Airy disk - Wikipedia

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class="vector-toc-link" href="#Focused_laser_beam"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Focused laser beam</span> </div> </a> <ul id="toc-Focused_laser_beam-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Aiming_sight" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Aiming_sight"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Aiming sight</span> </div> </a> <ul id="toc-Aiming_sight-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Conditions_for_observation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Conditions_for_observation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Conditions for observation</span> </div> </a> <ul id="toc-Conditions_for_observation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mathematical_formulation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mathematical_formulation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Mathematical formulation</span> </div> </a> <ul id="toc-Mathematical_formulation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Approximation_using_a_Gaussian_profile" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Approximation_using_a_Gaussian_profile"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Approximation using a Gaussian profile</span> </div> </a> <ul id="toc-Approximation_using_a_Gaussian_profile-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Obscured_Airy_pattern" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Obscured_Airy_pattern"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Obscured Airy pattern</span> </div> </a> <ul id="toc-Obscured_Airy_pattern-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Comparison_to_Gaussian_beam_focus" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Comparison_to_Gaussian_beam_focus"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Comparison to Gaussian beam focus</span> </div> </a> <ul id="toc-Comparison_to_Gaussian_beam_focus-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Elliptical_aperture" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Elliptical_aperture"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Elliptical aperture</span> </div> </a> <ul id="toc-Elliptical_aperture-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes_and_references" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes_and_references"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Notes and references</span> </div> </a> <ul id="toc-Notes_and_references-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Airy disk</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 16 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-16" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">16 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Disc_d%27Airy" title="Disc d'Airy – Catalan" lang="ca" hreflang="ca" data-title="Disc d'Airy" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Beugungsscheibchen" title="Beugungsscheibchen – German" lang="de" hreflang="de" data-title="Beugungsscheibchen" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Disco_de_Airy" title="Disco de Airy – Spanish" lang="es" hreflang="es" data-title="Disco de Airy" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Tache_d%27Airy" title="Tache d'Airy – French" lang="fr" hreflang="fr" data-title="Tache d'Airy" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%97%90%EC%96%B4%EB%A6%AC_%EC%9B%90%EB%B0%98" title="에어리 원반 – Korean" lang="ko" hreflang="ko" data-title="에어리 원반" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Disco_di_Airy" title="Disco di Airy – Italian" lang="it" hreflang="it" data-title="Disco di Airy" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%93%D7%99%D7%A1%D7%A7%D7%AA_%D7%90%D7%99%D7%99%D7%A8%D7%99" title="דיסקת איירי – Hebrew" lang="he" hreflang="he" data-title="דיסקת איירי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Airy-schijf" title="Airy-schijf – Dutch" lang="nl" hreflang="nl" data-title="Airy-schijf" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%A8%E3%82%A2%E3%83%AA%E3%83%BC%E3%83%87%E3%82%A3%E3%82%B9%E3%82%AF" title="エアリーディスク – Japanese" lang="ja" hreflang="ja" data-title="エアリーディスク" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Plamka_Airy%E2%80%99ego" title="Plamka Airy’ego – Polish" lang="pl" hreflang="pl" data-title="Plamka Airy’ego" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D0%B8%D1%81%D0%BA_%D0%AD%D0%B9%D1%80%D0%B8" title="Диск Эйри – Russian" lang="ru" hreflang="ru" data-title="Диск Эйри" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Airyjev_disk" title="Airyjev disk – Slovenian" lang="sl" hreflang="sl" data-title="Airyjev disk" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%88%E0%B8%B2%E0%B8%99%E0%B9%81%E0%B8%AD%E0%B8%A3%E0%B8%B5" title="จานแอรี – Thai" lang="th" hreflang="th" data-title="จานแอรี" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Airy_lekesi" title="Airy lekesi – Turkish" lang="tr" hreflang="tr" data-title="Airy lekesi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D0%B8%D1%81%D0%BA_%D0%95%D0%B9%D1%80%D1%96" title="Диск Ейрі – Ukrainian" lang="uk" hreflang="uk" data-title="Диск Ейрі" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%89%BE%E9%87%8C%E6%96%91" title="艾里斑 – Chinese" lang="zh" hreflang="zh" data-title="艾里斑" data-language-autonym="中文" data-language-local-name="Chinese" 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<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">Diffraction pattern in optics</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Airy-pattern.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/Airy-pattern.svg/220px-Airy-pattern.svg.png" decoding="async" width="220" height="163" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/Airy-pattern.svg/330px-Airy-pattern.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/14/Airy-pattern.svg/440px-Airy-pattern.svg.png 2x" data-file-width="283" data-file-height="210" /></a><figcaption>A computer-generated image of an Airy disk. The <a href="/wiki/Grayscale" title="Grayscale">grayscale</a> intensities have been adjusted to enhance the brightness of the outer rings of the Airy pattern.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Airy_disk_D65.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/Airy_disk_D65.png/220px-Airy_disk_D65.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/Airy_disk_D65.png/330px-Airy_disk_D65.png 1.5x, //upload.wikimedia.org/wikipedia/commons/4/40/Airy_disk_D65.png 2x" data-file-width="401" data-file-height="401" /></a><figcaption>A computer-generated Airy disk from diffracted white light (<a href="/wiki/Illuminant_D65" class="mw-redirect" title="Illuminant D65">D65 spectrum</a>). Note that the red component is diffracted more than the blue, so that the center appears slightly bluish.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Beugungsscheibchen.k.720.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Beugungsscheibchen.k.720.jpg/220px-Beugungsscheibchen.k.720.jpg" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Beugungsscheibchen.k.720.jpg/330px-Beugungsscheibchen.k.720.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Beugungsscheibchen.k.720.jpg/440px-Beugungsscheibchen.k.720.jpg 2x" data-file-width="5184" data-file-height="3888" /></a><figcaption>A real Airy disk created by passing a red <a href="/wiki/Laser" title="Laser">laser</a> beam through a 90-<a href="/wiki/Micrometre" title="Micrometre">micrometre</a> pinhole aperture with 27 orders of diffraction</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Rubinar-1000_plus_2x_K-1_telekonv_Airy_disk_1.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Rubinar-1000_plus_2x_K-1_telekonv_Airy_disk_1.jpg/220px-Rubinar-1000_plus_2x_K-1_telekonv_Airy_disk_1.jpg" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Rubinar-1000_plus_2x_K-1_telekonv_Airy_disk_1.jpg/330px-Rubinar-1000_plus_2x_K-1_telekonv_Airy_disk_1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/b/bd/Rubinar-1000_plus_2x_K-1_telekonv_Airy_disk_1.jpg 2x" data-file-width="400" data-file-height="400" /></a><figcaption>Airy disk captured by 2000 mm <a href="/wiki/Camera_lens" title="Camera lens">camera lens</a> at f/25 aperture. Image size: 1×1 mm.</figcaption></figure> <p>In <a href="/wiki/Optics" title="Optics">optics</a>, the <b>Airy disk</b> (or <b>Airy disc</b>) and <b>Airy pattern</b> are descriptions of the best-<a href="/wiki/Focus_(optics)" title="Focus (optics)">focused</a> <a href="/wiki/Point_source#Light" title="Point source">spot</a> of <a href="/wiki/Light" title="Light">light</a> that a perfect <a href="/wiki/Lens_(optics)" class="mw-redirect" title="Lens (optics)">lens</a> with a circular <a href="/wiki/Aperture" title="Aperture">aperture</a> can make, limited by the <a href="/wiki/Diffraction" title="Diffraction">diffraction</a> of light. The Airy disk is of importance in <a href="/wiki/Physics" title="Physics">physics</a>, <a href="/wiki/Optics" title="Optics">optics</a>, and <a href="/wiki/Astronomy" title="Astronomy">astronomy</a>. </p><p>The diffraction pattern resulting from a uniformly illuminated, circular aperture has a bright <a href="/wiki/Circle_of_confusion" title="Circle of confusion">central region</a>, known as the Airy disk, which together with the series of <a href="/wiki/Concentric_objects" title="Concentric objects">concentric</a> rings around is called the Airy pattern. Both are named after <a href="/wiki/George_Biddell_Airy" title="George Biddell Airy">George Biddell Airy</a>. The disk and rings phenomenon had been known prior to Airy; <a href="/wiki/John_Herschel" title="John Herschel">John Herschel</a> described the appearance of a bright <a href="/wiki/Star" title="Star">star</a> seen through a <a href="/wiki/Telescope" title="Telescope">telescope</a> under high magnification for an 1828 article on light for the <i><a href="/wiki/Encyclopedia_Metropolitana" class="mw-redirect" title="Encyclopedia Metropolitana">Encyclopedia Metropolitana</a></i>: </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>...the star is then seen (in favourable circumstances of tranquil atmosphere, uniform temperature, etc.) as a perfectly round, well-defined planetary disc, surrounded by two, three, or more alternately dark and bright rings, which, if examined attentively, are seen to be slightly coloured at their borders. They succeed each other nearly at equal intervals round the central disc....<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></p></blockquote> <p>Airy wrote the first full theoretical treatment explaining the phenomenon (his 1835 "On the Diffraction of an Object-glass with Circular Aperture").