5960: 5959: 5958: 5953: 5948: 5943: 5938: 5933: 5928: 5923: 5915: 5914: 5905: 5902: 5899: 5898: 5896: 5893: 5890: 5879: 5874: 5870: 5866: 5863: 5858: 5854: 5850: 5847: 5844: 5841: 5838: 5835: 5830: 5826: 5822: 5812: 5809: 5806: 5802: 5801: 5798: 5795: 5792: 5779: 5775: 5771: 5768: 5763: 5759: 5755: 5745: 5742: 5731: 5728: 5723: 5719: 5708: 5704: 5703: 5701: 5699: 5697: 5686: 5683: 5673: 5671: 5660: 5657: 5652: 5648: 5637: 5633: 5632: 5630: 5627: 5624: 5613: 5610: 5607: 5602: 5598: 5594: 5591: 5581: 5578: 5567: 5564: 5561: 5556: 5552: 5548: 5545: 5535: 5531: 5530: 5528: 5525: 5522: 5511: 5508: 5505: 5500: 5496: 5485: 5482: 5471: 5468: 5465: 5460: 5456: 5445: 5441: 5440: 5438: 5435: 5432: 5421: 5418: 5415: 5410: 5406: 5402: 5399: 5389: 5386: 5375: 5372: 5369: 5364: 5360: 5356: 5353: 5343: 5339: 5338: 5335: 5332: 5329: 5326: 5323: 5320: 5310: 5307: 5212: 5202: 5147: 5089: 5073: 5021: 5017: 5013: 5008: 5005: 4994: 4990: 4986: 4983: 4979: 4974: 4967: 4963: 4957: 4953: 4947: 4944: 4941: 4921: 4918: 4904: 4887: 4877: 4870: 4869: 4865: 4858: 4857: 4856: 4855: 4854: 4841: 4834: 4823: 4816: 4809: 4802: 4795: 4788: 4773: 4766: 4746: 4741:virtual object 4735: 4725: 4715: 4709: 4691: 4678: 4660:virtual object 4653: 4646: 4645: 4637: 4630: 4629: 4628: 4627: 4626: 4604: 4594: 4586: 4566: 4557: 4548: 4535: 4501: 4492: 4475: 4459: 4450: 4434:{\textstyle f} 4430: 4410:{\textstyle f} 4406: 4384: 4380: 4357: 4353: 4310: 4306: 4301: 4296: 4292: 4288: 4282: 4277: 4272: 4268: 4262: 4261: 4256: 4252: 4247: 4243: 4239: 4233: 4226: 4222: 4216: 4212: 4206: 4205: 4190: 4186: 4182: 4175: 4171: 4167: 4160: 4156: 4152: 4145: 4141: 4137: 4118: 4114: 4091: 4087: 4064: 4060: 4037: 4033: 4010: 4006: 3983: 3979: 3956: 3952: 3932: 3928: 3925: 3920: 3916: 3912: 3907: 3903: 3882: 3879: 3874: 3870: 3866: 3861: 3857: 3834: 3828: 3824: 3818: 3814: 3810: 3805: 3801: 3760: 3752: 3748: 3744: 3739: 3732: 3728: 3724: 3719: 3714: 3711: 3690: 3681: 3665: 3662: 3655: 3628: 3625: 3610: 3606: 3583: 3576: 3572: 3568: 3563: 3556: 3552: 3548: 3543: 3540: 3537: 3534: 3531: 3528: 3525: 3520: 3517: 3497:{\textstyle f} 3493: 3473: 3470: 3465: 3461: 3438: 3433: 3430: 3425: 3420: 3417: 3412: 3407: 3404: 3378: 3375: 3368: 3364: 3360: 3355: 3348: 3344: 3340: 3335: 3332: 3329: 3326: 3319: 3315: 3309: 3305: 3299: 3296: 3291: 3288: 3283: 3280: 3273: 3269: 3265: 3260: 3253: 3249: 3245: 3240: 3237: 3232: 3228: 3224: 3219: 3215: 3211: 3208: 3203: 3198: 3194: 3169: 3166: 3163: 3160: 3140: 3118: 3115: 3108: 3104: 3100: 3095: 3088: 3084: 3080: 3075: 3072: 3067: 3063: 3059: 3054: 3050: 3046: 3043: 3038: 3033: 3029: 3023: 3018: 3013: 3009: 2984: 2981: 2978: 2958: 2951: 2948: 2944: 2941: 2938: 2934: 2931: 2927: 2922: 2917: 2913: 2906: 2903: 2896: 2892: 2888: 2883: 2876: 2872: 2868: 2863: 2860: 2855: 2851: 2847: 2842: 2838: 2834: 