40:
2475:
4865:
215:
227:
6147:
3774:
1182:
701:
7445:
6014:, 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:
4853:
7554:
5239:
662:
610:
7412:
479:
4641:
647:
637:
6152:
6080:
600:
4625:
4566:
6019:
4324:
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.
4591:. 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:
4319:
6127:) 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
6071:. 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
6243:
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
364:
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
6529:
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.,
6759:
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
3647:
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
6066:
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
418:
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,
394:
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
1397:
628:
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.
451:
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.
2967:
7690:
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
6936:
6755:
1165:
4582:
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
3185:
4191:
8489:
5992:
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.
3127:
7440:
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.
7256:
1267:
2783:
2623:
657:
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.
5026:
1508:
39:
804:
5876:
2788:
8481:
6450:
5107:, 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:
6765:
6584:
1020:
7041:
9377:
2465:
7124:
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
8037:
9112:
7460:
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.
5099:
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
6119:, 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:
9038:
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:
7359:
5778:
1260:
7295:), 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
6131:
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.
7158:
5610:
5372:
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:
3835:
8713:
4314:{\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
5564:
5418:
3933:
7305:. 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:
6577:
5076:), 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:
3883:
6524:
4335:
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
5683:
4927:
2524:
267:
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
8519:
6046:-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
5728:
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
9374:
5508:
5468:
1420:
5657:
6236:
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:
7612:; 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
6964:
6365:
6335:
6305:
6275:
9201:
6372:
4387:
4360:
4121:
4094:
4067:
4040:
4013:
3986:
3959:
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
9398:
8029:
6472:
3149:
1543:
4883:) 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
9151:
8390:
7146:
6975:
6240:
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.
4431:
4407:
4069:
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
8743:
5223:, 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.
8815:
6206:
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.
850:
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
2346:
1857:
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}}}
6530:
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.,
5103:—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
5034:
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
4600:
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
7061:
5118:
Using an inappropriate measurement of magnification can be formally correct but yield a meaningless number. For instance, using a magnifying glass of
673:
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
3703:
5804:
3396:
7847:
is sometimes seen. While it is listed as an alternative spelling in some dictionaries, most mainstream dictionaries do not list it as acceptable.
8179:
8155:
7932:
Analyses the almost negligible frequency of use and concludes that the misspelling is a result of a wrong singularisation of the plural (lenses).
7487:
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.
2087:
means that rays reaching the surface have already passed the center of curvature. Consequently, for external lens surfaces as diagrammed above,
9223:
Schottner, G (May 2003). "Scratch and
Abrasion Resistant Coatings on Plastic Lenses—State of the Art, Current Developments and Perspectives".
4186:
in the right figure. The similarities give the following equations and combining these results gives the
Newtonian form of the lens equation.
6011:
7436:
have at least one surface that is neither spherical nor cylindrical. The more complicated shapes allow such lenses to form images with less
8905:
8682:
5969:
2080:
indicates a surface's center of curvature is further along in the direction of the ray travel (right, in the accompanying diagrams), while
3631:
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
1169:
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
9330:
8078:
Kriss, Timothy C.; Kriss, Vesna
Martich (April 1998). "History of the Operating Microscope: From Magnifying Glass to Microneurosurgery".
6134:
Different lens materials may also be used to minimise chromatic aberration, such as specialised coatings or lenses made from the crystal
9416:
4858:
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.
8705:
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:
9225:
399:
and experimentation was showing no single-element lens could bring all colours to a focus. This led to the invention of the compound
8264:
8452:"A Light to Lighten our Darkenss: Lighthouse Optics and the Later Development of Fresnel's Revolutionary Refracting Lens 1780-1900"
8129:
7363:
which is the ratio of the output beam width to the input beam width. Note the sign convention: a telescope with two convex lenses (
9435:
7310:
6931:{\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})}}.}
6750:{\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})}}.}
9344:
7534:
for microwaves. Improved versions have been made by other researchers. As of 2014 the superlens has not yet been demonstrated at
4604:. 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
458:
or sphero-cylindrical lenses have surfaces with two different radii of curvature in two orthogonal planes. They have a different
7056:, it can be seen that the powers of thin lenses in contact are additive. The general case of multiple thin lenses in contact is
6194:
Even if a lens is designed to minimize or eliminate the aberrations described above, the image quality is still limited by the
8857:
8511:
6010:
occurs because spherical surfaces are not the ideal shape for a lens, but are by far the simplest shape to which glass can be
9311:
9288:
9268:
8940:
Grbic, A.; Eleftheriades, G. V. (2004). "Overcoming the
Diffraction Limit with a Planar Left-handed Transmission-line Lens".
8888:
8787:
8657:
8629:
8599:
8550:
8418:
8365:
8331:
8292:
8233:
8206:
7900:
7866:
584:
types, its focal point lies close to the ball's surface. Because of the ball's curvature extremes compared to the lens size,
8298:
8239:
8007:
6760:
point. In this case, the 1st lens images the object at its focal point. So, the thin lens formula for the 2nd lens becomes
7301:, since it produces no net convergence or divergence of the beam. Two lenses at this separation form the simplest type of
3786:
2474:
9448:
7919:
7422:
have curvature along only one axis. They are used to focus light into a line, or to convert the elliptical light from a
7400:
7399:) produces a positive magnification and the image is upright. For further information on simple optical telescopes, see
4497:
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
9395:
9365:
9170:
5143:
is 5, meaning that the object appears 5 times larger to the eye than without the lens. When taking a picture of the
3643:
of light placed at the focal point is converted into a collimated beam by the lens. These two cases are examples of
5962:
8387:
2486:
is derived here with respect to the right figure. The 1st spherical lens surface (which meets the optical axis at
8735:
2069:
The signs of the lens' radii of curvature indicate whether the corresponding surfaces are convex or concave. The
231:
5788:
As an object moves to the lens, the virtual image also gets closer to the lens while the image size is reduced.
5737:
1203:
8804:
5038:
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
8985:
Valentine, J.; et al. (2008). "Three-dimensional optical metamaterial with a negative refractive index".
3654:
7617:
7589:
6182:
5923:
2035:
463:
17:
5573:
5335:
4739:
the location of its real image. In that case even a negative lens can project a real image, as is done by a
7823:
7523:
3154:
3122:{\displaystyle {\frac {n_{1}}{u}}+{\frac {n_{1}}{v}}=(n_{2}-n_{1})({\frac {1}{R_{1}}}-{\frac {1}{R_{2}}}).}
2064:
7624:
are designed to attenuate light; sunglass lenses that also correct visual impairments can be custom made.
214:
9486:
9461:
7513:
5955:
5092:), a converging lens would form an inverted image, whereas a diverging lens would form an upright image.
570:
99:
9390:
7696:
4864:
2297:
This convention seems to be mainly used for this article, although there is another convention such as
8176:
8152:
5527:
5381:
3888:
9355:
8765:
7722:
1400:
9060:
6533:
2972:
5100:
4735:. This happens when the lens is inserted into a converging beam (being focused by a previous lens)
3842:
2298:
389:
329:
7268:
given for thin lenses in contact. It can be shown that the same formula works for thick lenses if
1521:
The position of the focus of a spherical lens depends on the radii of curvature of the two facets.
711:
For a single refraction for a circular boundary, the relation between object and its image in the
8942:
7758:
7676:
7383:) produces a negative magnification, indicating an inverted image. A convex plus a concave lens (
6477:
5938:
5665:
1800:
8912:
8674:
7858:
7852:
7474:
to make images that have an illusion of depth or that change when viewed from different angles.
5700:
4126:
This
Newtonian form of the lens equation can be derived by using a similarity between triangles
1785:
535:
526:
depending on the curvature of the other surface. A lens with one convex and one concave side is
9055:
8451:
7251:{\displaystyle {\frac {1}{f}}={\frac {1}{f_{1}}}+{\frac {1}{f_{2}}}-{\frac {d}{f_{1}f_{2}}}\,.}
4434:
3669:
2483:
2014:
712:
9327:
8060:
7675:
where the lens's spherical aberration corrects the opposite aberration in the mirror (such as
5477:
5437:
447:(flat). The line joining the centres of the spheres making up the lens surfaces is called the
7818:
7729:
6176:
5629:
5228:
874:
is the on-axis image distance from the line. Due to paraxial approximation where the line of
31:
5186:
image on the film corresponds to an angular size of the moon seen from earth of about 0.5°.
9127:
9108:
9047:
8996:
8951:
7788:
7302:
6091:
6002:
5943:
5918:
991:
589:
378:
268:
195:
75:
9387:
FDTD Animation of Electromagnetic Propagation through Convex Lens (on- and off-axis) Video
7892:
7885:
7275:
If the separation distance between two lenses is equal to the sum of their focal lengths (
6244:
intermediate image is located within the focal length of the next lens or even beyond it.