<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Mathematically, the diffraction pattern is characterized by the wavelength of light illuminating the circular aperture, and the aperture's size. The <i>appearance</i> of the diffraction pattern is additionally characterized by the sensitivity of the eye or other detector used to observe the pattern. </p><p>The most important application of this concept is in <a href="/wiki/Camera" title="Camera">cameras</a>, <a href="/wiki/Microscope" title="Microscope">microscopes</a> and <a href="/wiki/Telescope" title="Telescope">telescopes</a>. Due to diffraction, the smallest point to which a lens or mirror can focus a beam of light is the size of the Airy disk. Even if one were able to make a perfect lens, there is still a limit to the resolution of an image created by such a lens. An optical system in which the resolution is no longer limited by imperfections in the lenses but only by diffraction is said to be <a href="/wiki/Diffraction_limited" class="mw-redirect" title="Diffraction limited">diffraction limited</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Size">Size</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Airy_disk&action=edit&section=1" title="Edit section: Size"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Far from the aperture, the angle at which the first minimum occurs, measured from the direction of incoming light, is given by the approximate formula: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta \approx 1.22{\frac {\lambda }{d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>≈</mo> <mn>1.22</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <mi>d</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta \approx 1.22{\frac {\lambda }{d}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7fdb76fcaf71828b5afa6180bfbfaf7ba4f098d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.757ex; height:5.509ex;" alt="{\displaystyle \sin \theta \approx 1.22{\frac {\lambda }{d}}}"></span> </p><p>or, for small angles, simply </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta \approx 1.22{\frac {\lambda }{d}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>≈</mo> <mn>1.22</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <mi>d</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta \approx 1.22{\frac {\lambda }{d}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f457db8c02895d5047d4af1b53fea00e07677f77" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.161ex; height:5.509ex;" alt="{\displaystyle \theta \approx 1.22{\frac {\lambda }{d}},}"></span> </p><p>where <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 \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> is in radians, <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 \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> is the wavelength of the light in meters, and <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 {d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/397fcfcbb193baab76d57c315c2897b494c914d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle {d}}"></span> is the diameter of the aperture in meters. The full width at half maximum is given by <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 \theta _{\mathrm {FWHM} }=1.029{\frac {\lambda }{d}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> <mi mathvariant="normal">W</mi> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">M</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>1.029</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <mi>d</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{\mathrm {FWHM} }=1.029{\frac {\lambda }{d}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea7c71317ba6e303de86e5990501539cd0fcd8bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.058ex; height:5.509ex;" alt="{\displaystyle \theta _{\mathrm {FWHM} }=1.029{\frac {\lambda }{d}}.}"></span> </p><p>Airy wrote this relation as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s={\frac {2.76}{a}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2.76</mn> <mi>a</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s={\frac {2.76}{a}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/848a755cb5b0d18aabd08411168b015e05952b95" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.806ex; height:5.176ex;" alt="{\displaystyle s={\frac {2.76}{a}},}"></span> </p><p>where <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 {s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {s}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80b078fead9e918b43cd541daba9ecf31e1c63c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle {s}}"></span> was the angle of first minimum in seconds of arc, <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 {a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecaee8e0957dc3d0a7a3c7da3b54def4bcd27062" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle {a}}"></span> was the radius of the aperture in inches, and the wavelength of light was assumed to be 0.000022 inches (560 nm; the mean of visible wavelengths).<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> This is equal to the <a href="/wiki/Angular_resolution" title="Angular resolution">angular resolution</a> of a circular aperture. The <a href="/wiki/Angular_resolution" title="Angular resolution">Rayleigh criterion</a> for barely resolving two objects that are point sources of light, such as stars seen through a telescope, is that the center of the Airy disk for the first object occurs at the first minimum of the Airy disk of the second. This means that the angular resolution of a diffraction-limited system is given by the same formulae. </p><p>However, while the angle at which the first minimum occurs (which is sometimes described as the radius of the Airy disk) depends only on wavelength and aperture size, the appearance of the diffraction pattern will vary with the intensity (brightness) of the light source. Because any detector (eye, film, digital) used to observe the diffraction pattern can have an intensity threshold for detection, the full diffraction pattern may not be apparent. In astronomy, the outer rings are frequently not apparent even in a highly magnified image of a star. It may be that none of the rings are apparent, in which case the star image appears as a disk (central maximum only) rather than as a full diffraction pattern. Furthermore, fainter stars will appear as smaller disks than brighter stars, because less of their central maximum reaches the threshold of detection.<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> While in theory all stars or other "point sources" of a given wavelength and seen through a given aperture have the same Airy disk radius characterized by the above equation (and the same diffraction pattern size), differing only in intensity, the appearance is that fainter sources appear as smaller disks, and brighter sources appear as larger disks.<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> This was described by Airy in his original work:<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> <blockquote> <p>The rapid decrease of light in the successive rings will sufficiently explain the visibility of two or three rings with a very bright star and the non-visibility of rings with a faint star. The difference of the diameters of the central spots (or spurious disks) of different stars ... is also fully explained. Thus the radius of the spurious disk of a faint star, where light of less than half the intensity of the central light makes no impression on the eye, is determined by [<i>s</i> = 1.17/<i>a</i>], whereas the radius of the spurious disk of a bright star, where light of 1/10 the intensity of the central light is sensible, is determined by [<i>s</i> = 1.97/<i>a</i>]. </p> </blockquote> <p>Despite this feature of Airy's work, the radius of the Airy disk is often given as being simply the angle of first minimum, even in standard textbooks.<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> In reality, the angle of first minimum is a limiting value for the size of the Airy disk, and not a definite radius. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Airy_disk&action=edit&section=2" title="Edit section: Examples"><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:Diffraction_limit_diameter_vs_angular_resolution.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Diffraction_limit_diameter_vs_angular_resolution.svg/220px-Diffraction_limit_diameter_vs_angular_resolution.svg.png" decoding="async" width="220" height="396" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Diffraction_limit_diameter_vs_angular_resolution.svg/330px-Diffraction_limit_diameter_vs_angular_resolution.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/38/Diffraction_limit_diameter_vs_angular_resolution.svg/440px-Diffraction_limit_diameter_vs_angular_resolution.svg.png 2x" data-file-width="512" data-file-height="922" /></a><figcaption>Log-log plot of aperture diameter vs angular resolution at the diffraction limit for various light wavelengths compared with various astronomical instruments. For example, the blue star shows that the Hubble Space Telescope is almost diffraction-limited in the visible spectrum at 0.1 arcsecs, whereas the red circle shows that the human eye should have a resolving power of 20 arcsecs in theory, though 20/20 vision resolves to only 60 arcsecs (1 arcminute)</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Cameras">Cameras</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Airy_disk&action=edit&section=3" title="Edit section: Cameras"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If two objects imaged by a camera are separated by an angle small enough that their Airy disks on the camera detector start overlapping, the objects cannot be clearly separated any more in the image, and they start blurring together. Two objects are said to be <i>just resolved</i> when the maximum of the first Airy pattern falls on top of the first minimum of the second Airy pattern (the <a href="/wiki/Angular_resolution#The_Rayleigh_criterion" title="Angular resolution">Rayleigh criterion</a>). </p><p>Therefore, the smallest angular separation two objects can have before they significantly blur together is given as stated above by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta =1.22\,{\frac {\lambda }{d}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mn>1.22</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <mi>d</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta =1.22\,{\frac {\lambda }{d}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8184067b267c264d2fedbdb38e10a2709d88707" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.791ex; height:5.509ex;" alt="{\displaystyle \sin \theta =1.22\,{\frac {\lambda }{d}}.