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class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Angle between converging or diverging light rays</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="/wiki/Vergence_(disambiguation)" class="mw-disambig" title="Vergence (disambiguation)">Vergence (disambiguation)</a>.</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Vergence.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Vergence.svg/220px-Vergence.svg.png" decoding="async" width="220" height="98" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Vergence.svg/330px-Vergence.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Vergence.svg/440px-Vergence.svg.png 2x" data-file-width="621" data-file-height="277" /></a><figcaption>Vergence of a beam. The vergence is inversely proportional to the distance from the focus in metres. If a (positive) lens is focussing the beam, it has to sit left of the focus, while a negative lens has to sit right of the focus to produce the appropriate vergence.</figcaption></figure> <p>In <a href="/wiki/Optics" title="Optics">optics</a>, <b>vergence</b> is the angle formed by <a href="/wiki/Ray_(optics)" title="Ray (optics)">rays</a> of light that are not perfectly <a href="/wiki/Parallel_(geometry)" title="Parallel (geometry)">parallel</a> to one another. Rays that move closer to the <a href="/wiki/Optical_axis" title="Optical axis">optical axis</a> as they propagate are said to be <i>converging</i>, while rays that move away from the axis are <a href="/wiki/Beam_divergence" title="Beam divergence"><i>diverging</i></a>. These imaginary rays are always perpendicular to the <a href="/wiki/Wavefront" title="Wavefront">wavefront</a> of the light, thus the vergence of the light is directly related to the <a href="/wiki/Radius_of_curvature_(optics)" title="Radius of curvature (optics)">radii of curvature</a> of the wavefronts. A convex <a href="/wiki/Lens" title="Lens">lens</a> or concave <a href="/wiki/Curved_mirror" title="Curved mirror">mirror</a> will cause parallel rays to focus, converging toward a point. Beyond that <a href="/wiki/Focus_(optics)" title="Focus (optics)">focal point</a>, the rays diverge. Conversely, a concave lens or convex mirror will cause parallel rays to diverge. </p><p>Light does not actually consist of imaginary rays and light sources are not single-point sources, thus vergence is typically limited to simple ray modeling of optical systems. In a real system, the vergence is a product of the diameter of a light source, its distance from the optics, and the curvature of the optical surfaces. An increase in curvature causes an increase in vergence and a decrease in <a href="/wiki/Focal_length" title="Focal length">focal length</a>, and the image or spot size (waist diameter) will be smaller. Likewise, a decrease in curvature decreases vergence, resulting in a longer focal length and an increase in image or spot diameter. This reciprocal relationship between vergence, focal length, and waist diameter are constant throughout an optical system, and is referred to as the <a href="/wiki/Optical_invariant" class="mw-redirect" title="Optical invariant">optical invariant</a>. A beam that is expanded to a larger diameter will have a lower degree of divergence, but if condensed to a smaller diameter the divergence will be greater. </p><p>The simple ray model fails for some situations, such as for <a href="/wiki/Laser" title="Laser">laser</a> light, where <a href="/wiki/Gaussian_beam" title="Gaussian beam">Gaussian beam</a> analysis must be used instead. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vergence_(optics)&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:PSM_V28_D191_Light_wave_divergence_traveling_through_different_focal_points.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/PSM_V28_D191_Light_wave_divergence_traveling_through_different_focal_points.jpg/220px-PSM_V28_D191_Light_wave_divergence_traveling_through_different_focal_points.jpg" decoding="async" width="220" height="88" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/PSM_V28_D191_Light_wave_divergence_traveling_through_different_focal_points.jpg/330px-PSM_V28_D191_Light_wave_divergence_traveling_through_different_focal_points.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6e/PSM_V28_D191_Light_wave_divergence_traveling_through_different_focal_points.jpg/440px-PSM_V28_D191_Light_wave_divergence_traveling_through_different_focal_points.jpg 2x" data-file-width="1639" data-file-height="655" /></a><figcaption>A simple <a href="/wiki/Telescope" title="Telescope">telescope</a>. Collimated (parallel) light waves converge through a lens, then diverge to be collimated by another lens, converging again through the lens of the eye.</figcaption></figure> <p>In <a href="/wiki/Geometrical_optics" title="Geometrical optics">geometrical optics</a>, <b>vergence</b> describes the <a href="/wiki/Curvature" title="Curvature">curvature</a> of optical wavefronts.