1 f = ( n − 1 ) ( 1 R 1 − 1 R 2 + ( n − 1 ) d n R 1 R 2 ) , {\displaystyle {\frac {1}{f}}=(n-1)\left({\frac {1}{R_{1}}}-{\frac {1}{R_{2}}}+{\frac {(n-1)d}{nR_{1}R_{2}}}\right),} where n is the refractive index of the lens medium. The quantity ⁠1/f⁠ is also known as the optical power of the lens.

Modifying the angle of view over time (known as zooming), is a frequently used cinematic technique, often combined with camera movement to produce a "dolly zoom" effect, made famous by the film Vertigo. Using a wide angle of view can exaggerate the camera's perceived speed, and is a common technique in tracking shots, phantom rides, and racing video games. See also Field of view in video games.

For a lens projecting a rectilinear image (focused at infinity, see derivation), the angle of view (α) can be calculated from the chosen dimension (d), and effective focal length (f) as follows:[4] α = 2 arctan ⁡ d 2 f {\displaystyle \alpha =2\arctan {\frac {d}{2f}}}

The focal length of a lens determines the magnification at which it images distant objects. It is equal to the distance between the image plane and a pinhole that images distant objects the same size as the lens in question. For rectilinear lenses (that is, with no image distortion), the imaging of distant objects is well modelled as a pinhole camera model.[7] This model leads to the simple geometric model that photographers use for computing the angle of view of a camera; in this case, the angle of view depends only on the ratio of focal length to film size. In general, the angle of view depends also on the distortion.[8]

The effective focal length is nearly equal to the stated focal length of the lens (F), except in macro photography where the lens-to-object distance is comparable to the focal length. In this case, the magnification factor (m) must be taken into account: f = F ⋅ ( 1 + m ) {\displaystyle f=F\cdot (1+m)}

The optical power of a lens or curved mirror is a physical quantity equal to the reciprocal of the focal length, expressed in metres. A dioptre is its unit of measurement with dimension of reciprocal length, equivalent to one reciprocal metre, 1 dioptre = 1 m−1. For example, a 2-dioptre lens brings parallel rays of light to focus at 1⁄2 metre. A flat window has an optical power of zero dioptres, as it does not cause light to converge or diverge.[10]

In most photography and all telescopy, where the subject is essentially infinitely far away, longer focal length (lower optical power) leads to higher magnification and a narrower angle of view; conversely, shorter focal length or higher optical power is associated with lower magnification and a wider angle of view. On the other hand, in applications such as microscopy in which magnification is achieved by bringing the object close to the lens, a shorter focal length (higher optical power) leads to higher magnification because the subject can be brought closer to the center of projection.

Camera lens focal lengths are usually specified in millimetres (mm), but some older lenses are marked in centimetres (cm) or inches.

When a lens is used to form an image of some object, the distance from the object to the lens u, the distance from the lens to the image v, and the focal length f are related by

A second effect which comes into play in macro photography is lens asymmetry (an asymmetric lens is a lens where the aperture appears to have different dimensions when viewed from the front and from the back). The lens asymmetry causes an offset between the nodal plane and pupil positions. The effect can be quantified using the ratio (P) between apparent exit pupil diameter and entrance pupil diameter. The full formula for angle of view now becomes:[7] α = 2 arctan ⁡ d 2 F ⋅ ( 1 + m / P ) {\displaystyle \alpha =2\arctan {\frac {d}{2F\cdot (1+m/P)}}}

Determining the focal length of a concave lens is somewhat more difficult. The focal length of such a lens is defined as the point at which the spreading beams of light meet when they are extended backwards. No image is formed during such a test, and the focal length must be determined by passing light (for example, the light of a laser beam) through the lens, examining how much that light becomes dispersed/ bent, and following the beam of light backwards to the lens's focal point.

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This table shows the diagonal, horizontal, and vertical angles of view, in degrees, for lenses producing rectilinear images, when used with 36 mm × 24 mm format (that is, 135 film or full-frame 35 mm digital using width 36 mm, height 24 mm, and diagonal 43.3 mm for d in the formula above).[16] Digital compact cameras sometimes state the focal lengths of their lenses in 35 mm equivalents, which can be used in this table.

