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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Pupil function</span></span>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>The <b>pupil function</b> or <b>aperture function</b> describes how a light wave is affected upon transmission through an optical imaging system such as a camera, microscope, or the human eye. More specifically, it is a <a href="Complex-valued_function" class="mw-redirect" title="Complex-valued function">complex function</a> of the position in the pupil<sup id="cite_ref-pupil_1-0" class="reference"><a href="#cite_note-pupil-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> or aperture (often an <a href="Diaphragm_(optics)" title="Diaphragm (optics)">iris</a>) that indicates the relative change in amplitude and phase of the light wave. Sometimes this function is referred to as the <i>generalized</i> pupil function, in which case pupil function only indicates whether light is transmitted or not.<sup id="cite_ref-Goodman2005_2-0" class="reference"><a href="#cite_note-Goodman2005-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Imperfections in the optics typically have a direct effect on the pupil function, it is therefore an important tool to study optical imaging systems and their performance.<sup id="cite_ref-Fisher2008_3-0" class="reference"><a href="#cite_note-Fisher2008-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Relationship_with_other_functions_in_optics">Relationship with other functions in optics</h2></div>

<p>The complex pupil function <span class="mwe-math-element mwe-math-element-inline"><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 {P} (u,v)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {P} (u,v)}</annotation>

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</math></span><img src="./fabbaef1ff39d940f2494e3c0d6450e68c0e17ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.883ex; height:2.843ex;" alt="{\displaystyle \mathrm {P} (u,v)}" loading="lazy"></span> can be written in <a href="Polar_coordinates" class="mw-redirect" title="Polar coordinates">polar coordinates</a> using two real functions:

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {P} (u,v)=\mathrm {A} (u,v)\cdot \mathrm {exp} (i\,\mathrm {\Theta } (u,v))}</annotation>

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</math></span><img src="./a69d57962efed0df2c45821b46ef9c39903c7de6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.365ex; height:2.843ex;" alt="{\displaystyle \mathrm {P} (u,v)=\mathrm {A} (u,v)\cdot \mathrm {exp} (i\,\mathrm {\Theta } (u,v))}" loading="lazy"></span>,</dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><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 {\Theta } (u,v)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {\Theta } (u,v)}</annotation>

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</math></span><img src="./b3853da3347f6218b6a643f1e830f47c92ae06c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.109ex; height:2.843ex;" alt="{\displaystyle \mathrm {\Theta } (u,v)}" loading="lazy"></span> is the phase change (in radians) introduced by the optics,<sup id="cite_ref-Fisher2008_3-1" class="reference"><a href="#cite_note-Fisher2008-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> or the surrounding medium.<sup id="cite_ref-Pawley2006_4-0" class="reference"><a href="#cite_note-Pawley2006-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> It captures all <a href="Optical_aberrations" class="mw-redirect" title="Optical aberrations">optical aberrations</a> that occur between the image plane and the focal plane in the scene or sample. The light may also be attenuated differently at different positions <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (u,v)}">

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<annotation encoding="application/x-tex">{\displaystyle (u,v)}</annotation>

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</math></span><img src="./eadf12294edccd7a29c99cfc1765e4a14bf47e58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.301ex; height:2.843ex;" alt="{\displaystyle (u,v)}" loading="lazy"></span> in the pupil, sometimes deliberately for the purpose of <a href="Apodization" title="Apodization">apodization</a>. Such change in amplitude of the light wave is described by the factor <span class="mwe-math-element mwe-math-element-inline"><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 {A} (u,v)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {A} (u,v)}</annotation>

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</math></span><img src="./675e1b3d4dd7f6a1fee92e222e0535fd4a226594.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.044ex; height:2.843ex;" alt="{\displaystyle \mathrm {A} (u,v)}" loading="lazy"></span>.

</p><p>The pupil function is also directly related to the <a href="Point_spread_function" title="Point spread function">point spread function</a> by its <a href="Fourier_transform" title="Fourier transform">Fourier transform</a>. As such, the effect of aberrations on the point spread function can be described mathematically using the concept of the pupil function.

</p><p>Since the (incoherent) point spread function is also related to the optical transfer function via a Fourier transform, a direct relationship exists between the pupil function and the optical transfer function. In the case of an incoherent optical imaging system, the optical transfer function is the auto correlation of the pupil function.<sup id="cite_ref-Goodman2005_2-1" class="reference"><a href="#cite_note-Goodman2005-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-OpticsCourseNotesOnOTF_5-0" class="reference"><a href="#cite_note-OpticsCourseNotesOnOTF-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<div class="mw-heading mw-heading3"><h3 id="In_focus">In focus</h3></div>

