Editing Lens (section)

== Imaging properties ==
As mentioned above, a positive or converging lens in air focuses a collimated beam travelling along the lens axis to a spot (known as the [[Focus (optics)|focal point]]) at a distance {{mvar|f}} from the lens. Conversely, a [[point source]] of light placed at the focal point is converted into a collimated beam by the lens. These two cases are examples of [[image]] formation in lenses. In the former case, an object at an infinite distance (as represented by a collimated beam of waves) is focused to an image at the focal point of the lens. In the latter, an object at the focal length distance from the lens is imaged at infinity. The plane perpendicular to the lens axis situated at a distance {{mvar|f}} from the lens is called the ''[[Cardinal point (optics)#Focal planes|{{dfn|focal plane}}]]''.

=== Lens equation ===
For [[Paraxial approximation|paraxial rays]], if the distances from an object to a spherical [[thin lens]] (a lens of negligible thickness) and from the lens to the image are {{math|''S''{{sub|1}}}} and {{math|''S''{{sub|2}}}} respectively, the distances are related by the (Gaussian) '''thin lens formula''':<ref>{{cite web |url=http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/lenseq.html |title=Thin Lens Equation |website=Hyperphysics |first=Carl R. |last=Nave |publisher=Georgia State University |access-date=March 17, 2015 |archive-date=12 October 2000 |archive-url=https://web.archive.org/web/20001012073640/http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/lenseq.html |url-status=live }}</ref><ref>{{cite web |url=http://dev.physicslab.org/Document.aspx?doctype=3&filename=GeometricOptics_ThinLensEquation.xml |title=Resource Lesson: Thin Lens Equation |website=PhysicsLab.org |first=Catharine H. |last=Colwell |access-date=March 17, 2015 |archive-date=2 April 2015 |archive-url=https://web.archive.org/web/20150402160324/http://dev.physicslab.org/Document.aspx?doctype=3&filename=GeometricOptics_ThinLensEquation.xml |url-status=live }}</ref><ref>{{cite web |url=http://www.physicsclassroom.com/class/refrn/Lesson-5/The-Mathematics-of-Lenses |title=The Mathematics of Lenses |website=The Physics Classroom |access-date=March 17, 2015 |archive-date=10 March 2015 |archive-url=https://web.archive.org/web/20150310061631/http://www.physicsclassroom.com/class/refrn/Lesson-5/The-Mathematics-of-Lenses |url-status=live }}</ref>

<math display="block">{1\over f} = {1\over S_1} + {1\over S_2}\,.</math>
<!--
 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 is probably using a different sign convention.
-->[[File:Single Lens Imaging, 2024-05-30.png|thumb|Single thin lens imaging with chief rays]]
The right figure shows how the image of an object point can be found by using three rays; the first ray parallelly incident on the lens and refracted toward the second focal point of it, the second ray crossing [[Cardinal point (optics)#Optical center|the optical center of the lens]] (so its direction does not change), and the third ray toward the first focal point and refracted to the direction parallel to the optical axis. This is a simple ray tracing method easily used. Two rays among the three are sufficient to locate the image point. By moving the object along the optical axis, it is shown that the second ray determines the image size while other rays help to locate the image location.

The lens equation can also be put into the "Newtonian" form:<ref name="Hecht-2017a">{{Cite book |last=Hecht |first=Eugene |title=Optics |publisher=Pearson |year=2017 |isbn=978-1-292-09693-3 |edition=5th |pages= |language=English |chapter=Finite Imagery}}</ref>

<math display="block">f^2 = x_1 x_2\,,</math>

where <math>x_1 = S_1-f</math> and <math>x_2 = S_2-f\,.</math> <math display="inline">x_1</math> is positive if it is left to the front focal point <math display="inline">F_1</math>, and <math display="inline">x_2</math> is positive if it is right to the rear focal point <math display="inline">F_2</math>. Because <math display="inline">f^2</math> is positive, an object point and the corresponding imaging point made by a lens are always in opposite sides with respect to their respective focal points. (<math display="inline">x_1</math> and <math display="inline">x_2</math> are either positive or negative.)   