}"></span> </p><p>Thus, the ability of the system to resolve detail is limited by the ratio of λ/<i>d</i>. The larger the aperture for a given wavelength, the finer the detail that can be distinguished in the image. </p><p>This can also be expressed as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {x}{f}}=1.22\,{\frac {\lambda }{d}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>f</mi> </mfrac> </mrow> <mo>=</mo> <mn>1.22</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <mi>d</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {x}{f}}=1.22\,{\frac {\lambda }{d}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e29f748217540389a26c285f86c6d99262aaa4c0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.624ex; height:5.843ex;" alt="{\displaystyle {\frac {x}{f}}=1.22\,{\frac {\lambda }{d}},}"></span> where <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> is the separation of the images of the two objects on the film, and <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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> is the distance from the lens to the film. If we take the distance from the lens to the film to be approximately equal to the <a href="/wiki/Focal_length" title="Focal length">focal length</a> of the lens, we find <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=1.22\,{\frac {\lambda \,f}{d}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>1.22</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>λ</mi> <mspace width="thinmathspace" /> <mi>f</mi> </mrow> <mi>d</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=1.22\,{\frac {\lambda \,f}{d}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/003d413a73086e705d7428f6bdb9056fdd331a1d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.453ex; height:5.509ex;" alt="{\displaystyle x=1.22\,{\frac {\lambda \,f}{d}},}"></span> but <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 {\frac {f}{d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>f</mi> <mi>d</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {f}{d}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e105b6f76d0c8f5861035fdc78bb091477051ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:2.115ex; height:5.509ex;" alt="{\displaystyle {\frac {f}{d}}}"></span> is the <a href="/wiki/F-number" title="F-number">f-number</a> of a lens. A typical setting for use on an overcast day would be <style data-mw-deduplicate="TemplateStyles:r1207775266">.mw-parser-output span.fnumber,.mw-parser-output .fnumber-fallback{display:inline-block;white-space:nowrap;width:max-content}.mw-parser-output span.fnumber::first-letter,.mw-parser-output .fnumber-fallback .first-letter{font-style:italic;font-family:Trebuchet MS,Candara,Georgia,Calibri,Corbel,serif}</style><a href="/wiki/F-number" title="F-number"><span class="fnumber-fallback"><span class="first-letter">f</span>/8</span></a> (see <a href="/wiki/Sunny_16_rule" title="Sunny 16 rule">Sunny 16 rule</a>). For violet, the shortest wavelength visible light, the wavelength λ is about 420 <a href="/wiki/Nanometer" class="mw-redirect" title="Nanometer">nanometers</a> (see <a href="/wiki/Cone_cell" title="Cone cell">cone cells</a> for sensitivity of S cone cells). This gives a value for <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> of about 4 μm. In a digital camera, making the pixels of the <a href="/wiki/Image_sensor" title="Image sensor">image sensor</a> smaller than half this value (one pixel for each object, one for each space between) would not significantly increase the captured <a href="/wiki/Image_resolution" title="Image resolution">image resolution</a>. However, it may improve the final image by over-sampling, allowing noise reduction. </p> <div class="mw-heading mw-heading3"><h3 id="The_human_eye">The human eye</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Airy_disk&action=edit&section=4" title="Edit section: The human eye"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Spherical-aberration-slice.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Spherical-aberration-slice.jpg/220px-Spherical-aberration-slice.jpg" decoding="async" width="220" height="331" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Spherical-aberration-slice.jpg/330px-Spherical-aberration-slice.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/10/Spherical-aberration-slice.jpg/440px-Spherical-aberration-slice.jpg 2x" data-file-width="1024" data-file-height="1542" /></a><figcaption>Longitudinal sections through a focused beam with (top) negative, (center) zero, and (bottom) positive spherical aberration. The lens is to the left.</figcaption></figure> <p>The <a href="/wiki/Lens_speed" title="Lens speed">fastest</a> <a href="/wiki/F-number" title="F-number">f-number</a> for the <a href="/wiki/Human_eye" title="Human eye">human eye</a> is about 2.1,<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> corresponding to a diffraction-limited <a href="/wiki/Point_spread_function" title="Point spread function">point spread function</a> with approximately 1 μm diameter. However, at this f-number, spherical aberration limits visual acuity, while a 3 mm pupil diameter (f/5.7) approximates the resolution achieved by the human eye.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> The maximum density of cones in the human <a href="/wiki/Fovea_centralis" title="Fovea centralis">fovea</a> is approximately 170,000 per square millimeter,<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> which implies that the cone spacing in the human eye is about 2.5 μm, approximately the diameter of the point spread function at f/5. </p> <div class="mw-heading mw-heading3"><h3 id="Focused_laser_beam">Focused laser beam</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Airy_disk&action=edit&section=5" title="Edit section: Focused laser beam"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A circular laser beam with uniform intensity across the circle (a flat-top beam) focused by a lens will form an Airy disk pattern at the focus. The size of the Airy disk determines the laser intensity at the focus. </p> <div class="mw-heading mw-heading3"><h3 id="Aiming_sight">Aiming sight</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Airy_disk&action=edit&section=6" title="Edit section: Aiming sight"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some weapon aiming sights (e.g. <a href="/wiki/FN_FNC" title="FN FNC">FN FNC</a>) require the user to align a peep sight (rear, nearby sight, i.e. which will be out of focus) with a tip (which should be focused and overlaid on the target) at the end of the barrel. When looking through the peep sight, the user will notice an Airy disk that will help center the sight over the pin.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Conditions_for_observation">Conditions for observation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Airy_disk&action=edit&section=7" title="Edit section: Conditions for observation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Light from a uniformly illuminated circular aperture (or from a uniform, flattop beam) will exhibit an Airy diffraction pattern far away from the aperture due to <a href="/wiki/Fraunhofer_diffraction" title="Fraunhofer diffraction">Fraunhofer diffraction</a> (far-field diffraction). </p><p>The conditions for being in the far field and exhibiting an Airy pattern are: the incoming light illuminating the aperture is a plane wave (no phase variation across the aperture), the intensity is constant over the area of the aperture, and the distance <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> from the aperture where the diffracted light is observed (the screen distance) is large compared to the aperture size, and the radius <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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> of the aperture is not too much larger than the wavelength <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 \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> of the light. The last two conditions can be formally written as <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 R>a^{2}/\lambda .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>></mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>λ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R>a^{2}/\lambda .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/769f2866af8245d2cfa709f7cca3b1f8d01beb2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.311ex; height:3.176ex;" alt="{\displaystyle R>a^{2}/\lambda .}"></span> </p><p>In practice, the conditions for uniform illumination can be met by placing the source of the illumination far from the aperture. If the conditions for far field are not met (for example if the aperture is large), the far-field Airy diffraction pattern can also be obtained on a screen much closer to the aperture by using a lens right after the aperture (or the lens itself can form the aperture). The Airy pattern will then be formed at the focus of the lens rather than at infinity. </p><p>Hence, the focal spot of a uniform circular laser beam (a flattop beam) focused by a lens will also be an Airy pattern. </p><p>In a camera or imaging system an object far away gets imaged onto the film or detector plane by the objective lens, and the far field diffraction pattern is observed at the detector. The resulting image is a convolution of the ideal image with the Airy diffraction pattern due to diffraction from the iris aperture or due to the finite size of the lens. This leads to the finite resolution of a lens system described above. </p> <div class="mw-heading mw-heading2"><h2 id="Mathematical_formulation">Mathematical formulation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Airy_disk&action=edit&section=8" title="Edit section: Mathematical formulation"><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:Circular_aperture_variables.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Circular_aperture_variables.svg/220px-Circular_aperture_variables.svg.png" decoding="async" width="220" height="183" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Circular_aperture_variables.svg/330px-Circular_aperture_variables.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Circular_aperture_variables.svg/440px-Circular_aperture_variables.svg.png 2x" data-file-width="300" data-file-height="250" /></a><figcaption>Diffraction from a circular aperture. The Airy pattern is observable when <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 R\gg a^{2}/\lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>≫</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\gg a^{2}/\lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/455d48c203d9071425eea274e1c9cbfa6d3a8cee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.18ex; height:3.176ex;" alt="{\displaystyle R\gg a^{2}/\lambda }"></span> (i.e. in the far field)</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Circular_aperture_with_lens.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Circular_aperture_with_lens.svg/220px-Circular_aperture_with_lens.svg.png" decoding="async" width="220" height="183" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Circular_aperture_with_lens.svg/330px-Circular_aperture_with_lens.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Circular_aperture_with_lens.svg/440px-Circular_aperture_with_lens.svg.png 2x" data-file-width="300" data-file-height="250" /></a><figcaption>Diffraction from an aperture with a lens. The far field image will (only) be formed at the screen one focal length away, where R=f (f=focal length). The observation angle <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 \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> stays the same as in the lensless case.