The main benefit of using optical power rather than focal length is that the thin lens formula has the object distance, image distance, and focal length all as reciprocals. Additionally, when relatively thin lenses are placed close together their powers approximately add. Thus, a thin 2.0-dioptre lens placed close to a thin 0.5-dioptre lens yields almost the same focal length as a single 2.5-dioptre lens.

The corresponding front focal distance is:[6] FFD = f ( 1 + ( n − 1 ) d n R 2 ) , {\displaystyle {\mbox{FFD}}=f\left(1+{\frac {(n-1)d}{nR_{2}}}\right),} and the back focal distance: BFD = f ( 1 − ( n − 1 ) d n R 1 ) . {\displaystyle {\mbox{BFD}}=f\left(1-{\frac {(n-1)d}{nR_{1}}}\right).}

fov是什么

The distinction between front/rear focal length and EFL is important for studying the human eye. The eye can be represented by an equivalent thin lens at an air/fluid boundary with front and rear focal lengths equal to those of the eye, or it can be represented by a different equivalent thin lens that is totally in air, with focal length equal to the eye's EFL.

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Note that the angle of view varies slightly when the focus is not at infinity (See breathing (lens)), given by S 2 = S 1 f S 1 − f {\displaystyle S_{2}={\frac {S_{1}f}{S_{1}-f}}} rearranging the lens equation.

field ofview中文

To project a sharp image of distant objects, S 2 {\displaystyle S_{2}} needs to be equal to the focal length, F {\displaystyle F} , which is attained by setting the lens for infinity focus. Then the angle of view is given by:

In the optical instrumentation industry the term field of view (FOV) is most often used, though the measurements are still expressed as angles.[8] Optical tests are commonly used for measuring the FOV of UV, visible, and infrared (wavelengths about 0.1–20 μm in the electromagnetic spectrum) sensors and cameras.

Due to the popularity of the 35 mm standard, camera–lens combinations are often described in terms of their 35 mm-equivalent focal length, that is, the focal length of a lens that would have the same angle of view, or field of view, if used on a full-frame 35 mm camera. Use of a 35 mm-equivalent focal length is particularly common with digital cameras, which often use sensors smaller than 35 mm film, and so require correspondingly shorter focal lengths to achieve a given angle of view, by a factor known as the crop factor.

From the definition of magnification, m = S 2 / S 1 {\displaystyle m=S_{2}/S_{1}} , we can substitute S 1 {\displaystyle S_{1}} and with some algebra find: S 2 = F ⋅ ( 1 + m ) {\displaystyle S_{2}=F\cdot (1+m)}

The table below shows the horizontal, vertical and diagonal angles of view, in degrees, when used with 22.2 mm × 14.8 mm format (that is Canon's DSLR APS-C frame size) and a diagonal of 26.7 mm.

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Consider a 35 mm camera with a lens having a focal length of F = 50 mm. The dimensions of the 35 mm image format are 24 mm (vertically) × 36 mm (horizontal), giving a diagonal of about 43.3 mm.

Field of view

To render closer objects in sharp focus, the lens must be adjusted to increase the distance between the rear principal plane and the film, to put the film at the image plane. The focal length f, the distance from the front principal plane to the object to photograph s1, and the distance from the rear principal plane to the image plane s2 are then related by:

(In photography m {\displaystyle m} is usually defined to be positive, despite the inverted image.) For example, with a magnification ratio of 1:2, we find f = 1.5 ⋅ F {\displaystyle f=1.5\cdot F} and thus the angle of view is reduced by 33% compared to focusing on a distant object with the same lens.

Now α / 2 {\displaystyle \alpha /2} is the angle between the optical axis of the lens and the ray joining its optical center to the edge of the film. Here α {\displaystyle \alpha } is defined to be the angle-of-view, since it is the angle enclosing the largest object whose image can fit on the film. We want to find the relationship between:

Because different lenses generally require a different camera–subject distance to preserve the size of a subject, changing the angle of view can indirectly distort perspective, changing the apparent relative size of the subject and foreground.