2628:
830:
is the refractive index of medium (the medium other than the spherical surface material),
8:
9176:
8261:
7803:
7672:
7484:
7471:
6943:
6203:
6100:
5933:
4840:
has no limit in increasing so its extremum is only the minimum, at which the derivate of
3454:
1764:
347:
226:
9131:
9051:
9000:
8955:
8127:
7880:
Reports "lense" as listed in some dictionaries, but not generally considered acceptable.
6340:
6310:
6280:
6250:
9432:
9189:
9143:
9089:
9020:
7980:
7972:
7793:
7778:
7695:
for at least 2400 years. A modern application is the use of relatively large lenses to
7680:
7437:
6457:
5983:
5908:
5899:
4532:
4365:
4338:
4099:
4072:
4045:
4018:
3991:
3964:
3937:
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:
9339:
8198:
The Complete Book of Fire: Building Campfires for Warmth, Light, Cooking, and Survival
4196:
878:
is close to the vertex of the spherical surface meeting the optical axis on the left,
9307:
9284:
9264:
9193:
9107:
Nielsen, R.B.; Thoreson, M.D.; Chen, W.; Kristensen, A.; Hvam, J.M.; Shalaev, V. M.;
9081:
9073:
9012:
8967:
8884:
8783:
8653:
8625:
8595:
8546:
8414:
8361:
8327:
8288:
8229:
8202:
8095:
8091:
7984:
7896:
7862:
7808:
7700:
7667:, or can be viewed on a screen. In these devices lenses are sometimes paired up with
7660:
7636:
7535:
7527:
6215:
6170:
5988:
Lenses do not form perfect images, and always introduce some degree of distortion or
5232:
4531:, as using an image distance different from that required by this formula produces a
170:
107:
9410:
9147:
9093:
8355:
6146:
5021:{\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}}}.}
9230:
9185:
9135:
9065:
9024:
9004:
8987:
8959:
8849:
8841:
8463:
8087:
7964:
7763:
7748:
7605:
7573:
7562:
7444:
7419:
7131:
6104:
6059:
4884:
4601:
4416:
4392:
3487:
1745:
971:
924:
901:
881:
857:
833:
542:
324:
154:
8963:
7703:, 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
1181:
700:
9481:
9465:
9452:
9439:
9402:
9381:
9359:
9348:
9334:
8408:
8394:
8321:
8282:
8268:
8223:
8196:
8183:
8159:
8144:
8133:
7999:
7798:
7714:
7706:
7648:
7593:
7495:
7463:
7427:
6227:
6164:
6120:
5112:
5108:
4456:
4410:
2070:
1778:
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
8440:
This paragraph is adapted from the 1888 edition of the Encyclopædia Britannica.
7813:
7783:
7686:
Convex lenses produce an image of an object at infinity at their focus; if the
7664:
7609:
6219:
5130:
from the object, produces a virtual image at infinity of infinite linear size:
4722:), according to the above formula. It is also possible for the object distance
4605:
3632:
2027:
of the lens. If the focal length is in metres, this gives the optical power in
1784:
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:
9386:
9234:
9139:
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
599:
486:
Lenses are classified by the curvature of the two optical surfaces. A lens is
9475:
9445:
9257:
9077:
8467:
8323:
Thinking about Life: The History and Philosophy of Biology and Other Sciences
7911:
7692:
7668:
7652:
7621:
7613:
7581:
7433:
7297:
6030:
5928:
5913:
5871:{\displaystyle \vert S_{2}\vert <\vert f\vert ,S_{1}>\vert S_{2}\vert }
5045:
This magnification formula provides two easy ways to distinguish converging (
4918:
4588:
2625:
For the imaging by second lens surface, by taking the above sign convention,
2024:
1188:
685:
581:
351:
127:
123:
48:
44:
9069:
8061:"On an account of a rock-crystal lens and decomposed glass found in Niniveh"
9252:
9197:
9085:
9016:
8971:
8116:
7718:
7597:
7531:
7499:
7477:
7457:
7449:
6445:{\displaystyle {\frac {1}{f_{j}}}={\frac {1}{s_{oj}}}+{\frac {1}{s_{ij}}},}
6223:
4498:
3640:
3587:{\displaystyle {\frac {1}{f}}=(n-1)({\frac {1}{R_{1}}}-{\frac {1}{R_{2}}})}
562:
412:
275:
247:
87:
8099:
6247:
For a two-lens system the object distances of each lens can be denoted as
2013:
is positive for converging lenses, and negative for diverging lenses. The
507:
438:
305:
8456:
The International Journal for the History of Engineering & Technology
7423:
6195:
6139:
5224:
5104:
4740:
4654:(focus of red ray path) into a magnified real image (green rays at focus)
4647:
467:
459:
382:
366:
264:
219:
174:
71:
9008:
499:
432:
183:
8812:
Physics 1C, Summer Session I, 2011 - University of California San Diego
8120:
7656:
7640:
7632:
7601:
7585:
7569:
7545:
A prototype flat ultrathin lens, with no curvature has been developed.
7507:
7036:{\displaystyle {\frac {1}{f}}={\frac {1}{f_{1}}}+{\frac {1}{f_{2}}}\,.}
6128:
4901:
4852:
4503:
455:
396:
381:
in 1608, both of which appeared in the spectacle-making centres in the
361:
355:
280:
142:
67:
63:
9113:"Toward superlensing with metal–dielectric composites and multilayers"
8125:. Translated by Hickie, William James. Project Gutenberg. EBook #2562.
7976:
7955:
Sines, George; Sakellarakis, Yannis A. (1987). "Lenses in antiquity".
7628:
7553:
7519:
7480:
has two or more, or a graduated, focal lengths ground into the lens.
7467:
4512:
3673:
2333:
2136:
1760:
681:
577:
558:
297:
138:
119:
103:
8764:
There are always 3 "easy rays". For the third ray in this case, see
8221:
7561:
A single convex lens mounted in a frame with a handle or stand is a
7264:
tends to zero, the focal length of the system tends to the value of
4569:
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.
8590:
Hecht, Eugene (2017). "Chapter 6.1 Thick Lenses and Lens Systems".
7968:
7773:
7768:
7742:
7539:
6199:
6135:
5278:
5238:
2140:
2041:
988:
toward the left infinity leads to the second or image focal length
5189:
In the extreme case where an object is an infinite distance away,
4459:
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
8853:
7526:
and claim to produce images at spatial resolutions exceeding the
7411:
4678:, the lens thus being used as a magnifying glass (rather than if
2028:
1517:
408:
370:
333:
319:
301:
150:
95:
8984:
8541:
Hecht, Eugene (2017). "5.2.2 Refraction at Spherical Surfaces".
8319:
7119:{\displaystyle {\frac {1}{f}}=\sum _{k=1}^{N}{\frac {1}{f_{k}}}}
6111:, with the wavelength of light. Since, from the formulae above,
4750:, the minimum distance between an object and the real image is 4
7644:
7577:
7491:
6367:. If the lenses are thin, each satisfies the thin lens formula
4613:
4516:
4508:
4488:
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:
9458:
8067:(in German). Deutsche Physikalische Gesellschaft. p. 355.
7616:; their curvature may not be axially symmetric to correct for
6067:
focused to a ring-shaped image in the focal plane, known as a
6050:. Rays that pass through the centre of a lens of focal length
5242:
Images of black letters in a thin convex lens of focal length
4640:
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.
9106:
7753:
7710:
6043:
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:
8222:
Glick, Thomas F.; Steven John Livesey; Faith Wallis (2005).
7155:, then the focal length for the combined system is given by
6940:
A simplest case is where thin lenses are placed in contact (
6151:
6079:
2111:
surfaces (used to converge light in a positive lens), while
478:
9409:
8225:
Medieval science, technology, and medicine: an encyclopedia
6138:. This naturally occurring substance has the highest known
5144:
2478:
A Diagram for a Spherical Lens Equation with Paraxial Rays.
346:
became available in Latin translation in the 12th century (
293:
9352:
9328:
A chapter from an online textbook on refraction and lenses
5732:
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
7687:
7272:
is taken as the distance between their principal planes.
7151:
If two thin lenses are separated in air by some distance
5296:
5151:
lens, one is not concerned with the linear magnification
4565:
110:
to the required shape. A lens can focus light to form an
7732:-resistant coatings are available to help control this.
7557:
A watch with a plano-convex lens over the date indicator
7426:
into a round beam. They are also used in motion picture
6018:
4731:
to be negative, in which case the lens sees a so-called
4624:
4323:
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
8320:
Paul S. Agutter; Denys N. Wheatley (12 December 2008).
7530:. The first superlenses were made in 2004 using such a
2146:
1803:
of the lens, and the locations of the principal planes
8879:
Hecht, Eugene (2017). "Back and Front Focal Lengths".
7134:
6480:
6343:
6313:
6283:
6253:
5246:
are shown in red. Selected rays are shown for letters
4419:
4395:
4368:
4341:
4102:
4075:
4048:
4021:
3994:
3967:
3940:
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
7313:
7161:
7064:
6978:
6946:
6768:
6587:
6579:. So, the thin lens formula for the 1st lens becomes
6536:
6526:. (The 2nd lens images the image of the first lens.)