This Newtonian form of the lens equation can be derived by using a similarity between triangles ''P''<sub>1</sub>''P''<sub>O1</sub>''F''<sub>1</sub> and ''L''<sub>3</sub>''L''<sub>2</sub>''F''<sub>1</sub> and another similarity between triangles ''L''<sub>1</sub>''L''<sub>2</sub>''F''<sub>2</sub> and ''P''<sub>2</sub>''P''<sub>02</sub>''F''<sub>2</sub> in the right figure. The similarities give the following equations and combining these results gives the Newtonian form of the lens equation.

<math display="block">\begin{array}{lcr} \frac{y_1}{x_1} = \frac{\left\vert y_2 \right\vert}{f}
\\ \frac{y_1}{f} = \frac{\left\vert y_2 \right\vert}{x_2} \end{array}
</math>
[[File:Single Thick Lens Imaging, 2024-10-07.png|thumb|A diagram of imaging with a single thick lens imaging. ''H''<sub>1</sub> and ''H''<sub>2</sub> are principal points where [[Cardinal point (optics)#Principal planes and points|principal planes]] of the thick lens cross the optical axis. If the object and image spaces are the same medium, then these points are also [[Cardinal point (optics)#Nodal points|nodal points]].]]
[[File:lens3.svg|thumb|A camera lens forms a ''real image'' of a distant object.]]
The above equations also hold for thick lenses (including a compound lens made by multiple lenses, that can be treated as a thick lens) in air or vacuum (which refractive index can be treated as 1) if <math display="inline">S_1</math>, <math display="inline">S_2</math>, and <math display="inline">f</math> are with respect to the [[principal plane]]s of the lens (<math display="inline">f</math> is the [[effective focal length]] in this case).<ref name="Hecht-2017" /> This is because of triangle similarities like the thin lens case above; similarity between triangles ''P''<sub>1</sub>''P''<sub>O1</sub>''F''<sub>1</sub> and ''L''<sub>3</sub>''H''<sub>1</sub>''F''<sub>1</sub> and another similarity between triangles ''L''<sub>1</sub>'''H''<sub>2</sub>''F''<sub>2</sub> and ''P''<sub>2</sub>''P''<sub>02</sub>''F''<sub>2</sub> in the right figure. If distances {{math|''S''{{sub|1}}}} or {{math|''S''{{sub|2}}}} pass through a [[Medium (optics)|medium]] other than air or vacuum, then a more complicated analysis is required.

If an object is placed at a distance {{math|''S''{{sub|1}} > ''f''}} from a positive lens of focal length {{mvar|f}}, we will find an image at a distance {{math|''S''{{sub|2}}}} according to this formula. If a screen is placed at a distance {{math|''S''{{sub|2}}}} on the opposite side of the lens, an image is formed on it. This sort of image, which can be projected onto a screen or [[image sensor]], is known as a ''[[real image]]''. This is the principle of the [[camera]], and also of the [[human eye]], in which the [[retina]] serves as the image sensor.

The focusing adjustment of a camera adjusts {{math|''S''{{sub|2}}}}, as using an image distance different from that required by this formula produces a [[Defocus aberration|defocused]] (fuzzy) image for an object at a distance of {{math|''S''{{sub|1}}}} from the camera. Put another way, modifying {{math|''S''{{sub|2}}}} causes objects at a different {{math|''S''{{sub|1}}}} to come into perfect focus.

[[File:lens3b.svg|thumb|Virtual image formation using a positive lens as a magnifying glass.<ref>There are always 3 "easy rays". For the third ray in this case, see [[:File:Lens3b third ray.svg]].</ref>]]
In some cases, {{math|''S''{{sub|2}}}} is negative, indicating that the image is formed on the opposite side of the lens from where those rays are being considered. Since the diverging light rays emanating from the lens never come into focus, and those rays are not physically present at the point where they {{em|appear}} to form an image, this is called a [[virtual image]]. Unlike real images, a virtual image cannot be projected on a screen, but appears to an observer looking through the lens as if it were a real object at the location of that virtual image. Likewise, it appears to a subsequent lens as if it were an object at that location, so that second lens could again focus that light into a real image, {{math|''S''{{sub|1}}}} then being measured from the virtual image location behind the first lens to the second lens. This is exactly what the eye does when looking through a [[magnifying glass]]. The magnifying glass creates a (magnified) virtual image behind the magnifying glass, but those rays are then re-imaged by the [[Lens (anatomy)|lens of the eye]] to create a ''real image'' on the [[retina]].