For an optical system in a medium other than air or vacuum, the front and rear focal lengths are equal to the EFL times the refractive index of the medium in front of or behind the lens (n1 and n2 in the diagram above). The term "focal length" by itself is ambiguous in this case. The historical usage was to define the "focal length" as the EFL times the index of refraction of the medium.[2][4] For a system with different media on both sides, such as the human eye, the front and rear focal lengths are not equal to one another, and convention may dictate which one is called "the focal length" of the system. Some modern authors avoid this ambiguity by instead defining "focal length" to be a synonym for EFL.[1]

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The target's angular extent is: α = 2 arctan ⁡ L 2 f c {\displaystyle \alpha =2\arctan {\frac {L}{2f_{c}}}} where L {\displaystyle L} is the dimension of the target and f c {\displaystyle f_{c}} is the focal length of collimator.

UV/visible light from an integrating sphere (and/or other source such as a black body) is focused onto a square test target at the focal plane of a collimator (the mirrors in the diagram), such that a virtual image of the test target will be seen infinitely far away by the camera under test. The camera under test senses a real image of the virtual image of the target, and the sensed image is displayed on a monitor.[9]

Defining f = S 2 {\displaystyle f=S_{2}} as the "effective focal length", we get the formula presented above: α = 2 arctan ⁡ d 2 f {\displaystyle \alpha =2\arctan {\frac {d}{2f}}} where f = F ⋅ ( 1 + m ) {\displaystyle f=F\cdot (1+m)} .

For an optical system in air the effective focal length, front focal length, and rear focal length are all the same and may be called simply "focal length".

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Because this is a trigonometric function, the angle of view does not vary quite linearly with the reciprocal of the focal length. However, except for wide-angle lenses, it is reasonable to approximate α ≈ d f {\displaystyle \alpha \approx {\frac {d}{f}}} radians or 180 d π f {\displaystyle {\frac {180d}{\pi f}}} degrees.

The focal length of a thin convex lens can be easily measured by using it to form an image of a distant light source on a screen. The lens is moved until a sharp image is formed on the screen. In this case ⁠1/u⁠ is negligible, and the focal length is then given by

Zoom lenses are a special case wherein the focal length, and hence angle of view, of the lens can be altered mechanically without removing the lens from the camera.

Monochromatic aberrations include spherical aberration, coma, astigmatism, and field curvature and image distortion. Spherical aberration. Spherical aberrations.

Focal length (f) and field of view (FOV) of a lens are inversely proportional. For a standard rectilinear lens, F O V = 2 arctan ⁡ ( x 2 f ) {\textstyle \mathrm {FOV} =2\arctan {\left({x \over 2f}\right)}} , where x is the width of the film or imaging sensor.

Consider a rectilinear lens in a camera used to photograph an object at a distance S 1 {\displaystyle S_{1}} , and forming an image that just barely fits in the dimension, d {\displaystyle d} , of the frame (the film or image sensor). Treat the lens as if it were a pinhole at distance S 2 {\displaystyle S_{2}} from the image plane (technically, the center of perspective of a rectilinear lens is at the center of its entrance pupil):[6]

Focal length

A lens with a focal length about equal to the diagonal size of the film or sensor format is known as a normal lens; its angle of view is similar to the angle subtended by a large-enough print viewed at a typical viewing distance of the print diagonal, which therefore yields a normal perspective when viewing the print;[9] this angle of view is about 53 degrees diagonally. For full-frame 35 mm-format cameras, the diagonal is 43 mm and a typical "normal" lens has a 50 mm focal length. A lens with a focal length shorter than normal is often referred to as a wide-angle lens (typically 35 mm and less, for 35 mm-format cameras), while a lens significantly longer than normal may be referred to as a telephoto lens (typically 85 mm and more, for 35 mm-format cameras). Technically, long focal length lenses are only "telephoto" if the focal length is longer than the physical length of the lens, but the term is often used to describe any long focal length lens.

