# Focal Length And Distance From Subject

By | 24/10/2022

Measure of how strongly an optical system converges or diverges lite

The 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 optical system is a measure out of how strongly the system converges or diverges light; it is the changed of the organisation’s optical power. A positive focal length indicates that a system converges low-cal, while a negative focal length indicates that the organisation 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 apace. For the special case of a thin lens in air, a positive focal length is the altitude over which initially collimated (parallel) rays are brought to a focus, or alternatively a negative focal length indicates how far in front end 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 ability.

In almost photography and all telescopy, where the subject is essentially infinitely far abroad, 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 exist brought closer to the middle of projection.

## Thin lens approximation

For a sparse lens in air, the focal length is the distance from the center of the lens to the chief 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 axle 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 altitude to the indicate from which a collimated axle 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
five, and the focal length
f
are related by

${\displaystyle {\frac {1}{f}}={\frac {1}{u}}+{\frac {1}{v}}\ .}$

one
f

=

1
u

+

1
5

.

{\displaystyle {\frac {ane}{f}}={\frac {1}{u}}+{\frac {1}{5}}\ .}

The focal length of a thin
convex
lens can be easily measured by using it to class an epitome of a afar light source on a screen. The lens is moved until a sharp image is formed on the screen. In this case

1
/

u

is negligible, and the focal length is so given by

${\displaystyle f\approx v\ .}$

f

v

.

{\displaystyle f\approx v\ .}

Determining the focal length of a
concave
lens is somewhat more hard. The focal length of such a lens is considered that point at which the spreading beams of light would see before the lens if the lens were not there. No image is formed during such a exam, and the focal length must be determined by passing light (for example, the light of a light amplification by stimulated emission of radiation beam) through the lens, examining how much that light becomes dispersed/ aptitude, and following the beam of light backwards to the lens’s focal indicate.

## General optical systems

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

• Front focal length
(FFL) or
front focal distance
(FFD) (due south
F) is the distance from the front focal bespeak of the system (F) to the vertex of the
first optical surface
(Sone).[i]
[2]
• Back focal length
(BFL) or
back focal distance
(BFD) (south′
F′) is the distance from the vertex of the
last optical surface
of the system (S2) to the rear focal bespeak (F′).[i]
[2]

For an optical arrangement in air, the constructive focal length (f
and
f′) gives the distance from the front and rear principal planes (H and H′) to the respective focal points (F and F′). If the surrounding medium is not air, then the distance is multiplied by the refractive index of the medium (north
is the refractive alphabetize of the substance from which the lens itself is made;
due north
1
is the refractive index of any medium in front of the lens;
n
2
is that of any medium in dorsum of information technology). Some authors telephone call these distances the front/rear focal
lengths, distinguishing them from the front/rear focal
distances, divers above.[1]

In general, the focal length or EFL is the value that describes the ability of the optical system to focus calorie-free, and is the value used to summate the magnification of the system. The other parameters are used in determining where an image volition be formed for a given object position.

For the case of a lens of thickness
d
in air (
north
1
=
n
two
= 1
), and surfaces with radii of curvature
R
ane
and
R
2, the effective focal length
f
is given past the Lensmaker’s equation:

${\displaystyle {\frac {1}{f}}=(n-1)\left({\frac {1}{R_{1}}}-{\frac {1}{R_{2}}}+{\frac {(n-1)d}{nR_{1}R_{2}}}\right),}$

i
f

=
(
north

ane
)

(

1

R

one

1

R

two

+

(
n

1
)
d

n

R

i

R

2

)

,

{\displaystyle {\frac {i}{f}}=(north-ane)\left({\frac {i}{R_{1}}}-{\frac {1}{R_{2}}}+{\frac {(due north-1)d}{nR_{one}R_{2}}}\right),}

where
northward
is the refractive index of the lens medium. The quantity

1
/

f

is also known every bit the optical power of the lens.

The corresponding front focal distance is:[3]

${\displaystyle {\mbox{FFD}}=f\left(1+{\frac {(n-1)d}{nR_{2}}}\right),}$

FFD

=
f

(

1
+

(
n

1
)
d

n

R

2

)

,

{\displaystyle {\mbox{FFD}}=f\left(one+{\frac {(northward-one)d}{nR_{2}}}\right),}

and the dorsum focal distance:

${\displaystyle {\mbox{BFD}}=f\left(1-{\frac {(n-1)d}{nR_{1}}}\right).}$

BFD

=
f

(

one

(
n

1
)
d

n

R

one

)

.

{\displaystyle {\mbox{BFD}}=f\left(one-{\frac {(n-1)d}{nR_{1}}}\right).}

In the sign convention used here, the value of
R
ane
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 dissimilar 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 2. The focal length is positive for a concave mirror, and negative for a convex mirror. In the sign convention used in optical blueprint, a concave mirror has negative radius of curvature, so

${\displaystyle f=-{R \over 2},}$

f
=

R
ii

,

{\displaystyle f=-{R \over ii},}

where
R
is the radius of curvature of the mirror’s surface.

See Radius of curvature (eyes) for more than data on the sign convention for radius of curvature used here.

