View source for Focal length

{{Short description|Measure of how strongly an optical system converges or diverges light}}
{{Redirect|Rear focal distance|lens to film distance in a camera|Flange focal distance}}
{{more citations|date=November 2021}}
[[File:Focal-length.svg|right|thumb|The [[Focus (optics)|focal point]] '''F''' and focal length ''f'' of a positive (convex) [[lens]], a negative (concave) lens, a [[concave mirror]], and a [[convex mirror]].]]

The '''focal length''' of an [[Optics|optical]] system is a measure of how strongly the system converges or diverges [[light]]; it is the [[Multiplicative inverse|inverse]] of the system's [[optical power]]. A positive focal length indicates that a system [[Convergence (optics)|converges]] light, while a negative focal length indicates that the system [[Divergence (optics)|diverges]] light. A system with a shorter focal length bends the [[Ray (optics)|ray]]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]] in air, a positive focal length is the distance over which initially [[Collimated beam|collimated]] (parallel) rays are brought to a [[Focus (optics)|focus]], or alternatively a negative focal length indicates how far in front of the lens a [[point source]] 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]] and all [[telescopy]], where the subject is essentially infinitely far away, 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 lower magnification and 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.

== Thin lens approximation ==
For a thin lens in air, the focal length is the distance from the center of the [[Lens (optics)|lens]] to the principal foci (or ''focal points'') of the lens. For a converging lens (for example a [[Lens (optics)#Types of simple lenses|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 [[Lens (optics)#Types of simple lenses|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.

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
:<math>\frac{1}{f} =\frac{1}{u}+\frac{1}{v}\ .</math>

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 {{Sfrac|''u''}} is negligible, and the focal length is then given by 
:<math>f \approx v\ .</math>

Determining the focal length of a ''concave'' lens is somewhat more difficult.  The focal length of such a lens is defined as the point at which the spreading beams of light meet when they are extended backwards. 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 or bent, and following the beam of light backwards to the lens's focal point.

== General optical systems ==
[[File:Thick Lens Diagram.svg|thumb|upright=1.5|Thick lens diagram]]
For a ''thick'' lens (one which has a non-negligible thickness), or an imaging system consisting of several lenses or [[curved mirror|mirror]]s (e.g. a [[photographic lens]] or a [[telescope]]), there are several related concepts that are referred to as focal lengths:
;Effective focal length (EFL){{anchor|EFL}}: The effective focal length is the inverse of the [[optical power]] of an optical system, and is the value used to calculate the [[magnification]] of the system.<ref name="Greivenkamp">{{cite book |last= Grievenkamp |first= John E. |date= 2004 |title= Field Guide to Geometrical Optics |publisher= [[SPIE Press]] |pages= 6–9 |isbn= 978-0-8194-5294-8}}</ref> The imaging properties of the optical system can be modeled by replacing the system with an ideal thin lens with the same EFL.<ref name="efl" /> The EFL also provides a simple method for finding the [[nodal point]]s without tracing any rays. It was previously called ''equivalent focal length'' (not to be confused with [[35&nbsp;mm-equivalent focal length]]).
;Front focal length (FFL): The front focal length {{mvar|f}} is the distance from the front focal point {{math|F}} to the front [[principal plane]] {{math|H}}.
;Rear focal length (RFL): The rear focal length {{mvar|{{prime|f}}}} is the distance from the rear principal plane {{math|{{prime|H}}}} to the rear focal point {{math|{{prime|F}}}}.
;Front focal distance (FFD): The front focal distance (FFD) ({{math|''s''{{sub|F}}}}) is the distance from the front focal point of the system ({{math|F}}) to the [[surface vertex|vertex]] of the ''first optical surface'' ({{math|S{{sub|1}}}}).<ref name = "Greivenkamp" /><ref name="Hecht1">{{cite book |first=Eugene |last=Hecht |year=2002 |title=Optics |edition=4th |publisher=[[Addison Wesley]] |isbn=978-0805385663 |page=168}}</ref> Some authors refer to this as "front focal length".
;Back focal distance (BFD): Back focal distance (BFD) ({{math|''{{prime|s}}''{{sub|{{prime|F}}}}}}) is the distance from the vertex of the ''last optical surface'' of the system ({{math|S{{sub|2}}}}) to the rear focal point ({{math|{{prime|F}}}}).<ref name = "Greivenkamp" /><ref name="Hecht1" /> Some authors refer to this as "back focal length".
	