It is important to distinguish the angle of view from the angle of coverage, which describes the angle range that a lens can image. Typically the image circle produced by a lens is large enough to cover the film or sensor completely, possibly including some vignetting toward the edge. If the angle of coverage of the lens does not fill the sensor, the image circle will be visible, typically with strong vignetting toward the edge, and the effective angle of view will be limited to the angle of coverage.

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d {\displaystyle d} represents the size of the film (or sensor) in the direction measured (see below: sensor effects). For example, for 35 mm film which is 36 mm wide and 24 mm high, d = 36 m m {\displaystyle d=36\,\mathrm {mm} } would be used to obtain the horizontal angle of view and d = 24 m m {\displaystyle d=24\,\mathrm {mm} } for the vertical angle.

As noted above, a camera's angle level of view depends not only on the lens, but also on the sensor used. Digital sensors are usually smaller than 35 mm film, causing the lens to usually behave as a longer focal length lens would behave, and have a narrower angle of view than with 35 mm film, by a constant factor for each sensor (called the crop factor). In everyday digital cameras, the crop factor can range from around 1 (professional digital SLRs), to 1.6 (mid-market SLRs), to around 3 to 6 for compact cameras. So a standard 50 mm lens for 35 mm photography acts like a 50 mm standard "film" lens even on a professional digital SLR, but would act closer to a 75 mm (1.5×50 mm Nikon) or 80 mm lens (1.6×50mm Canon) on many mid-market DSLRs, and the 40-degree angle of view of a standard 50 mm lens on a film camera is equivalent to a 28–35 mm lens on many digital SLRs.

As s1 is decreased, s2 must be increased. For example, consider a normal lens for a 35 mm camera with a focal length of f = 50 mm. To focus a distant object (s1 ≈ ∞), the rear principal plane of the lens must be located a distance s2 = 50 mm from the film plane, so that it is at the location of the image plane. To focus an object 1 m away (s1 = 1,000 mm), the lens must be moved 2.6 mm farther away from the film plane, to s2 = 52.6 mm.

A camera's angle of view depends not only on the lens, but also on the sensor. Digital sensors are usually smaller than 35 mm film, and this causes the lens to have a narrower angle of view than with 35 mm film, by a constant factor for each sensor (called the crop factor). In everyday digital cameras, the crop factor can range from around 1 (professional digital SLRs), to 1.6 (consumer SLR), to 2 (Micro Four Thirds ILC) to 6 (most compact cameras). So a standard 50 mm lens for 35 mm photography acts like a 50 mm standard "film" lens on a professional digital SLR, but would act closer to an 80 mm lens (1.6×50mm) on many mid-market DSLRs, and the 40-degree angle of view of a standard 50 mm lens on a film camera is equivalent to an 80 mm lens on many digital SLRs.

For a thin lens in air, the focal length is the distance from the center of the lens to the principal foci (or focal points) of the lens. For a converging lens (for example a convex lens), the focal length is positive and is the distance at which a beam of collimated light will be focused to a single spot. For a diverging lens (for example a concave lens), the focal length is negative and is the distance to the point from which a collimated beam appears to be diverging after passing through the lens.

FOV tofocal lengthcalculator

The total field of view is then approximately: F O V = α D d {\displaystyle \mathrm {FOV} =\alpha {\frac {D}{d}}} or more precisely, if the imaging system is rectilinear: F O V = 2 arctan ⁡ L D 2 f c d {\displaystyle \mathrm {FOV} =2\arctan {\frac {LD}{2f_{c}d}}}

For a spherically-curved mirror in air, the magnitude of the focal length is equal to the radius of curvature of the mirror divided by two. The focal length is positive for a concave mirror, and negative for a convex mirror. In the sign convention used in optical design, a concave mirror has negative radius of curvature, so

For a given camera–subject distance, longer lenses magnify the subject more. For a given subject magnification (and thus different camera–subject distances), longer lenses appear to compress distance; wider lenses appear to expand the distance between objects.

