The focal length of an optical system is a measure of how strongly the system converges (focuses) or diverges (defocuses) light. For an optical system in air, it is the distance over which initially collimated rays are brought to a focus. A system with a shorter focal length has greater optical power than one with a long focal length; that is, it bends the rays more strongly, bringing them to a focus in a shorter distance.

In telescopy and most photography, longer focal length or lower optical power is associated with larger magnification of distant objects, and a narrower angle of view. Conversely, shorter focal length or higher optical power is associated with a wider angle of view. In microscopy, on the other hand, a shorter objective lens focal length leads to higher magnification.

Thin lens approximation

General optical systems

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 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 of the system. The other parameters are used in determining where an 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 R₁ and R₂, the effective focal length f is given by:

:$\backslash frac\{1\}\{f\}\; =\; (n-1)\; \backslash left[\; \backslash frac\{1\}\{R\_1\}\; -\; \backslash frac\{1\}\{R\_2\}\; +\; \backslash frac\{(n-1)d\}\{n\; R\_1\; R\_2\}\; \backslash right],$ where n is the refractive index of the lens medium. The quantity 1/f is also known as the optical power of the lens.

The corresponding front focal distance is: :$\backslash mbox\{FFD\}\; =\; f\; \backslash left(\; 1\; +\; \backslash frac\{\; (n-1)\; d\}\{n\; R\_2\}\; \backslash right),$ and the back focal distance: :$\backslash mbox\{BFD\}\; =\; f\; \backslash left(\; 1\; -\; \backslash frac\{\; (n-1)\; d\}\{n\; R\_1\}\; \backslash right).$

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 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 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

where $R$ is the radius of curvature of the mirror's surface.

See Radius of curvature (optics) 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 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 ($f$ ), 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:

:$\backslash frac\{1\}\{S\_1\}\; +\; \backslash frac\{1\}\{S\_2\}\; =\; \backslash frac\{1\}\{f\}$.

As $S\_1$ is decreased, $S\_2$ must be increased. For example, consider a normal lens for a 35 mm camera with a focal length of $f=50\; \backslash text\{\; mm\}$. To focus a distant object ($S\_1\backslash approx\; \backslash infty$), the rear nodal point of the lens must be located a distance $S\_2=50\; \backslash text\{\; mm\}$ from the image plane. To focus an object 1 m away ($S\_1=1000\; \backslash text\{\; mm\}$), the lens must be moved 2.6 mm further away from the image plane, to $S\_2=52.6\; \backslash text\{\; mm\}$.

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 lenses (that is, with no image distortion), the imaging of distant objects is well modeled as a pinhole camera model. 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.

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; 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 (typically 35 mm and less, for 35mm-format cameras), while a lens significantly longer than normal may be referred to as a telephoto lens (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, 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.

See also

References

Category:Geometrical optics Category:Length Category:Science of photography