For an optical system in air the effective focal length, front focal length, and rear focal length are all the same and may be called simply "focal length". 

[[File:Wiki efl.jpg|thumb|Sketch of human eye showing rear focal length {{math|{{prime|f}}}} and EFL]]
For an optical system in a medium other than air or vacuum, the front and rear focal lengths are equal to the EFL times the [[refractive index]] of the medium in front of or behind the lens ({{math|''n''{{sub|1}}}} and {{math|''n''{{sub|2}}}} in the diagram above). The term "focal length" by itself is ambiguous in this case. The historical usage was to define the "focal length" as the EFL times the index of refraction of the medium.<ref name = "efl">{{cite journal |last= Simpson |first= Michael J. |date= 24 February 2023 |title= Focal Length, EFL, and the Eye |journal= Applied Optics |volume= 62 |issue= 7 |pages= 1853–1857 |doi= 10.1364/AO.481805|pmid= 37132938 |bibcode= 2023ApOpt..62.1853S }}</ref><ref name = "Nodal">{{cite journal |last= Simpson |first= Michael J. |date= 28 March 2022 |title= Nodal points and the eye |journal= Applied Optics |volume= 61 |issue= 10 |pages= 2797–2804 |doi= 10.1364/AO.455464|pmid= 35471355 |bibcode= 2022ApOpt..61.2797S }}</ref> For a system with different media on both sides, such as the human eye, the front and rear focal lengths are not equal to one another, and convention may dictate which one is called "the focal length" of the system. Some modern authors avoid this ambiguity by instead defining "focal length" to be a synonym for EFL.<ref name="Greivenkamp"/> 

The distinction between front/rear focal length and EFL is important for studying the human eye. The eye can be represented by an equivalent thin lens at an air/fluid boundary with front and rear focal lengths equal to those of the eye, or it can be represented by a {{em|different}} equivalent thin lens that is totally in air, with focal length equal to the eye's EFL.

For the case of a lens of thickness {{mvar|d}} in air ({{math|1=''n''{{sub|1}} = ''n''{{sub|2}} = 1}}), and surfaces with [[Radius of curvature (optics)|radii of curvature]] {{math|''R''{{sub|1}}}} and {{math|''R''{{sub|2}}}}, the effective focal length {{mvar|f}} is given by the [[Lensmaker's equation]]:<ref>{{Cite book |last=Hecht |first=Eugene |title=Optics |publisher=Pearson |year=2017 |isbn=978-1-292-09693-3 |edition=5th |pages=257 |language=English |chapter=6.1 thick Lenses and Lens systems}}</ref>

<math display="block">\frac{1}{f} = (n-1) \left( \frac{1}{R_1} - \frac{1}{R_2} + \frac{(n-1)d}{n R_1 R_2} \right),</math>where {{mvar|n}} is the refractive index of the lens medium. The quantity {{math|{{Sfrac|''f''}}}} is also known as the [[optical power]] of the lens.
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The corresponding front focal distance is:<ref name=Hecht2>{{cite book |first=Eugene |last=Hecht |date=2002 |title=Optics |edition=4th |publisher=[[Addison Wesley]] |isbn=978-0805385663 |pages=244–245}}</ref><math display="block">\mbox{FFD} = f \left( 1 + \frac{ (n-1) d}{n R_2} \right), </math>
and the back focal distance:
<math display="block">\mbox{BFD} = f \left( 1 - \frac{ (n-1) d}{n R_1} \right). </math>
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In the [[Radius of curvature (optics)|sign convention]] used here, the value of {{math|''R''{{sub|1}}}} will be positive if the first lens surface is convex, and negative if it is concave. The value of {{math|''R''{{sub|2}}}} is negative if the second surface is convex, and positive if concave. Sign conventions vary between different authors, which results in different forms of these equations depending on the convention used.

