Editing Lens (section)

=== 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)
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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.
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