2831: 2826: 2821: 2817: 2811: 2806: 2801: 2797: 2774: 2767: 2763: 2756: 2752: 2748: 2743: 2739: 2732: 2727: 2722: 2718: 2712: 2706: 2703: 2699: 2696: 2692: 2686: 2682: 2659: 2656: 2652: 2649: 2645: 2642: 2638: 2635: 2614: 2607: 2603: 2596: 2592: 2588: 2583: 2579: 2572: 2566: 2563: 2557: 2553: 2547: 2542: 2537: 2533: 2500: 2496: 2471: 2468: 2456: 2451: 2439: 2435: 2427: 2422: 2411: 2407: 2399: 2393: 2388: 2384: 2381: 2378: 2374: 2370: 2361: 2354: 2332: 2327: 2318: 2306: 2303: 2293: 2292: 2289: 2286: 2281: 2274: 2269: 2265: 2260: 2259: 2256: 2253: 2250: 2246: 2241: 2240: 2237: 2234: 2231: 2227: 2222: 2221: 2218: 2215: 2212: 2206: 2205: 2204:Virtual image 2202: 2199: 2196: 2192: 2187: 2186: 2183: 2180: 2177: 2173: 2168: 2167: 2164: 2161: 2158: 2148: 2145: 2126: 2116: 2102: 2092: 2063:Main article: 2060: 2056: 2047: 2040: 1994: 1989: 1985: 1979: 1976: 1973: 1970: 1967: 1964: 1961: 1955: 1952: 1947: 1943: 1920: 1915: 1911: 1905: 1902: 1899: 1896: 1893: 1890: 1887: 1881: 1878: 1873: 1869: 1844: 1840: 1817: 1813: 1790: 1789: 1779: 1774: 1768: 1755: 1749: 1726: 1719: 1710: 1706: 1700: 1696: 1689: 1681: 1675: 1672: 1669: 1666: 1663: 1654: 1643: 1639: 1631: 1626: 1615: 1611: 1603: 1597: 1593: 1590: 1587: 1584: 1581: 1578: 1569: 1562: 1534: 1514: 1511: 1499: 1493: 1487: 1483: 1479: 1474: 1470: 1463: 1458: 1453: 1449: 1443: 1438: 1433: 1429: 1382: 1379: 1374: 1371: 1369: 1367: 1364: 1361: 1358: 1357: 1352: 1349: 1344: 1341: 1339: 1337: 1334: 1331: 1328: 1325: 1322: 1319: 1316: 1315: 1310: 1307: 1302: 1299: 1297: 1295: 1292: 1289: 1286: 1283: 1280: 1277: 1274: 1273: 1251: 1247: 1244: 1241: 1236: 1232: 1228: 1225: 1222: 1219: 1214: 1210: 1178: 1175: 1152: 1144: 1140: 1136: 1131: 1127: 1120: 1116: 1110: 1107: 1105: 1101: 1097: 1093: 1092: 1089: 1086: 1078: 1074: 1070: 1065: 1061: 1054: 1050: 1044: 1041: 1039: 1035: 1031: 1027: 1026: 1002: 998: 981:{\textstyle u} 977: 955: 951: 934:{\textstyle v} 930: 911:{\textstyle v} 907: 891:{\textstyle u} 887: 867:{\textstyle v} 863: 843:{\textstyle u} 839: 826: 817: 793: 787: 783: 779: 774: 770: 763: 758: 753: 749: 743: 738: 733: 729: 697: 694: 653: 652: 643: 640:Biconcave lens 616: 615: 606: 528:convex-concave 475: 472: 424: 421: 352:reading stones 339:Book of Optics 222:, 13th century 210: 209: 189: 187: 166: 163: 26: 9: 6: 4: 3: 2: 9511: 9500: 9497: 9495: 9492: 9491: 9489: 9479: 9475: 9472: 9469: 9466: 9462: 9459: 9456: 9453: 9449: 9446: 9443: 9442: 9431: 9430: 9424: 9418: 9416: 9412: 9409: 9406: 9404: 9400: 9397: 9395: 9391: 9388: 9384: 9382: 9381: 9376: 9373: 9369: 9366: 9362: 9358: 9355: 9354: 9350: 9348: 9344: 9341: 9338: 9337: 9327: 9321: 9317: 9313: 9308: 9304: 9298: 9294: 9289: 9284: 9278: 9273: 9272: 9266: 9265:Hecht, Eugene 9262: 9261: 9248: 9244: 9240: 9239: 9231: 9215: 9211: 9207: 9203: 9199: 9195: 9191: 9190: 9184: 9176: 9165: 9161: 9157: 9153: 9149: 9145: 9141: 9137: 9133: 9126: 9122: 9115: 9107: 9103: 9099: 9095: 9091: 9087: 9083: 9079: 9074: 9069: 9065: 9061: 9058:(5891): 930. 