6460:
6375:
5807:
5740:
5703:
5668:
5632:
5576:
5530:
5480:
5440:
5384:
5338:
4930:
4922:
of an imaging system using a single lens is given by
4194:
3891:
3845:
3789:
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
9396:
The Use of Magnifying Lenses in the Classical World
8400:
8262:
The Galileo Project > Science > The Telescope
7608:.) Most lenses used for other purposes have strict
7401:
Refracting telescope § Refracting telescope designs
6142:, indicating that the material has low dispersion.
4015:is positive if it is right to the rear focal point
3961: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
9256:
9169:
8939:
8805:"Lecture 9 Notes: 07 / 13 - Multiple-lens systems"
7907:Lists "lense" as an acceptable alternate spelling.
7884:
7353:
7250:
7140:
7118:
7035:
6958:
6930:
6749:
6571:
6518:
6466:
6444:
6359:
6329:
6299:
6269:
5870:
5772:
5722:
5677:
5651:
5604:
5558:
5502:
5462:
5412:
5366:
5020:
4425:
4401:
4381:
4354:
4313:
4115:
4088:
4061:
4034:
4007:
3980:
3953:
3927:
3877:
3829:
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
9420:. Vol. 16 (11th ed.). pp. 421–427.
7954:
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
9473:
9455:– Open source lens simulator (downloadable java)
8778:Hecht, Eugene (2017). "Thin-Lens Combinations".
8357:Renaissance Vision from Spectacles to Telescopes
7655:when applied to the human eye; others produce a
5059:) lenses: For an object very close to the lens (
3781:This can also be put into the "Newtonian" form:
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:
8360:. American Philosophical Society. p. 210.
5292:) has a double-size, virtual and upright image.
5178: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 1758. You can help by
8903:
8643:
8641:
8615:
8613:
8611:
8410:Stargazer: The Life and Times of the Telescope
8353:
5422:Inverted (opposite to the object orientation)
5269:) 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.
9037:
8648:Hecht, Eugene (2017). "Thin-Lens Equations".
8585:
8583:
8581:
8579:
8280:
8271:. Galileo.rice.edu. Retrieved on 6 June 2012.
7857:. Franklin, Beedle & Associates. p.
5963:
5302:Images of Real Objects Formed by Thin Lenses
4535:(fuzzy) image for an object at a distance of
2252:The height of an image from the optical axis
1170:
689:
8978:
8115:
8111:
8109:
7354:{\displaystyle M=-{\frac {f_{2}}{f_{1}}}\,,}
6157:
5865:
5852:
5833:
5827:
5821:
5808:
5754:
5741:
4544:from the camera. Put another way, modifying
4522:The focusing adjustment of a camera adjusts
9297:
9171:"Good-Bye to Curved Lens: New Lens Is Flat"
8638:
8608:
8565:
8406:
8058:
5376:Real (rays converging to each image point)
2179:The distance between an object and a lens.
2152:Sign convention for Gaussian lens equation
245:
8576:
8397:, Nobel Foundation. Retrieved 3 April 2009
8287:. Courier Dover Publications. p. 27.
8077:
7997:
7991:
7627:Other uses are in imaging systems such as
7510:, or transforms a laser beam into a ring.
6454:If the distance between the two lenses is
6198:of light passing through the lens' finite
6103:of the lens material—the variation of its
5970:
5956:
5773:{\displaystyle \vert S_{2}\vert >S_{1}}
4634: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 "
9226:Journal of Sol-Gel Science and Technology
9222:
9100:
9059:
8106:
7728:Lenses can become scratched and abraded.
7347:
7244:
7052:is the power of a lens with focal length
7029:
5258:in blue, green and orange, respectively.
4691:as for a camera). Using a negative lens (
4156:and another similarity between triangles
3921:
3823:
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
8935:
8933:
8802:
8620:Hecht, Eugene (2017). "Finite Imagery".
7552:
7443:
7410:
5237:
5227:places a lower limit on the size of the
4904:may also be viewed when put on a screen.
4564:
4322:
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:
8897:
6189:
6085:
5996:
5782:Erect (same to the object orientation)
5182:, from which one can conclude that the
4746: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:
9474:
8883:(5th ed.). Pearson. p. 181.
8782:(5th ed.). Pearson. p. 178.
8545:(5th ed.). Pearson. p. 164.
8194:
7850:
7651:. Some of these instruments produce a
7406:
5605:{\displaystyle \infty >S_{2}>2f}
5367:{\displaystyle \infty >S_{1}>2f}
5297:Table for thin lens imaging properties
5042:is positive, so the image is upright.
4661:Using a positive lens of focal length
4479:, 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
9278:
9251:
9167:
8930:
8878:
8839:
8777:
8706:"Resource Lesson: Thin Lens Equation"
8647:
8619:
8589:
8570:
8540:
8492:from the original on 26 November 2022
8449:
6054:are focused at a point with distance
4475:from a positive lens of focal length
4462:If an object is placed at a distance
4123: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
8685:from the original on 12 October 2000
8522:from the original on 26 October 2021
8242:from the original on 20 January 2023
8071:
8040:from the original on 19 October 2012
8010:from the original on 1 February 2009
7887:Merriam-Webster's Medical Dictionary
4804:by the lens equation (or expressing
4712:) can only produce a virtual image (
2482:The spherical thin lens equation in
2147:Sign convention for other parameters
178:
79:
8703:
8281:Henry C. King (28 September 2003).
8201:. Menasha Ridge Press. p. 25.
7912:"Lens or Lense – Which is Correct?"
5330:Converging lens (or positive lens)
4587:to form an image, this is called a
3830:{\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:
9190:10.1038/scientificamerican0515-22b
8860:from the original on 29 April 2024
8821:from the original on 18 April 2024
8746:from the original on 10 March 2015
8512:"Refraction at Spherical Surfaces"
7922:from the original on 21 April 2018
6966:). Then the combined focal length
6209:
6162:Other kinds of aberration include
5793:Diverging lens (or negative lens)
5672:
5577:
5339:
5115:according to their focal lengths.
4818:), and equating the derivative of
3167:
2785:Adding these two equations yields,
645:
635:
608:
598:
437:(bulging outwards from the lens),
25:
9498:
9321:
9300:Field Guide to Geometrical Optics
8716:from the original on 2 April 2015
8470:– via Taylor & Francis.
8413:. Allen & Unwin. p. 55.
8030:"The Nimrud lens/The Layard lens"
7998:Whitehouse, David (1 July 1999).
7891:. Merriam-Webster. 1995. p.
6307:, and the image distances as and
6232:Simple lenses are subject to the
3648:lens axis situated at a distance
9283:(4th ed.). Addison Wesley.
9263:(2nd ed.). Addison Wesley.
9229:. Vol. 27. pp. 71–79.
9204:from the original on 19 May 2015
8672:
8301:from the original on 2 July 2023
8092:10.1097/00006123-199804000-00116
6233:
6150:
6145:
6078:
6017:
5559:{\displaystyle f<S_{1}<2f}
5413:{\displaystyle f<S_{2}<2f}
4911:
4863:
4851:
4639:
4623:
3928:{\displaystyle x_{2}=S_{2}-f\,.}
3663:
182:
9245:
9216:
9161:
9031:
8872:
8833:
8796:
8771:
8758:
8728:
8697:
8666:
8559:
8534:
8504:
8474:
8443:
8434:
8381:
8347:
8313:
8274:
8254:
8215:
8188:
7957:American Journal of Archaeology
4665:, a virtual image results when
4507:. This is the principle of the
2469:
9426:
8407:Fred Watson (1 October 2007).
8165:
8138:
8052:
8022:
7948:
7837:
6919:
6893:
6882:
6863:
6835:
6738:
6712:
6701:
6682:
6654:
6572:{\displaystyle s_{i1}=d-f_{2}}
5891:
4772:). This is derived by letting
4553:causes objects at a different
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:
9459:Animations demonstrating lens
9298:Greivenkamp, John E. (2004).
8964:10.1103/PhysRevLett.92.117403
8450:Julia, Elton (18 July 2013).
7941:
7745:treatment of optical surfaces
7498:optical surface. It images a
3878:{\displaystyle x_{1}=S_{1}-f}
3639:from the lens. Conversely, a
1525:The (effective) focal length
1176:
423:Construction of simple lenses
8681:. Georgia State University.
8284:The History of the Telescope
7824:Ray transfer matrix analysis
7524:negative index metamaterials
7514:Diffractive optical elements
6519:{\textstyle s_{o2}=d-s_{i1}}
6042:, derives its name from the
4562:to come into perfect focus.
4519:serves as the image sensor.
4437: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:
9442:– Concave and Convex Lenses
8736:"The Mathematics of Lenses"
8482:"4.4: Spherical Refractors"
8065:Die Fortschritte der Physik
8000:"World's oldest telescope?"