For a [[curved mirror|spherically-curved mirror]] in air, the magnitude of the focal length is equal to the [[Radius of curvature (optics)|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

<math display="block">f = -{R \over 2},</math>
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  CAUTION TO EDITORS: This equation depends on an arbitrary sign convention
  (explained on the page). If the signs don't match your textbook, your book
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where {{mvar|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 ==
{{Multiple image |align=center |width=150 |image1=Angleofview 28mm f4.jpg |caption1=28&nbsp;mm lens |image2=Angleofview 50mm f4.jpg |caption2=50&nbsp;mm lens |image3=Angleofview 70mm f4.jpg |caption3=70&nbsp;mm lens |image4=Angleofview 210mm f4.jpg |caption4=210&nbsp;mm lens |footer=An example of how lens choice affects angle of view. The photos above were taken by a [[135 film|35&nbsp;mm]] camera at a fixed distance from the subject. |footer_align=center}}
[[File:Thin lens images.svg|thumb|upright=1.5|Images of black letters in a thin convex lens of focal length {{mvar|f}} are shown in red. Selected rays are shown for letters '''E''', '''I''' and '''K''' in blue, green and orange, respectively. '''E''' (at {{math|2''f''}}) has an equal-size, real and inverted image; '''I''' (at {{mvar|f}}) has its image at infinity; and '''K''' (at {{math|1={{Sfrac|''f''|2}}}}) has a double-size, virtual and upright image.]]
[[File:Camera focal length distance house animation.gif|thumb|upright=1.5|In this computer simulation, adjusting the field of view (by changing the focal length) while keeping the subject in frame (by changing accordingly the position of the camera) results in vastly differing images. At focal lengths approaching infinity (0 degrees of angle of view), the light rays are nearly parallel to each other, resulting in the subject looking "flattened". At small focal lengths (bigger field of view), the subject appears "foreshortened".]]

Camera lens focal lengths are usually specified in millimetres (mm), but some older lenses are marked in centimetres (cm) or inches.

Focal length {{(--}}{{mvar|f}}{{--)}} and [[field of view]] (FOV) of a lens are inversely proportional. For a standard [[rectilinear lens]], <math display=inline>\mathrm{FOV} = 2\arctan{\left({x\over2f}\right)}</math>, where {{mvar|x}} is the width of the film or imaging sensor.

When a photographic lens is set to "infinity", its rear [[principal plane]] is separated from the sensor or film, which is then situated 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 principal plane and the film, to put the film at the image plane. The focal length {{mvar|f}}, the distance from the front principal plane to the object to photograph {{math|''s''{{sub|1}}}}, and the distance from the rear principal plane to the image plane {{math|''s''{{sub|2}}}} are then related by:

<math display=block>\frac{1}{s_1} + \frac{1}{s_2} = \frac{1}{f}\,.</math>

As {{math|''s''{{sub|1}}}} is decreased, {{math|''s''{{sub|2}}}} must be increased. For example, consider a [[normal lens]] for a [[35mm format|35&nbsp;mm]] camera with a focal length of {{math|1=''f'' {{=}}}} 50&nbsp;mm. To focus a distant object ({{math|1=''s''{{sub|1}} ≈ ∞}}), the rear principal plane of the lens must be located a distance {{math|1=''s''{{sub|2}} {{=}}}} 50&nbsp;mm from the film plane, so that it is at the location of the image plane. To focus an object 1&nbsp;m away ({{math|1=''s''{{sub|1}} {{=}}}} 1,000&nbsp;mm), the lens must be moved 2.6&nbsp;mm farther away from the film plane, to {{math|1=''s''{{sub|2}} {{=}}}} 52.6&nbsp;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 camera|pinhole that images]] distant objects the same size as the lens in question.  For [[rectilinear lens]]es (that is, with no [[image distortion]]), the imaging of distant objects is well modelled as a [[pinhole camera model]].<ref>
{{cite book
 | title = Practical astrophotography
 | edition = 
 | first = Jeffrey | last = Charles
 | publisher = Springer
 | year = 2000
 | isbn = 978-1-85233-023-1
 | pages = [https://archive.org/details/practicalastroph00char/page/63 63]–66
 | url = https://archive.org/details/practicalastroph00char
 | url-access = registration
 }}</ref>  
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 format|film size]].  In general, the angle of view depends also on the distortion.<ref>
{{cite book
 | title = The Focal encyclopedia of photography
 | edition = 3rd
 | first1 = Leslie | last1 = Stroebel
 | first2 = Richard D. | last2 = Zakia
 | publisher = [[Focal Press]]
 | year = 1993
 | isbn = 978-0-240-51417-8
 | page = [https://archive.org/details/focalencyclopedi00lesl/page/27 27]
 | url = https://archive.org/details/focalencyclopedi00lesl
 | url-access = registration
 }}</ref>