9057: 9053: 9046: 9038: 9034: 9030: 9026: 9022: 9018: 9014: 9010: 9006: 9002: 9001: 8993: 8985: 8981: 8977: 8973: 8969: 8965: 8961: 8957: 8956: 8948: 8946: 8926: 8919: 8912: 8904: 8898: 8894: 8887: 8871: 8867: 8863: 8859: 8855: 8848: 8829: 8825: 8818: 8811: 8803: 8797: 8793: 8786: 8779: 8773: 8757: 8753: 8749: 8743: 8727: 8723: 8719: 8712: 8696: 8692: 8688: 8681: 8673: 8667: 8663: 8656: 8654: 8645: 8639: 8635: 8628: 8626: 8624: 8615: 8609: 8605: 8598: 8596: 8594: 8592: 8584: 8579: 8574: 8566: 8560: 8556: 8549: 8533: 8529: 8525: 8519: 8503: 8499: 8495: 8489: 8481: 8477: 8473: 8469: 8465: 8458: 8449: 8434: 8428: 8424: 8423: 8415: 8408: 8404: 8401: 8396: 8381: 8375: 8371: 8370: 8362: 8347: 8341: 8337: 8336: 8328: 8312: 8308: 8302: 8298: 8297: 8289: 8282: 8278: 8275: 8269: 8253: 8249: 8243: 8239: 8238: 8230: 8222: 8216: 8212: 8211: 8203: 8197: 8193: 8190: 8186: 8180: 8173: 8169: 8166: 8162: 8158: 8153: 8147: 8143: 8140: 8136: 8135: 8130: 8124: 8122: 8113: 8109: 8105: 8101: 8097: 8093: 8086: 8078: 8074: 8067: 8051: 8047: 8043: 8037: 8021: 8017: 8013: 8006: 7998: 7994: 7990: 7986: 7982: 7978: 7974: 7970: 7963: 7959: 7933: 7929: 7925: 7921: 7916: 7910: 7906: 7901: 7900: 7894: 7882: 7876: 7872: 7868: 7867: 7861: 7860: 7858: 7852: 7848: 7837: 7834: 7832: 7829: 7827: 7824: 7822: 7819: 7817: 7814: 7812: 7809: 7807: 7804: 7802: 7799: 7797: 7794: 7792: 7789: 7787: 7784: 7782: 7779: 7777: 7774: 7772: 7769: 7767: 7764: 7762: 7759: 7756: 7753: 7752: 7745: 7743: 7738: 7736: 7732: 7728: 7724: 7720: 7716: 7714: 7710: 7706: 7701: 7696: 7695:correctors). 7694: 7690: 7686: 7682: 7678: 7674: 7670: 7666: 7665:virtual image 7662: 7658: 7654: 7650: 7646: 7642: 7637: 7635: 7631: 7627: 7623: 7619: 7615: 7611: 7607: 7603: 7599: 7595: 7594:hypermetropia 7591: 7587: 7583: 7578: 7576: 7567: 7558: 7555: 7554:wavelengths. 7553: 7549: 7545: 7541: 7537: 7533: 7529: 7527: 7523: 7521: 7515: 7513: 7509: 7505: 7500: 7498: 7493: 7491: 7487: 7485: 7481: 7477: 7473: 7471: 7463: 7458: 7454: 7451: 7447: 7443: 7441: 7437: 7433: 7425: 7416: 7414: 7406: 7399: 7389: 7379: 7373: 7360: 7352: 7348: 7342: 7338: 7332: 7329: 7326: 7318: 7316: 7312: 7311: 7310:afocal system 7302: 7295: 7291: 7285: 7270: 7257: 7248: 7244: 7238: 7234: 7229: 7224: 7217: 7213: 7209: 7204: 7197: 7193: 7189: 7184: 7179: 7176: 7161: 7147: 7138: 7121: 7117: 7113: 7106: 7101: 7098: 7095: 7091: 7087: 7082: 7079: 7069: 7062: 7055: 7042: 7034: 7030: 7026: 7021: 7014: 7010: 7006: 7001: 6996: 6993: 6983: 6965: 6962: 6959: 6950: 6937: 6926: 6922: 6918: 6913: 6909: 6902: 6899: 6889: 6885: 6881: 6878: 6870: 6866: 6859: 6856: 6853: 6850: 6839: 6835: 6831: 6828: 6824: 6819: 