Metrogon is a high resolution, low-distortion, extra-wide field (90 degree field of view ) photographic lens design, popularized by Bausch and Lomb . Variations of this design were used extensively by the US military for aerial photography , fitted to the T-11 and other aerial cameras.
90-532: The most common Metrogon lenses have a f number of 6.3 and a focal length of 6 inches. The company name (Bausch and Lomb) and the US Patent number (2031792) are prominently inscribed on the front of the lens barrel. However, this patent is for a family of symmetric wide-angle lenses designed by Robert Richter of Carl Zeiss AG , which was filed in 1934 and sold by Zeiss as the Topogon . For this reason, it
180-458: A T-stop of 2.3: T = 2.0 0.75 = 2.309... {\displaystyle T={\frac {2.0}{\sqrt {0.75}}}=2.309...} Since real lenses have transmittances of less than 100%, a lens's T-stop number is always greater than its f-number. With 8% loss per air-glass surface on lenses without coating, multicoating of lenses is the key in lens design to decrease transmittance losses of lenses. Some reviews of lenses do measure
270-405: A biconcave or plano-concave lens in a lower-index medium, a collimated beam of light passing through the lens is diverged (spread); the lens is thus called a negative or diverging lens. The beam, after passing through the lens, appears to emanate from a particular point on the axis in front of the lens. For a thin lens in air, the distance from this point to the lens is the focal length, though it
360-458: A doubling of sensitivity is represented by a doubling of the number, and a logarithmic number. In the ISO system, a 3° increase in the logarithmic number corresponds to a doubling of sensitivity. Doubling or halving the sensitivity is equal to a difference of one T-stop in terms of light transmittance. Most electronic cameras allow to amplify the signal coming from the pickup element. This amplification
450-490: A factor of one-half. The one-stop unit is also known as the EV ( exposure value ) unit. On a camera, the aperture setting is traditionally adjusted in discrete steps, known as f-stops . Each " stop " is marked with its corresponding f-number, and represents a halving of the light intensity from the previous stop. This corresponds to a decrease of the pupil and aperture diameters by a factor of 1/ √ 2 or about 0.7071, and hence
540-603: A few conventional differences in their numbers ( 1 ⁄ 15 , 1 ⁄ 30 , and 1 ⁄ 60 second instead of 1 ⁄ 16 , 1 ⁄ 32 , and 1 ⁄ 64 ). In practice the maximum aperture of a lens is often not an integral power of √ 2 (i.e., √ 2 to the power of a whole number), in which case it is usually a half or third stop above or below an integral power of √ 2 . Modern electronically controlled interchangeable lenses, such as those used for SLR cameras, have f-stops specified internally in 1 ⁄ 8 -stop increments, so
630-435: A film twice as sensitive, has the same effect on the exposed image. For all practical purposes extreme accuracy is not required (mechanical shutter speeds were notoriously inaccurate as wear and lubrication varied, with no effect on exposure). It is not significant that aperture areas and shutter speeds do not vary by a factor of precisely two. Photographers sometimes express other exposure ratios in terms of 'stops'. Ignoring
720-642: A focus. This led to the invention of the compound achromatic lens by Chester Moore Hall in England in 1733, an invention also claimed by fellow Englishman John Dollond in a 1758 patent. Developments in transatlantic commerce were the impetus for the construction of modern lighthouses in the 18th century, which utilize a combination of elevated sightlines, lighting sources, and lenses to provide navigational aid overseas. With maximal distance of visibility needed in lighthouses, conventional convex lenses would need to be significantly sized which would negatively affect
810-412: A great deal of experimentation with lens shapes in the 17th and early 18th centuries by those trying to correct chromatic errors seen in lenses. Opticians tried to construct lenses of varying forms of curvature, wrongly assuming errors arose from defects in the spherical figure of their surfaces. Optical theory on refraction and experimentation was showing no single-element lens could bring all colours to
900-581: A half stop ( 1 ⁄ 2 EV) series would be ( 2 ) 0 2 , ( 2 ) 1 2 , ( 2 ) 2 2 , ( 2 ) 3 2 , ( 2 ) 4 2 , … {\displaystyle ({\sqrt {2}})^{\frac {0}{2}},\ ({\sqrt {2}})^{\frac {1}{2}},\ ({\sqrt {2}})^{\frac {2}{2}},\ ({\sqrt {2}})^{\frac {3}{2}},\ ({\sqrt {2}})^{\frac {4}{2}},\ \ldots } The steps in