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In the sign convention used here, the value of R1 will be positive if the first lens surface is convex, and negative if it is concave. The value of R2 is negative if the second surface is convex, and positive if concave. Sign conventions vary between different authors, which results in different forms of these equations depending on the convention used.

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Field of view vsangleof view

For lenses projecting rectilinear (non-spatially-distorted) images of distant objects, the effective focal length and the image format dimensions completely define the angle of view. Calculations for lenses producing non-rectilinear images are much more complex and in the end not very useful in most practical applications. (In the case of a lens with distortion, e.g., a fisheye lens, a longer lens with distortion can have a wider angle of view than a shorter lens with low distortion)[3] Angle of view may be measured horizontally (from the left to right edge of the frame), vertically (from the top to bottom of the frame), or diagonally (from one corner of the frame to its opposite corner).

In photography, angle of view (AOV)[1] describes the angular extent of a given scene that is imaged by a camera. It is used interchangeably with the more general term field of view.

The focal length of an optical system is a measure of how strongly the system converges or diverges light; it is the inverse of the system's optical power. A positive focal length indicates that a system converges light, while a negative focal length indicates that the system diverges light. A system with a shorter focal length bends the rays more sharply, bringing them to a focus in a shorter distance or diverging them more quickly. For the special case of a thin lens in air, a positive focal length is the distance over which initially collimated (parallel) rays are brought to a focus, or alternatively a negative focal length indicates how far in front of the lens a point source must be located to form a collimated beam. For more general optical systems, the focal length has no intuitive meaning; it is simply the inverse of the system's optical power.

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For macro photography, we cannot neglect the difference between S 2 {\displaystyle S_{2}} and F {\displaystyle F} . From the thin lens formula, 1 F = 1 S 1 + 1 S 2 . {\displaystyle {\frac {1}{F}}={\frac {1}{S_{1}}}+{\frac {1}{S_{2}}}.}

fov和焦距的关系

The sensed image, which includes the target, is displayed on a monitor, where it can be measured. Dimensions of the full image display and of the portion of the image that is the target are determined by inspection (measurements are typically in pixels, but can just as well be inches or cm).

fov参数

If the subject image size remains the same, then at any given aperture all lenses, wide angle and long lenses, will give the same depth of field.[15]

The collimator's distant virtual image of the target subtends a certain angle, referred to as the angular extent of the target, that depends on the collimator focal length and the target size. Assuming the sensed image includes the whole target, the angle seen by the camera, its FOV, is this angular extent of the target times the ratio of full image size to target image size.[10]

The purpose of this test is to measure the horizontal and vertical FOV of a lens and sensor used in an imaging system, when the lens focal length or sensor size is not known (that is, when the calculation above is not immediately applicable). Although this is one typical method that the optics industry uses to measure the FOV, there exist many other possible methods.

Using basic trigonometry, we find: tan ⁡ ( α / 2 ) = d / 2 S 2 . {\displaystyle \tan(\alpha /2)={\frac {d/2}{S_{2}}}.} which we can solve for α, giving: α = 2 arctan ⁡ d 2 S 2 {\displaystyle \alpha =2\arctan {\frac {d}{2S_{2}}}}

For the case of a lens of thickness d in air (n1 = n2 = 1), and surfaces with radii of curvature R1 and R2, the effective focal length f is given by the Lensmaker's equation:[5]

For a thick lens (one which has a non-negligible thickness), or an imaging system consisting of several lenses or mirrors (e.g. a photographic lens or a telescope), there are several related concepts that are referred to as focal lengths:

When a photographic lens is set to "infinity", its rear principal plane is separated from the sensor or film, which is then situated at the focal plane, by the lens's focal length. Objects far away from the camera then produce sharp images on the sensor or film, which is also at the image plane.

Another result of using a wide angle lens is a greater apparent perspective distortion when the camera is not aligned perpendicularly to the subject: parallel lines converge at the same rate as with a normal lens, but converge more due to the wider total field. For example, buildings appear to be falling backwards much more severely when the camera is pointed upward from ground level than they would if photographed with a normal lens at the same distance from the subject, because more of the subject building is visible in the wide-angle shot.