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The focal length of an
optical Optics is the branch of physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities ...

optical
system is a measure of how strongly the system converges or diverges
light Light or visible light is electromagnetic radiation within the portion of the electromagnetic spectrum that can be visual perception, perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 ...

light
; it is the
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when add ...

inverse
of the system's
optical power Optical power (also referred to as dioptric power, refractive power, focusing power, or convergence power) is the degree to which a lens A lens is a transmissive optical Optics is the branch of physics Physics (from grc, φυσικ ...
. 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
ray Ray may refer to: Science and mathematics * Ray (geometry), half of a line proceeding from an initial point * Ray (graph theory), an infinite sequence of vertices such that each vertex appears at most once in the sequence and each two consecutive ...
s more sharply, bringing them to a focus in a shorter distance or diverging them more quickly. For the special case of a
thin lens A lens may be considered a thin lens if its thickness is much less than the radii of curvature of its surfaces ( and )., 300px In optics, a thin lens is a lens (optics), lens with a thickness (distance along the optical axis between the two surfa ...
in air, a positive focal length is the distance over which initially
collimated A collimated beam of light Light or visible light is electromagnetic radiation within the portion of the electromagnetic spectrum that can be visual perception, perceived by the human eye. Visible light is usually defined as having wavelength ...
(parallel) rays are brought to a
focus FOCUS is a fourth-generation programming language (4GL) computer programming programming language, language and development environment that is used to build database queries. Produced by Information Builders Inc., it was originally developed for d ...
, or alternatively a negative focal length indicates how far in front of the lens a
point source A point source is a single identifiable ''localised'' source of something. A point source has negligible extent, distinguishing it from other source geometries. Sources are called point sources because in mathematical modeling, these sources can us ...
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. In most
photography Photography is the art Art is a diverse range of (products of) human activities Humans (''Homo sapiens'') are the most populous and widespread species of primates, characterized by bipedality, opposable thumbs, hairlessness, and int ...

photography
and all telescopy, where the subject is essentially infinitely far away, longer focal length (lower optical power) leads to higher
magnification Magnification is the process of enlarging the apparent size, not physical size, of something. This enlargement is quantified by a calculated number also called "magnification". When this number is less than one, it refers to a reduction in size, ...
and a narrower
angle of view The angle of view is the decisive variable for the visual perception of the size or projection of the size of an object. Angle of view and perception of size The perceived size of an object depends on the size of the image projected onto the ...

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 Microscopy is the technical field of using microscope A microscope (from grc, μικρός ''mikrós'' 'small' and ''skopeîn'' 'to look (at); examine, inspect') is a laboratory instrument used to examine objects that are too small to b ...

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.


Thin lens approximation

For a thin lens in air, the focal length is the distance from the center of the
lens A lens is a transmissive optical Optics is the branch of physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, s ...

lens
to the principal foci (or ''focal points'') of the lens. For a converging lens (for example a
convex lens A lens is a transmissive optics, optical device which focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a #Compound lenses, compound lens consists of several simpl ...
), the focal length is positive and is the distance at which a beam of
collimated light A collimated beam of light or other electromagnetic radiation has parallel rays, and therefore will spread minimally as it propagates. A perfectly collimated light beam, with no beam divergence, divergence, would not disperse with distance. Howe ...
will be focused to a single spot. For a diverging lens (for example a
concave lens A lens is a transmissive optics, 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 lenses, compound lens consists of several simple ...
), 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. 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 :\frac =\frac+\frac\ . 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 is negligible, and the focal length is then given by :f \approx v\ . Determining the focal length of a ''concave'' lens is somewhat more difficult. The focal length of such a lens is considered that point at which the spreading beams of light would meet before the lens if the lens were not there. 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.


General optical systems

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 A camera lens (also known as photographic lens or photographic objective) is an optical lens or assembly of lenses used in conjunction with a camera body and mechanism to make images of objects either on photographic film Photographic film i ...
or a
telescope A telescope is an optical instrument An optical instrument (or "optic" for short) is a device that processes light waves (or photons), either to enhance an image for viewing or to analyze and determine their characteristic properties. Common ...