990-529: A half-stop or a one-third-stop system; sometimes f /1.3 and f /3.2 and other differences are used for the one-third stop scale. An H-stop (for hole, by convention written with capital letter H) is an f-number equivalent for effective exposure based on the area covered by the holes in the diffusion discs or sieve aperture found in Rodenstock Imagon lenses. A T-stop (for transmission stops, by convention written with capital letter T)
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#17327909226411080-406: A halving of the area of the pupil. Most modern lenses use a standard f-stop scale, which is an approximately geometric sequence of numbers that corresponds to the sequence of the powers of the square root of 2 : f /1 , f /1.4 , f /2 , f /2.8 , f /4 , f /5.6 , f /8 , f /11 , f /16 , f /22 , f /32 , f /45 , f /64 , f /90 , f /128 , etc. Each element in
1170-537: A lens in air, f is then given by 1 f ≈ ( n − 1 ) [ 1 R 1 − 1 R 2 ] . {\displaystyle \ {\frac {1}{\ f\ }}\approx \left(n-1\right)\left[\ {\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}\ \right]~.} The spherical thin lens equation in paraxial approximation
1260-558: A lower f-number is "opening up" the lens. Selecting a higher f-number is "closing" or "stopping down" the lens. Depth of field increases with f-number, as illustrated in the image here. This means that photographs taken with a low f-number (large aperture) will tend to have subjects at one distance in focus, with the rest of the image (nearer and farther elements) out of focus. This is frequently used for nature photography and portraiture because background blur (the aesthetic quality known as ' bokeh ') can be aesthetically pleasing and puts
1350-579: A magnifying glass, or a burning glass. Others have suggested that certain Egyptian hieroglyphs depict "simple glass meniscal lenses". The oldest certain reference to the use of lenses is from Aristophanes ' play The Clouds (424 BCE) mentioning a burning-glass. Pliny the Elder (1st century) confirms that burning-glasses were known in the Roman period. Pliny also has the earliest known reference to
1440-545: A result, smaller formats will have a deeper field than larger formats at the same f-number for the same distance of focus and same angle of view since a smaller format requires a shorter focal length (wider angle lens) to produce the same angle of view, and depth of field increases with shorter focal lengths. Therefore, reduced–depth-of-field effects will require smaller f-numbers (and thus potentially more difficult or complex optics) when using small-format cameras than when using larger-format cameras. Beyond focus, image sharpness
1530-493: A single photograph. By 1944, the military had charted 3,000,000 sq mi (7,800,000 km) in a single year using the tri- Metrogon , a triple aerial camera system fitted to the noses of aircraft including the P-38 Lightning and B-25 Mitchell . In 1943, Bausch and Lomb was granted an independent patent for a similar lens design with 5 elements and the same f /6.3 maximum aperture, showing less distortion than
1620-831: A spherical lens in air or vacuum for paraxial rays can be calculated from the lensmaker's equation : 1 f = ( n − 1 ) [ 1 R 1 − 1 R 2 + ( n − 1 ) d n R 1 R 2 ] , {\displaystyle {\frac {1}{\ f\ }}=\left(n-1\right)\left[\ {\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}+{\frac {\ \left(n-1\right)\ d~}{\ n\ R_{1}\ R_{2}\ }}\ \right]\ ,} where The focal length f {\textstyle \ f\ }
1710-575: A third stop ( 1 ⁄ 3 EV) series would be ( 2 ) 0 3 , ( 2 ) 1 3 , ( 2 ) 2 3 , ( 2 ) 3 3 , ( 2 ) 4 3 , … {\displaystyle ({\sqrt {2}})^{\frac {0}{3}},\ ({\sqrt {2}})^{\frac {1}{3}},\ ({\sqrt {2}})^{\frac {2}{3}},\ ({\sqrt {2}})^{\frac {3}{3}},\ ({\sqrt {2}})^{\frac {4}{3}},\ \ldots } As in
1800-403: Is biconvex (or double convex , or just convex ) if both surfaces are convex . If both surfaces have the same radius of curvature, the lens is equiconvex . A lens with two concave surfaces is biconcave (or just concave ). If one of the surfaces is flat, the lens is plano-convex or plano-concave depending on the curvature of the other surface. A lens with one convex and one concave side
1890-407: Is convex-concave or meniscus . Convex-concave lenses are most commonly used in corrective lenses , since the shape minimizes some aberrations. For a biconvex or plano-convex lens in a lower-index medium, a collimated beam of light passing through the lens converges to a spot (a focus ) behind the lens. In this case, the lens is called a positive or converging lens. For a thin lens in air,
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#17327909226411980-445: Is h ), and v {\textstyle v} is the on-axis image distance from the line. Due to paraxial approximation where the line of h is close to the vertex of the spherical surface meeting the optical axis on the left, u {\textstyle u} and v {\textstyle v} are also considered distances with respect to the vertex. Moving v {\textstyle v} toward