</figcaption></figure> <p>The <a href="/wiki/Intensity_(physics)" title="Intensity (physics)">intensity</a> of the Airy pattern follows the <a href="/wiki/Fraunhofer_diffraction" title="Fraunhofer diffraction">Fraunhofer diffraction</a> pattern of a circular aperture, given by the <a href="/wiki/Squared_modulus" class="mw-redirect" title="Squared modulus">squared modulus</a> of the <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a> of the circular aperture: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(\theta )=I_{0}\left[{\frac {2J_{1}(k\,a\sin \theta )}{k\,a\sin \theta }}\right]^{2}=I_{0}\left[{\frac {2J_{1}(x)}{x}}\right]^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>k</mi> <mspace width="thinmathspace" /> <mi>a</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>k</mi> <mspace width="thinmathspace" /> <mi>a</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>x</mi> </mfrac> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(\theta )=I_{0}\left[{\frac {2J_{1}(k\,a\sin \theta )}{k\,a\sin \theta }}\right]^{2}=I_{0}\left[{\frac {2J_{1}(x)}{x}}\right]^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546efea8efe27906221b7f353b24a9228ff19887" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.624ex; height:6.676ex;" alt="{\displaystyle I(\theta )=I_{0}\left[{\frac {2J_{1}(k\,a\sin \theta )}{k\,a\sin \theta }}\right]^{2}=I_{0}\left[{\frac {2J_{1}(x)}{x}}\right]^{2}}"></span> </p><p>where <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 I_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/893d08e90ea73781dc133414d661529d0651ca80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.077ex; height:2.509ex;" alt="{\displaystyle I_{0}}"></span> is the maximum intensity of the pattern at the Airy disc center, <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 J_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/260ffe7da7c858cf114ad89a6c794944ea4e760f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.344ex; height:2.509ex;" alt="{\displaystyle J_{1}}"></span> is the <a href="/wiki/Bessel_function" title="Bessel function">Bessel function</a> of the first kind of order one, <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 k={2\pi }/{\lambda }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k={2\pi }/{\lambda }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e067e2ff643f0577d40777af05c01872c38c1d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.322ex; height:2.843ex;" alt="{\displaystyle k={2\pi }/{\lambda }}"></span> is the wavenumber, <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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> is the radius of the aperture, and <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 \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> is the angle of observation, i.e. the angle between the axis of the circular aperture and the line between aperture center and observation point. <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 x=ka\sin \theta ={\frac {2\pi a}{\lambda }}{\frac {q}{R}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>k</mi> <mi>a</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>a</mi> </mrow> <mi>λ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>q</mi> <mi>R</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=ka\sin \theta ={\frac {2\pi a}{\lambda }}{\frac {q}{R}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/897d8017f2714cf4aeda3dce5706909230090761" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.495ex; height:5.343ex;" alt="{\displaystyle x=ka\sin \theta ={\frac {2\pi a}{\lambda }}{\frac {q}{R}},}"></span> where <i>q</i> is the radial distance from the observation point to the optical axis and <i>R</i> is its distance to the aperture. Note that the Airy disk as given by the above expression is only valid for large <i>R</i>, where <a href="/wiki/Fraunhofer_diffraction" title="Fraunhofer diffraction">Fraunhofer diffraction</a> applies; calculation of the shadow in the near-field must rather be handled using <a href="/wiki/Fresnel_diffraction" title="Fresnel diffraction">Fresnel diffraction</a>. </p><p>However the exact Airy pattern <i>does</i> appear at a finite distance if a lens is placed at the aperture. Then the Airy pattern will be perfectly focussed at the distance given by the lens's focal length (assuming <a href="/wiki/Collimated" class="mw-redirect" title="Collimated">collimated</a> light incident on the aperture) given by the above equations. </p><p>The zeros of <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 J_{1}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{1}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b27a53b7793512b449dd4ffe7ab0b80c7a322e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.483ex; height:2.843ex;" alt="{\displaystyle J_{1}(x)}"></span> are at <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 x=ka\sin \theta \approx 3.8317,7.0156,10.1735,13.3237,16.4706\dots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>k</mi> <mi>a</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>≈</mo> <mn>3.8317</mn> <mo>,</mo> <mn>7.0156</mn> <mo>,</mo> <mn>10.1735</mn> <mo>,</mo> <mn>13.3237</mn> <mo>,</mo> <mn>16.4706</mn> <mo>…</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=ka\sin \theta \approx 3.8317,7.0156,10.1735,13.3237,16.4706\dots .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7076262b6c9b0bce7c0740caa318bfcfd24e0bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:58.751ex; height:2.509ex;" alt="{\displaystyle x=ka\sin \theta \approx 3.8317,7.0156,10.1735,13.3237,16.4706\dots .}"></span> From this, it follows that the first dark ring in the diffraction pattern occurs where <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 ka\sin {\theta }=3.8317\dots ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mi>a</mi> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> <mo>=</mo> <mn>3.8317</mn> <mo>…</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ka\sin {\theta }=3.8317\dots ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/739a1405473fdab32b1fe3a25f005714523efeb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.863ex; height:2.509ex;" alt="{\displaystyle ka\sin {\theta }=3.8317\dots ,}"></span> or </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta \approx {\frac {3.83}{ka}}={\frac {3.83\lambda }{2\pi a}}=1.22{\frac {\lambda }{2a}}=1.22{\frac {\lambda }{d}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3.83</mn> <mrow> <mi>k</mi> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3.83</mn> <mi>λ</mi> </mrow> <mrow> <mn>2</mn> <mi>π</mi> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1.22</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1.22</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <mi>d</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta \approx {\frac {3.83}{ka}}={\frac {3.83\lambda }{2\pi a}}=1.22{\frac {\lambda }{2a}}=1.22{\frac {\lambda }{d}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f67559b735cc78101714bdfed012de9a665da67" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:42.358ex; height:5.509ex;" alt="{\displaystyle \sin \theta \approx {\frac {3.83}{ka}}={\frac {3.83\lambda }{2\pi a}}=1.22{\frac {\lambda }{2a}}=1.22{\frac {\lambda }{d}}.}"></span> </p><p>If a lens is used to focus the Airy pattern at a finite distance, then the radius <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 q_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9daa41f6e8f78ea6bb5711d7ac97901ce564b94e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.091ex; height:2.009ex;" alt="{\displaystyle q_{1}}"></span> of the first dark ring on the focal plane is solely given by the <a href="/wiki/Numerical_aperture" title="Numerical aperture">numerical aperture</a> <i>A</i> (closely related to the <a href="/wiki/F-number" title="F-number">f-number</a>) by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{1}=R\sin \theta _{1}\approx 1.22{R}{\frac {\lambda }{d}}=1.22{\frac {\lambda }{2A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>R</mi> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≈</mo> <mn>1.22</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <mi>d</mi> </mfrac> </mrow> <mo>=</mo> <mn>1.22</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <mrow> <mn>2</mn> <mi>A</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{1}=R\sin \theta _{1}\approx 1.22{R}{\frac {\lambda }{d}}=1.22{\frac {\lambda }{2A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aa541b53bc93548a35492ed001dcbb7a0730db6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:34.891ex; height:5.509ex;" alt="{\displaystyle q_{1}=R\sin \theta _{1}\approx 1.22{R}{\frac {\lambda }{d}}=1.22{\frac {\lambda }{2A}}}"></span> </p><p>where the numerical aperture <i>A</i> is equal to the aperture's radius <i>d</i>/2 divided by R', the distance from the center of the Airy pattern to the <i>edge</i> of the aperture. Viewing the aperture of radius <i>d</i>/2 and lens as a camera (see diagram above) projecting an image onto a focal plane at distance <i>f</i>, the numerical aperture <i>A</i> is related to the commonly-cited f-number <i>N= f/d</i> (ratio of the focal length to the lens diameter) according to </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\frac {r}{R'}}={\frac {r}{\sqrt {f^{2}+r^{2}}}}={\frac {1}{\sqrt {4N^{2}+1}}};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>r</mi> <msup> <mi>R</mi> <mo>′</mo> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>r</mi> <msqrt> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>4</mn> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </msqrt> </mfrac> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {r}{R'}}={\frac {r}{\sqrt {f^{2}+r^{2}}}}={\frac {1}{\sqrt {4N^{2}+1}}};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f74421c1fe1478616ed7797817851b9810241fe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:36.95ex; height:6.509ex;" alt="{\displaystyle A={\frac {r}{R'}}={\frac {r}{\sqrt {f^{2}+r^{2}}}}={\frac {1}{\sqrt {4N^{2}+1}}};}"></span> </p><p>for <i>N</i>≫1 it is simply approximated as <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="{\textstyle A\approx 1/2N.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>A</mi> <mo>≈</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mi>N</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle A\approx 1/2N.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22e95a74c04562b5b9fb2b28799cd68bb9126d4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.039ex; height:2.843ex;" alt="{\textstyle A\approx 1/2N.}"></span> This shows that the best possible image <a href="/wiki/Optical_resolution" title="Optical resolution">resolution</a> of a camera is <a href="/wiki/Angular_resolution#Explanation" title="Angular resolution">limited</a> by the numerical aperture (and thus f-number) of its lens due to <a href="/wiki/Diffraction" title="Diffraction">diffraction</a>. </p><p>The half maximum of the central Airy disk (where <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 2J_{1}(x)/x=1/{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2J_{1}(x)/x=1/{\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ede264b83ff3e3523c5296fc19c1706938b2002" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.66ex; height:3.176ex;" alt="{\displaystyle 2J_{1}(x)/x=1/{\sqrt {2}}}"></span>) occurs at <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 x=1.61633995\dots ;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>1.61633995</mn> <mo>…</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=1.61633995\dots ;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/712e367d46c7840b7337ac45c1e57086fbe28fe8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.681ex; height:2.509ex;" alt="{\displaystyle x=1.61633995\dots ;}"></span> the 1/e<sup>2</sup> point (where <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 2J_{1}(x)/x=1/{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2J_{1}(x)/x=1/{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a0455f1e92618ce522d2519bee5c3338d1b24ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.645ex; height:2.843ex;" alt="{\displaystyle 2J_{1}(x)/x=1/{e}}"></span>) occurs at <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 x=2.58383899\dots ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>2.58383899</mn> <mo>…</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=2.58383899\dots ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3904fd72e353d3a4ca29ada7230e01f87ebb8041" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.681ex; height:2.509ex;" alt="{\displaystyle x=2.58383899\dots ,}"></span> and the maximum of the first ring occurs at <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 x=5.13562230\dots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>5.13562230</mn> <mo>…</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=5.13562230\dots .