telescope
), the focal length is often called the ''effective focal length'' (EFL), to distinguish it from other commonly used parameters: * ''Front focal length'' (FFL) or ''front focal distance'' (FFD) (''s''F) is the distance from the front focal point of the system (F) to the
vertex Vertex (Latin: peak; plural vertices or vertexes) means the "top", or the highest geometric point of something, usually a curved surface or line, or a point where any two geometric sides or edges meet regardless of elevation; as opposed to an Apex ( ...
of the ''first optical surface'' (S1). * ''Back focal length'' (BFL) or ''back focal distance'' (BFD) (''s′''F′) is the distance from the vertex of the ''last optical surface'' of the system (S2) to the rear focal point (F′). For an optical system in air, the effective focal length (''f'' and ''f′'') gives the distance from the front and rear principal planes (H and H′) to the corresponding focal points (F and F′). If the surrounding medium is not air, then the distance is multiplied by the
refractive index In optics, the refractive index (also known as refraction index or index of refraction) of a optical medium, material is a dimensionless number that describes how fast EM radiation, light travels through the material. It is defined as :n = \frac ...

refractive index
of the medium (''n'' is the refractive index of the substance from which the lens itself is made; ''n''1 is the refractive index of any medium in front of the lens; ''n''2 is that of any medium in back of it). Some authors call these distances the front/rear focal ''lengths'', distinguishing them from the front/rear focal ''distances'', defined above. In general, the focal length or EFL is the value that describes the ability of the optical system to focus light, and is the value used to calculate the
magnification Magnification is the process of enlarging the apparent size, not physical size, of something. This enlargement is quantified by a calculated number also called "magnification". When this number is less than one, it refers to a reduction in size, ...
of the system. The other parameters are used in determining where an
image An image (from la, imago) is an artifact that depicts visual perception Visual perception is the ability to interpret the surrounding environment Environment most often refers to: __NOTOC__ * Natural environment, all living and non- ...

image
will be formed for a given object position. For the case of a lens of thickness ''d'' in air (), and surfaces with radii of curvature ''R''1 and ''R''2, the effective focal length ''f'' is given by the
Lensmaker's equation 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' ...
: :\frac = (n-1) \left( \frac - \frac + \frac \right), where ''n'' is the
refractive index In optics, the refractive index (also known as refraction index or index of refraction) of a optical medium, material is a dimensionless number that describes how fast EM radiation, light travels through the material. It is defined as :n = \frac ...

refractive index
of the lens medium. The quantity is also known as the optical power of the lens. The corresponding front focal distance is: :\mbox = f \left( 1 + \frac \right), and the back focal distance: :\mbox = f \left( 1 - \frac \right). In the
sign conventionIn physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Spac ...
used here, the value of ''R''1 will be positive if the first lens surface is convex, and negative if it is concave. The value of ''R''2 is negative if the second surface is convex, and positive if concave. Note that sign conventions vary between different authors, which results in different forms of these equations depending on the convention used. For a curved
mirror A mirror is an object that Reflection (physics), reflects an image. Light that bounces off a mirror will show an image of whatever is in front of it, when focused through the lens of the eye or a camera. Mirrors reverse the direction of the ...

mirror
in air, the magnitude of the focal length is equal to the
radius of curvature In differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differentia ...
of the mirror divided by two. The focal length is positive for a
concave mirror A curved mirror is a mirror with a curved reflecting surface. The surface may be either ''convex'' (bulging outward) or ''concave'' (recessed inward). Most curved mirrors have surfaces that are shaped like part of a sphere of a sphere A spher ...

concave mirror
, and negative for a
convex mirror A curved mirror is a mirror Grange, East Yorkshire, UK, from World War I. The mirror magnified the sound of approaching enemy Zeppelins for a microphone placed at the Focus (geometry), focal point. A mirror is an object that Reflection (p ...

convex mirror
. In the sign convention used in optical design, a concave mirror has negative radius of curvature, so :f = -, where ''R'' is the radius of curvature of the mirror's surface. See
Radius of curvature (optics) Radius of curvature (ROC) has specific meaning and sign convention in optical design. A spherical lens or mirror surface has a center of curvature located either along or decentered from the system local optical axis 300px, Optical axis (co ...
for more information on the sign convention for radius of curvature used here.