2070-431: Is also known as the focal ratio , f-ratio , or f-stop , and it is key in determining the depth of field , diffraction , and exposure of a photograph. The f-number is dimensionless and is usually expressed using a lower-case hooked f with the format f / N , where N is the f-number. The f-number is also known as the inverse relative aperture , because it is the inverse of the relative aperture , defined as
2160-538: Is an f-number adjusted to account for light transmission efficiency ( transmittance ). A lens with a T-stop of N projects an image of the same brightness as an ideal lens with 100% transmittance and an f-number of N . A particular lens's T-stop, T , is given by dividing the f-number by the square root of the transmittance of that lens: T = N transmittance . {\displaystyle T={\frac {N}{\sqrt {\text{transmittance}}}}.} For example, an f /2.0 lens with transmittance of 75% has
2250-602: Is believed the Metrogon lenses marked with this patent are a licensed version of the popular and very similar (if not identical) Topogon design. An aerial camera fitted with a Metrogon lens deployed by the United States Army Air Corps was featured in a 1941 article in Popular Science , which noted the lens gave the camera a 93° field of view, doubling the area that could be captured in
2340-416: Is completely round. When used in novelty photography it is often called a "lensball". A ball-shaped lens has the advantage of being omnidirectional, but for most optical glass types, its focal point lies close to the ball's surface. Because of the ball's curvature extremes compared to the lens size, optical aberration is much worse than thin lenses, with the notable exception of chromatic aberration . For
2430-466: Is customary to write f-numbers preceded by " f / ", which forms a mathematical expression of the entrance pupil's diameter in terms of f and N . For example, if a lens's focal length were 100 mm and its entrance pupil's diameter were 50 mm , the f-number would be 2. This would be expressed as " f /2 " in a lens system. The aperture diameter would be equal to f /2 . Camera lenses often include an adjustable diaphragm , which changes
2520-794: Is derived here with respect to the right figure. The 1st spherical lens surface (which meets the optical axis at V 1 {\textstyle \ V_{1}\ } as its vertex) images an on-axis object point O to the virtual image I , which can be described by the following equation, n 1 u + n 2 v ′ = n 2 − n 1 R 1 . {\displaystyle \ {\frac {\ n_{1}\ }{\ u\ }}+{\frac {\ n_{2}\ }{\ v'\ }}={\frac {\ n_{2}-n_{1}\ }{\ R_{1}\ }}~.} For
2610-460: Is further along in the direction of the ray travel (right, in the accompanying diagrams), while negative R means that rays reaching the surface have already passed the center of curvature. Consequently, for external lens surfaces as diagrammed above, R 1 > 0 and R 2 < 0 indicate convex surfaces (used to converge light in a positive lens), while R 1 < 0 and R 2 > 0 indicate concave surfaces. The reciprocal of
2700-504: Is made possible by an error correction system which includes secondary and tertiary mirrors, a three element refractive system and active mounting and optics. The camera equation, or G#, is the ratio of the radiance reaching the camera sensor to the irradiance on the focal plane of the camera lens : G # = 1 + 4 N 2 τ π , {\displaystyle G\#={\frac {1+4N^{2}}{\tau \pi }}\,,} where τ
2790-407: Is negative with respect to the focal length of a converging lens. The behavior reverses when a lens is placed in a medium with higher refractive index than the material of the lens. In this case a biconvex or plano-convex lens diverges light, and a biconcave or plano-concave one converges it. Convex-concave (meniscus) lenses can be either positive or negative, depending on the relative curvatures of
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2880-433: Is positive for converging lenses, and negative for diverging lenses. The reciprocal of the focal length, 1 f , {\textstyle \ {\tfrac {1}{\ f\ }}\ ,} is the optical power of the lens. If the focal length is in metres, this gives the optical power in dioptres (reciprocal metres). Lenses have the same focal length when light travels from
2970-443: Is related to f-number through two different optical effects: aberration , due to imperfect lens design, and diffraction which is due to the wave nature of light. The blur-optimal f-stop varies with the lens design. For modern standard lenses having 6 or 7 elements, the sharpest image is often obtained around f /5.6 – f /8 , while for older standard lenses having only 4 elements ( Tessar formula ) stopping to f /11 will give