7735:
5678:{\displaystyle \pm \infty }
571:extended hemispherical lens
10:
9503:
8228:. Routledge. p. 167.
7725:into a collector antenna.
7415:An aspheric biconvex lens.
6970:of the lenses is given by
6213:
6089:
6028:
6000:
5981:
5723:{\displaystyle S_{1}<f}
5319:Lateral Image Orientation
2062:
1200:on the spherical surface,
387:
168:
164:
29:
9302:. SPIE Field Guides vol.
9140:10.1007/s00340-010-4065-z
8803:Vlasenko, Alexey (2011).
8766:File:Lens3b third ray.svg
8652:(5th ed.). Pearson.
8624:(5th ed.). Pearson.
8594:(5th ed.). Pearson.
8182:28 September 2008 at the
7723:electromagnetic radiation
7148:is the number of lenses.
6158:Other types of aberration
2341: is then given by
2299:Cartesian sign convention
1401:small angle approximation
9401:13 November 2017 at the
9333:17 December 2009 at the
8468:10.1179/175812109X449612
8326:. Springer. p. 17.
7854:Common Errors in English
7830:
7697:concentrate solar energy
7659:that can be captured on
7516:can function as lenses.
7448:Close-up view of a flat
5503:{\displaystyle S_{2}=2f}
5463:{\displaystyle S_{1}=2f}
4409:are with respect to the
390:History of the telescope
252:, the Latin name of the
9438:21 January 2010 at the
9433:Learning by Simulations
9417:Encyclopædia Britannica
9375:Ancient Egyptian lenses
9235:10.1023/A:1022684011222
9070:10.1126/science.1157566
8943:Physical Review Letters
8904:Proteep Mallik (2005).
8840:Paschotta, Dr Rüdiger.
8354:Vincent Ilardi (2007).
7759:Cardinal point (optics)
7548:
6024:
5939:Petzval field curvature
5652:{\displaystyle S_{1}=f}
2305:Thin lens approximation
1799:is with respect to the
696:For a spherical surface
498:) if both surfaces are
9451:6 October 2010 at the
9341:Thin Spherical Lenses
9279:Hecht, Eugene (2002).
9168:Patel, Prachi (2015).
8704:Colwell, Catharine H.
8393:9 January 2010 at the
8388:Microscopes: Time Line
8175:(trans. John Bostock)
8158:4 October 2008 at the
8151:(trans. John Bostock)
8036:. The British Museum.
7572:for the correction of
7558:
7453:
7416:
7355:
7252:
7142:
7120:
7098:
7037:
6960:
6932:
6751:
6573:
6520:
6468:
6446:
6361:
6331:
6301:
6271:
5872:
5774:
5724:
5679:
5653:
5606:
5560:
5504:
5464:
5414:
5368:
5293:
5147:using a camera with a
5022:
4836:) to zero. (Note that
4570:
4435:effective focal length
4427:
4403:
4383:
4356:
4332:
4327:A camera lens forms a
4315:
4117:
4090:
4063:
4036:
4009:
3982:
3955:
3929:
3879:
3831:
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:
9408:Henker, Otto (1911).
9347:13 March 2020 at the
8740:The Physics Classroom
8195:Tilton, Buck (2005).
8177:Book XXXVII, Chap. 16
8153:Book XXXVII, Chap. 10
7851:Brians, Paul (2003).
7843:The variant spelling
7556:
7447:
7414:
7356:
7253:
7143:
7121:
7078:
7038:
6961:
6933:
6752:
6574:
6521:
6469:
6447:
6362:
6332:
6302:
6272:
6177:pincushion distortion
5873:
5775:
5725:
5680:
5654:
5607:
5561:
5505:
5465:
5415:
5369:
5241:
5231:. This is called the
5229:point spread function
5140:angular magnification
5101:angular magnification
5095:Linear magnification
5023:
4568:
4428:
4404:
4384:
4357:
4326:
4316:
4118:
4091:
4064:
4037:
4010:
3983:
3956:
3930:
3880:
3832:
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)
9464:4 April 2012 at the
8846:www.rp-photonics.com
8675:"Thin Lens Equation"
8516:personal.math.ubc.ca
8267:23 June 2004 at the
8132:28 June 2017 at the
8119:(22 January 2013) .
8059:D. Brewster (1852).
7916:writingexplained.org
7789:List of lens designs
7717:, commonly called a
7699:on relatively small
7311:
7159:
7132:
7062:
6976:
6944:
6766:
6585:
6534:
6478:
6458:
6373:
6341:
6311:
6281:
6251:
6190:Aperture diffraction
6097:Chromatic aberration
6092:Chromatic aberration
6086:Chromatic aberration
6008:Spherical aberration
6003:Spherical aberration
5997:Spherical aberration
5944:Chromatic aberration
5919:Spherical aberration
5805:
5738:
5701:
5666:
5630:
5574:
5528:
5478:
5438:
5382:
5336:
5322:Image Magnification
4928:
4417:
4393:
4366:
4339:
4331:of a distant object.
4192:
4100:
4073:
4046:
4019:
3992:
3965:
3938:
3889:
3843:
3787:
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
9380:25 May 2022 at the
9358:14 May 2017 at the
9275:Chapters 5 & 6.
9177:Scientific American
9132:2010ApPhB.100...93N
9052:2008Sci...321..930Y
9009:10.1038/nature07247
9001:2008Natur.455..376V
8956:2004PhRvL..92k7403G
8918:on 23 November 2009
8173:The Natural History
8149:The Natural History
8034:Collection database
7804:Optical lens design
7673:catadioptric system
7568:Lenses are used as
7485:gradient index lens
7472:lenticular printing
7407:Non spherical types
6959:{\displaystyle d=0}
6360:{\textstyle s_{i2}}
6330:{\textstyle s_{i1}}
6300:{\textstyle s_{o2}}
6270:{\textstyle s_{o1}}
6234:optical aberrations
6204:diffraction-limited
6012:ground and polished
5303:
5277:) has its image at
5122:focal length, held
2153:
1765:radius of curvature
1171:lensmaker's formula
348:Eugenius of Palermo
9487:Optical components
9422:(with 21 diagrams)
9368:digitalartform.com
8486:Physics LibreTexts
7794:Numerical aperture
7779:Gravitational lens
7713:systems often use
7701:photovoltaic cells
7622:Sunglasses' lenses
7559:
7454:
7420:Cylindrical lenses
7417:
7351:
7248:
7138:
7116:
7033:
6956:
6928:
6747:
6569:
6516:
6464:
6442:
6357:
6327:
6297:
6267:
6121:achromatic doublet
6115:is dependent upon
6040:comatic aberration
5984:Optical aberration
5900:Optical aberration
5868:
5770:
5720:
5675:
5649:
5602:
5556:
5500:
5460:
5410:
5364:
5301:
5294:
5018:
4571:
4511:, and also of the
4423:
4399:
4382:{\textstyle S_{2}}
4379:
4355:{\textstyle S_{1}}
4352:
4333:
4311:
4309:
4116:{\textstyle x_{2}}
4113:
4089:{\textstyle x_{1}}
4086:
4062:{\textstyle f^{2}}
4059:
4035:{\textstyle F_{2}}
4032:
4008:{\textstyle x_{2}}
4005:
3981:{\textstyle F_{1}}
3978:
3954:{\textstyle x_{1}}
3951:
3925:
3875:
3827:
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
9313:978-0-8194-5294-8
9290:978-0-321-18878-6
9270:978-0-201-11609-0
9120:Applied Physics B
8890:978-1-292-09693-3
8789:978-1-292-09693-3
8659:978-1-292-09693-3
8631:978-1-292-09693-3
8601:978-1-292-09693-3
8552:978-1-292-09693-3
8420:978-1-74175-383-7
8367:978-0-87169-259-7
8333:978-1-4020-8865-0
8294:978-0-486-43265-6
8235:978-0-415-96930-7
8208:978-0-89732-633-9
8171:Pliny the Elder,
7918:. 30 April 2017.
7902:978-0-87779-914-6
7868:978-1-887902-89-2
7809:Photochromic lens
7715:dielectric lenses
7661:photographic film
7574:refractive errors
7528:diffraction limit
7470:that are used in
7464:Lenticular lenses
7428:anamorphic lenses
7345:
7303:optical telescope
7242:
7210:
7190:
7170:
7114:
7073:
7027:
7007:
6987:
6923:
6833:
6805:
6784:
6742:
6652:
6624:
6603:
6467:{\displaystyle d}
6437:
6414:
6391:
6216:Photographic lens
6099:is caused by the
5980:
5979:
5889:
5888:
5233:diffraction limit
5126:from the eye and
5113:wide-angle lenses
5109:long-focus lenses
5052:) and diverging (
5016:
4994:
4990:
4962:
4305:
4270:
4249:
4221:
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
9494:
9446:OpticalRayTracer
9421:
9413:
9366:Lens article at
9317:
9294:
9274:
9262:
9239:
9238:
9220:
9214:
9213:
9211:
9209:
9173:
9165:
9159:
9158:
9157:on 9 March 2013.