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a773275c89f7b63822d2f6ae6797fc7db6d24f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:19.681ex; height:2.176ex;" alt="{\displaystyle x=5.13562230\dots .}"></span> </p><p>The intensity <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 I_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/893d08e90ea73781dc133414d661529d0651ca80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.077ex; height:2.509ex;" alt="{\displaystyle I_{0}}"></span> at the center of the diffraction pattern is related to the total power <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 P_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/671bd891701e0d6cfa6da0114a5dd64233b58709" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{0}}"></span> incident on the aperture by<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{0}={\frac {\mathrm {E} _{A}^{2}A^{2}}{2R^{2}}}={\frac {P_{0}A}{\lambda ^{2}R^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>A</mi> </mrow> <mrow> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{0}={\frac {\mathrm {E} _{A}^{2}A^{2}}{2R^{2}}}={\frac {P_{0}A}{\lambda ^{2}R^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0495d02427e7dfe0d38f98ae5a0dd9c4936dda3e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.019ex; height:6.343ex;" alt="{\displaystyle I_{0}={\frac {\mathrm {E} _{A}^{2}A^{2}}{2R^{2}}}={\frac {P_{0}A}{\lambda ^{2}R^{2}}}}"></span> </p><p>where <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 \mathrm {E} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {E} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be1811407dea8b43727d28dbe8da7251985b03e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle \mathrm {E} }"></span> is the source strength per unit area at the aperture, A is the area of the aperture (<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 A=\pi a^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>π</mi> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\pi a^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52bf0461601168629eba3a93831f1cd56c4b367d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.458ex; height:2.676ex;" alt="{\displaystyle A=\pi a^{2}}"></span>) and R is the distance from the aperture. At the focal plane of a lens, <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 I_{0}=(P_{0}A)/(\lambda ^{2}f^{2}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>A</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{0}=(P_{0}A)/(\lambda ^{2}f^{2}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab47205d5f53fbead7d1bac8304b3bf4a7a8a015" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.678ex; height:3.176ex;" alt="{\displaystyle I_{0}=(P_{0}A)/(\lambda ^{2}f^{2}).}"></span> The intensity at the maximum of the first ring is about 1.75% of the intensity at the center of the Airy disk. </p><p>The expression for <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 I(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e3621a13a1f446a54ba41c15f7ad7e164099c0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.071ex; height:2.843ex;" alt="{\displaystyle I(\theta )}"></span> above can be integrated to give the total power contained in the diffraction pattern within a circle of given size: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(\theta )=P_{0}[1-J_{0}^{2}(ka\sin \theta )-J_{1}^{2}(ka\sin \theta )]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">[</mo> <mn>1</mn> <mo>−</mo> <msubsup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>k</mi> <mi>a</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msubsup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>k</mi> <mi>a</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(\theta )=P_{0}[1-J_{0}^{2}(ka\sin \theta )-J_{1}^{2}(ka\sin \theta )]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43e77cf6912164bcf59ec7e82c503a752d59df73" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:41.528ex; height:3.176ex;" alt="{\displaystyle P(\theta )=P_{0}[1-J_{0}^{2}(ka\sin \theta )-J_{1}^{2}(ka\sin \theta )]}"></span> </p><p>where <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 J_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/750fead02065cb001fe30d98d309652b642f8e8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.344ex; height:2.509ex;" alt="{\displaystyle J_{0}}"></span> and <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 J_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/260ffe7da7c858cf114ad89a6c794944ea4e760f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.344ex; height:2.509ex;" alt="{\displaystyle J_{1}}"></span> are <a href="/wiki/Bessel_function" title="Bessel function">Bessel functions</a>. Hence the fractions of the total power contained within the first, second, and third dark rings (where <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 J_{1}(ka\sin \theta )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>k</mi> <mi>a</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{1}(ka\sin \theta )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc719916324d8010772f27dc30e2c6b1f015cca0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.576ex; height:2.843ex;" alt="{\displaystyle J_{1}(ka\sin \theta )=0}"></span>) are 83.8%, 91.0%, and 93.8% respectively.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>The Airy disk and diffraction pattern can be computed numerically from first principles using Feynman path integrals.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <table style="margin:auto;"> <tbody><tr> <td><figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:Airy_Pattern.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Airy_Pattern.svg/400px-Airy_Pattern.svg.png" decoding="async" width="400" height="256" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Airy_Pattern.svg/600px-Airy_Pattern.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Airy_Pattern.svg/800px-Airy_Pattern.svg.png 2x" data-file-width="720" data-file-height="460" /></a><figcaption>The Airy Pattern on the interval <i>ka</i>sin<i>θ</i> = [−10, 10]</figcaption></figure> </td> <td><figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:Airy_Pattern_Intensity_and_Encircled_Power.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Airy_Pattern_Intensity_and_Encircled_Power.svg/400px-Airy_Pattern_Intensity_and_Encircled_Power.svg.png" decoding="async" width="400" height="256" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Airy_Pattern_Intensity_and_Encircled_Power.svg/600px-Airy_Pattern_Intensity_and_Encircled_Power.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Airy_Pattern_Intensity_and_Encircled_Power.svg/800px-Airy_Pattern_Intensity_and_Encircled_Power.svg.png 2x" data-file-width="720" data-file-height="460" /></a><figcaption>The encircled power graphed next to the intensity.</figcaption></figure> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Approximation_using_a_Gaussian_profile">Approximation using a Gaussian profile</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Airy_disk&action=edit&section=9" title="Edit section: Approximation using a Gaussian profile"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Airy_vs_gaus.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Airy_vs_gaus.svg/400px-Airy_vs_gaus.svg.png" decoding="async" width="400" height="275" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Airy_vs_gaus.svg/600px-Airy_vs_gaus.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Airy_vs_gaus.svg/800px-Airy_vs_gaus.svg.png 2x" data-file-width="502" data-file-height="345" /></a><figcaption>A radial cross-section through the Airy pattern (solid curve) and its Gaussian profile approximation (dashed curve). The abscissa is given in units of the wavelength <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 \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> times the f-number of the optical system.</figcaption></figure> <p>The Airy pattern falls rather slowly to zero with increasing distance from the center, with the outer rings containing a significant portion of the integrated intensity of the pattern. As a result, the <a href="/wiki/Root_mean_square" title="Root mean square">root mean square</a> (RMS) spotsize is undefined (i.e. infinite). An alternative measure of the spot size is to ignore the relatively small outer rings of the Airy pattern and to approximate the central lobe with a <a href="/wiki/Gaussian_function" title="Gaussian function">Gaussian</a> profile, such that </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(q)\approx I'_{0}\exp \left({\frac {-2q^{2}}{\omega _{0}^{2}}}\right)\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <msubsup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mo>′</mo> </msubsup> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>2</mn> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(q)\approx I'_{0}\exp \left({\frac {-2q^{2}}{\omega _{0}^{2}}}\right)\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06af08cb3c95cbb9ed18b9650ffebe235433eaca" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:24.015ex; height:7.509ex;" alt="{\displaystyle I(q)\approx I'_{0}\exp \left({\frac {-2q^{2}}{\omega _{0}^{2}}}\right)\ ,}"></span> </p><p>where <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 I'_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I'_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e1b1fbd58101b691d5f00be42a15f2f17dd9836" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.077ex; height:2.843ex;" alt="{\displaystyle I'_{0}}"></span> is the irradiance at the center of the pattern, <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 q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> represents the radial distance from the center of the pattern, and <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="{\textstyle \omega _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \omega _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71944cee35d88da5965de9c8e923b0ad5b99b23f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\textstyle \omega _{0}}"></span> is the Gaussian RMS width (in one dimension). If we equate the peak amplitude of the Airy pattern and Gaussian profile, that is, <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 I'_{0}=I_{0},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I'_{0}=I_{0},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/689b6ddbb09e393f134a964c7551b4180166b274" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.9ex; height:2.843ex;" alt="{\displaystyle I'_{0}=I_{0},}"></span> and find the value of <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="{\textstyle \omega _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \omega _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71944cee35d88da5965de9c8e923b0ad5b99b23f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\textstyle \omega _{0}}"></span> giving the optimal approximation to the pattern, we obtain<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </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="{\textstyle \omega _{0}\approx 0.84\lambda N\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≈</mo> <mn>0.84</mn> <mi>λ</mi> <mi>N</mi> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \omega _{0}\approx 0.84\lambda N\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe324a4ffaaadf21bb0270fc7a7847eb48cc21d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.379ex; height:2.509ex;" alt="{\textstyle \omega _{0}\approx 0.84\lambda N\ ,}"></span></dd></dl> <p>where <i>N</i> is the <a href="/wiki/F-number" title="F-number">f-number</a>. If, on the other hand, we wish to enforce that the Gaussian profile has the same volume as does the Airy pattern, then this becomes </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="{\textstyle \omega _{0}\approx 0.90\lambda N\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≈</mo> <mn>0.90</mn> <mi>λ</mi> <mi>N</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \omega _{0}\approx 0.90\lambda N\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f7982d5eaf6b1bb08fc53264a8e121f59084d02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.379ex; height:2.509ex;" alt="{\textstyle \omega _{0}\approx 0.90\lambda N\ .