3060-428: Is the aperture setting that limits the brightness of the image by restricting the input pupil size, while a field stop is a stop intended to cut out light that would be outside the desired field of view and might cause flare or other problems if not stopped. In photography, stops are also a unit used to quantify ratios of light or exposure, with each added stop meaning a factor of two, and each subtracted stop meaning
3150-400: Is the radius of the spherical surface, n 2 is the refractive index of the material of the surface, n 1 is the refractive index of medium (the medium other than the spherical surface material), u {\textstyle u} 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
3240-519: Is the transmission coefficient of the lens, and the units are in inverse steradians (sr ). Lens (optics) A lens is a transmissive optical device that focuses or disperses a light beam by means of refraction . A simple lens consists of a single piece of transparent material , while a compound lens consists of several simple lenses ( elements ), usually arranged along a common axis . Lenses are made from materials such as glass or plastic and are ground , polished , or molded to
3330-429: Is usually called gain and is measured in decibels. Every 6 dB of gain is equivalent to one T-stop in terms of light transmittance. Many camcorders have a unified control over the lens f-number and gain. In this case, starting from zero gain and fully open iris, one can either increase f-number by reducing the iris size while gain remains zero, or one can increase gain while iris remains fully open. An example of
3420-1135: Is with respect to the principal planes of the lens, and the locations of the principal planes h 1 {\textstyle \ h_{1}\ } and h 2 {\textstyle \ h_{2}\ } 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. h 1 = − ( n − 1 ) f d n R 2 {\displaystyle \ h_{1}=-\ {\frac {\ \left(n-1\right)f\ d~}{\ n\ R_{2}\ }}\ } h 2 = − ( n − 1 ) f d n R 1 {\displaystyle \ h_{2}=-\ {\frac {\ \left(n-1\right)f\ d~}{\ n\ R_{1}\ }}\ } The focal length f {\displaystyle \ f\ }
3510-497: The Wild Aviotar and Zeiss Biogon in the 1950s rendered the Metrogon obsolete, and Metrogon lenses were sold to the public as surplus starting in the early 1960s. F number An f-number is a measure of the light-gathering ability of an optical system such as a camera lens . It is calculated by dividing the system's focal length by the diameter of the entrance pupil ("clear aperture "). The f-number
3600-479: The field of view of the instrument and the scale of the image that is presented at the focal plane to an eyepiece , film plate, or CCD . For example, the SOAR 4-meter telescope has a small field of view (about f /16 ) which is useful for stellar studies. The LSST 8.4 m telescope, which will cover the entire sky every three days, has a very large field of view. Its short 10.3 m focal length ( f /1.2 )
3690-480: The 11th and 13th century " reading stones " were invented. These were primitive plano-convex lenses initially made by cutting a glass sphere in half. The medieval (11th or 12th century) rock crystal Visby lenses may or may not have been intended for use as burning glasses. Spectacles were invented as an improvement of the "reading stones" of the high medieval period in Northern Italy in the second half of
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3780-512: The 13th century. This was the start of the optical industry of grinding and polishing lenses for spectacles, first in Venice and Florence in the late 13th century, and later in the spectacle-making centres in both the Netherlands and Germany . Spectacle makers created improved types of lenses for the correction of vision based more on empirical knowledge gained from observing the effects of
3870-979: The Gaussian thin lens equation is 1 u + 1 v = 1 f . {\displaystyle \ {\frac {1}{\ u\ }}+{\frac {1}{\ v\ }}={\frac {1}{\ f\ }}~.} For the thin lens in air or vacuum where n 1 = 1 {\textstyle \ n_{1}=1\ } can be assumed, f {\textstyle \ f\ } becomes 1 f = ( n − 1 ) ( 1 R 1 − 1 R 2 ) {\displaystyle \ {\frac {1}{\ f\ }}=\left(n-1\right)\left({\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}\right)\ } where
3960-484: The Latin name of the lentil (a seed of a lentil plant), because a double-convex lens is lentil-shaped. The lentil also gives its name to a geometric figure . Some scholars argue that the archeological evidence indicates that there was widespread use of lenses in antiquity, spanning several millennia. The so-called Nimrud lens is a rock crystal artifact dated to the 7th century BCE which may or may not have been used as
4050-471: The Latin translation of an incomplete and very poor Arabic translation. The book was, however, received by medieval scholars in the Islamic world, and commented upon by Ibn Sahl (10th century), who was in turn improved upon by Alhazen ( Book of Optics , 11th century). The Arabic translation of Ptolemy's Optics became available in Latin translation in the 12th century ( Eugenius of Palermo 1154). Between