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;<ref>
{{cite book
 | title = View Camera Technique
 | first = Leslie D. | last = Stroebel
 | publisher = [[Focal Press]]
 | year = 1999
 | pages = 135–138
 | isbn = 978-0-240-80345-6
 | url = https://books.google.com/books?id=71zxDuunAvMC&q=appear-normal+focal-length-lens+print-size+diagonal+viewer+distance&pg=PA136  
 }}</ref> 
this angle of view is about 53 degrees diagonally. For [[Full-frame digital SLR|full-frame]] 35&nbsp;mm-format cameras, the diagonal is 43&nbsp;mm and a typical "normal" lens has a 50&nbsp;mm focal length.  A lens with a focal length shorter than normal is often referred to as a [[wide-angle lens]] (typically 35&nbsp;mm and less, for 35&nbsp;mm-format cameras), while a lens significantly longer than normal may be referred to as a [[telephoto lens]] (typically 85&nbsp;mm and more, for 35&nbsp;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 [[135 film|35&nbsp;mm standard]], camera–lens combinations are often described in terms of their 35&nbsp;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&nbsp;mm camera.  Use of a 35&nbsp;mm-equivalent focal length is particularly common with [[digital camera]]s, which often use sensors smaller than 35&nbsp;mm film, and so require correspondingly shorter focal lengths to achieve a given angle of view, by a factor known as the [[crop factor]].

==Optical power==
[[File:optical_power_of_a_lens.svg|thumb|Illustration of the relationship between optical power and focal length]]
The [[optical power]] of a [[lens (optics)|lens]] or curved [[mirror]] is a [[physical quantity]] equal to the [[Multiplicative inverse|reciprocal]] of the focal length, expressed in [[metre]]s. A [[dioptre]] is its [[unit of measurement]] with [[Dimension (physics)|dimension]] of [[reciprocal length]], equivalent to one [[reciprocal metre]], 1 dioptre = 1 m<sup>−1</sup>. For example, a 2-dioptre lens brings parallel [[ray (optics)|rays]] of light to focus at {{frac|2}} metre. A flat window has an optical power of zero dioptres, as it does not cause light to converge or diverge.<ref>{{cite book | first=John E. | last=Greivenkamp | year=2004 | title=Field Guide to Geometrical Optics | publisher=SPIE | others=SPIE Field Guides vol. '''FG01''' | isbn=0-8194-5294-7 |page=7 }}</ref>

The main benefit of using optical power rather than focal length is that the [[thin lens formula]] has the object distance, image distance, and focal length all as reciprocals. Additionally, when [[thin lens|relatively thin lenses]] are placed close together their powers approximately add. Thus, a thin 2.0-dioptre lens placed close to a thin 0.5-dioptre lens yields almost the same focal length as a single 2.5-dioptre lens.

== See also ==
* [[Depth of field]]
* [[Dioptre]]
* [[f-number]] or focal ratio

== References ==
{{Commons category|Focal length}}
{{Reflist}}

{{photography subject}}
{{Authority control}}

[[Category:Geometrical optics]]
[[Category:Length]]
[[Category:Science of photography]]
[[Category:Optical quantities]]