{{multiple image
| width = 180
| image1 = lens4.svg
| caption1 = A ''negative'' lens produces a demagnified virtual image.
| image2 = Barlow lens.svg
| caption2 = A [[Barlow lens]] (B) reimages a ''virtual object'' (focus of red ray path) into a magnified real image (green rays at focus)
| align = 
| direction = 
| total_width = 
| alt1 = 
}}
Using a positive lens of focal length {{mvar|f}}, a virtual image results when {{math|''S''{{sub|1}} < ''f''}}, the lens thus being used as a magnifying glass (rather than if {{math|''S''{{sub|1}} ≫ ''f''}} as for a camera). Using a negative lens ({{math|''f'' < 0}}) with a {{em|real object}} ({{math|''S''{{sub|1}} > 0}}) can only produce a virtual image ({{math|''S''{{sub|2}} < 0}}), according to the above formula. It is also possible for the object distance {{math|''S''{{sub|1}}}} to be negative, in which case the lens sees a so-called ''virtual object''. This happens when the lens is inserted into a converging beam (being focused by a previous lens) {{em|before}} the location of its real image. In that case even a negative lens can project a real image, as is done by a [[Barlow lens]].

For a given lens with the focal length ''f'', the minimum distance between an object and the real image is 4''f'' (''S''<sub>1</sub> = ''S''<sub>2</sub> = 2''f''). This is derived by letting ''L'' = ''S''<sub>1</sub> + ''S''<sub>2</sub>, expressing ''S''<sub>2</sub> in terms of ''S''<sub>1</sub> by the lens equation (or expressing ''S''<sub>1</sub> in terms of ''S''<sub>2</sub>), and equating the derivative of ''L'' with respect to ''S''<sub>1</sub> (or ''S''<sub>2</sub>) to zero. (Note that ''L'' has no limit in increasing so its extremum is only the minimum, at which the derivate of ''L'' is zero.) 

{{multiple image
| width2            = 250
| image1            = Reflectionprojection.jpg
| caption1          = Real image of a lamp is projected onto a screen (inverted). Reflections of the lamp from both surfaces of the biconvex lens are visible.
| image2            = Convex lens (magnifying glass) and upside-down image.jpg
| caption2          = A convex lens ({{math|''f'' ≪ ''S''{{sub|1}}}}) forming a real, inverted image (as the image formed by the objective lens of a telescope or binoculars) rather than the upright, virtual image as seen in a [[magnifying glass]] ({{math|''f'' > ''S''{{sub|1}}}}). This [[real image]] may also be viewed when put on a screen.
| align             = 
| direction         = 
| total_width       = 
| alt1              = 
}}

=== Magnification ===
The linear ''[[magnification]]'' of an imaging system using a single lens is given by

<math display="block"> M = - \frac{S_2}{S_1} = \frac{f}{f - S_1}\ = - \frac{f}{x_1}</math>
<!--
 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
 is probably using a different sign convention.
-->

where {{mvar|M}} is the magnification factor defined as the ratio of the size of an image compared to the size of the object. The sign convention here dictates that if {{mvar|M}} is negative, as it is for real images, the image is upside-down with respect to the object. For virtual images {{mvar|M}} is positive, so the image is upright.

This magnification formula provides two easy ways to distinguish converging ({{math|''f'' > 0}}) and diverging ({{math|''f'' < 0}}) lenses: For an object very close to the lens ({{math|1=0 < ''S''{{sub|1}} < {{abs|''f''}}}}), a converging lens would form a magnified (bigger) virtual image, whereas a diverging lens would form a demagnified (smaller) image; For an object very far from the lens ({{math|1=''S''{{sub|1}} > {{abs|''f''}} > 0}}), a converging lens would form an inverted image, whereas a diverging lens would form an upright image.