6813: 6810: 6807: 6803: 6798: 6791: 6787: 6783: 6773: 6769: 6756: 6745: 6741: 6737: 6732: 6728: 6721: 6718: 6708: 6704: 6700: 6697: 6689: 6685: 6678: 6675: 6672: 6669: 6658: 6654: 6650: 6647: 6643: 6638: 6632: 6629: 6626: 6622: 6617: 6610: 6606: 6602: 6592: 6576: 6572: 6568: 6565: 6562: 6557: 6554: 6550: 6539: 6523: 6520: 6516: 6512: 6509: 6506: 6501: 6498: 6494: 6473: 6464: 6451: 6444: 6441: 6437: 6433: 6428: 6421: 6418: 6414: 6410: 6405: 6398: 6394: 6390: 6380: 6364: 6361: 6357: 6334: 6331: 6327: 6304: 6301: 6297: 6274: 6271: 6267: 6257: 6253: 6251: 6250:compound lens 6247: 6241: 6237: 6233: 6229: 6219: 6217: 6213: 6209: 6199: 6197: 6196: 6191: 6190: 6185: 6184: 6179: 6178: 6167: 6165: 6160: 6155: 6153: 6149: 6144: 6142: 6138: 6134: 6118: 6114: 6110: 6105: 6095: 6093: 6088: 6086: 6082: 6076: 6073: 6070: 6057: 6053: 6049: 6044: 6043:Coma (optics) 6034: 6032: 6027: 6025: 6021: 6016: 6006: 6003: 5997: 5985: 5980: 5978: 5973: 5971: 5966: 5965: 5963: 5962: 5957: 5954: 5952: 5949: 5947: 5944: 5942: 5939: 5937: 5934: 5932: 5929: 5927: 5924: 5922: 5919: 5918: 5917: 5916: 5913: 5910: 5909: 5897: 5894: 5891: 5872: 5868: 5861: 5856: 5852: 5848: 5842: 5836: 5828: 5824: 5813: 5810: 5807: 5804: 5803: 5799: 5796: 5793: 5777: 5773: 5769: 5761: 5757: 5746: 5743: 5729: 5726: 5721: 5717: 5709: 5706: 5705: 5702: 5700: 5698: 5681: 5674: 5672: 5658: 5655: 5650: 5646: 5638: 5635: 5634: 5631: 5628: 5625: 5611: 5608: 5605: 5600: 5596: 5592: 5582: 5579: 5565: 5562: 5559: 5554: 5550: 5546: 5543: 5536: 5533: 5532: 5529: 5526: 5523: 5509: 5506: 5503: 5498: 5494: 5486: 5483: 5469: 5466: 5463: 5458: 5454: 5446: 5443: 5442: 5439: 5436: 5433: 5419: 5416: 5413: 5408: 5404: 5400: 5397: 5390: 5387: 5373: 5370: 5367: 5362: 5358: 5354: 5344: 5341: 5340: 5336: 5333: 5330: 5327: 5324: 5321: 5318: 5317: 5303: 5297: 5292: 5288: 5280: 5275: 5269: 5265: 5261: 5257: 5248: 5244: 5242: 5238: 5234: 5228: 5224: 5218: 5211: 5201: 5195: 5163: 5154: 5150: 5142: 5124: 5122: 5118: 5114: 5110: 5101: 5098:| > 0 5096: 5088: 5080: 5072: 5064: 5057: 5051: 5036: 5019: 5015: 5011: 5006: 5003: 4992: 4988: 4984: 4981: 4977: 4972: 4965: 4961: 4955: 4951: 4945: 4942: 4939: 4931: 4929: 4928: 4927:magnification 4920:Magnification 4911: 4903: 4899: 4894: 4886: 4882: 4874: 4862: 4853: 4851: 4847: 4840: 4833: 4829: 4822: 4815: 4808: 4801: 4798:, expressing 4794: 4787: 4783: 4779: 4772: 4765: 4761: 4757: 4752: 4750: 4744: 4742: 4734: 4724: 4714: 4707: 4703: 4697: 4690: 4684: 4677: 4661: 4657: 4650: 4641: 4634: 4625: 4623: 4619: 4615: 4611: 4603: 4598: 4597:virtual image 4592: 4585: 4575: 4571: 4565: 4556: 4547: 4542: 4534: 4528: 4526: 4522: 4518: 4514: 4513: 4508: 4500: 4491: 4481: 4474: 4468: 4466: 4458: 4449: 4444: 4428: 4421:of the lens ( 4420: 4404: 4382: 4378: 4355: 4351: 4338: 4333: 4329: 4308: 4304: 4299: 4294: 4290: 