Focal length
Overview
 
The focal length of an optical
Optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...

 system is a measure of how strongly the system converges (focuses) or diverges (defocuses) light
Light
Light or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...

. For an optical system in air, it is the distance over which initially collimated rays are brought to a focus
Focus (optics)
In geometrical optics, a focus, also called an image point, is the point where light rays originating from a point on the object converge. Although the focus is conceptually a point, physically the focus has a spatial extent, called the blur circle. This non-ideal focusing may be caused by...

. A system with a shorter focal length has greater optical power
Optical power
Optical power is the degree to which a lens, mirror, or other optical system converges or diverges light. It is equal to the reciprocal of the focal length of the device. The dioptre is the most common unit of measurement of optical power...

 than one with a long focal length; that is, it bends the ray
Ray (optics)
In optics, a ray is an idealized narrow beam of light. Rays are used to model the propagation of light through an optical system, by dividing the real light field up into discrete rays that can be computationally propagated through the system by the techniques of ray tracing. This allows even very...

s more strongly, bringing them to a focus in a shorter distance.

In most photography
Photography
Photography is the art, science and practice of creating durable images by recording light or other electromagnetic radiation, either electronically by means of an image sensor or chemically by means of a light-sensitive material such as photographic film...

 and all telescopy, where the subject is essentially infinitely far away and magnification is achieved by moving the image plane away from the center of projection, longer focal length (lower optical power) leads to higher magnification
Magnification
Magnification is the process of enlarging something only in appearance, not in physical size. This enlargement is quantified by a calculated number also called "magnification"...

 and a narrower angle of view
Angle of view
In photography, angle of view 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....

; conversely, shorter focal length or higher optical power is associated with a wider angle of view.
Encyclopedia
The focal length of an optical
Optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...

 system is a measure of how strongly the system converges (focuses) or diverges (defocuses) light
Light
Light or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...

. For an optical system in air, it is the distance over which initially collimated rays are brought to a focus
Focus (optics)
In geometrical optics, a focus, also called an image point, is the point where light rays originating from a point on the object converge. Although the focus is conceptually a point, physically the focus has a spatial extent, called the blur circle. This non-ideal focusing may be caused by...

. A system with a shorter focal length has greater optical power
Optical power
Optical power is the degree to which a lens, mirror, or other optical system converges or diverges light. It is equal to the reciprocal of the focal length of the device. The dioptre is the most common unit of measurement of optical power...

 than one with a long focal length; that is, it bends the ray
Ray (optics)
In optics, a ray is an idealized narrow beam of light. Rays are used to model the propagation of light through an optical system, by dividing the real light field up into discrete rays that can be computationally propagated through the system by the techniques of ray tracing. This allows even very...

s more strongly, bringing them to a focus in a shorter distance.

In most photography
Photography
Photography is the art, science and practice of creating durable images by recording light or other electromagnetic radiation, either electronically by means of an image sensor or chemically by means of a light-sensitive material such as photographic film...

 and all telescopy, where the subject is essentially infinitely far away and magnification is achieved by moving the image plane away from the center of projection, longer focal length (lower optical power) leads to higher magnification
Magnification
Magnification is the process of enlarging something only in appearance, not in physical size. This enlargement is quantified by a calculated number also called "magnification"...

 and a narrower angle of view
Angle of view
In photography, angle of view 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....

; conversely, shorter focal length or higher optical power is associated with a wider angle of view. On the other hand, in applications such as microscopy
Microscopy
Microscopy is the technical field of using microscopes to view samples and objects that cannot be seen with the unaided eye...

 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
Thin lens
[Image:Lens1.svg|thumb|A lens can be considered a thin lens if d [Image:Lens1.svg|thumb|A lens can be considered a thin lens if d [Image:Lens1.svg|thumb|A lens can be considered a thin lens if d...

 in air, the focal length is the distance from the center of the lens
Lens (optics)
A lens is an optical device with perfect or approximate axial symmetry which transmits and refracts light, converging or diverging the beam. A simple lens consists of a single optical element...

 to the principal foci
Focus (optics)
In geometrical optics, a focus, also called an image point, is the point where light rays originating from a point on the object converge. Although the focus is conceptually a point, physically the focus has a spatial extent, called the blur circle. This non-ideal focusing may be caused by...

