![Focal-length](https://upload.wikimedia.org/wikipedia/commons/8/8b/Focal-length.svg)
The focal length of an
optical
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
system is a measure of how strongly the system converges or diverges
light
Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 tera ...
; it is the
inverse of the system's
optical power
In optics, optical power (also referred to as dioptric power, refractive power, focusing power, or convergence power) is the degree to which a lens (optics), lens, mirror, or other optical system converges or diverges light. It is equal to the Mu ...
. 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 optics, a thin lens is a lens with a thickness (distance along the optical axis between the two surfaces of the lens) that is negligible compared to the radii of curvature of the lens surfaces. Lenses whose thickness is not negligible are so ...
in air, a positive focal length is the distance over which initially
collimated
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 divergence, would not disperse with distance. However, diffraction pr ...
(parallel) rays are brought to a
focus
Focus, or its plural form foci may refer to:
Arts
* Focus or Focus Festival, former name of the Adelaide Fringe arts festival in South Australia Film
*''Focus'', a 1962 TV film starring James Whitmore
* ''Focus'' (2001 film), a 2001 film based ...
, 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 ...
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, 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. It is employed ...
and all
telescopy
A telescope is a device used to observe distant objects by their emission, absorption, or reflection of electromagnetic radiation. Originally meaning only an optical instrument using lenses, curved mirrors, or a combination of both to observe ...
, 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 siz ...
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 ...
; 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 microscopes to view objects and areas of objects that cannot be seen with the naked eye (objects that are not within the resolution range of the normal eye). There are three well-known branches of micr ...
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 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 lens consists of several simple lenses (''elements''), ...
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 simp ...
), 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 divergence, would not disperse with distance. However, diffraction pr ...
will be focused to a single spot. For a diverging lens (for example a
concave lens
A lens is a transmissive 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 lens consists of several simple lenses (''elements''), ...
), 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
:
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
:
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
![Thick Lens Diagram](https://upload.wikimedia.org/wikipedia/commons/f/fc/Thick_Lens_Diagram.svg)
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 or on other media capab ...
or a
telescope
A telescope is a device used to observe distant objects by their emission, absorption, or reflection of electromagnetic radiation. Originally meaning only an optical instrument using lenses, curved mirrors, or a combination of both to observe ...
), 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, vertices or vertexes may refer to:
Science and technology Mathematics and computer science
*Vertex (geometry), a point where two or more curves, lines, or edges meet
*Vertex (computer graphics), a data structure that describes the position ...
of the ''first optical surface'' (S
1).
* ''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 (S
2) 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 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'' ...
s (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 (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium.
The refractive index determines how much the path of light is bent, or ...
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 siz ...
of the system. The other parameters are used in determining where an
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
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 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 lens consists of several simple lenses (''elements''), ...
:
:
where ''n'' is the
refractive index
In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium.
The refractive index determines how much the path of light is bent, or ...
of the lens medium. The quantity is also known as the optical power of the lens.
The corresponding front focal distance is:
:
and the back focal distance:
:
In the
sign convention
In physics, a sign convention is a choice of the physical significance of signs (plus or minus) for a set of quantities, in a case where the choice of sign is arbitrary. "Arbitrary" here means that the same physical system can be correctly describ ...
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
spherically curved
mirror
A mirror or looking glass 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 ...
in air, the magnitude of the focal length is equal to the
radius of curvature
In differential geometry, the radius of curvature, , is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius o ...
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, but other shapes are ...
, and negative for a
convex 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, but other shapes are ...
. In the sign convention used in optical design, a concave mirror has negative radius of curvature, so
:
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. The vertex of the lens s ...
for more information on the sign convention for radius of curvature used here.
In photography
![Thin lens images](https://upload.wikimedia.org/wikipedia/commons/4/47/Thin_lens_images.svg)
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 seen at any given moment. In the case of optical instruments or sensors it is a solid angle through which a detector is sensitive to electromagnetic radiation.
Humans a ...
(FOV) of a lens are inversely proportional. For a standard
rectilinear lens
In photography, a rectilinear lens is a photographic lens that yields images where straight features, such as the edges of walls of buildings, appear with straight lines, as opposed to being curved. In other words, it is a lens with little or no b ...
, FOV = 2 arctan , where ''x'' is the diagonal of the film.
When a photographic lens is set to "infinity", its rear
principal 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'' ...
is separated from the sensor or film, which is then situated 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'' ...
, 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 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 (''s''
1), and the distance from the rear principal plane to the image plane (''s''
2) are then related by:
:
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 lens that reproduces a field of view that appears "natural" to a human observer. In contrast, depth compression and expansion with shorter or longer focal lengths introduces noticeable, and som ...
for a
35 mm camera with a focal length of ''f'' = 50 mm. To focus a distant object (''s''
1 ≈ ∞), the rear principal plane of the lens must be located a distance ''s''
2 = 50 mm from the film plane, so that it is at location of 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 film 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
pinhole that images distant objects the same size as the lens in question. For
rectilinear lens
In photography, a rectilinear lens is a photographic lens that yields images where straight features, such as the edges of walls of buildings, appear with straight lines, as opposed to being curved. In other words, it is a lens with little or no b ...
es (that is, with no
image distortion
In geometric optics, distortion is a deviation from rectilinear projection; a projection in which straight lines in a scene remain straight in an image. It is a form of optical aberration.
Radial distortion
Although distortion can be irreg ...
), the imaging of distant objects is well modelled as a
pinhole camera model
The pinhole camera model describes the mathematical relationship between the coordinates of a point in three-dimensional space and its projection onto the image plane of an ''ideal'' pinhole camera, where the camera aperture is described as a poi ...
.
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 ...
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 lens that reproduces a field of view that appears "natural" to a human observer. In contrast, depth compression and expansion with shorter or longer focal lengths introduces noticeable, and som ...
; 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 and cinematography, a wide-angle lens refers to a lens whose focal length is substantially smaller than the focal length of a normal lens for a given film plane. This type of lens allows more of the scene to be included in the pho ...
(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 and cinematography, is a specific type of a long-focus lens in which the physical length of the lens is shorter than the focal length. This is achieved by incorporating a special lens group known as a ''telephoto ...
(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 that captures photographs in digital memory. Most cameras produced today are digital, largely replacing those that capture images on photographic film. Digital cameras are now widely incorporated into mobile device ...
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 dimensions of a camera's imaging area compared to a reference format; most often, this term is applied to digital ca ...
.
See also
*
Depth of field
The depth of field (DOF) is the distance between the nearest and the furthest objects that are in acceptably sharp focus in an image captured with a camera.
Factors affecting depth of field
For cameras that can only focus on one object dist ...
*
Dioptre
A dioptre (British spelling) or diopter (American spelling) is a unit of measurement with dimension of reciprocal length, equivalent to one reciprocal metre, 1 dioptre = 1 m−1. It is normally used to express the optical power of a lens or curv ...
*
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 ...
or focal ratio
References
{{Authority control
Geometrical optics
Length
Science of photography