9156:
9150:. Archived from
9117:
9104:
9098:
9097:
9063:
9035:
9029:
9028:
8982:
8976:
8975:
8937:
8928:
8927:
8925:
8923:
8917:
8911:. Archived from
8910:
8901:
8895:
8894:
8876:
8870:
8869:
8867:
8865:
8842:"focal distance"
8837:
8831:
8830:
8828:
8826:
8820:
8809:
8800:
8794:
8793:
8775:
8769:
8762:
8756:
8755:
8753:
8751:
8732:
8726:
8725:
8723:
8721:
8701:
8695:
8694:
8692:
8690:
8670:
8664:
8663:
8645:
8636:
8635:
8617:
8606:
8605:
8587:
8574:
8566:Greivenkamp 2004
8563:
8557:
8556:
8538:
8532:
8531:
8529:
8527:
8508:
8502:
8501:
8499:
8497:
8478:
8472:
8471:
8447:
8441:
8438:
8432:
8431:
8429:
8427:
8404:
8398:
8385:
8379:
8378:
8376:
8374:
8351:
8345:
8344:
8342:
8340:
8317:
8311:
8310:
8308:
8306:
8278:
8272:
8258:
8252:
8251:
8249:
8247:
8219:
8213:
8212:
8192:
8186:
8169:
8163:
8142:
8136:
8126:
8113:
8104:
8103:
8075:
8069:
8068:
8056:
8050:
8049:
8047:
8045:
8026:
8020:
8019:
8017:
8015:
7995:
7989:
7988:
7952:
7935:
7931:
7929:
7927:
7906:
7890:
7879:
7877:
7875:
7841:
7799:Optical coatings
7764:Caustic (optics)
7749:Back focal plane
7606:intraocular lens
7563:magnifying glass
7398:
7382:
7372:
7360:
7358:
7357:
7352:
7346:
7344:
7343:
7334:
7333:
7324:
7294:
7271:
7267:
7263:
7257:
7255:
7254:
7249:
7243:
7241:
7240:
7239:
7230:
7229:
7216:
7211:
7209:
7208:
7196:
7191:
7189:
7188:
7176:
7171:
7163:
7154:
7147:
7145:
7144:
7139:
7125:
7123:
7122:
7117:
7115:
7113:
7112:
7100:
7097:
7092:
7074:
7066:
7055:
7051:
7042:
7040:
7039:
7034:
7028:
7026:
7025:
7013:
7008:
7006:
7005:
6993:
6988:
6980:
6969:
6965:
6963:
6962:
6957:
6937:
6935:
6934:
6929:
6924:
6922:
6918:
6917:
6905:
6904:
6885:
6881:
6880:
6862:
6861:
6851:
6834:
6832:
6831:
6830:
6811:
6806:
6804:
6790:
6785:
6783:
6782:
6770:
6756:
6754:
6753:
6748:
6743:
6741:
6737:
6736:
6724:
6723:
6704:
6700:
6699:
6681:
6680:
6670:
6653:
6651:
6650:
6649:
6630:
6625:
6623:
6609:
6604:
6602:
6601:
6589:
6578:
6576:
6575:
6570:
6568:
6567:
6549:
6548:
6525:
6523:
6522:
6517:
6515:
6514:
6493:
6492:
6473:
6471:
6470:
6465:
6451:
6449:
6448:
6443:
6438:
6436:
6435:
6420:
6415:
6413:
6412:
6397:
6392:
6390:
6389:
6377:
6366:
6364:
6363:
6358:
6356:
6355:
6336:
6334:
6333:
6328:
6326:
6325:
6306:
6304:
6303:
6298:
6296:
6295:
6276:
6274:
6273:
6268:
6266:
6265:
6210:Compound lenses
6154:
6149:
6118:
6114:
6110:
6105:refractive index
6082:
6065:
6053:
6049:
6021:
5972:
5965:
5958:
5896:
5895:
5877:
5875:
5874:
5869:
5864:
5863:
5848:
5847:
5820:
5819:
5779:
5777:
5776:
5771:
5769:
5768:
5753:
5752:
5729:
5727:
5726:
5721:
5713:
5712:
5695:Converging lens
5684:
5682:
5681:
5676:
5658:
5656:
5655:
5650:
5642:
5641:
5624:Converging lens
5611:
5609:
5608:
5603:
5592:
5591:
5565:
5563:
5562:
5557:
5546:
5545:
5522:Converging lens
5509:
5507:
5506:
5501:
5490:
5489:
5469:
5467:
5466:
5461:
5450:
5449:
5432:Converging lens
5419:
5417:
5416:
5411:
5400:
5399:
5373:
5371:
5370:
5365:
5354:
5353:
5310:Object Location
5304:
5300:
5291:
5276:
5268:
5245:
5222:
5211:
5198:
5185:
5181:
5177:
5175:
5173:
5167:
5165:
5159:
5150:
5136:
5129:
5125:
5121:
5098:
5091:
5089:
5075:
5073:
5058:
5051:
5041:
5037:
5033:
5027:
5025:
5024:
5019:
5017:
5015:
5014:
5002:
4992:
4991:
4989:
4988:
4987:
4968:
4963:
4961:
4960:
4951:
4950:
4941:
4899:
4885:magnifying glass
4882:
4867:
4855:
4822:with respect to
4730:
4721:
4711:
4697:
4690:
4677:
4664:
4643:
4627:
4602:magnifying glass
4599:
4581:
4561:
4552:
4543:
4530:
4501:, is known as a
4496:
4487:
4478:
4474:
4454:
4445:
4432:
4430:
4429:
4424:
4411:principal planes
4408:
4406:
4405:
4400:
4388:
4386:
4385:
4380:
4378:
4377:
4361:
4359:
4358:
4353:
4351:
4350:
4320:
4318:
4317:
4312:
4310:
4306:
4304:
4303:
4294:
4290:
4289:
4276:
4271:
4266:
4265:
4256:
4250:
4245:
4241:
4240:
4227:
4222:
4220:
4219:
4210:
4209:
4200:
4122:
4120:
4119:
4114:
4112:
4111:
4095:
4093:
4092:
4087:
4085:
4084:
4068:
4066:
4065:
4060:
4058:
4057:
4041:
4039:
4038:
4033:
4031:
4030:
4014:
4012:
4011:
4006:
4004:
4003:
3987:
3985:
3984:
3979:
3977:
3976:
3960:
3958:
3957:
3952:
3950:
3949:
3934:
3932:
3931:
3926:
3914:
3913:
3901:
3900:
3884:
3882:
3881:
3876:
3868:
3867:
3855:
3854:
3836:
3834:
3833:
3828:
3822:
3821:
3812:
3811:
3799:
3798:
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:
9502:
9501:
9497:
9496:
9495:
9493:
9492:
9491:
9472:
9471:
9466:Wayback Machine
9453:Wayback Machine
9440:Wayback Machine
9429:
9407:
9403:Wayback Machine
9382:Wayback Machine
9360:Wayback Machine
9353:Project PHYSNET
9349:Wayback Machine
9335:Wayback Machine
9324:
9314:
9291:
9271:
9248:
9243:
9242:
9221:
9217:
9207:
9205:
9166:
9162:
9154:
9115:
9105:
9101:
9061:10.1.1.716.4426
9036:
9032:
8995:(7211): 376–9.
8983:
8979:
8938:
8931:
8921:
8919:
8915:
8908:
8902:
8898:
8891:
8877:
8873:
8863:
8861:
8838:
8834:
8824:
8822:
8818:
8807:
8801:
8797:
8790:
8776:
8772:
8763:
8759:
8749:
8747:
8734:
8733:
8729:
8719:
8717:
8702:
8698:
8688:
8686:
8671:
8667:
8660:
8646:
8639:
8632:
8618:
8609:
8602:
8588:
8577:
8569:
8564:
8560:
8553:
8539:
8535:
8525:
8523:
8510:
8509:
8505:
8495:
8493:
8488:. 2 July 2019.
8480:
8479:
8475:
8448:
8444:
8439:
8435:
8425:
8423:
8421:
8405:
8401:
8395:Wayback Machine
8386:
8382:
8372:
8370:
8368:
8352:
8348:
8338:
8336:
8334:
8318:
8314:
8304:
8302:
8295:
8279:
8275:
8269:Wayback Machine
8260:Al Van Helden.