 (or focal points
Focus (optics)
In geometrical optics, a focus, also called an image point, is the point where light rays originating from a point on the object converge. Although the focus is conceptually a point, physically the focus has a spatial extent, called the blur circle. This non-ideal focusing may be caused by...

) 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
Collimated light
Collimated light is light whose rays are parallel, and therefore will spread slowly as it propagates. The word is related to "collinear" and implies light that does not disperse with distance , or that will disperse minimally...

 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.

General optical systems

For a thick lens (one which has a non-negligible
Negligible
Negligible refers to the quantities so small that they can be ignored when studying the larger effect. Although related to the more mathematical concepts of infinitesimal, the idea of negligibility is particularly useful in practical disciplines like physics, chemistry, mechanical and electronic...

 thickness), or an imaging system consisting of several lenses and/or mirrors (e.g., a photographic lens
Photographic lens
A camera lens 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 capable of storing an image chemically or electronically.While in principle a simple convex lens will suffice, in...

 or a telescope
Telescope
A telescope is an instrument that aids in the observation of remote objects by collecting electromagnetic radiation . The first known practical telescopes were invented in the Netherlands at the beginning of the 1600s , using glass lenses...

), 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) is the distance from the front focal point of the system to the vertex of the first optical surface.
  • Back focal length (BFL) or Back focal distance (BFD) is the distance from the vertex of the last optical surface of the system to the rear focal point.


For an optical system in air, the effective focal length gives the distance from the front and rear principal planes to the corresponding focal points. If the surrounding medium is not air, then the distance is multiplied by the refractive index
Refractive index
In optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....

 of the medium. Some authors call this distance the front (rear) focal length, distinguishing it from the front (rear) focal distance, 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
Magnification is the process of enlarging something only in appearance, not in physical size. This enlargement is quantified by a calculated number also called "magnification"...

 of the system. The other parameters are used in determining where an image
Image
An image is an artifact, for example a two-dimensional picture, that has a similar appearance to some subject—usually a physical object or a person.-Characteristics:...

 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
Radius of curvature (optics)
Radius of curvature has specific meaning and sign convention in optical design. A spherical lens or mirror surface has a center of curvature located in either along or decentered from the system local optical axis. The vertex of the lens surface is located on the local optical axis...

 R1 and R2, the effective focal length f is given by:

where n is the refractive index
Refractive index
In optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....

 of the lens medium. The quantity 1/f is also known as the optical power
Optical power
Optical power is the degree to which a lens, mirror, or other optical system converges or diverges light. It is equal to the reciprocal of the focal length of the device. The dioptre is the most common unit of measurement of optical power...

 of the lens.
The corresponding front focal distance is:
and the back focal distance:
In the sign convention
Sign convention
In physics, a sign convention is a choice of the physical significance of signs 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 described using different choices for the signs, as long as one set of...

 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 positive if the second surface is concave, and negative if convex. Note that sign conventions vary between different authors, which results in different forms of these equations depending on the convention used.

For a spherically
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

 curved mirror
Mirror
A mirror is an object that reflects light or sound in a way that preserves much of its original quality prior to its contact with the mirror. Some mirrors also filter out some wavelengths, while preserving other wavelengths in the reflection...

 in air, the magnitude of the focal length is equal to the radius of curvature
Radius of curvature (optics)
Radius of curvature has specific meaning and sign convention in optical design. A spherical lens or mirror surface has a center of curvature located in either along or decentered from the system local optical axis. The vertex of the lens surface is located on the local optical axis...

 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
,
where is the radius of curvature of the mirror's surface.

See Radius of curvature (optics)
Radius of curvature (optics)
Radius of curvature has specific meaning and sign convention in optical design. A spherical lens or mirror surface has a center of curvature located in either along or decentered from the system local optical axis. The vertex of the lens surface is located on the local optical axis...

 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.

When a photographic lens
Photographic lens
A camera lens 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 capable of storing an image chemically or electronically.While in principle a simple convex lens will suffice, in...

 is set to "infinity", its rear nodal point is separated from the sensor or film, 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.
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 ( ), the distance from the front nodal point to the object to photograph (), and the distance from the rear nodal point to the image plane () are then related by:
.