4140-598: The T-stop or transmission rate in their benchmarks. T-stops are sometimes used instead of f-numbers to more accurately determine exposure, particularly when using external light meters . Lens transmittances of 60%–95% are typical. T-stops are often used in cinematography, where many images are seen in rapid succession and even small changes in exposure will be noticeable. Cinema camera lenses are typically calibrated in T-stops instead of f-numbers. In still photography, without
4230-462: The amount of light admitted by the f /2 lens is four times that of the f /4 lens. To obtain the same photographic exposure , the exposure time must be reduced by a factor of four. A 200 mm focal length f /4 lens has an entrance pupil diameter of 50 mm . The 200 mm lens's entrance pupil has four times the area of the 100 mm f /4 lens's entrance pupil, and thus collects four times as much light from each object in
4320-412: The aperture diameter divided by focal length. The relative aperture indicates how much light can pass through the lens at a given focal length. A lower f-number means a larger relative aperture and more light entering the system, while a higher f-number means a smaller relative aperture and less light entering the system. The f-number is related to the numerical aperture (NA) of the system, which measures
4410-407: The aperture scale usually had a click stop at every whole and half stop. On modern cameras, especially when aperture is set on the camera body, f-number is often divided more finely than steps of one stop. Steps of one-third stop ( 1 ⁄ 3 EV) are the most common, since this matches the ISO system of film speeds . Half-stop steps are used on some cameras. Usually the full stops are marked, and
4500-405: The back to the front as when light goes from the front to the back. Other properties of the lens, such as the aberrations are not the same in both directions. The signs of the lens' radii of curvature indicate whether the corresponding surfaces are convex or concave. The sign convention used to represent this varies, but in this article a positive R indicates a surface's center of curvature
4590-432: The cameras' 1 ⁄ 3 -stop settings are approximated by the nearest 1 ⁄ 8 -stop setting in the lens. Including aperture value AV: N = 2 AV {\displaystyle N={\sqrt {2^{\text{AV}}}}} Conventional and calculated f-numbers, full-stop series: Sometimes the same number is included on several scales; for example, an aperture of f /1.2 may be used in either
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#17327909226414680-421: 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. Most lenses are spherical lenses : their two surfaces are parts of the surfaces of spheres. Each surface can be convex (bulging outwards from
4770-405: The diameter of an aperture stop in the system: N = f D → × D f = N D {\displaystyle N={\frac {f}{D}}\quad {\xrightarrow {\times D}}\quad f=ND} Even though the principles of focal ratio are always the same, the application to which the principle is put can differ. In photography the focal ratio varies
4860-403: The distance from the lens to the spot is the focal length of the lens, which is commonly represented by f in diagrams and equations. An extended hemispherical lens is a special type of plano-convex lens, in which the lens's curved surface is a full hemisphere and the lens is much thicker than the radius of curvature. Another extreme case of a thick convex lens is a ball lens , whose shape
4950-1116: The earlier DIN and ASA film-speed standards, the ISO speed is defined only in one-third stop increments, and shutter speeds of digital cameras are commonly on the same scale in reciprocal seconds. A portion of the ISO range is the sequence … 16 / 13 ∘ , 20 / 14 ∘ , 25 / 15 ∘ , 32 / 16 ∘ , 40 / 17 ∘ , 50 / 18 ∘ , 64 / 19 ∘ , 80 / 20 ∘ , 100 / 21 ∘ , 125 / 22 ∘ , … {\displaystyle \ldots 16/13^{\circ },\ 20/14^{\circ },\ 25/15^{\circ },\ 32/16^{\circ },\ 40/17^{\circ },\ 50/18^{\circ },\ 64/19^{\circ },\ 80/20^{\circ },\ 100/21^{\circ },\ 125/22^{\circ },\ \ldots } while shutter speeds in reciprocal seconds have
5040-416: The edges for large apertures. Photojournalists have a saying, " f /8 and be there ", meaning that being on the scene is more important than worrying about technical details. Practically, f /8 (in 35 mm and larger formats) allows adequate depth of field and sufficient lens speed for a decent base exposure in most daylight situations. Computing the f-number of the human eye involves computing
5130-426: The effect of the lens' thickness. For a single refraction for a circular boundary, the relation between object and its image in the paraxial approximation is given by n 1 u + n 2 v = n 2 − n 1 R {\displaystyle {\frac {n_{1}}{u}}+{\frac {n_{2}}{v}}={\frac {n_{2}-n_{1}}{R}}} where R