}"></span></dd></dl> <p>In <a href="/wiki/Optical_aberration" title="Optical aberration">optical aberration</a> theory, it is common to describe an imaging system as <i>diffraction-limited</i> if the Airy disk radius is larger than the RMS spotsize determined from geometric ray tracing (see <a href="/wiki/Optical_lens_design" title="Optical lens design">Optical lens design</a>). The Gaussian profile approximation provides an alternative means of comparison: using the approximation above shows that the Gaussian waist <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="{\textstyle \omega _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \omega _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71944cee35d88da5965de9c8e923b0ad5b99b23f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\textstyle \omega _{0}}"></span> of the Gaussian approximation to the Airy disk is about two-third the Airy disk radius, i.e. <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 0.84\lambda N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0.84</mn> <mi>λ</mi> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0.84\lambda N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a5de6d82a3b9a9c01e36820209a0f35498bd580" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.553ex; height:2.176ex;" alt="{\displaystyle 0.84\lambda N}"></span> as opposed to <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 1.22\lambda N.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1.22</mn> <mi>λ</mi> <mi>N</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1.22\lambda N.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b870dafefca4581ed9c9ff253409891b24313097" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.2ex; height:2.176ex;" alt="{\displaystyle 1.22\lambda N.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Obscured_Airy_pattern">Obscured Airy pattern</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Airy_disk&action=edit&section=10" title="Edit section: Obscured Airy pattern"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Similar equations can also be derived for the obscured Airy diffraction pattern<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Mahajan_17-0" class="reference"><a href="#cite_note-Mahajan-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> which is the diffraction pattern from an annular aperture or beam, i.e. a uniform circular aperture (beam) obscured by a circular block at the center. This situation is relevant to many common reflector telescope designs that incorporate a secondary mirror, including <a href="/wiki/Newtonian_telescope" title="Newtonian telescope">Newtonian telescopes</a> and <a href="/wiki/Schmidt%E2%80%93Cassegrain_telescope" title="Schmidt–Cassegrain telescope">Schmidt–Cassegrain telescopes</a>. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(R)={\frac {I_{0}}{(1-\epsilon ^{2})^{2}}}\left({\frac {2J_{1}(x)}{x}}-{\frac {2\epsilon J_{1}(\epsilon x)}{x}}\right)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>x</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>ϵ</mi> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ϵ</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>x</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(R)={\frac {I_{0}}{(1-\epsilon ^{2})^{2}}}\left({\frac {2J_{1}(x)}{x}}-{\frac {2\epsilon J_{1}(\epsilon x)}{x}}\right)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f377c0985e9273d6a50f4317f1fb5137315a082d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:41.712ex; height:6.843ex;" alt="{\displaystyle I(R)={\frac {I_{0}}{(1-\epsilon ^{2})^{2}}}\left({\frac {2J_{1}(x)}{x}}-{\frac {2\epsilon J_{1}(\epsilon x)}{x}}\right)^{2}}"></span> </p><p>where <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 \epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3837cad72483d97bcdde49c85d3b7b859fb3fd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }"></span> is the annular aperture obscuration ratio, or the ratio of the diameter of the obscuring disk and the diameter of the aperture (beam). <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 \left(0\leq \epsilon <1\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>≤</mo> <mi>ϵ</mi> <mo><</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(0\leq \epsilon <1\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10801ba728d9374deba92bbd113021c8a2179a82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.309ex; height:2.843ex;" alt="{\displaystyle \left(0\leq \epsilon <1\right),}"></span> and x is defined as above: <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 x=ka\sin(\theta )\approx {\frac {\pi R}{\lambda N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>k</mi> <mi>a</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <mi>R</mi> </mrow> <mrow> <mi>λ</mi> <mi>N</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=ka\sin(\theta )\approx {\frac {\pi R}{\lambda N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c04914f977560cabf08b6ba0df331db8baf9ae60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.365ex; height:5.343ex;" alt="{\displaystyle x=ka\sin(\theta )\approx {\frac {\pi R}{\lambda N}}}"></span> where <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> is the radial distance in the focal plane from the optical axis, <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 \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> is the wavelength and <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 N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> is the <a href="/wiki/F-number" title="F-number">f-number</a> of the system. The fractional encircled energy (the fraction of the total energy contained within a circle of radius <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> centered at the optical axis in the focal plane) is then given by: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(R)={\frac {1}{(1-\epsilon ^{2})}}\left(1-J_{0}^{2}(x)-J_{1}^{2}(x)+\epsilon ^{2}\left[1-J_{0}^{2}(\epsilon x)-J_{1}^{2}(\epsilon x)\right]-4\epsilon \int _{0}^{x}{\frac {J_{1}(t)J_{1}(\epsilon t)}{t}}\,dt\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msubsup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msubsup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>−</mo> <msubsup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>ϵ</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msubsup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>ϵ</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mo>−</mo> <mn>4</mn> <mi>ϵ</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ϵ</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mi>t</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(R)={\frac {1}{(1-\epsilon ^{2})}}\left(1-J_{0}^{2}(x)-J_{1}^{2}(x)+\epsilon ^{2}\left[1-J_{0}^{2}(\epsilon x)-J_{1}^{2}(\epsilon x)\right]-4\epsilon \int _{0}^{x}{\frac {J_{1}(t)J_{1}(\epsilon t)}{t}}\,dt\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72d72a2dc536d89d33a879a3ff54808eb9662ba" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:90.086ex; height:6.509ex;" alt="{\displaystyle E(R)={\frac {1}{(1-\epsilon ^{2})}}\left(1-J_{0}^{2}(x)-J_{1}^{2}(x)+\epsilon ^{2}\left[1-J_{0}^{2}(\epsilon x)-J_{1}^{2}(\epsilon x)\right]-4\epsilon \int _{0}^{x}{\frac {J_{1}(t)J_{1}(\epsilon t)}{t}}\,dt\right)}"></span> </p><p>For <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 \epsilon \rightarrow 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon \rightarrow 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68af83da5cb4596259c9ec2e76260e0bacee12ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.721ex; height:2.176ex;" alt="{\displaystyle \epsilon \rightarrow 0}"></span> the formulas reduce to the unobscured versions above. </p><p>The practical effect of having a central obstruction in a telescope is that the central disc becomes slightly smaller, and the first bright ring becomes brighter at the expense of the central disc. This becomes more problematic with short focal length telescopes which require larger secondary mirrors.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Comparison_to_Gaussian_beam_focus">Comparison to Gaussian beam focus</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Airy_disk&action=edit&section=11" title="Edit section: Comparison to Gaussian beam focus"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A circular laser beam with uniform intensity profile, focused by a lens, will form an Airy pattern at the focal plane of the lens. The intensity at the center of the focus will be <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 I_{0,Airy}=(P_{0}A)/(\lambda ^{2}f^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mi>A</mi> <mi>i</mi> <mi>r</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>A</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{0,Airy}=(P_{0}A)/(\lambda ^{2}f^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8fe96d4a135d5a548f16e7053cac4682ed931fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.847ex; height:3.343ex;" alt="{\displaystyle I_{0,Airy}=(P_{0}A)/(\lambda ^{2}f^{2})}"></span> where <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 P_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/671bd891701e0d6cfa6da0114a5dd64233b58709" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{0}}"></span> is the total power of the beam, <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 A=\pi D^{2}/4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>π</mi> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\pi D^{2}/4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b9625426b77fe664529e252c91ef0cacbf14a7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.477ex; height:3.176ex;" alt="{\displaystyle A=\pi D^{2}/4}"></span> is the area of the beam (<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 D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> is the beam diameter), <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 \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> is the wavelength, and <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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> is the focal length of the lens. </p><p>A Gaussian beam transmitted through a hard aperture will be clipped. Energy is lost and edge diffraction occurs, effectively increasing the divergence. Because of these effects there is a Gaussian beam diameter which maximizes the intensity in the far field. This occurs when the <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 1/e^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/e^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a80f1d525f47366360aa0eb60e951c721b29205" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.463ex; height:3.176ex;" alt="{\displaystyle 1/e^{2}}"></span> diameter of the Gaussian is 89% of the aperture diameter, and the on axis intensity in the far field will be 81% of that produced by a uniform intensity profile.