As is decreased, must be increased. For example, consider a normal lens
Normal lens
In photography and cinematography a normal lens, also called a standard lens, is a lens that reproduces perspective that generally looks "natural" to a human observer under normal viewing conditions, as compared with lenses with longer or shorter focal lengths which produce an expanded or...

 for a 35 mm camera with a focal length of . To focus a distant object (), the rear nodal point of the lens must be located a distance from the image plane. To focus an object 1 m away (), the lens must be moved 2.6 mm further away from the image plane, to .

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
Pinhole camera
A pinhole camera is a simple camera without a lens and with a single small aperture – effectively a light-proof box with a small hole in one side. Light from a scene passes through this single point and projects an inverted image on the opposite side of the box...

 distant objects the same size as the lens in question. For rectilinear lens
Rectilinear lens
In photography, a rectilinear lens is a photographic lens that yields images where straight features, such as the walls of buildings, appear with straight lines, as opposed to being curved. In other words, it is a lens with little or no barrel or pincushion distortion...

es (that is, with no image distortion), the imaging of distant objects is well modeled as a pinhole camera model
Pinhole camera model
The pinhole camera model describes the mathematical relationship between the coordinates of a 3D point and its projection onto the image plane of an ideal pinhole camera, where the camera aperture is described as a point and no lenses are used to focus light...

.
This model leads to the simple geometric model that photographers use for computing the angle of view
Angle of view
In photography, angle of view 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....

 of a camera; in this case, the angle of view depends only on the ratio of focal length to film size
Film format
A film format is a technical definition of a set of standard characteristics regarding image capture on photographic film, for either stills or movies. It can also apply to projected film, either slides or movies. The primary characteristic of a film format is its size and shape.In the case of...

. 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
Normal lens
In photography and cinematography a normal lens, also called a standard lens, is a lens that reproduces perspective that generally looks "natural" to a human observer under normal viewing conditions, as compared with lenses with longer or shorter focal lengths which produce an expanded or...

; 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 35mm-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
Wide-angle lens
From a design perspective, a wide angle lens is one that projects a substantially larger image circle than would be typical for a standard design lens of the same focal length; this enables either large tilt & shift movements with a view camera, or lenses with wide fields of view.More informally,...

 (typically 35 mm and less, for 35mm-format cameras), while a lens significantly longer than normal may be referred to as a telephoto lens
Telephoto lens
In photography and cinematography, a telephoto lens 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 group that extends the light path to create a long-focus...

 (typically 85 mm and more, for 35mm-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
135 film
The term 135 was introduced by Kodak in 1934 as a designation for cartridge film wide, specifically for still photography. It quickly grew in popularity, surpassing 120 film by the late 1960s to become the most popular photographic film format...

, camera–lens combinations are often described in terms of their 35 mm equivalent focal length
35 mm equivalent focal length
In photography, the 35 mm equivalent focal length is a measure that indicates the angle of view of a particular combination of a camera lens and film or sensor size...

, that is, the focal length of a lens that would have the same angle of view
Angle of view
In photography, angle of view 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....

, 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
Digital camera
A digital camera is a camera that takes video or still photographs, or both, digitally by recording images via an electronic image sensor. It is the main device used in the field of digital photography...

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
Crop factor
In digital photography, a crop factor is related to the ratio of the dimensions of a camera's imaging area compared to a reference format; most often, this term is applied to digital cameras, relative to 35 mm film format as a reference. In the case of digital cameras, the imaging device would be a...

.

See also

  • Depth of field
    Depth of field
    In optics, particularly as it relates to film and photography, depth of field is the distance between the nearest and farthest objects in a scene that appear acceptably sharp in an image...

  • f-number
    F-number
    In optics, the f-number of an optical system expresses the diameter of the entrance pupil in terms of the focal length of the lens; in simpler terms, the f-number is the focal length divided by the "effective" aperture diameter...

     or focal ratio
  • Dioptre
    Dioptre
    A dioptre, or diopter, is a unit of measurement of the optical power of a lens or curved mirror, which is equal to the reciprocal of the focal length measured in metres . It is thus a unit of reciprocal length. For example, a 3-dioptre lens brings parallel rays of light to focus at metre...

  • Focus (optics)
    Focus (optics)
    In geometrical optics, a focus, also called an image point, is the point where light rays originating from a point on the object converge. Although the focus is conceptually a point, physically the focus has a spatial extent, called the blur circle. This non-ideal focusing may be caused by...

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