8259:
8255:
8245:
8243:
8236:
8220:
8216:
8209:
8193:
8189:
8184:Wayback Machine
8170:
8166:
8160:Wayback Machine
8145:Pliny the Elder
8143:
8139:
8134:Wayback Machine
8114:
8107:
8076:
8072:
8057:
8053:
8043:
8041:
8028:
8027:
8023:
8013:
8011:
7996:
7992:
7953:
7949:
7944:
7939:
7938:
7925:
7923:
7910:
7903:
7883:
7873:
7871:
7869:
7842:
7838:
7833:
7828:
7738:
7707:Radio astronomy
7693:burning-glasses
7594:corrective lens
7551:
7434:Aspheric lenses
7409:
7397:
7390:
7384:
7380:
7374:
7370:
7364:
7339:
7335:
7329:
7325:
7323:
7312:
7309:
7308:
7293:
7286:
7276:
7269:
7265:
7261:
7235:
7231:
7225:
7221:
7220:
7215:
7204:
7200:
7195:
7184:
7180:
7175:
7162:
7160:
7157:
7156:
7152:
7133:
7130:
7129:
7108:
7104:
7099:
7093:
7082:
7065:
7063:
7060:
7059:
7053:
7046:
7021:
7017:
7012:
7001:
6997:
6992:
6979:
6977:
6974:
6973:
6967:
6945:
6942:
6941:
6913:
6909:
6900:
6896:
6886:
6876:
6872:
6857:
6853:
6852:
6850:
6826:
6822:
6815:
6810:
6794:
6789:
6778:
6774:
6769:
6767:
6764:
6763:
6732:
6728:
6719:
6715:
6705:
6695:
6691:
6676:
6672:
6671:
6669:
6645:
6641:
6634:
6629:
6613:
6608:
6597:
6593:
6588:
6586:
6583:
6582:
6563:
6559:
6541:
6537:
6535:
6532:
6531:
6507:
6503:
6485:
6481:
6479:
6476:
6475:
6459:
6456:
6455:
6428:
6424:
6419:
6405:
6401:
6396:
6385:
6381:
6376:
6374:
6371:
6370:
6348:
6344:
6342:
6339:
6338:
6318:
6314:
6312:
6309:
6308:
6288:
6284:
6282:
6279:
6278:
6258:
6254:
6252:
6249:
6248:
6230:
6228:Achromatic lens
6212:
6192:
6165:field curvature
6160:
6116:
6112:
6108:
6094:
6088:
6055:
6051:
6047:
6033:
6027:
6005:
5999:
5986:
5976:
5894:
5859:
5855:
5843:
5839:
5815:
5811:
5806:
5803:
5802:
5764:
5760:
5748:
5744:
5739:
5736:
5735:
5708:
5704:
5702:
5699:
5698:
5667:
5664:
5663:
5637:
5633:
5631:
5628:
5627:
5587:
5583:
5575:
5572:
5571:
5541:
5537:
5529:
5526:
5525:
5485:
5481:
5479:
5476:
5475:
5445:
5441:
5439:
5436:
5435:
5395:
5391:
5383:
5380:
5379:
5349:
5345:
5337:
5334:
5333:
5316:Image Location
5299:
5286:
5274:
5263:
5243:
5213:
5206:
5200:
5196:
5190:
5183:
5179:
5171:
5169:
5163:
5161:
5157:
5152:
5148:
5141:
5131:
5127:
5123:
5119:
5096:
5085:
5083:
5077:
5069:
5067:
5060:
5053:
5046:
5039:
5035:
5031:
5010:
5006:
5001:
4983:
4979:
4972:
4967:
4956:
4952:
4946:
4942:
4940:
4929:
4926:
4925:
4914:
4909:
4908:
4907:
4906:
4905:
4898:
4888:
4881:
4871:
4870:A convex lens (
4868:
4860:
4859:
4856:
4835:
4828:
4817:
4810:
4803:
4796:
4789:
4782:
4767:
4760:
4729:
4723:
4719:
4713:
4709:
4703:
4692:
4685:
4679:
4672:
4666:
4662:
4659:
4658:
4657:
4656:
4655:
4650:(B) reimages a
4644:
4636:
4635:
4628:
4606:lens of the eye
4598:
4592:
4580:
4574:
4573:In some cases,
4560:
4554:
4551:
4545:
4542:
4536:
4529:
4523:
4515:, in which the
4495:
4489:
4486:
4480:
4476:
4469:
4463:
4455:pass through a
4453:
4447:
4444:
4438:
4418:
4415:
4414:
4394:
4391:
4390:
4373:
4369:
4367:
4364:
4363:
4346:
4342:
4340:
4337:
4336:
4308:
4307:
4299:
4295:
4285:
4281:
4277:
4275:
4261:
4257:
4255:
4252:
4251:
4236:
4232:
4228:
4226:
4215:
4211:
4205:
4201:
4199:
4195:
4193:
4190:
4189:
4184:
4180:
4176:
4169:
4165:
4161:
4154:
4150:
4146:
4139:
4135:
4131:
4107:
4103:
4101:
4098:
4097:
4080:
4076:
4074:
4071:
4070:
4053:
4049:
4047:
4044:
4043:
4026:
4022:
4020:
4017:
4016:
3999:
3995:
3993:
3990:
3989:
3972:
3968:
3966:
3963:
3962:
3945:
3941:
3939:
3936:
3935:
3909:
3905:
3896:
3892:
3890:
3887:
3886:
3863:
3859:
3850:
3846:
3844:
3841:
3840:
3817:
3813:
3807:
3803:
3794:
3790:
3788:
3785:
3784:
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:
9500:
9490:
9489:
9484:
9470:
9469:
9456:
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640:Biconcave lens
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352:reading stones
339:Book of Optics
222:, 13th century
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7683:correctors).
7682:
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7653:virtual image
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7582:hypermetropia
7579:
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7542:wavelengths.
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7283:
7279:
7273:
7258:
7245:
7236:
7232:
7226:
7222:
7217:
7212:
7205:
7201:
7197:
7192:
7185:
7181:
7177:
7172:
7167:
7164:
7149:
7135:
7126:
7109:
7105:
7101:
7094:
7089:
7086:
7083:
7079:
7075:
7070:
7067:
7057:
7050:
7043:
7030:
7022:
7018:
7014:
7009:
7002:
6998:
6994:
6989:
6984:
6981:
6971:
6953:
6950:
6947:
6938:
6925:
6914:
6910:
6906:
6901:
6897:
6890:
6887:
6877:
6873:
6869:
6866:
6858:
6854:
6847:
6844:
6841:
6838:
6827:
6823:
6819:
6816:
6812:
6807:
6801:
6798:
6795:
6791:
6786:
6779:
6775:
6771:
6761:
6757:
6744:
6733:
6729:
6725:
6720:
6716:
6709:
6706:
6696:
6692:
6688:
6685:
6677:
6673:
6666:
6663:
6660:
6657:
6646:
6642:
6638:
6635:
6631:
6626:
6620:
6617:
6614:
6610:
6605:
6598:
6594:
6590:
6580:
6564:
6560:
6556:
6553:
6550:
6545:
6542:
6538:
6527:
6511:
6508:
6504:
6500:
6497:
6494:
6489:
6486:
6482:
6461:
6452:
6439:
6432:
6429:
6425:
6421:
6416:
6409:
6406:
6402:
6398:
6393:
6386:
6382:
6378:
6368:
6352:
6349:
6345:
6322:
6319:
6315:
6292:
6289:
6285:
6262:
6259:
6255:
6245:
6241:
6239:
6238:compound lens
6235:
6229:
6225:
6221:
6217:
6207:
6205:
6201:
6197:
6187:
6185:
6184:
6179:
6178:
6173:
6172:
6167:
6166:
6155:
6153:
6148:
6143:
6141:
6137:
6132:
6130:
6126:
6122:
6106:
6102:
6098:
6093:
6083:
6081:
6076:
6074:
6070:
6064:
6061:
6058:
6045:
6041:
6037:
6032:
6031:Coma (optics)
6022:
6020:
6015:
6013:
6009:
6004:
5994:
5991:
5985:
5973:
5968:
5966:
5961:
5959:
5954:
5953:
5951:
5950:
5945:
5942:
5940:
5937:
5935:
5932:
5930:
5927:
5925:
5922:
5920:
5917:
5915:
5912:
5910:
5907:
5906:
5905:
5904:
5901:
5898:
5897:
5885:
5882:
5879:
5860:
5856:
5849:
5844:
5840:
5836:
5830:
5824:
5816:
5812:
5801:
5798:
5795:
5792:
5791:
5787:
5784:
5781:
5765:
5761:
5757:
5749:
5745:
5734:
5731:
5717:
5714:
5709:
5705:
5697:
5694:
5693:
5690:
5688:
5686:
5669:
5662:
5660:
5646:
5643:
5638:
5634:
5626:
5623:
5622:
5619:
5616:
5613:
5599:
5596:
5593:
5588:
5584:
5580:
5570:
5567:
5553:
5550:
5547:
5542:
5538:
5534:
5531:
5524:
5521:
5520:
5517:
5514:
5511:
5497:
5494:
5491:
5486:
5482:
5474:
5471:
5457:
5454:
5451:
5446:
5442:
5434:
5431:
5430:
5427:
5424:
5421:
5407:
5404:
5401:
5396:
5392:
5388:
5385:
5378:
5375:
5361:
5358:
5355:
5350:
5346:
5342:
5332:
5329:
5328:
5324:
5321:
5318:
5315:
5312:
5309:
5306:
5305:
5289:
5284:
5280:
5272:
5267:
5261:
5257:
5253:
5249:
5240:
5236:
5234:
5230:
5226:
5220:
5216:
5210:
5203:
5193:
5187:
5155:
5146:
5142:
5134:
5116:
5114:
5110:
5106:
5102:
5093:
5090:| > 0
5088:
5080:
5072:
5064:
5056:
5049:
5043:
5028:
5011:
5007:
5003:
4998:
4995:
4984:
4980:
4976:
4973:
4969:
4964:
4957:
4953:
4947:
4943:
4937:
4934:
4931:
4923:
4921:
4920:
4919:magnification
4912:Magnification
4903:
4895:
4891:
4886:
4878:
4874:
4866:
4854:
4845:
4843:
4839:
4832:
4825:
4821:
4814:
4807:
4800:
4793:
4790:, expressing
4786:
4779:
4775:
4771:
4764:
4757:
4753:
4749:
4744:
4742:
4736:
4734:
4726:
4716:
4706:
4699:
4695:
4689:
4682:
4676:
4669:
4653:
4649:
4642:
4633:
4626:
4617:
4615:
4611:
4607:
4603:
4595:
4590:
4589:virtual image
4584:
4577:
4567:
4563:
4557:
4548:
4539:
4534:
4526:
4520:
4518:
4514:
4510:
4506:
4505:
4500:
4492:
4483:
4473:
4466:
4460:
4458:
4450:
4441:
4436:
4420:
4413:of the lens (
4412:
4396:
4374:
4370:
4347:
4343:
4330:
4325:
4321:
4300:
4296:
4291:
4286:
4282:
4278:
4272:
4267:
4262:
4258:
4246:
4242:
4237:
4233:
4229:
4223:
4216:
4212:
4206:
4202:
4187:
4185:
4170:
4155:
4140:
4124:
4108:
4104:
4081:
4077:
4054:
4050:
4027:
4023:
4000:
3996:
3973:
3969:
3946:
3942:
3922:
3918:
3915:
3910:
3906:
3902:
3897:
3893:
3872:
3869:
3864:
3860:
3856:
3851:
3847:
3837:
3824:
3818:
3814:
3808:
3804:
3800:
3795:
3791:
3782:
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:
18:Compound lens
9415:
9411:"Lens"
9367:
9340:
9303:
9299:
9280:
9258:
9246:Bibliography
9224:
9218:
9206:. Retrieved
9181:
9175:
9163:
9152:the original
9123:
9119:
9102:
9043:
9039:
9033:
8992:
8986:
8980:
8947:
8941:
8920:. Retrieved
8913:the original
8906:"The Axicon"
8899:
8880:
8874:
8862:. Retrieved
8854:10.61835/6as
8845:
8835:
8823:. Retrieved
8811:
8798:
8779:
8773:
8760:
8748:. Retrieved
8739:
8730:
8718:. Retrieved
8709:
8699:
8687:. Retrieved
8679:Hyperphysics
8678:
8668:
8649:
8621:
8591:
8573:, § 6.1
8568:, p. 14
8561:
8542:
8536:
8524:. Retrieved
8515:
8506:
8494:. Retrieved
8485:
8476:
8462:(2): 72–76.
8459:
8455:
8445:
8436:
8424:. Retrieved
8409:
8402:
8383:
8371:. Retrieved
8356:
8349:
8337:. Retrieved
8322:
8315:
8303:. Retrieved
8283:
8276:
8256:
8244:. Retrieved
8224:
8217:
8197:
8190:
8172:
8167:
8148:
8140:
8121:
8117:Aristophanes
8083:
8080:Neurosurgery
8079:
8073:
8064:
8054:
8042:. Retrieved
8033:
8024:
8012:. Retrieved
8003:
7993:
7960:
7956:
7950:
7924:. Retrieved
7915:
7886:
7872:. Retrieved
7853:
7844:
7839:
7743:Anti-fogging
7727:
7719:lens antenna
7705:
7685:
7626:
7598:contact lens
7567:
7560:
7544:
7532:metamaterial
7518:
7512:
7502:into a line
7500:point source
7489:
7482:
7478:Bifocal lens
7476:
7462:
7458:Fresnel lens
7455:
7450:Fresnel lens
7432:
7418:
7392:
7391:> 0 >
7385:
7375:
7365:
7362:
7307:
7296:
7288:
7281:
7277:
7274:
7259:
7150:
7127:
7058:
7048:
7044:
6972:
6939:
6762:
6758:
6581:
6528:
6453:
6369:
6246:
6242:
6237:
6231:
6224:Triplet lens
6193:
6181:
6175:
6169:
6163:
6161:
6144:
6133:
6124:
6096:
6095:
6077:
6072:
6068:
6062:
6056:
6039:
6035:
6034:
6016:
6007:
6006:
5989:
5987:
5287:
5282:
5270:
5265:
5259:
5255:
5251:
5247:
5218:
5214:
5208:
5201:
5191:
5188:
5153:
5138:
5132:
5117:
5094:
5086:
5078:
5070:
5062:
5054:
5047:
5044:
5029:
4924:
4917:
4915:
4893:
4889:
4876:
4872:
4841:
4837:
4830:
4823:
4819:
4812:
4811:in terms of
4805:
4798:
4797:in terms of
4791:
4784:
4777:
4773:
4769:
4762:
4755:
4751:
4747:
4745:
4732:
4724:
4714:
4704:
4693:
4687:
4680:
4674:
4667:
4660:
4651:
4631:
4609:
4608:to create a
4593:
4575:
4572:
4555:
4546:
4537:
4524:
4521:
4502:
4499:image sensor
4490:
4481:
4471:
4464:
4461:
4448:
4439:
4334:
4328:
4188:
4172:
4157:
4142:
4127:
4125:
3838:
3783:
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:
9427:Simulations
9373:Article on
8922:22 November
8044:25 November
7819:Ray tracing
7721:to refract
7641:microscopes
7618:astigmatism
7590:astigmatism
7570:prosthetics
7520:Superlenses
7468:microlenses
7424:laser diode
6196:diffraction
6183:astigmatism
6140:Abbe number
5924:Astigmatism
5892:Aberrations
5313:Image Type
5225:diffraction
5184:0.5 mm
5158:−50 mm
5105:plate scale
5084:> |
5068:< |
4916:The linear
4844:is zero.)
4741:Barlow lens
4700:real object
4648: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
9476:Categories
9351:(.pdf) on
8571:Hecht 1987
8122:The Clouds
7942:References
7671:to make a
7657:real image
7649:projectors
7637:telescopes
7633:binoculars
7629:monoculars
7602:eyeglasses
7586:presbyopia
7508:optic axis
7438:aberration
6214:See also:
6129:apochromat
6101:dispersion
5990:aberration
5934:Distortion
5883:Magnified
5785:Magnified
5617:Magnified
5515:Same size
5307:Lens Type
5149:50 mm
5137:. But the
5124:20 cm
4902:real image
4610:real image
4504:real image
4329:real image
4042:. 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
9184:(5): 22.
9126:(1): 93.
9078:0036-8075
9056:CiteSeerX
7985:191384703
7321:−
7213:−
7080:∑
6891:−
6870:−
6836:→
6820:−
6710:−
6689:−
6655:→
6639:−
6557:−
6501:−
5796:Anywhere
5673:∞
5670:±
5614:Inverted
5578:∞
5512:Inverted
5425:Minified
5340:∞
5128:5 cm
5120:5 cm
4999:−
4977:−
4938:−
4698:) with a
4533:defocused
4513:human eye
3916:−
3870:−
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
9462:Archived
9449:Archived
9436:Archived
9399:Archived
9378:Archived
9356:Archived
9345:Archived
9331:Archived
9306:. SPIE.
9255:(1987).
9202:Archived
9198:26336702
9148:39903291
9111:(2010).
9094:20978013
9086:18703734
9017:18690249
8972:15089166
8864:29 April
8858:Archived
8825:19 April
8816:Archived
8750:17 March
8744:Archived
8720:17 March
8714:Archived
8689:17 March
8683:Archived
8520:Archived
8490:Archived
8391:Archived
8299:Archived
8265:Archived
8246:24 April
8240:Archived
8180:Archived
8156:Archived
8130:Archived
8038:Archived
8008:Archived
8004:BBC News
7926:21 April
7920:Archived
7774:F-number
7769:Eyepiece
7736:See also
7730:Abrasion
7681:meniscus
7614:eyeballs
7576:such as
7540:infrared
7538:or near-
6200:aperture
6136:fluorite
6125:achromat
6075:lenses.