Linear magnification {{mvar|M}} is not always the most useful measure of magnifying power. For instance, when characterizing a visual telescope or binoculars that produce only a virtual image, one would be more concerned with the [[Magnification#Angular magnification|angular magnification]]—which expresses how much larger a distant object appears through the telescope compared to the naked eye. In the case of a camera one would quote the [[plate scale]], which compares the apparent (angular) size of a distant object to the size of the real image produced at the focus. The plate scale is the reciprocal of the focal length of the camera lens; lenses are categorized as [[long-focus lens]]es or [[wide-angle lens]]es according to their focal lengths.

Using an inappropriate measurement of magnification can be formally correct but yield a meaningless number. For instance, using a magnifying glass of {{val|5|u=cm}} focal length, held {{val|20|u=cm}} from the eye and {{val|5|u=cm}} from the object, produces a virtual image at infinity of infinite linear size: {{math|1=''M'' = ∞}}. But the ''{{dfn|angular magnification}}'' is 5, meaning that the object appears 5 times larger to the eye than without the lens. When taking a picture of the [[moon]] using a camera with a {{val|50|u=mm}} lens, one is not concerned with the linear magnification {{math|1=''M'' ≈ {{val|-50|u=mm}} / {{val|380000|u=km}} = {{val|-1.3|e=-10}}.}} Rather, the plate scale of the camera is about {{val|1|u=°|up=mm}}, from which one can conclude that the {{val|0.5|u=mm}} image on the film corresponds to an angular size of the moon seen from earth of about 0.5°.

In the extreme case where an object is an infinite distance away, {{math|1=''S''{{sub|1}} = ∞}}, {{math|1=''S''{{sub|2}} = ''f''}} and {{math|1=''M'' = −''f''/∞ = 0}}, indicating that the object would be imaged to a single point in the focal plane. In fact, the diameter of the projected spot is not actually zero, since [[diffraction]] places a lower limit on the size of the [[point spread function]]. This is called the [[diffraction limit]].

[[File:Thin lens images.svg|thumb|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|''f''/2}}) has a double-size, virtual and upright image.

Note that the images of letters H, I, J, and i are located far away from the lens such that they are not shown here. What is also shown here that the ray that is parallelly incident on the lens and refracted toward the second focal point ''f'' determines the image size while other rays help to locate the image location.]]

=== Table for thin lens imaging properties ===
{| class="wikitable"
|+Images of Real Objects Formed by Thin Lenses<ref name="Hecht-2017a" /> 
!Lens Type
!Object Location
!Image Type
!Image Location
!Lateral Image Orientation
!Image Magnification
!Remark
|-
|Converging lens (or positive lens)
|<math>\infty > S_1 > 2f</math>
|Real (rays converging to each image point)
|<math>f < S_2 < 2f</math>
|Inverted (opposite to the object orientation)
|Diminished
|
|-
|Converging lens
|<math>S_1 = 2f</math>
|Real
|<math>S_2 = 2f</math>
|Inverted
|Same size
|
|-
|Converging lens
|<math>f < S_1 < 2f</math>
|Real
|<math>\infty > S_2 > 2f</math>
|Inverted
|Magnified
|
|-
|Converging lens
|<math>S_1 = f</math>
|
|<math>\plusmn \infty</math>
|
|
|
|-
|Converging lens
|<math>S_1 < f</math>
|Virtual (rays apparently diverging from each image point)
|<math>\vert S_2 \vert  > S_1 </math>
|Erect (same to the object orientation)
|Magnified
|As an object moves to the lens, the virtual image also gets closer to the lens while the image size is reduced.
|-
|Diverging lens (or negative lens)
|Anywhere
|Virtual
|<math>\vert S_2 \vert < \vert f \vert, S_1 > \vert S_2 \vert</math>
|Erect
|Diminished
|
|}