Editing Lens (section)

== Construction of simple lenses<span class="anchor" id="simple_lens_anchor"></span>==
Most lenses are ''spherical lenses'': their two surfaces are parts of the surfaces of spheres. Each surface can be [[wikt:convex|''convex'']] (bulging outwards from the lens), [[wikt:concave|''concave'']] (depressed into the lens), or ''planar'' (flat). The line joining the centres of the spheres making up the lens surfaces is called the ''axis'' of the lens. Typically the lens axis passes through the physical centre of the lens, because of the way they are manufactured. Lenses may be cut or ground after manufacturing to give them a different shape or size. The lens axis may then not pass through the physical centre of the lens.

[[Toric lens|Toric]] or sphero-cylindrical lenses have surfaces with two different radii of curvature in two orthogonal planes. They have a different [[focal power]] in different meridians. This forms an [[Astigmatism (optical systems)|astigmatic]] lens. An example is eyeglass lenses that are used to correct [[astigmatism]] in someone's eye.

===Types of simple lenses=== <!--Many redirects point to this section title-->

[[File:Lenses en.svg|Types of lenses|alt=Types of lenses|thumb]]
Lenses are classified by the curvature of the two optical surfaces. A lens is ''biconvex'' (or ''double convex'', or just ''convex'') if both surfaces are [[wikt:convex|convex]]. If both surfaces have the same radius of curvature, the lens is ''equiconvex''. A lens with two [[wikt:concave|concave]] surfaces is ''biconcave'' (or just ''concave''). If one of the surfaces is flat, the lens is ''plano-convex'' or ''plano-concave'' depending on the curvature of the other surface. A lens with one convex and one concave side is ''convex-concave'' or ''meniscus''. Convex-concave lenses are most commonly used in [[corrective lenses#Lens shape|corrective lens]]es, since the shape minimizes some aberrations.

For a biconvex or plano-convex lens in a lower-index medium, a [[collimated light|collimated]] beam of light passing through the lens converges to a spot (a ''focus'') behind the lens. In this case, the lens is called a ''positive'' or ''converging'' lens. For a [[thin lens]] in air, the distance from the lens to the spot is the [[focal length]] of the lens, which is commonly represented by  {{mvar|f}}  in diagrams and equations. An [[extended hemispherical lens]] is a special type of plano-convex lens, in which the lens's curved surface is a full hemisphere and the lens is much thicker than the radius of curvature.

Another extreme case of a thick convex lens is a [[ball lens]], whose shape is completely round. When used in novelty photography it is often called a "lensball". A ball-shaped lens has the advantage of being omnidirectional, but for most [[optical glass]] types, its focal point lies close to the ball's surface. Because of the ball's curvature extremes compared to the lens size, [[optical aberration]] is much worse than thin lenses, with the notable exception of [[chromatic aberration]].

{|
|-
|[[File:lens1.svg|left|390px|Biconvex lens]]
|[[File:Large convex lens.jpg|right|250px]]
|}
{{clear}}

For a biconcave or plano-concave lens in a lower-index medium, a collimated beam of light passing through the lens is diverged (spread); the lens is thus called a ''negative'' or ''diverging'' lens. The beam, after passing through the lens, appears to emanate from a particular point on the axis in front of the lens. For a thin lens in air, the distance from this point to the lens is the focal length, though it is negative with respect to the focal length of a converging lens.
{|
|-
|[[File:lens1b.svg|left|390px|Biconcave lens]]
|[[File:concave lens.jpg|right|250px]]
|}
{{clear}}

The behavior reverses when a lens is placed in a medium with higher refractive index than the material of the lens. In this case a biconvex or plano-convex lens diverges light, and a biconcave or plano-concave one converges it.

[[File:Meniscus lenses.svg|thumb|right|upright=0.6|Meniscus lenses: negative (top) and positive (bottom)]]
Convex-concave (meniscus) lenses can be either positive or negative, depending on the relative curvatures of the two surfaces. A ''negative meniscus'' lens has a steeper concave surface (with a shorter radius than the convex surface) and is thinner at the centre than at the periphery. Conversely, a ''positive meniscus'' lens has a steeper convex surface (with a shorter radius than the concave surface) and is thicker at the centre than at the periphery.