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Elliptical_aperture">Elliptical aperture</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Airy_disk&action=edit&section=12" title="Edit section: Elliptical aperture"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Fourier integral of the circular cross section of radius <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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> is </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 \int _{0}^{a}rdr\int _{0}^{2\pi }d\phi e^{i{\vec {k}}\cdot {\vec {r}}}=\int _{0}^{a}rdr\int _{0}^{2\pi }d\phi e^{ikr\cos \phi }=2\int _{0}^{a}rdr\int _{0}^{\pi }d\phi \cos(kr\cos \phi )=2\pi \int _{0}^{a}rdrJ_{0}(kr)=2\pi {\frac {a}{k}}J_{1}(kr).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mi>r</mi> <mi>d</mi> <mi>r</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <mi>d</mi> <mi>ϕ</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> </msup> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mi>r</mi> <mi>d</mi> <mi>r</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <mi>d</mi> <mi>ϕ</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> </mrow> </msup> <mo>=</mo> <mn>2</mn> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mi>r</mi> <mi>d</mi> <mi>r</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <mi>d</mi> <mi>ϕ</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>π</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mi>r</mi> <mi>d</mi> <mi>r</mi> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>k</mi> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>k</mi> </mfrac> </mrow> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>k</mi> <mi>r</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{a}rdr\int _{0}^{2\pi }d\phi e^{i{\vec {k}}\cdot {\vec {r}}}=\int _{0}^{a}rdr\int _{0}^{2\pi }d\phi e^{ikr\cos \phi }=2\int _{0}^{a}rdr\int _{0}^{\pi }d\phi \cos(kr\cos \phi )=2\pi \int _{0}^{a}rdrJ_{0}(kr)=2\pi {\frac {a}{k}}J_{1}(kr).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e98eaf2a5cdbf804ddec21217c97938d140b76c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:110.983ex; height:6.176ex;" alt="{\displaystyle \int _{0}^{a}rdr\int _{0}^{2\pi }d\phi e^{i{\vec {k}}\cdot {\vec {r}}}=\int _{0}^{a}rdr\int _{0}^{2\pi }d\phi e^{ikr\cos \phi }=2\int _{0}^{a}rdr\int _{0}^{\pi }d\phi \cos(kr\cos \phi )=2\pi \int _{0}^{a}rdrJ_{0}(kr)=2\pi {\frac {a}{k}}J_{1}(kr).}"></span></dd></dl> <p>This is the special case of the Fourier integral of the elliptical cross section with half axes <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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> and <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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>:<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </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 \int _{x^{2}/a^{2}+y^{2}/b^{2}\leq 1}e^{ik_{x}x}e^{ik_{y}y}dxdy=2\pi {\frac {ab}{c}}J_{1}(c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≤</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>x</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mi>y</mi> </mrow> </msup> <mi>d</mi> <mi>x</mi> <mi>d</mi> <mi>y</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mi>b</mi> </mrow> <mi>c</mi> </mfrac> </mrow> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{x^{2}/a^{2}+y^{2}/b^{2}\leq 1}e^{ik_{x}x}e^{ik_{y}y}dxdy=2\pi {\frac {ab}{c}}J_{1}(c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cab72b71e6bddf97fd02880e3e847c8d6d3d19b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:41.323ex; height:6.176ex;" alt="{\displaystyle \int _{x^{2}/a^{2}+y^{2}/b^{2}\leq 1}e^{ik_{x}x}e^{ik_{y}y}dxdy=2\pi {\frac {ab}{c}}J_{1}(c)}"></span></dd></dl> <p>where </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 c\equiv {\sqrt {(k_{x}a)^{2}+(k_{y}b)^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\equiv {\sqrt {(k_{x}a)^{2}+(k_{y}b)^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f8687f97b0f7a834effcd7df6bb9ddb60b03157" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:22.515ex; height:4.843ex;" alt="{\displaystyle c\equiv {\sqrt {(k_{x}a)^{2}+(k_{y}b)^{2}}}.}"></span></dd></dl> <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=Airy_disk&action=edit&section=13" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Amateur_astronomy" title="Amateur astronomy">Amateur astronomy</a></li> <li><a href="/wiki/Apodization" title="Apodization">Apodization</a></li> <li><a href="/wiki/Fraunhofer_diffraction" title="Fraunhofer diffraction">Fraunhofer diffraction</a></li> <li><a href="/wiki/Bloom_(shader_effect)" title="Bloom (shader effect)">Bloom (shader effect)</a></li> <li><a href="/wiki/Newton%27s_rings" title="Newton's rings">Newton's rings</a></li> <li><a href="/wiki/Optical_unit" title="Optical unit">Optical unit</a></li> <li><a href="/wiki/Point_spread_function" title="Point spread function">Point spread function</a></li> <li><a href="/wiki/Powder_diffraction" title="Powder diffraction">Debye-Scherrer ring</a></li> <li><a href="/wiki/Strehl_ratio" title="Strehl ratio">Strehl ratio</a></li> <li><a href="/wiki/Speckle_pattern" class="mw-redirect" title="Speckle pattern">Speckle pattern</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes_and_references">Notes and references</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Airy_disk&action=edit&section=14" title="Edit section: Notes and 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 reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></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="CITEREFHerschel1828" class="citation encyclopaedia cs1">Herschel, J. F. W. (1828). <a rel="nofollow" class="external text" href="https://archive.org/details/treatisesonphysi00hersrich">"Light"</a>. <i>Transactions Treatises on physical astronomy, light and sound contributed to the Encyclopaedia Metropolitana</i>. Richard Griffin & Co. p. 491.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Light&rft.btitle=Transactions+Treatises+on+physical+astronomy%2C+light+and+sound+contributed+to+the+Encyclopaedia+Metropolitana&rft.pages=491&rft.pub=Richard+Griffin+%26+Co.&rft.date=1828&rft.aulast=Herschel&rft.aufirst=J.+F.+W.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftreatisesonphysi00hersrich&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAiry+disk" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAiry1835" class="citation journal cs1">Airy, G. B. (1835). <a rel="nofollow" class="external text" href="https://archive.org/details/transactionsofca05camb">"On the Diffraction of an Object-glass with Circular Aperture"</a>. <i>Transactions of the Cambridge Philosophical Society</i>. <b>5</b>: 283–91. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1835TCaPS...5..283A">1835TCaPS...5..283A</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+Cambridge+Philosophical+Society&rft.atitle=On+the+Diffraction+of+an+Object-glass+with+Circular+Aperture&rft.volume=5&rft.pages=283-91&rft.date=1835&rft_id=info%3Abibcode%2F1835TCaPS...5..283A&rft.aulast=Airy&rft.aufirst=G.+B.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftransactionsofca05camb&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAiry+disk" class="Z3988"></span></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">Airy, G. B., "On the Diffraction of an Object-glass with Circular Aperture", <a rel="nofollow" class="external text" href="https://archive.org/details/transactionsofca05camb"><i>Transactions of the Cambridge Philosophical Society</i>, Vol. 5</a>, 1835, p. 287.</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">Sidgwick, J. B., <a rel="nofollow" class="external text" href="https://books.google.com/books?id=UN_gJSexgrwC"><i>Amateur Astronomer's Handbook</i></a>, Dover Publications, 1980, pp. 39–40.</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGraney2009" class="citation journal cs1">Graney, Christopher M. (2009). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110514050058/http://scitation.aip.org/tpt/">"Objects in Telescope Are Farther Than They Appear – How diffraction tricked Galileo into mismeasuring distances to the stars"</a>. <i>The Physics Teacher</i>. <b>47</b> (6): 362–365. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.3204117">10.1119/1.3204117</a>. Archived from <a rel="nofollow" class="external text" href="http://scitation.aip.org/tpt/">the original</a> on 2011-05-14.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Physics+Teacher&rft.atitle=Objects+in+Telescope+Are+Farther+Than+They+Appear+%E2%80%93+How+diffraction+tricked+Galileo+into+mismeasuring+distances+to+the+stars&rft.volume=47&rft.issue=6&rft.pages=362-365&rft.date=2009&rft_id=info%3Adoi%2F10.1119%2F1.3204117&rft.aulast=Graney&rft.aufirst=Christopher+M.&rft_id=http%3A%2F%2Fscitation.aip.org%2Ftpt%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAiry+disk" class="Z3988"></span></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">Airy, G. B., "On the Diffraction of an Object-glass with Circular Aperture", <a rel="nofollow" class="external text" href="https://archive.org/details/transactionsofca05camb"><i>Transactions of the Cambridge Philosophical Society</i>, Vol. 5</a>, 1835, p. 288.</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">Giancoli, D. C., <a rel="nofollow" class="external text" href="https://books.google.com/books?lr=&cd=3&id=maQeAQAAIAAJ&dq=giancoli&q="><i>Physics for Scientists and Engineers (3rd edition)</i></a>, Prentice-Hall, 2000, p. 896.</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHecht1987" class="citation book cs1">Hecht, Eugene (1987). <i>Optics</i> (2nd ed.). <a href="/wiki/Addison_Wesley" class="mw-redirect" title="Addison Wesley">Addison Wesley</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-11609-X" title="Special:BookSources/0-201-11609-X"><bdi>0-201-11609-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Optics&rft.edition=2nd&rft.pub=Addison+Wesley&rft.date=1987&rft.isbn=0-201-11609-X&rft.aulast=Hecht&rft.aufirst=Eugene&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAiry+disk" class="Z3988"></span> Sect. 5.7.1</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSteve_Chapman2000" class="citation book cs1">Steve Chapman, ed. (2000). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=byx2Ne9cD1IC&q=eye+resolution+line-pairs+1.7&pg=PA164"><i>Optical System Design</i></a>. <a href="/wiki/McGraw-Hill_Professional" class="mw-redirect" title="McGraw-Hill Professional">McGraw-Hill Professional</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-134916-2" title="Special:BookSources/0-07-134916-2"><bdi>0-07-134916-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Optical+System+Design&rft.pub=McGraw-Hill+Professional&rft.date=2000&rft.isbn=0-07-134916-2&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dbyx2Ne9cD1IC%26q%3Deye%2Bresolution%2Bline-pairs%2B1.7%26pg%3DPA164&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAiry+disk" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</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="http://www.wdv.com/Various/Eye/EyeBandwidth/index.html">"Eye Receptor Density"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-12-18</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Eye+Receptor+Density&rft_id=http%3A%2F%2Fwww.wdv.com%2FVarious%2FEye%2FEyeBandwidth%2Findex.