5220-403: The f-number markings, the f-stops make a logarithmic scale of exposure intensity. Given this interpretation, one can then think of taking a half-step along this scale, to make an exposure difference of a "half stop". Most twentieth-century cameras had a continuously variable aperture, using an iris diaphragm , with each full stop marked. Click-stopped aperture came into common use in the 1960s;
5310-413: The focal-plane illuminance (or optical power per unit area in the image) and is used to control variables such as depth of field . When using an optical telescope in astronomy, there is no depth of field issue, and the brightness of stellar point sources in terms of total optical power (not divided by area) is a function of absolute aperture area only, independent of focal length. The focal length controls
5400-1499: The imaging by second lens surface, by taking the above sign convention, u ′ = − v ′ + d {\textstyle \ u'=-v'+d\ } and n 2 − v ′ + d + n 1 v = n 1 − n 2 R 2 . {\displaystyle \ {\frac {n_{2}}{\ -v'+d\ }}+{\frac {\ n_{1}\ }{\ v\ }}={\frac {\ n_{1}-n_{2}\ }{\ R_{2}\ }}~.} Adding these two equations yields n 1 u + n 1 v = ( n 2 − n 1 ) ( 1 R 1 − 1 R 2 ) + n 2 d ( v ′ − d ) v ′ . {\displaystyle \ {\frac {\ n_{1}\ }{u}}+{\frac {\ n_{1}\ }{v}}=\left(n_{2}-n_{1}\right)\left({\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}\right)+{\frac {\ n_{2}\ d\ }{\ \left(\ v'-d\ \right)\ v'\ }}~.} For
5490-592: The intermediate positions click but are not marked. As an example, the aperture that is one-third stop smaller than f /2.8 is f /3.2 , two-thirds smaller is f /3.5 , and one whole stop smaller is f /4 . The next few f-stops in this sequence are: f / 4.5 , f / 5 , f / 5.6 , f / 6.3 , f / 7.1 , f / 8 , … {\displaystyle f/4.5,\ f/5,\ f/5.6,\ f/6.3,\ f/7.1,\ f/8,\ \ldots } To calculate
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#17327909226415580-549: The lens in the Carl Zeiss patent. The Bausch and Lomb patent also compares the distortion of their design favorably to a similar 5 element lens, patented in 1938, which has a slightly wider maximum f-number of f /5.6 . It is not certain whether Bausch and Lomb incorporated their own design instead of the Zeiss design when producing Metrogon lenses after 1943. The introduction of faster lenses with equivalent coverage including
5670-480: The lens's field of view. But compared to the 100 mm lens, the 200 mm lens projects an image of each object twice as high and twice as wide, covering four times the area, and so both lenses produce the same illuminance at the focal plane when imaging a scene of a given luminance. The word stop is sometimes confusing due to its multiple meanings. A stop can be a physical object: an opaque part of an optical system that blocks certain rays. The aperture stop
5760-417: The lens), concave (depressed into the lens), or planar (flat). The line joining the centres of the spheres making up the lens surfaces is called the axis of the lens. Typically the lens axis passes through the physical centre of the lens, because of the way they are manufactured. Lenses may be cut or ground after manufacturing to give them a different shape or size. The lens axis may then not pass through
5850-399: The lens. These two cases are examples of image formation in lenses. In the former case, an object at an infinite distance (as represented by a collimated beam of waves) is focused to an image at the focal point of the lens. In the latter, an object at the focal length distance from the lens is imaged at infinity. The plane perpendicular to the lens axis situated at a distance f from the lens
5940-402: The lenses (probably without the knowledge of the rudimentary optical theory of the day). The practical development and experimentation with lenses led to the invention of the compound optical microscope around 1595, and the refracting telescope in 1608, both of which appeared in the spectacle-making centres in the Netherlands . With the invention of the telescope and microscope there was
6030-519: The light-refracting properties of the liquids in the eye be taken into account. Treating the eye as an ordinary air-filled camera and lens results in an incorrect focal length and f-number. In astronomy, the f-number is commonly referred to as the focal ratio (or f-ratio ) notated as N {\displaystyle N} . It is still defined as the focal length f {\displaystyle f} of an objective divided by its diameter D {\displaystyle D} or by
6120-411: The need for rigorous consistency of all lenses and cameras used, slight differences in exposure are less important; however, T-stops are still used in some kinds of special-purpose lenses such as Smooth Trans Focus lenses by Minolta and Sony . Photographic film 's and electronic camera sensor's sensitivity to light is often specified using ASA/ISO numbers . Both systems have a linear number where