## In photography

28 mm lens

50 mm lens

seventy mm lens

210 mm lens

An example of how lens pick affects angle of view. The photos above were taken past a 35 mm camera at a fixed altitude from the subject.

Images of blackness letters in a sparse convex lens of focal length
f
are shown in cerise. Selected rays are shown for letters
E,
I
and
K
in blue, greenish and orange, respectively. Note that
East
(at twof) has an equal-size, real and inverted image;
I
(at
f) has its image at infinity; and
K
(at

f

/
2
) has a double-size, virtual and upright image.

In this figurer simulation, adjusting the field of view (by irresolute the focal length) while keeping the subject in frame (by changing appropriately the position of the camera) results in vastly differing images. At focal lengths budgeted infinity (0 degrees of field of view), the lite rays are nearly parallel to each other, resulting in the subject looking “flattened”. At small focal lengths (bigger field of view), the field of study appears “foreshortened”.

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

Focal length (f) and field of view (FOV) of a lens are inversely proportional. For a standard rectilinear lens, FOV = 2 arctan

x

/
2f

, where
x
is the diagonal of the moving-picture show.

When a photographic lens is set up to “infinity”, its rear principal airplane is separated from the sensor or film, which is then situated at the focal plane, past the lens’south focal length. Objects far away from the camera then produce sharp images on the sensor or picture show, which is also at the image airplane.

To render closer objects in sharp focus, the lens must exist adjusted to increase the altitude between the rear main plane and the film, to put the film at the image plane. The focal length (f), the distance from the forepart principal aeroplane to the object to photograph (due south
1), and the distance from the rear principal airplane to the epitome plane (s
two) are so related by:

${\displaystyle {\frac {1}{s_{1}}}+{\frac {1}{s_{2}}}={\frac {1}{f}}.}$

ane

s

1

+

1

s

2

=

1
f

.

{\displaystyle {\frac {1}{s_{i}}}+{\frac {1}{s_{2}}}={\frac {1}{f}}.}

Every bit
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 mm. To focus a distant object (s
1 ≈ ∞), the rear chief plane of the lens must be located a altitude
due south
ii = 50 mm from the film aeroplane, so that it is at location of the image plane. To focus an object 1 m abroad (s
i = 1,000 mm), the lens must be moved 2.6 mm farther away from the film plane, to
s
2 = 52.half dozen mm.

The focal length of a lens determines the magnification at which it images distant objects. Information technology is equal to the distance between the image aeroplane and a pinhole that images distant objects the aforementioned size as the lens in question. For rectilinear lenses (that is, with no prototype baloney), the imaging of distant objects is well modelled as a pinhole photographic camera model.[four]
This model leads to the elementary geometric model that photographers use for calculating 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 full general, the bending of view depends also on the baloney.[5]

A lens with a focal length near equal to the diagonal size of the film or sensor format is known equally a normal lens; its angle of view is similar to the bending subtended past a big-enough print viewed at a typical viewing distance of the impress diagonal, which therefore yields a normal perspective when viewing the print;[6]
this angle of view is nigh 53 degrees diagonally. For full-frame 35 mm-format cameras, the diagonal is 43 mm and a typical “normal” lens has a fifty mm focal length. A lens with a focal length shorter than normal is oft referred to as a wide-angle lens (typically 35 mm and less, for 35 mm-format cameras), while a lens significantly longer than normal may exist referred to equally a telephoto lens (typically 85 mm and more, for 35 mm-format cameras). Technically, long focal length lenses are just “telephoto” if the focal length is longer than the physical length of the lens, but the term is ofttimes used to describe whatsoever long focal length lens.

Due to the popularity of the 35 mm standard, photographic 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 accept the same angle of view, or field of view, if used on a total-frame 35 mm camera. Apply of a 35 mm-equivalent focal length is peculiarly common with digital cameras, which often use sensors smaller than 35 mm film, so require correspondingly shorter focal lengths to attain a given angle of view, past a factor known equally the crop factor.

## See as well

• Depth of field
• Dioptre
• f-number or focal ratio

## References

1. ^

a

b

c

John E. Greivenkamp (2004).
Field Guide to Geometrical Optics. SPIE Press. pp. 6–9. ISBN978-0-8194-5294-8.

2. ^

a

b

Hecht, Eugene (2002).
Eyes
(4th ed.). Addison Wesley. p. 168. ISBN978-0805385663.

3. ^

Hecht, Eugene (2002).
Optics
(4th ed.). Addison Wesley. pp. 244–245. ISBN978-0805385663.

4. ^

Charles, Jeffrey (2000).

Practical astrophotography
. Springer. pp. 63–66. ISBN978-ane-85233-023-one.

5. ^

Stroebel, Leslie; Zakia, Richard D. (1993).

The Focal encyclopedia of photography

(3rd ed.). Focal Printing. p. 27. ISBN978-0-240-51417-viii.

6. ^

Stroebel, Leslie D. (1999).
View Camera Technique. Focal Printing. pp. 135–138. ISBN978-0-240-80345-6.

Source: https://en.wikipedia.org/wiki/Focal_length