4286: 4280: 4275: 4270: 4266: 4254: 4250: 4245: 4241: 4237: 4231: 4224: 4220: 4214: 4210: 4195: 4193: 4178: 4163: 4148: 4132: 4116: 4112: 4089: 4085: 4062: 4058: 4035: 4031: 4008: 4004: 3981: 3977: 3954: 3950: 3930: 3926: 3923: 3918: 3914: 3910: 3905: 3901: 3880: 3877: 3872: 3868: 3864: 3859: 3855: 3845: 3832: 3826: 3822: 3816: 3812: 3808: 3803: 3799: 3790: 3787: 3785: 3775: 3771: 3758: 3750: 3746: 3742: 3737: 3730: 3726: 3722: 3717: 3712: 3709: 3699: 3697: 3689: 3680: 3675: 3671: 3670:paraxial rays 3664:Lens equation 3661: 3659: 3658: 3646: 3642: 3634: 3624: 3608: 3604: 3594: 3574: 3570: 3566: 3561: 3554: 3550: 3546: 3535: 3532: 3529: 3523: 3518: 3515: 3505: 3491: 3471: 3468: 3463: 3459: 3449: 3436: 3431: 3428: 3423: 3418: 3415: 3410: 3405: 3402: 3392: 3389: 3376: 3366: 3362: 3358: 3353: 3346: 3342: 3338: 3327: 3324: 3317: 3313: 3307: 3303: 3294: 3289: 3286: 3271: 3267: 3263: 3258: 3251: 3247: 3243: 3230: 3226: 3222: 3217: 3213: 3206: 3201: 3196: 3192: 3181: 3164: 3158: 3138: 3129: 3116: 3106: 3102: 3098: 3093: 3086: 3082: 3078: 3065: 3061: 3057: 3052: 3048: 3041: 3036: 3031: 3027: 3021: 3016: 3011: 3007: 2996: 2982: 2976: 2956: 2949: 2946: 2939: 2936: 2932: 2929: 2920: 2915: 2911: 2904: 2894: 2890: 2886: 2881: 2874: 2870: 2866: 2853: 2849: 2845: 2840: 2836: 2829: 2824: 2819: 2815: 2809: 2804: 2799: 2795: 2772: 2765: 2761: 2754: 2750: 2746: 2741: 2737: 2730: 2725: 2720: 2716: 2710: 2704: 2701: 2697: 2694: 2690: 2684: 2680: 2657: 2654: 2650: 2647: 2643: 2640: 2636: 2633: 2612: 2605: 2601: 2594: 2590: 2586: 2581: 2577: 2570: 2564: 2561: 2555: 2551: 2545: 2540: 2535: 2531: 2520: 2516: 2498: 2494: 2485: 2476: 2467: 2454: 2449: 2437: 2433: 2425: 2420: 2409: 2405: 2397: 2391: 2386: 2382: 2379: 2376: 2372: 2368: 2359: 2352: 2342: 2335: 2326: 2317: 2302: 2300: 2290: 2287: 2284: 2277: 2270: 2268: 2262: 2261: 2257: 2254: 2251: 2249: 2243: 2242: 2238: 2236:Erect object 2235: 2232: 2230: 2224: 2223: 2219: 2216: 2213: 2211: 2208: 2207: 2203: 2200: 2197: 2195: 2189: 2188: 2184: 2181: 2178: 2176: 2170: 2169: 2165: 2162: 2159: 2156: 2155: 2144: 2142: 2138: 2134: 2125: 2115: 2110: 2101: 2091: 2083: 2076: 2072: 2066: 2055: 2046: 2039: 2037: 2032: 2030: 2026: 2025:optical power 2021: 2016: 2008: 1992: 1987: 1983: 1977: 1971: 1968: 1965: 1959: 1953: 1950: 1945: 1941: 1918: 1913: 1909: 1903: 1897: 1894: 1891: 1885: 1879: 1876: 1871: 1867: 1858: 1842: 1838: 1815: 1811: 1802: 1793: 1787: 1780: 1773: 1769: 1766: 1762: 1754: 1750: 1747: 1740: 1739: 1738: 1724: 1717: 1708: 1704: 1698: 1694: 1687: 1679: 1670: 1667: 1664: 1652: 1641: 1637: 1629: 1624: 1613: 1609: 1601: 1595: 1588: 1585: 1582: 1576: 1567: 1560: 1550: 1548: 1532: 1519: 1510: 1497: 1491: 1485: 1481: 1477: 1472: 1468: 1461: 1456: 1451: 1447: 1441: 1436: 1431: 1427: 1416: 1402: 1380: 1377: 1372: 1370: 1365: 1362: 1359: 