<sup id="cite_ref-Intro_to_Geomet_Optics_1-0" class="reference"><a href="#cite_note-Intro_to_Geomet_Optics-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Vergence is defined as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V={\frac {n}{r}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>r</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\frac {n}{r}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/106d036b1f944b6f55828dae919fa86dc55adbbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.763ex; height:4.676ex;" alt="{\displaystyle V={\frac {n}{r}},}"></span></dd></dl> <p>where <i>n</i> is the medium's refractive index and <i>r</i> is the distance from the point source to the wavefront. Vergence is measured in units of <a href="/wiki/Dioptre" title="Dioptre">dioptres</a> (D) which are equivalent to m<sup>−1</sup>.<sup id="cite_ref-Intro_to_Geomet_Optics_1-1" class="reference"><a href="#cite_note-Intro_to_Geomet_Optics-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> This describes the vergence in terms of <a href="/wiki/Optical_power" title="Optical power">optical power</a>. For optics like convex lenses, the converging point of the light exiting the lens is on the input side of the <a href="/wiki/Focal_plane" class="mw-redirect" title="Focal plane">focal plane</a>, and is positive in optical power. For concave lenses, the <a href="/wiki/Focus_(optics)" title="Focus (optics)">focal point</a> is on the back side of the lens, or the output side of the focal plane, and is negative in power. A lens with no optical power is called an <a href="/wiki/Optical_window" title="Optical window">optical window</a>, having flat, parallel faces. The optical power directly relates to how large positive images will be magnified, and how small negative images will be diminished. </p><p>All light sources produce some degree of divergence, as the waves exiting these sources always have some degree of curvature. At the proper distance, these waves can be straightened by use of a lens or mirror, creating <a href="/wiki/Collimated" class="mw-redirect" title="Collimated">collimated</a> beams with minimal divergence, but some degree of divergence will remain, depending on the diameter of the beam versus the focal length.<sup id="cite_ref-Shukla_2-0" class="reference"><a href="#cite_note-Shukla-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> When the distance between the point source and wavefront becomes very large, the vergence goes to zero meaning that the wavefronts are planar and no longer have any detectable curvature. Light from distant stars have such a large radius that any curvature of the wavefronts are undetectable, and have no vergence.<sup id="cite_ref-Shukla_2-1" class="reference"><a href="#cite_note-Shukla-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>The light can also be pictured as consisting of a bundle of lines radiating in the direction of propagation, which are always perpendicular to the wavefront, called "rays". These imaginary lines of infinitely small thicknesses are separated by only the angle between them. In <a href="/wiki/Ray_tracing_(physics)" title="Ray tracing (physics)">ray tracing</a>, the vergence can then be pictured as the angle between any two rays. For imaging or beams, the vergence is often described as the angle between the outermost rays in the bundle (<a href="/wiki/Marginal_ray" class="mw-redirect" title="Marginal ray">marginal rays</a>), at the edge (verge) of a cone of light, and the <a href="/wiki/Optical_axis" title="Optical axis">optical axis</a>. This slope is typically measured in <a href="/wiki/Radians" class="mw-redirect" title="Radians">radians</a>. Thus, in this case the convergence of the rays transmitted by a lens is equal to the radius of the light source divided by its distance from the optics. This limits the size of an image or the minimum spot diameter that can be produced by any focusing optics, which is determined by the reciprocal of that equation; the divergence of the light source multiplied by the distance. This relationship between vergence, focal length, and the minimum spot diameter (also called the "waist diameter") remains constant through all space, and is commonly referred to as the <a href="/wiki/Optical_invariant" class="mw-redirect" title="Optical invariant">optical invariant</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>This angular relationship becomes especially important with laser operations such as <a href="/wiki/Laser_cutting" title="Laser cutting">laser cutting</a> or <a href="/wiki/Laser_welding" class="mw-redirect" title="Laser welding">laser welding</a>, as there is always a trade off between spot diameter, which affects the intensity of the energy, and the distance to the object. When low divergence in the beam is desired, then a larger diameter beam is necessary, but if a smaller beam is needed one must settle for greater divergence, and no change in the position of the lens will alter this. The only way to achieve a smaller spot is to use a lens with a shorter focal-length, or expand the beam to a larger diameter.