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAiry+disk" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">See <a class="external free" href="https://en.wikibooks.org/wiki/Marksmanship">http://en.wikibooks.org/wiki/Marksmanship</a>, "Sight Alignment"</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">E. Hecht, <i>Optics</i>, Addison Wesley (2001)</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">M. Born and E. Wolf, <i><a href="/wiki/Principles_of_Optics" title="Principles of Optics">Principles of Optics</a></i> (Pergamon Press, New York, 1965)</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDerenzo2023" class="citation journal cs1">Derenzo, S. E. (2023). "Feynman photon path integral calculations of optical reflection, diffraction, and scattering from conduction electrons". <i>Nuclear Instruments and Methods A</i>. <b>1056</b>: 168679. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2309.09827">2309.09827</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2023NIMPA105668679D">2023NIMPA105668679D</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.nima.2023.168679">10.1016/j.nima.2023.168679</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nuclear+Instruments+and+Methods+A&rft.atitle=Feynman+photon+path+integral+calculations+of+optical+reflection%2C+diffraction%2C+and+scattering+from+conduction+electrons&rft.volume=1056&rft.pages=168679&rft.date=2023&rft_id=info%3Aarxiv%2F2309.09827&rft_id=info%3Adoi%2F10.1016%2Fj.nima.2023.168679&rft_id=info%3Abibcode%2F2023NIMPA105668679D&rft.aulast=Derenzo&rft.aufirst=S.+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAiry+disk" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZhangZerubiaOlivo-Marin2007" class="citation journal cs1">Zhang, Bo; Zerubia, Josiane; Olivo-Marin, Jean-Christophe (2007-04-01). <a rel="nofollow" class="external text" href="https://www.osapublishing.org/abstract.cfm?uri=ao-46-10-1819">"Gaussian approximations of fluorescence microscope point-spread function models"</a>. <i>Applied Optics</i>. <b>46</b> (10): 1819–1829. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2007ApOpt..46.1819Z">2007ApOpt..46.1819Z</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1364%2FAO.46.001819">10.1364/AO.46.001819</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2155-3165">2155-3165</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/17356626">17356626</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Applied+Optics&rft.atitle=Gaussian+approximations+of+fluorescence+microscope+point-spread+function+models&rft.volume=46&rft.issue=10&rft.pages=1819-1829&rft.date=2007-04-01&rft_id=info%3Adoi%2F10.1364%2FAO.46.001819&rft.issn=2155-3165&rft_id=info%3Apmid%2F17356626&rft_id=info%3Abibcode%2F2007ApOpt..46.1819Z&rft.aulast=Zhang&rft.aufirst=Bo&rft.au=Zerubia%2C+Josiane&rft.au=Olivo-Marin%2C+Jean-Christophe&rft_id=https%3A%2F%2Fwww.osapublishing.org%2Fabstract.cfm%3Furi%3Dao-46-10-1819&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAiry+disk" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRivolta1986" class="citation journal cs1">Rivolta, Claudio (1986). 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A</i>. <b>3</b> (4): 470. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1986JOSAA...3..470M">1986JOSAA...3..470M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1364%2FJOSAA.3.000470">10.1364/JOSAA.3.000470</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Opt.+Soc.+Am.+A&rft.atitle=Uniform+versus+Gaussian+beams%3A+a+comparison+of+the+effects+of+diffraction%2C+osbscuration%2C+and+aberrations&rft.volume=3&rft.issue=4&rft.pages=470&rft.date=1986&rft_id=info%3Adoi%2F10.1364%2FJOSAA.3.000470&rft_id=info%3Abibcode%2F1986JOSAA...3..470M&rft.aulast=Mahajan&rft.aufirst=Virendra+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAiry+disk" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSacek2006" class="citation web cs1">Sacek, Vladimir (July 14, 2006). <a rel="nofollow" class="external text" href="http://www.telescope-optics.net/obstruction.htm">"Chapter 7 Obstruction effects (7.1. Central obstruction effect)"</a>. <i>7. Notes on amateur telescope optics</i><span class="reference-accessdate">. Retrieved <span class="nowrap">May 18,</span> 2013</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=7.+Notes+on+amateur+telescope+optics&rft.atitle=Chapter+7+Obstruction+effects+%287.1.+Central+obstruction+effect%29&rft.date=2006-07-14&rft.aulast=Sacek&rft.aufirst=Vladimir&rft_id=http%3A%2F%2Fwww.telescope-optics.net%2Fobstruction.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAiry+disk" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text">A.E. Siegman, Lasers, Se. 18.4, University Science Books, Mill Valley, CA, 1989</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDe1955" class="citation journal cs1">De, M. (1955). "The influence of astigmatism on the response function of an optical system". <i>Proc. Roy. Soc. Lond. A</i>. <b>233</b> (1192): 91–104. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.1955.0248">10.1098/rspa.1955.0248</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proc.+Roy.+Soc.+Lond.+A&rft.atitle=The+influence+of+astigmatism+on+the+response+function+of+an+optical+system.&rft.volume=233&rft.issue=1192&rft.pages=91-104&rft.date=1955&rft_id=info%3Adoi%2F10.1098%2Frspa.1955.0248&rft.aulast=De&rft.aufirst=M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAiry+disk" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTheunissen2012" class="citation journal cs1">Theunissen, R. (2012). "Modelling of imaged ellipse intensity profiles using euclidean geometry". <i>The Imaging Sci. J</i>. <b>63</b> (6): 321–331. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1179%2F1743131X15Y.0000000013">10.1179/1743131X15Y.0000000013</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/1983%2Fb43884b1-6e90-4f7f-a766-86306d0c7464">1983/b43884b1-6e90-4f7f-a766-86306d0c7464</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Imaging+Sci.+J.&rft.atitle=Modelling+of+imaged+ellipse+intensity+profiles+using+euclidean+geometry&rft.volume=63&rft.issue=6&rft.pages=321-331&rft.date=2012&rft_id=info%3Ahdl%2F1983%2Fb43884b1-6e90-4f7f-a766-86306d0c7464&rft_id=info%3Adoi%2F10.1179%2F1743131X15Y.0000000013&rft.aulast=Theunissen&rft.aufirst=R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAiry+disk" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Airy_disk&action=edit&section=15" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMichael_W._Davidson" class="citation web cs1"><a href="/wiki/Michael_W._Davidson" title="Michael W. Davidson">Michael W. Davidson</a>. <a rel="nofollow" class="external text" href="http://www.microscopyu.com/articles/formulas/formulasresolution.html">"Concepts and Formulas in Microscopy: Resolution"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Concepts+and+Formulas+in+Microscopy%3A+Resolution&rft.au=Michael+W.+Davidson&rft_id=http%3A%2F%2Fwww.microscopyu.com%2Farticles%2Fformulas%2Fformulasresolution.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAiry+disk" class="Z3988"></span> Nikon MicroscopyU (website). <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKenneth_R._SpringBrian_O._FlynnMichael_W._Davidson" class="citation web cs1">Kenneth R. Spring; Brian O. Flynn & Michael W. Davidson. <a rel="nofollow" class="external text" href="http://micro.magnet.fsu.edu/primer/java/imageformation/airyna/index.html">"Image Formation: Numerical Aperture and Image Resolution"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">June 15,</span> 2006</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Image+Formation%3A+Numerical+Aperture+and+Image+Resolution&rft.au=Kenneth+R.+Spring&rft.au=Brian+O.+Flynn&rft.au=Michael+W.+Davidson&rft_id=http%3A%2F%2Fmicro.magnet.fsu.edu%2Fprimer%2Fjava%2Fimageformation%2Fairyna%2Findex.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAiry+disk" class="Z3988"></span> (Interactive Java Tutorial) <i>Molecular Expressions</i> (website).</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKenneth_R._SpringBrian_O._FlynnMichael_W._Davidson" class="citation web cs1">Kenneth R. Spring; Brian O. Flynn & Michael W. Davidson. <a rel="nofollow" class="external text" href="http://micro.magnet.fsu.edu/primer/java/imageformation/airydiskformation/index.html">"Image Formation: Airy Pattern Formation"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">June 15,</span> 2006</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Image+Formation%3A+Airy+Pattern+Formation&rft.au=Kenneth+R.+Spring&rft.au=Brian+O.+Flynn&rft.au=Michael+W.+Davidson&rft_id=http%3A%2F%2Fmicro.magnet.fsu.edu%2Fprimer%2Fjava%2Fimageformation%2Fairydiskformation%2Findex.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAiry+disk" class="Z3988"></span>(Interactive Java Tutorial) <i>Molecular Expressions</i>.</li></ul></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPaul_Padley" class="citation web cs1">Paul Padley. <a rel="nofollow" class="external text" href="http://cnx.org/content/m13097/latest/">"Diffraction from a Circular Aperture"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Diffraction+from+a+Circular+Aperture&rft.au=Paul+Padley&rft_id=http%3A%2F%2Fcnx.org%2Fcontent%2Fm13097%2Flatest%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAiry+disk" class="Z3988"></span>, <i>Connexions</i> (website), November 8, 2005. – Mathematical details to derive the above formula.</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120911020523/http://www.oldham-optical.co.uk/Airy%20Disk.htm">"The Airy Disk: An Explanation Of What It Is, And Why You Can't Avoid It"</a>, <i>Oldham Optical UK</i>.</li> <li><span class="citation mathworld" id="Reference-Mathworld-Bessel_Function_Zeros"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/BesselFunctionZeros.html">"Bessel Function Zeros"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Bessel+Function+Zeros&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FBesselFunctionZeros.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAiry+disk" class="Z3988"></span></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.nijboerzernike.nl">"Extended Nijboer-Zernike (ENZ) Analysis and Aberration Retrieval"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Extended+Nijboer-Zernike+%28ENZ%29+Analysis+and+Aberration+Retrieval&rft_id=http%3A%2F%2Fwww.nijboerzernike.nl&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAiry+disk" class="Z3988"></span></li></ul>    </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=Airy_disk&oldid=1259437616">https://en.wikipedia.org/w/index.php?title=Airy_disk&oldid=1259437616</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">Categories</a>: <ul><li><a href="/wiki/Category:Physical_optics" title="Category:Physical optics">Physical optics</a></li><li><a href="/wiki/Category:Diffraction" title="Category:Diffraction">Diffraction</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_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches 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 25 November 2024, at 04:15<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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