Editing Lens (section)

===Lensmaker's equation===<!--Lensmaker's equation redirects here-->
[[File:Spherical Lens.gif|thumb|alt=Simulation of the effect of lenses with different curvatures of the two facets on a collimated Gaussian beam.|The position of the focus of a spherical lens depends on the radii of curvature of the two facets.]]
The (effective) focal length <math>f</math> of a spherical lens in air or vacuum for paraxial rays can be calculated from the '''lensmaker's equation''':<ref>{{harvnb|Greivenkamp|2004|p=14}}<br/>{{harvnb|Hecht|1987|loc=§&nbsp;6.1}}</ref><ref name="Hecht-2017" />

<math display="block"> \frac{ 1 }{\ f\ } = \left( n - 1 \right) \left[\ \frac{ 1 }{\ R_1\ } - \frac{ 1 }{\ R_2\ } + \frac{\ \left( n - 1 \right)\ d ~}{\ n\ R_1\ R_2\ }\ \right]\ ,</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.
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where

* <math display="inline">\ n\ </math> is the [[refractive index]] of the lens material;
* <math display="inline">\ R_1\ </math> is the (signed, see [[#Sign convention for radii of curvature R1 and R2|below]]) [[radius of curvature]] of the lens surface closer to the light source;
* <math display="inline">\ R_2\ </math> is the radius of curvature of the lens surface farther from the light source; and
* <math display="inline">\ d\ </math> is the thickness of the lens (the distance along the lens axis between the two [[surface vertex#Surface vertices|surface vertices]]).



The focal length <math display="inline">\ f\ </math> is with respect to the [[Cardinal point (optics)|principal planes]] of the lens, and the locations of the principal planes <math display="inline">\ h_1\ </math> and <math display="inline">\ h_2\ </math> with respect to the respective lens vertices are given by the following formulas, where it is a positive value if it is right to the respective vertex.<ref name="Hecht-2017">{{Cite book |last=Hecht |first=Eugene |title=Optics |publisher=Pearson |year=2017 |isbn=978-1-292-09693-3 |edition=5th |pages= |chapter=Chapter 6.1 Thick Lenses and Lens Systems}}</ref>

<math display="block">\ h_1 = -\ \frac{\ \left( n - 1 \right) f\ d ~}{\ n\ R_2\ }\ </math><math display="block">\ h_2 = -\ \frac{\ \left( n - 1 \right) f\ d ~}{\ n\ R_1\ }\ </math>

The focal length <math>\ f\ </math> is positive for converging lenses, and negative for diverging lenses. The [[multiplicative inverse|reciprocal]] of the focal length, <math display="inline">\ \tfrac{ 1 }{\ f\ }\ ,</math> is the [[optical power]] of the lens. If the focal length is in metres, this gives the optical power in [[dioptre]]s (reciprocal metres).

Lenses have the same focal length when light travels from the back to the front as when light goes from the front to the back. Other properties of the lens, such as the [[Aberration in optical systems|aberrations]] are not the same in both directions.

==== Sign convention for radii of curvature {{math|''R''{{sub|1}}}} and {{math|''R''{{sub|2}}}} <span class="anchor" id="sign convention"></span>====
{{Main|Radius of curvature (optics)}}
<!-- [[Spherical aberration]] links here -->
The signs of the lens' radii of curvature indicate whether the corresponding surfaces are convex or concave. The [[sign convention]] used to represent this varies,<ref>{{Cite web |title=Rule sign for concave and convex lens? |url=https://physics.stackexchange.com/questions/211345/rule-sign-for-concave-and-convex-lens |access-date=2024-10-27 |website=Physics Stack Exchange |language=en}}</ref> but in this article a ''positive'' {{mvar|R}} indicates a surface's center of curvature is further along in the direction of the ray travel (right, in the accompanying diagrams), while ''negative'' {{mvar|R}} means that rays reaching the surface have already passed the center of curvature. Consequently, for external lens surfaces as diagrammed above, {{math|''R''{{sub|1}} > 0}} and {{math|''R''{{sub|2}} < 0}} indicate ''convex'' surfaces (used to converge light in a positive lens), while {{math|''R''{{sub|1}} < 0}} and {{math|''R''{{sub|2}} > 0}} indicate ''concave'' surfaces. The reciprocal of the radius of curvature is called the [[curvature]]. A flat surface has zero curvature, and its radius of curvature is [[infinity|infinite]].

