Focal length

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 most photography 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 and a narrower angle 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 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.

Contents

Thin lens approximation

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.

General optical systems

For a thick lens (one which has a non-negligible thickness), or an imaging system consisting of several lenses and/or mirrors (e.g., a photographic lens or a telescope), the focal length is often called the effective focal length (EFL), to distinguish it from other commonly-used parameters:

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.[1]

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:

\frac{1}{f} = (n-1) \left[ \frac{1}{R_1} - \frac{1}{R_2} %2B \frac{(n-1)d}{n R_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.

The corresponding front focal distance is:

\mbox{FFD} = f \left( 1 %2B \frac{ (n-1) d}{n R_2} \right),

and the back focal distance:

\mbox{BFD} = f \left( 1 - \frac{ (n-1) d}{n R_1} \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

f = -{R \over 2},

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

28 mm lens
50 mm lens
70 mm lens
210 mm lens
An example of how lens choice affects angle of view. The photos above were taken by a 35 mm camera at a constant distance from the subject.

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:

\frac{1}{S_1} %2B \frac{1}{S_2} = \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 \text{ mm}. To focus a distant object (S_1\approx \infty), the rear nodal point of the lens must be located a distance S_2=50 \text{ mm} from the image plane. To focus an object 1 m away (S_1=1000 \text{ mm}), the lens must be moved 2.6 mm further away from the image plane, to S_2=52.6 \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.[3] 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.[4]

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;[5] 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

  1. ^ a b c John E. Greivenkamp (2004). Field Guide to Geometrical Optics. SPIE Press. p. 6–9. ISBN 9780819452948. http://books.google.com/?id=1YfZNWZAwCAC&pg=PA6&dq=%22focal+length%22+%22rear+principal%22+intitle:Geometrical#v=onepage&q=%22focal%20length%22%20%22rear%20principal%22%20intitle%3AGeometrical&f=false. 
  2. ^ a b Hecht, Eugene (1987). Optics (2nd ed.). Addison Wesley. pp. 148–9. ISBN 0-201-11609-X. 
  3. ^ Jeffrey Charles (2000). Practical astrophotography. Springer. p. 63–66. ISBN 9781852330231. http://books.google.com/books?id=KuQErr3Olf0C&pg=PA63. 
  4. ^ Leslie Stroebel and Richard D. Zakia (1993). The Focal encyclopedia of photography (3rd ed.). Focal Press. p. 27. ISBN 9780240514178. http://books.google.com/books?id=CU7-2ZLGFpYC&pg=PA27. 
  5. ^ Leslie D. Stroebel (1999). View Camera Technique. Focal Press. p. 135–138. ISBN 9780240803456. http://books.google.com/?id=71zxDuunAvMC&pg=PA136&dq=appear-normal+focal-length-lens+print-size+diagonal+viewer+distance.