1350: 1347: 1342: 1340: 1332: 1329: 1326: 1320: 1317: 1308: 1305: 1300: 1298: 1290: 1287: 1284: 1278: 1275: 1262: 1249: 1245: 1242: 1239: 1234: 1230: 1226: 1223: 1220: 1217: 1212: 1208: 1199: 1190: 1183: 1174: 1172: 1167: 1150: 1142: 1138: 1134: 1129: 1125: 1118: 1114: 1108: 1106: 1099: 1095: 1087: 1084: 1076: 1072: 1068: 1063: 1059: 1052: 1048: 1042: 1040: 1033: 1029: 1016: 1000: 996: 975: 953: 949: 928: 919: 905: 885: 877: 861: 853: 837: 825: 816: 806: 791: 785: 781: 777: 772: 768: 761: 756: 751: 747: 741: 736: 731: 727: 716: 714: 707: 702: 693: 691: 687: 686:optical power 683: 678: 676: 672: 663: 659: 648: 644: 638: 634: 633: 630: 627: 623: 611: 607: 603:Biconvex lens 601: 597: 596: 593: 591: 587: 583: 582:optical glass 579: 574: 572: 564: 560: 556: 552: 548: 544: 539: 537: 533: 529: 525: 524:plano-concave 521: 517: 513: 509: 505: 501: 497: 493: 492:double convex 489: 480: 471: 469: 465: 461: 457: 453: 450: 446: 442: 441: 436: 435: 430: 420: 416: 414: 410: 406: 402: 398: 391: 386: 384: 380: 376: 372: 368: 363: 359: 357: 353: 349: 345: 341: 340: 335: 331: 327: 326: 321: 317: 315: 311: 307: 303: 299: 295: 291: 287: 283: 282: 277: 272: 270: 266: 261: 259: 255: 250: 249: 243: 242: 233: 228: 221: 216: 206: 197: 193: 190:This section 188: 185: 181: 180: 176: 172: 162: 160: 159:hypermetropia 156: 152: 148: 144: 140: 135: 133: 129: 125: 121: 117: 113: 109: 105: 101: 97: 93: 89: 85: 81: 80:compound lens 77: 73: 69: 65: 61: 57: 50: 49:biconvex lens 46: 41: 37: 33: 19: 9427: 9423:"Lens" 9379: 9352: 9315: 9311: 9292: 9270: 9258:Bibliography 9236: 9230: 9218:. Retrieved 9193: 9187: 9175: 9164:the original 9135: 9131: 9114: 9055: 9051: 9045: 9004: 8998: 8992: 8959: 8953: 8932:. Retrieved 8925:the original 8918:"The Axicon" 8911: 8892: 8886: 8874:. Retrieved 8866:10.61835/6as 8857: 8847: 8835:. Retrieved 8823: 8810: 8791: 8785: 8772: 8760:. Retrieved 8751: 8742: 8730:. Retrieved 8721: 8711: 8699:. Retrieved 8691:Hyperphysics 8690: 8680: 8661: 8633: 8603: 8585:, § 6.1 8580:, p. 14 8573: 8554: 8548: 8536:. Retrieved 8527: 8518: 8506:. Retrieved 8497: 8488: 8474:(2): 72–76. 8471: 8467: 8457: 8448: 8436:. Retrieved 8421: 8414: 8395: 8383:. Retrieved 8368: 8361: 8349:. Retrieved 8334: 8327: 8315:. Retrieved 8295: 8288: 8268: 8256:. Retrieved 8236: 8229: 8209: 8202: 8184: 8179: 8160: 8152: 8133: 8129:Aristophanes 8095: 8092:Neurosurgery 8091: 8085: 8076: 8066: 8054:. Retrieved 8045: 8036: 8024:. Retrieved 8015: 8005: 7972: 7968: 7962: 7936:. Retrieved 7927: 7898: 7884:. Retrieved 7865: 7856: 7851: 7755:Anti-fogging 7739: 7731:lens antenna 7717: 7697: 7638: 7610:contact lens 7579: 7572: 7556: 7544:metamaterial 7530: 7524: 7514:into a line 7512:point source 7501: 7494: 7490:Bifocal lens 7488: 7474: 7470:Fresnel lens 7467: 7462:Fresnel lens 7444: 7430: 7404: 7403:> 0 > 