==== Sign convention for other parameters ====
{| class="wikitable sortable mw-collapsible"
|+ Sign convention for Gaussian lens equation<ref name="Hecht-2017a" />
! Parameter
! Meaning
! + Sign
! − Sign
|-
|align=center| {{mvar|s}}<sub>o</sub>
| The distance between an object and a lens.
| Real object
| Virtual object
|-
|align=center| {{mvar|s}}{{sub|i}}
| The distance between an image and a lens.
| Real image
| Virtual image
|-
|align=center| {{mvar|f}}
| The focal length of a lens.
| Converging lens
| Diverging lens
|-
|align=center| {{mvar|y}}{{sub|o}}
| The height of an object from the optical axis.
| Erect object
| Inverted object
|-
|align=center| {{mvar|y}}{{sub|i}}
| The height of an image from the optical axis
| Erect image
| Inverted image
|-
|align=center| {{mvar|M}}{{sub|T}}
| The transverse magnification in imaging (&nbsp;{{math|{{=}}}} the ratio of {{mvar|y}}{{sub|i}} to {{mvar|y}}{{sub|o}}&nbsp;).
| Erect image
| Inverted image
|}
This convention is used in this article. Other conventions such as the [http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/lenseq.html#c2 Cartesian sign convention] change the form of the equations.

==== Thin lens approximation ====
If {{mvar|d}} is small compared to {{math|''R''{{sub|1}}}} and {{math|''R''{{sub|2}}}} then the {{dfn|[[thin lens]]}} approximation can be made. For a lens in air, {{mvar|f}}&thinsp; is then given by<ref name="Hecht-2017b">{{Cite book |last=Hecht |first=Eugene |title=Optics |publisher=Pearson |year=2017 |isbn=978-1-292-09693-3 |edition=5th |language=English |chapter=Thin-Lens Equations}}</ref>

<math display="block">\ \frac{ 1 }{\ f\ } \approx \left( n - 1 \right) \left[\ \frac{ 1 }{\ R_1\ } - \frac{ 1 }{\ R_2\ }\ \right] ~.</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.
-->
==== Derivation ====
[[File:A Diagram for a Spherical Lens Equation with Paraxial Rays, 2024-08-27.png|thumb|A Diagram for a Spherical Lens Equation with Paraxial Rays.]]
The spherical thin lens equation in [[paraxial approximation]] is derived here with respect to the right figure.<ref name="Hecht-2017b" /> The 1st spherical lens surface (which meets the optical axis at <math display="inline">\ V_1\ </math> as its vertex) images an on-axis object point ''O'' to the virtual image ''I'', which can be described by the following equation,<math display="block">\ \frac{\ n_1\ }{\ u\ } + \frac{\ n_2\ }{\ v'\ } = \frac{\ n_2 - n_1\ }{\ R_1\ } ~.</math> For the imaging by second lens surface, by taking the above sign convention, <math display="inline">\ u' = - v' + d\ </math> and <math display="block">\ \frac{ n_2 }{\ -v' + d\ } + \frac{\ n_1\ }{\ v\ } = \frac{\ n_1 - n_2\ }{\ R_2\ } ~.</math> Adding these two equations yields <math display="block">\ \frac{\ n_1\ }{ u } + \frac{\ n_1\ }{ v } = \left( n_2 - n_1 \right) \left( \frac{ 1 }{\ R_1\ } - \frac{ 1 }{\ R_2\ } \right) + \frac{\ n_2\ d\ }{\ \left(\ v' - d\ \right)\ v'\ } ~.</math> For the thin lens approximation where <math>\ d \rightarrow 0\ ,</math> the 2nd term of the RHS (Right Hand Side) is gone, so

<math display="block">\ \frac{\ n_1\ }{ u } + \frac{\ n_1\ }{ v } = \left( n_2 - n_1 \right) \left( \frac{ 1 }{\ R_1\ } - \frac{ 1 }{\ R_2\ } \right) ~.</math>

The focal length <math>\ f\ </math> of the thin lens is found by limiting <math>\ u \rightarrow - \infty\ ,</math>

<math display="block">\ \frac{\ n_1\ }{\ f\ } = \left( n_2 - n_1 \right) \left( \frac{ 1 }{\ R_1\ } - \frac{ 1 }{\ R_2\ } \right) \rightarrow \frac{ 1 }{\ f\ } = \left( \frac{\ n_2\ }{\ n_1\ } - 1 \right) \left( \frac{ 1 }{\ R_1\ } - \frac{ 1 }{\ R_2\ } \right) ~.</math>

So, the Gaussian thin lens equation is

<math display="block">\ \frac{ 1 }{\ u\ } + \frac{ 1 }{\ v\ } = \frac{ 1 }{\ f\ } ~.</math>

For the thin lens in air or vacuum where <math display="inline">\ n_1 = 1\ </math> can be assumed, <math display="inline">\ f\ </math> becomes

<math display="block">\ \frac{ 1 }{\ f\ } = \left( n - 1 \right)\left(\frac{ 1 }{\ R_1\ } - \frac{ 1 }{\ R_2\ } \right)\ </math>

where the subscript of 2 in <math display="inline">\ n_2\ </math> is dropped.