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>However, this measure of the curvature of wavefronts is only fully valid in <a href="/wiki/Geometrical_optics" title="Geometrical optics">geometrical optics</a>, not in <a href="/wiki/Gaussian_beam" title="Gaussian beam">Gaussian beam</a> optics or in <a href="/wiki/Wave_optics" class="mw-redirect" title="Wave optics">wave optics</a>, where the wavefront at the focus is <a href="/wiki/Wavelength" title="Wavelength">wavelength</a>-dependent and the curvature is not proportional to the distance from the focus. In this case, the <a href="/wiki/Diffraction" title="Diffraction">diffraction</a> of the light starts to play a very active role, often limiting the spot size to even larger diameters, especially in the <a href="/wiki/Far_field" class="mw-redirect" title="Far field">far field</a>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> For non-circular light sources, the divergence may differ depending on the cross-sectional position of the rays from the optical axis. <a href="/wiki/Diode_laser" class="mw-redirect" title="Diode laser">Diode lasers</a>, for example, have greater divergence across the parallel direction (fast axis) than the perpendicular (slow axis), producing beams with rectangular profiles. This type of difference in divergence can be reduced by beam-shaping methods, such as using a <a href="/wiki/Rod_lens" class="mw-redirect" title="Rod lens">rod lens</a> that only affects divergence along a single cross-sectional direction.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Convergence,_divergence,_and_sign_convention"><span id="Convergence.2C_divergence.2C_and_sign_convention"></span>Convergence, divergence, and sign convention</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vergence_(optics)&action=edit&section=2" title="Edit section: Convergence, divergence, and sign convention"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Wavefronts propagating toward a single point yield positive vergence. This is also referred to as <i>convergence</i> since the wavefronts are all converging to the same point of focus. Contrarily, wavefronts propagating away from a single source point give way to negative vergence. Negative vergence is also called <i>divergence</i>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vergence_(optics)&action=edit&section=3" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Optical_power" title="Optical power">Optical power</a></li> <li><a href="/wiki/Optical_resolution" title="Optical resolution">Optical resolution</a></li> <li><a href="/wiki/Cardinal_point_(optics)" title="Cardinal point (optics)">Cardinal point (optics)</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vergence_(optics)&action=edit&section=4" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Intro_to_Geomet_Optics-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Intro_to_Geomet_Optics_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Intro_to_Geomet_Optics_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFKatz2002" class="citation book cs1">Katz, Milton (2002). <i>Introduction to Geometrical Optics</i>. World Scientific. p. 85. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-238-202-3" title="Special:BookSources/978-981-238-202-3"><bdi>978-981-238-202-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Geometrical+Optics&rft.pages=85&rft.pub=World+Scientific&rft.date=2002&rft.isbn=978-981-238-202-3&rft.aulast=Katz&rft.aufirst=Milton&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVergence+%28optics%29" class="Z3988"></span></span> </li> <li id="cite_note-Shukla-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Shukla_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Shukla_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><i>Ophthalmic Medical Personnel: A Guide to Laws, Formulae, Calculations, and Clinical Applications</i> by Aaron V. Shukla – Slack Inc. 2009 Page 73–76</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><i>Clinical Optics and Refraction</i> by Andrew Keirl, Caroline Christie – Elsevier 2007 Page 11–15</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><i>Handbook of Ophthalmology</i> by Amar Agarwal – Slack Inc. 2006 Page 597</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><i>Encyclopedia of Modern Optics</i> by Bob D. Guenther, Duncan Steel – Elsevier 2018 Page 113</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><i>Laser Materials Processing (Manufacturing, Engineering, and Materials Processing)</i> by Leonard R. Migliore – CRC Press 2018 Page 50</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.newport.com/n/focusing-and-collimating">"Focusing and Collimating"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Focusing+and+Collimating&rft_id=https%3A%2F%2Fwww.newport.com%2Fn%2Ffocusing-and-collimating&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVergence+%28optics%29" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><i>Laser Beam Shaping Applications</i> by Fred M. Dickey, Todd E. Lizotte – CRC Press 2017 Page 76–77</span> </li> </ol></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Vergence_(optics)&oldid=1079711532">https://en.wikipedia.org/w/index.php?title=Vergence_(optics)&oldid=1079711532</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Category</a>: <ul><li><a href="/wiki/Category:Geometrical_optics" title="Category:Geometrical optics">Geometrical optics</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 28 March 2022, at 06:50<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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