In photography

Camera lens focal lengths are usually specified in millimetres (mm), but some older lenses are marked in centimetres (cm) or inches. Focal length (''f'') and
field of view The field of view (FoV) is the extent of the observable world that is at any given moment. In the case of s or sensors it is a through which a detector is sensitive to . Humans and animals In the context of human and primate vision, th ...

field of view
(FOV) of a lens are inversely proportional. For a standard
rectilinear lens In photography Photography is the art, application, and practice of creating durable images by recording light, either electronically by means of an image sensor, or chemically by means of a light-sensitive material such as photographic film. I ...
, FOV = 2 arctan , where ''x'' is the diagonal of the film. When a photographic lens is set to "infinity", its rear
nodal point In Gaussian optics, the cardinal points consist of three pairs of Point (geometry), points located on the optical axis of a Rotational symmetry, rotationally symmetric, focal, optical system. These are the Focus (optics), focal points, the princip ...
is separated from the sensor or film, at the
focal plane In Gaussian optics, the cardinal points consist of three pairs of points located on the optical axis of a rotationally symmetric, focal, optical system. These are the focal points, the principal points, and the nodal points. For ''ideal'' sys ...

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. To render closer objects in sharp focus, the lens must be adjusted to increase the distance between the rear nodal point and the film, to put the film at the image plane. The focal length (''f''), the distance from the front nodal point to the object to photograph (''s''1), and the distance from the rear nodal point to the image plane (''s''2) are then related by: :\frac + \frac = \frac . As ''s''1 is decreased, ''s''2 must be increased. For example, consider a
normal lens In photography and cinematography, a normal lens is a photographic lens, lens that reproduces a Angle of view, field of view that appears "natural" to a human observer. In contrast, depth compression and expansion with shorter or longer focal lengt ...

normal lens
for a
35 mm35 mm may refer to: * 135 film, a type of still photography format commonly referred to as 35 mm film * 35 mm movie film, a type of motion picture film stock * Ryan Scott Oliver#35MM, 35MM, a "musical exhibition" by Ryan Scott Oliver that ...
camera with a focal length of ''f'' = 50 mm. To focus a distant object (''s''1 ≈ ∞), the rear nodal point of the lens must be located a distance ''s''2 = 50 mm from the image plane. To focus an object 1 m away (''s''1 = 1,000 mm), the lens must be moved 2.6 mm farther away from the image plane, to ''s''2 = 52.6 mm. 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 distant objects the same size as the lens in question. For
rectilinear lens In photography Photography is the art, application, and practice of creating durable images by recording light, either electronically by means of an image sensor, or chemically by means of a light-sensitive material such as photographic film. I ...
es (that is, with no
image distortion Wine glasses creating a non-uniform distortion of their background In geometric optics Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of t ...
), the imaging of distant objects is well modelled as a
pinhole camera model . The pinhole camera model describes the mathematical relationship between the coordinate In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithm ...
. This model leads to the simple geometric model that photographers use for computing the
angle of view The angle of view is the decisive variable for the visual perception of the size or projection of the size of an object. Angle of view and perception of size The perceived size of an object depends on the size of the image projected onto 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. A lens with a focal length about equal to the diagonal size of the film or sensor format is known as a
normal lens In photography and cinematography, a normal lens is a photographic lens, lens that reproduces a Angle of view, field of view that appears "natural" to a human observer. In contrast, depth compression and expansion with shorter or longer focal lengt ...

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; 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 In photography Photography is the art Art is a diverse range of (products of) human activities involving creative imagination to express technical proficiency, beauty, emotional power, or conceptual ideas. There is no generally ag ...
(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 A telephoto lens, in photography Photography is the art, application, and practice of creating durable images by recording light, either electronically by means of an image sensor, or chemically by means of a light-sensitive material such as p ...
(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. 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 camera A digital camera is a camera A camera is an optical Optics is the branch of physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behav ...

digital camera
s, 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 In digital photography, the crop factor, format factor, or focal length multiplier of an image sensor format is the ratio of the Image sensor format, dimensions of a camera's imaging area compared to a reference format; most often, this term is app ...
.


See also

*
Depth of field For many cameras, depth of field (DOF) is the distance between the nearest and the farthest objects that are in acceptably sharp focus in an image. The depth of field can be calculated based on focal length The focal length of an optical ...

Depth of field
*
Dioptre A dioptre (British spelling Despite the various English dialects spoken from country to country and within different regions of the same country, there are only slight regional variations in English orthography English orthography is the ...

Dioptre
*
f-number In optics, the f-number of an optical system such as a camera lens is the ratio of the system's focal length to the diameter of the entrance pupil ("clear aperture").Smith, Warren ''Modern Optical Engineering'', 4th Ed., 2007 McGraw-Hill Prof ...
or focal ratio


References

{{Authority control Geometrical optics Length Science of photography