6210-403: The physical aperture and focal length of the eye. Typically, the pupil can dilate to be as large as 6–7 mm in darkness, which translates into the maximal physical aperture. Some individuals' pupils can dilate to over 9 mm wide. The f-number of the human eye varies from about f /8.3 in a very brightly lit place to about f /2.1 in the dark. Computing the focal length requires that
6300-406: The physical centre of the lens. Toric or sphero-cylindrical lenses have surfaces with two different radii of curvature in two orthogonal planes. They have a different focal power in different meridians. This forms an astigmatic lens. An example is eyeglass lenses that are used to correct astigmatism in someone's eye. Lenses are classified by the curvature of the two optical surfaces. A lens
6390-410: The projected image ( illuminance ) relative to the brightness of the scene in the lens's field of view ( luminance ) decreases with the square of the f-number. A 100 mm focal length f /4 lens has an entrance pupil diameter of 25 mm . A 100 mm focal length f /2 lens has an entrance pupil diameter of 50 mm . Since the area is proportional to the square of the pupil diameter,
6480-414: The radius of curvature is called the curvature . A flat surface has zero curvature, and its radius of curvature is infinite . This convention seems to be mainly used for this article, although there is another convention such as Cartesian sign convention requiring different lens equation forms. If d is small compared to R 1 and R 2 then the thin lens approximation can be made. For
6570-444: The range of angles over which light can enter or exit the system. The numerical aperture takes into account the refractive index of the medium in which the system is working, while the f-number does not. The f-number N is given by: N = f D {\displaystyle N={\frac {f}{D}}\ } where f is the focal length , and D is the diameter of the entrance pupil ( effective aperture ). It
6660-574: The required shape. A lens can focus light to form an image , unlike a prism , which refracts light without focusing. Devices that similarly focus or disperse waves and radiation other than visible light are also called "lenses", such as microwave lenses, electron lenses , acoustic lenses , or explosive lenses . Lenses are used in various imaging devices such as telescopes , binoculars , and cameras . They are also used as visual aids in glasses to correct defects of vision such as myopia and hypermetropia . The word lens comes from lēns ,
6750-775: The right infinity leads to the first or object focal length f 0 {\textstyle f_{0}} for the spherical surface. Similarly, u {\textstyle u} toward the left infinity leads to the second or image focal length f i {\displaystyle f_{i}} . f 0 = n 1 n 2 − n 1 R , f i = n 2 n 2 − n 1 R {\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}}} Applying this equation on
6840-453: The same way as one f-stop corresponds to a factor of two in light intensity, shutter speeds are arranged so that each setting differs in duration by a factor of approximately two from its neighbour. Opening up a lens by one stop allows twice as much light to fall on the film in a given period of time. Therefore, to have the same exposure at this larger aperture as at the previous aperture, the shutter would be opened for half as long (i.e., twice
6930-919: The sequence is one stop lower than the element to its left, and one stop higher than the element to its right. The values of the ratios are rounded off to these particular conventional numbers, to make them easier to remember and write down. The sequence above is obtained by approximating the following exact geometric sequence: f / 1 = f ( 2 ) 0 , f / 1.4 = f ( 2 ) 1 , f / 2 = f ( 2 ) 2 , f / 2.8 = f ( 2 ) 3 , … {\displaystyle f/1={\frac {f}{({\sqrt {2}})^{0}}},\ f/1.4={\frac {f}{({\sqrt {2}})^{1}}},\ f/2={\frac {f}{({\sqrt {2}})^{2}}},\ f/2.8={\frac {f}{({\sqrt {2}})^{3}}},\ \ldots } In
7020-437: The sharpest image. The larger number of elements in modern lenses allow the designer to compensate for aberrations, allowing the lens to give better pictures at lower f-numbers. At small apertures, depth of field and aberrations are improved, but diffraction creates more spreading of the light, causing blur. Light falloff is also sensitive to f-stop. Many wide-angle lenses will show a significant light falloff ( vignetting ) at
7110-410: The sign) would have zero optical power (as its focal length becomes infinity as shown in the lensmaker's equation ), meaning that it would neither converge nor diverge light. All real lenses have a nonzero thickness, however, which makes a real lens with identical curved surfaces slightly positive. To obtain exactly zero optical power, a meniscus lens must have slightly unequal curvatures to account for