7397: 7387: 7377: 7374: 7319: 7308: 7300: 7293: 7289: 7286: 7271: 7162: 7139: 7070: 7060: 7056: 6984: 6951: 6774: 6770: 6593: 6540: 6465: 6381: 6258: 6254: 6249: 6243: 6236:Triplet lens 6205: 6193: 6187: 6181: 6175: 6173: 6156: 6145: 6136: 6108: 6107: 6089: 6084: 6080: 6074: 6068: 6051: 6047: 6046: 6028: 6019: 6018: 6001: 5999: 5301: 5295: 5290: 5278: 5273: 5267: 5263: 5259: 5255: 5226: 5222: 5216: 5209: 5199: 5196: 5161: 5146: 5140: 5125: 5102: 5094: 5086: 5078: 5070: 5062: 5055: 5052: 5037: 4932: 4925: 4923: 4901: 4897: 4884: 4880: 4849: 4845: 4838: 4831: 4827: 4820: 4819:in terms of 4813: 4806: 4805:in terms of 4799: 4792: 4785: 4781: 4777: 4770: 4763: 4759: 4755: 4753: 4740: 4732: 4722: 4712: 4701: 4695: 4688: 4682: 4675: 4668: 4659: 4639: 4617: 4616:to create a 4601: 4583: 4580: 4563: 4554: 4545: 4532: 4529: 4510: 4507:image sensor 4498: 4489: 4479: 4472: 4469: 4456: 4447: 4342: 4336: 4196: 4180: 4165: 4150: 4135: 4133: 3846: 3791: 3788: 3780: 3700: 3695: 3687: 3678: 3667: 3653: 3641:point source 3630: 3623:is dropped. 3595: 3506: 3450: 3393: 3390: 3182: 3130: 2997: 2518: 2514: 2481: 2343: 2324: 2315: 2308: 2296: 2288:Erect image 2279: 2272: 2263: 2255:Erect image 2244: 2225: 2209: 2190: 2182:Real object 2171: 2132: 2123: 2113: 2108: 2099: 2089: 2081: 2074: 2068: 2053: 2044: 2033: 2019: 2009: 1859: 1794: 1791: 1771: 1752: 1551: 1546: 1524: 1417: 1399:, and using 1263: 1195: 1168: 1017: 920: 875: 851: 823: 814: 807: 717: 715:is given by 710: 679: 674: 670: 668: 656: 625: 621: 619: 575: 563:focal length 557:lens. For a 554: 550: 546: 540: 531: 527: 523: 520:plano-convex 519: 515: 511: 510:surfaces is 503: 495: 491: 487: 485: 454: 448: 444: 439: 433: 428: 426: 417: 413:John Dollond 393: 360: 356:Visby lenses 343: 337: 323: 318: 304:(presumably 298:gladiatorial 279: 276:Aristophanes 273: 262: 239: 237: 203:January 2012 200: 196:adding to it 191: 136: 83: 66:by means of 55: 53: 36: 9439:Simulations 9385:Article on 8934:22 November 8056:25 November 7831:Ray tracing 7733:to refract 7653:microscopes 7630:astigmatism 7602:astigmatism 7582:prosthetics 7532:Superlenses 7480:microlenses 7436:laser diode 6208:diffraction 6195:astigmatism 6152:Abbe number 5936:Astigmatism 5904:Aberrations 5325:Image Type 5233:diffraction 5192:0.5 mm 5166:−50 mm 5113:plate scale 5092:> | 5076:< | 4924:The linear 4852:is zero.) 4749:Barlow lens 4708:real object 4656:Barlow lens 3656:focal plane 3633:focal point 2201:Real image 2036:aberrations 1198:Snell's law 468:astigmatism 460:focal power 383:Netherlands 367:Netherlands 265:Nimrud lens 244:comes from 220:Roger Bacon 175:Camera lens 114:, unlike a 72:simple lens 9488:Categories 9363:(.pdf) on 8583:Hecht 1987 8134:The Clouds 7954:References 7683:to make a 7669:real image 7661:projectors 7649:telescopes 7645:binoculars 7641:monoculars 