6073:bestform
5799:Virtual
5279:infinity
5166: km
4900:). This
4632: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
9391:YouTube
9128:Bibcode
9048:Bibcode
9040:Science
9025:4314138
8997:Bibcode
8952:Bibcode
8100:9574655
7874:28 June
7677:Schmidt
7645:cameras
7592:. (See
7536:visible
7496:conical
6474:, then
5909:Defocus
5325:Remark
5061:0 <
4612:on the
4433:is the
2166:- Sign
2163:+ Sign
2133:concave
1744:is the
921:Moving
854:), and
516:concave
508:concave
440:concave
409:England
371:Germany
334:Alhazen
320:Ptolemy
306:concave
302:emerald
278:' play
165:History
151:glasses
147:cameras
96:plastic
60:optical
9482:Lenses
9468:by QED
9310:
9287:
9281:Optics
9267:
9259:Optics
9208:16 May
9196:
9146:
9092:
9084:
9076:
9058:
9023:
9015:
8988:Nature
8970:
8887:
8881:Optics
8786:
8780:Optics
8656:
8650:Optics
8628:
8622:Optics
8598:
8592:Optics
8549:
8543:Optics
8526:2 July
8496:2 July
8426:6 June
8417:
8373:6 June
8364:
8339:6 June
8330:
8305:6 June
8291:
8232:
8205:
8098:
8014:10 May
7983:
7977:505216
7975:
7899:
7865:
7663:or an
7588:, and
7578:myopia
7494:has a
7492:axicon
7381:> 0
7371:> 0
7128:where
7045:Since
6226:, and
6180:, and
6171:barrel
5880:Erect
5281:; and
5221:/∞ = 0
5074:|
5057:< 0
5050:> 0
5030:where
4993:
4737:before
4720:< 0
4710:> 0
4696:< 0
4614:retina
4585:appear
4517:retina
4509:camera
4457:medium
4389:, and
3988:, and
3839:where
2443:
2430:
2415:
2402:
2363:
2357:
2129:> 0
2119:< 0
2109:convex
2105:< 0
2095:> 0
1737:where
1722:
1691:
1683:
1677:
1659:
1647:
1634:
1619:
1606:
1571:
1565:
1411:, and
808:where
706:Desmos
500:convex
496:convex
445:planar
434:convex
344:Optics
325:Optics
254:lentil
155:myopia
145:, and
108:molded
100:ground
9155:(PDF)
9144:S2CID
9116:(PDF)
9090:S2CID
9021:S2CID
8916:(PDF)
8909:(PDF)
8819:(PDF)
8808:(PDF)
7981:S2CID
7973:JSTOR
7845:lense
7831:Notes
7754:Bokeh
7711:radar
7504:along
6044:comet
6038:, or
5568:Real
5472:Real
5180:1°/mm
4892:>
4673:<
4470:>
3645:image
1761:below
547:focus
456:Toric
130:, or
116:prism
112:image
106:, or
92:glass
9308:ISBN
9304:FG01
9285:ISBN
9265:ISBN
9210:2015
9194:PMID
9082:PMID
9074:ISSN
9013:PMID
8968:PMID
8924:2007
8885:ISBN
8866:2024
8827:2024
8784:ISBN
8752:2015
8722:2015
8691:2015
8654:ISBN
8626:ISBN
8596:ISBN
8547:ISBN
8528:2023
8498:2023
8428:2012
8415:ISBN
8375:2012
8362:ISBN
8341:2012
8328:ISBN
8307:2012
8289:ISBN
8248:2011
8230:ISBN
8203:ISBN
8096:PMID
8046:2012
8016:2008
7928:2018
7897:ISBN
7876:2009
7863:ISBN
7709:and
7679:and
7647:and
7549:Uses
7506:the
6337:and
6277:and
6202:. A
6174:and
6123:(or
6036:Coma
6025:Coma
5929:Coma
5914:Tilt
5850:>
5825:<
5758:>
5715:<
5594:>
5581:>
5548:<
5535:<
5402:<
5389:<
5356:>
5343:>
5285:(at
5273:(at
5262:(at
5254:and
5212:and
5170:−1.3
5145:moon
4829:(or
4171:and
4141:and
4096:and
3885:and
3685:and
3668:For
2322:and
2121:and
2097:and
2051:and
1830:and
898:and
490:(or
449:axis
369:and
294:Nero
248:lēns
241:lens
232:LSST
173:and
157:and
88:axis
70:. A
56:lens
9389:on
9231:doi
9186:doi
9182:312
9136:doi
9124:100
9066:doi
9044:321
9005:doi
8993:455
8960:doi
8850:doi
8464:doi
8088:doi
7965:doi
7893:368
7859:125
7688:sun
7490:An
7260:As
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5217:= −
5197:= ∞
5164:000
5162:380
5135:= ∞
5111:or
4768:= 2
4446:or
2670:and
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2285:).
2278:to
1360:sin
1318:tan
1276:tan
1240:sin
1218:sin
624:or
553:or
530:or
522:or
407:in
403:by
198:.
94:or
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3268:R
3264:1
3252:1
3248:R
3244:1
3239:(
3236:)
3231:1
3227:n
3218:2
3214:n
3210:(
3207:=
3202:f
3197:1
3193:n
3159:u
3139:f
3117:.
3114:)
3107:2
3103:R
3099:1
3087:1
3083:R
3079:1
3074:(
3071:)
3066:1
3062:n
3053:2
3049:n
3045:(
3042:=
3037:v
3032:1
3028:n
3022:+
3017:u
3012:1
3008:n
2983:0
2977:d
2957:.
2947:v
2943:)
2940:d
2930:v
2926:(
2921:d
2916:2
2912:n
2905:+
2902:)
2895:2
2891:R
2887:1
2875:1
2871:R
2867:1
2862:(
2859:)
2854:1
2850:n
2841:2
2837:n
2833:(
2830:=
2825:v
2820:1
2816:n
2810:+
2805:u
2800:1
2796:n
2773:.
2766:2
2762:R
2755:2
2751:n
2742:1
2738:n
2731:=
2726:v
2721:1
2717:n
2711:+
2705:d
2702:+
2695:v
2685:2
2681:n
2658:d
2655:+
2648:v
2641:=
2634:u
2613:.
2606:1
2602:R
2595:1
2591:n
2582:2
2578:n
2571:=
2562:v
2556:2
2552:n
2546:+
2541:u
2536:1
2532:n
2515:O
2499:1
2495:V
2455:.
2450:]
2438:2
2434:R
2426:1
2410:1
2406:R
2398:1
2392:[
2387:)
2383:1
2377:n
2373:(
2360:f
2353:1
2339:f
2328:2
2325:R
2319:1
2316:R
2311:d
2282:o
2280:y
2275:i
2273:y
2266:T
2264:M
2247:i
2245:y
2228:o
2226:y
2210:f
2193:i
2191:s
2174:o
2172:s
2127:2
2124:R
2117:1
2114:R
2103:2
2100:R
2093:1
2090:R
2085:R
2078:R
2057:2
2054:R
2048:1
2045:R
2020:f
2011:f
1993:n
1988:1
1984:R
1978:d
1975:)
1972:1
1966:n
1963:(
1960:f
1951:=
1946:2
1942:h
1919:n
1914:2
1910:R
1904:d
1901:)
1898:1
1892:n
1889:(
1886:f
1877:=
1872:1
1868:h
1843:2
1839:h
1816:1
1812:h
1797:f
1782:d
1775:2
1772:R
1756:1
1753:R
1742:n
1725:,
1718:]
1709:2
1705:R
1699:1
1695:R
1688:n
1680:d
1674:)
1671:1
1665:n
1662:(
1653:+
1642:2
1638:R
1630:1
1614:1
1610:R
1602:1
1596:[
1592:)
1589:1
1583:n
1580:(
1577:=
1568:f
1561:1
1533:f
1498:.
1492:R
1486:1
1482:n
1473:2
1469:n
1462:=
1457:u
1452:1
1448:n
1442:+
1437:v
1432:2
1428:n
1413:θ
1409:r
1405:i
1381:R
1378:h
1373:=
1351:v
1348:h
1343:=
1336:)
1333:r
1324:(
1309:u
1306:h
1301:=
1294:)
1285:i
1282:(
1250:.
1246:r
1235:2
1231:n
1227:=
1224:i
1213:1
1209:n
1151:R
1143:1
1139:n
1130:2
1126:n
1119:2
1115:n
1109:=
1100:i
1096:f
1088:,
1085:R
1077:1
1073:n
1064:2
1060:n
1053:1
1049:n
1043:=
1034:0
1030:f
1001:i
997:f
976:u
954:0
950:f
929:v
906:v
886:u
876:h
862:v
852:h
838:u
827:1
824:n
818:2
815:n
810:R
792:R
786:1
782:n
773:2
769:n
762:=
757:v
752:2
748:n
742:+
737:u
732:1
728:n
567:f
336:(
205:)
201:(
34:.
20:)
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