An ideal [[thin lens]] with two surfaces of equal curvature (also equal in the sign) would have zero [[optical power]] (as its focal length becomes infinity as shown in the [[#Lensmaker's equation|lensmaker's equation]]), meaning that it would neither converge nor diverge light. All real lenses have a nonzero thickness, however, which makes a real lens with identical curved surfaces slightly positive. To obtain exactly zero optical power, a meniscus lens must have slightly unequal curvatures to account for the effect of the lens' thickness.

{{clear}}

=== For a spherical surface ===
[[File:Refraction at spherical surface.svg|thumb|Simulation of refraction at spherical surface at [https://www.desmos.com/calculator/ax4rsqdot0 Desmos]]]
For a single refraction for a circular boundary, the relation between object and its image in the [[paraxial approximation]] is given by<ref>{{Cite web |date=2019-07-02 |title=4.4: Spherical Refractors |url=https://phys.libretexts.org/Courses/University_of_California_Davis/UCD%3A_Physics_9B__Waves_Sound_Optics_Thermodynamics_and_Fluids/04%3A_Geometrical_Optics/4.04%3A_Spherical_Refractors |access-date=2023-07-02 |website=Physics LibreTexts |language=en |archive-date=26 November 2022 |archive-url=https://web.archive.org/web/20221126132929/https://phys.libretexts.org/Courses/University_of_California_Davis/UCD%3A_Physics_9B__Waves_Sound_Optics_Thermodynamics_and_Fluids/04%3A_Geometrical_Optics/4.04%3A_Spherical_Refractors |url-status=live }}</ref><ref>{{Cite web |title=Refraction at Spherical Surfaces |url=https://personal.math.ubc.ca/~cass/courses/m309-01a/chu/MirrorsLenses/refraction-curved.htm |access-date=2023-07-02 |website=personal.math.ubc.ca |archive-date=26 October 2021 |archive-url=https://web.archive.org/web/20211026211612/https://personal.math.ubc.ca/~cass/courses/m309-01a/chu/MirrorsLenses/refraction-curved.htm |url-status=live }}</ref>

<math display="block">\frac {n_1}u + \frac {n_2}v = \frac {n_2-n_1}R</math>

where {{mvar|R}} is the radius of the spherical surface, {{math|''n''{{sub|2}}}} is the refractive index of the material of the surface, {{math|''n''{{sub|1}}}} is the refractive index of medium (the medium other than the spherical surface material), <math display="inline">u</math> is the on-axis (on the optical axis) object distance from the line perpendicular to the axis toward the refraction point on the surface (which height is ''h''), and <math display="inline">v</math> is the on-axis image distance from the line. Due to paraxial approximation where the line of ''h'' is close to the vertex of the spherical surface meeting the optical axis on the left, <math display="inline">u</math> and <math display="inline">v</math> are also considered distances with respect to the vertex.

Moving <math display="inline">v</math> toward the right infinity leads to the first or object focal length <math display="inline">f_0</math> for the spherical surface. Similarly, <math display="inline">u</math> toward the left infinity leads to the second or image focal length <math>f_i</math>.<ref>{{Cite book |last=Hecht |first=Eugene |title=Optics |publisher=Pearson |year=2017 |isbn=978-1-292-09693-3 |edition=5th |pages=164 |language=English |chapter=5.2.2 Refraction at Spherical Surfaces}}</ref>

<math display="block">\begin{align}
 f_0 &= \frac{n_1}{n_2 - n_1} R,\\
 f_i &= \frac{n_2}{n_2 - n_1} R
\end{align}</math>

Applying this equation on the two spherical surfaces of a lens and approximating the lens thickness to zero (so a thin lens) leads to the [[#Lensmaker's equation|lensmaker's formula]].

==== Derivation ====
[[File:Refraction in spherical surface.svg|thumb]]
[[File:Four spherical refractions.png|thumb|The four cases of spherical refraction]]
Applying [[Snell's law]] on the spherical surface, <math>n_1 \sin i = n_2 \sin r\,.</math>

Also in the diagram,<math display="block">\begin{align}
\tan (i - \theta) &= \frac hu \\
\tan (\theta - r) &= \frac hv \\
\sin \theta &= \frac hR
\end{align}</math>, and using [[Small-angle approximation|small angle approximation]] (paraxial approximation) and eliminating {{mvar|i}}, {{mvar|r}}, and {{mvar|θ}},

<math display="block">\frac {n_2}v + \frac {n_1}u = \frac {n_2-n_1}R\,.</math>

===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.
-->
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.