7614:eyeglasses 7598:presbyopia 7520:optic axis 7450:aberration 6226:See also: 6141:apochromat 6113:dispersion 6002:aberration 5946:Distortion 5895:Magnified 5797:Magnified 5629:Magnified 5527:Same size 5319:Lens Type 5157:50 mm 5145:. But the 5132:20 cm 4910:real image 4618:real image 4512:real image 4337:real image 4050:. Because 2470:Derivation 2157:Parameter 2015:reciprocal 1177:Derivation 555:converging 543:collimated 504:equiconvex 494:, or just 464:astigmatic 397:refraction 362:Spectacles 281:The Clouds 169:See also: 143:binoculars 139:telescopes 78:, while a 68:refraction 64:light beam 9196:(5): 22. 9138:(1): 93. 9090:0036-8075 9068:CiteSeerX 7997:191384703 7333:− 7225:− 7092:∑ 6903:− 6882:− 6848:→ 6832:− 6722:− 6701:− 6667:→ 6651:− 6569:− 6513:− 5808:Anywhere 5685:∞ 5682:± 5626:Inverted 5590:∞ 5524:Inverted 5437:Minified 5352:∞ 5136:5 cm 5128:5 cm 5007:− 4985:− 4946:− 4706:) with a 4541:defocused 4521:human eye 3924:− 3878:− 3674:thin lens 3562:− 3533:− 3354:− 3325:− 3282:→ 3259:− 3223:− 3168:∞ 3165:− 3162:→ 3094:− 3058:− 2980:→ 2937:− 2882:− 2846:− 2747:− 2691:− 2644:− 2587:− 2421:− 2380:− 2369:≈ 2334:thin lens 2331:then the 2137:curvature 2131:indicate 2107:indicate 2023:, is the 1969:− 1954:− 1895:− 1880:− 1668:− 1625:− 1586:− 1478:− 1366:θ 1363:⁡ 1330:− 1327:θ 1321:⁡ 1291:θ 1288:− 1279:⁡ 1243:⁡ 1221:⁡ 1196:Applying 1135:− 1069:− 778:− 682:thin lens 680:An ideal 626:diverging 578:ball lens 559:thin lens 514:(or just 512:biconcave 238:The word 230:Lens for 120:microwave 9474:Archived 9461:Archived 9448:Archived 9411:Archived 9390:Archived 9368:Archived 9357:Archived 9343:Archived 9318:. SPIE. 9267:(1987). 9214:Archived 9210:26336702 9160:39903291 9123:(2010). 9106:20978013 9098:18703734 9029:18690249 8984:15089166 8876:29 April 8870:Archived 8837:19 April 8828:Archived 8762:17 March 8756:Archived 8732:17 March 8726:Archived 8701:17 March 8695:Archived 8532:Archived 8502:Archived 8403:Archived 8311:Archived 8277:Archived 8258:24 April 8252:Archived 8192:Archived 8168:Archived 8142:Archived 8050:Archived 8020:Archived 8016:BBC News 7938:21 April 7932:Archived 7786:F-number 7781:Eyepiece 7748:See also 7742:Abrasion 7693:meniscus 7626:eyeballs 7588:such as 7552:infrared 7550:or near- 6212:aperture 6148:fluorite 6137:achromat 6087:lenses. 6085:bestform 5811:Virtual 5287:infinity 5174: km 4908:). This 4640:negative 3504:becomes 2950:′ 2933:′ 2698:′ 2651:′ 2637:′ 2565:′ 2160:Meaning 2141:infinite 2082:negative 2075:positive 2029:dioptres 622:negative 551:positive 532:meniscus 488:biconvex 330:Ibn Sahl 122:lenses, 104:polished 98:and are 84:elements 9403:YouTube 9140:Bibcode 9060:Bibcode 9052:Science 9037:4314138 9009:Bibcode 8964:Bibcode 8112:9574655 7886:28 June 7689:Schmidt 7657:cameras 7604:. 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