7200-408: The size of the aperture stop and thus the entrance pupil size. This allows the user to vary the f-number as needed. The entrance pupil diameter is not necessarily equal to the aperture stop diameter, because of the magnifying effect of lens elements in front of the aperture. Ignoring differences in light transmission efficiency, a lens with a greater f-number projects darker images. The brightness of
7290-404: The speed). The film will respond equally to these equal amounts of light, since it has the property of reciprocity . This is less true for extremely long or short exposures, where there is reciprocity failure . Aperture, shutter speed, and film sensitivity are linked: for constant scene brightness, doubling the aperture area (one stop), halving the shutter speed (doubling the time open), or using
7380-477: The steps in a full stop (1 EV) one could use ( 2 ) 0 , ( 2 ) 1 , ( 2 ) 2 , ( 2 ) 3 , ( 2 ) 4 , … {\displaystyle ({\sqrt {2}})^{0},\ ({\sqrt {2}})^{1},\ ({\sqrt {2}})^{2},\ ({\sqrt {2}})^{3},\ ({\sqrt {2}})^{4},\ \ldots } The steps in
7470-408: The subscript of 2 in n 2 {\textstyle \ n_{2}\ } is dropped. As mentioned above, a positive or converging lens in air focuses a collimated beam travelling along the lens axis to a spot (known as the focal point ) at a distance f from the lens. Conversely, a point source of light placed at the focal point is converted into a collimated beam by
7560-862: The thin lens approximation where d → 0 , {\displaystyle \ d\rightarrow 0\ ,} the 2nd term of the RHS (Right Hand Side) is gone, so n 1 u + n 1 v = ( n 2 − n 1 ) ( 1 R 1 − 1 R 2 ) . {\displaystyle \ {\frac {\ n_{1}\ }{u}}+{\frac {\ n_{1}\ }{v}}=\left(n_{2}-n_{1}\right)\left({\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}\right)~.} The focal length f {\displaystyle \ f\ } of
7650-1110: The thin lens is found by limiting u → − ∞ , {\displaystyle \ u\rightarrow -\infty \ ,} n 1 f = ( n 2 − n 1 ) ( 1 R 1 − 1 R 2 ) → 1 f = ( n 2 n 1 − 1 ) ( 1 R 1 − 1 R 2 ) . {\displaystyle \ {\frac {\ n_{1}\ }{\ f\ }}=\left(n_{2}-n_{1}\right)\left({\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}\right)\rightarrow {\frac {1}{\ f\ }}=\left({\frac {\ n_{2}\ }{\ n_{1}\ }}-1\right)\left({\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}\right)~.} So,
7740-1258: The two spherical surfaces of a lens and approximating the lens thickness to zero (so a thin lens) leads to the lensmaker's formula . Applying Snell's law on the spherical surface, n 1 sin i = n 2 sin r . {\displaystyle n_{1}\sin i=n_{2}\sin r\,.} Also in the diagram, tan ( i − θ ) = h u tan ( θ − r ) = h v sin θ = h R {\displaystyle {\begin{aligned}\tan(i-\theta )&={\frac {h}{u}}\\\tan(\theta -r)&={\frac {h}{v}}\\\sin \theta &={\frac {h}{R}}\end{aligned}}} , and using small angle approximation (paraxial approximation) and eliminating i , r , and θ , n 2 v + n 1 u = n 2 − n 1 R . {\displaystyle {\frac {n_{2}}{v}}+{\frac {n_{1}}{u}}={\frac {n_{2}-n_{1}}{R}}\,.} The (effective) focal length f {\displaystyle f} of
7830-418: The two surfaces. A negative meniscus lens has a steeper concave surface (with a shorter radius than the convex surface) and is thinner at the centre than at the periphery. Conversely, a positive meniscus lens has a steeper convex surface (with a shorter radius than the concave surface) and is thicker at the centre than at the periphery. An ideal thin lens with two surfaces of equal curvature (also equal in
7920-467: The use of a corrective lens when he mentions that Nero was said to watch the gladiatorial games using an emerald (presumably concave to correct for nearsightedness , though the reference is vague). Both Pliny and Seneca the Younger (3 BC–65 AD) described the magnifying effect of a glass globe filled with water. Ptolemy (2nd century) wrote a book on Optics , which however survives only in
8010-436: The use of f-numbers in photography is the sunny 16 rule : an approximately correct exposure will be obtained on a sunny day by using an aperture of f /16 and the shutter speed closest to the reciprocal of the ISO speed of the film; for example, using ISO 200 film, an aperture of f /16 and a shutter speed of 1 ⁄ 200 second. The f-number may then be adjusted downwards for situations with lower light. Selecting
8100-427: The viewer's focus on the main subject in the foreground. The depth of field of an image produced at a given f-number is dependent on other parameters as well, including the focal length , the subject distance, and the format of the film or sensor used to capture the image. Depth of field can be described as depending on just angle of view, subject distance, and entrance pupil diameter (as in von Rohr's method ). As
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