<p>In a homogeneous medium, a point source emits light with spherical wave fronts. A lens that is focused onto the point source will have optics that change the spherical wave front into a planar wave before it passes through the pupil or aperture stop. Often, additional lens element refocus the light onto a sensor or photographic film, by converting the planar wave front to a spherical wave front, centered onto the image plane. The pupil function of such an ideal system is equal to one at every point within the pupil, and zero out with it. In case of a circular pupil, this can be written mathematically as:

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {P} (u,v)=1,\forall u,v:{\sqrt {u^{2}+v^{2}}}\leq R}</annotation>

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</math></span><img src="./75fe6b2dc17665e061747127c98b6d3a4d717fe4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.492ex; height:3.509ex;" alt="{\displaystyle \mathrm {P} (u,v)=1,\forall u,v:{\sqrt {u^{2}+v^{2}}}\leq R}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {P} (u,v)=0,\forall u,v:{\sqrt {u^{2}+v^{2}}}>R,}</annotation>

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</math></span><img src="./9034ab7950e04ba6be9ecaece5900228bcc7a598.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.138ex; height:3.509ex;" alt="{\displaystyle \mathrm {P} (u,v)=0,\forall u,v:{\sqrt {u^{2}+v^{2}}}>R,}" loading="lazy"></span></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><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}">

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<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

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</math></span><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" 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}" loading="lazy"></span> is the pupil radius.

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<div class="mw-heading mw-heading3"><h3 id="Out_of_focus">Out of focus</h3></div>

<p>When the point source is out of focus, the spherical wave will not be completely made planar by the optics, but will have an approximately parabolic wave front: <span class="mwe-math-element mwe-math-element-inline"><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(u^{2}+v^{2})}">

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<annotation encoding="application/x-tex">{\displaystyle k(u^{2}+v^{2})}</annotation>

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</math></span><img src="./060a16c90dcf7175331cde1839dfbb987f170291.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.427ex; height:3.176ex;" alt="{\displaystyle k(u^{2}+v^{2})}" loading="lazy"></span>. Such a variation in optical path length corresponds to a radial variation in the complex argument of the pupil function:

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {P} (u,v)=\mathrm {exp} (i\,k(u^{2}+v^{2})),\forall u,v:{\sqrt {u^{2}+v^{2}}}\leq R}</annotation>

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</math></span><img src="./65e62ca1d44afe6041e83ad8bb12c63797c1bb76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.307ex; height:3.509ex;" alt="{\displaystyle \mathrm {P} (u,v)=\mathrm {exp} (i\,k(u^{2}+v^{2})),\forall u,v:{\sqrt {u^{2}+v^{2}}}\leq R}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {P} (u,v)=0,}</annotation>

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</math></span><img src="./af257f4ffa3d67099ffa72a3af19a62e976bee8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.791ex; height:2.843ex;" alt="{\displaystyle \mathrm {P} (u,v)=0,}" loading="lazy"></span> otherwise.</dd></dl>

<p>It is thus possible to deduce the point-spread function of the out of focus point source as the Fourier transform of the pupil function.

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<div class="mw-heading mw-heading3"><h3 id="Aberrated_Optics">Aberrated Optics</h3></div>

<p>The spherical wave could also be deformed by imperfect optics to an approximately cylindrical wave front: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ku^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle ku^{2}}</annotation>

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</math></span><img src="./784d0bf1ea796f298a1e83039589971fc3fa4a4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.595ex; height:2.676ex;" alt="{\displaystyle ku^{2}}" loading="lazy"></span>.

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {P} (u,v)=\mathrm {exp} (i\,ku^{2}),\forall u,v:{\sqrt {u^{2}+v^{2}}}\leq R}</annotation>

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</math></span><img src="./7b02dcceabe289caab3c6f0a6eefb042ac5edbab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.476ex; height:3.509ex;" alt="{\displaystyle \mathrm {P} (u,v)=\mathrm {exp} (i\,ku^{2}),\forall u,v:{\sqrt {u^{2}+v^{2}}}\leq R}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {P} (u,v)=0,}</annotation>

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</math></span><img src="./af257f4ffa3d67099ffa72a3af19a62e976bee8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.791ex; height:2.843ex;" alt="{\displaystyle \mathrm {P} (u,v)=0,}" loading="lazy"></span> otherwise.</dd></dl>

<p>Such a variation in optical path length will create an image that is blurred only in one dimension as is typical of systems with <a href="Astigmatism_(optical_systems)" title="Astigmatism (optical systems)">astigmatism</a>.

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<ul><li><a href="Fourier_optics" title="Fourier optics">Fourier optics</a></li>

<li><a href="Point_spread_function" title="Point spread function">Point spread function</a></li>

<li><a href="Optical_transfer_function" title="Optical transfer function">Optical transfer function</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFKidger2001" class="citation book cs1">Kidger, Michael J. (2001). <a rel="nofollow" class="external text" href="https://spie.org/x33111.xml"><i>Fundamental Optical Design</i></a>. SPIE Press, Bellingham, WA<span class="reference-accessdate">. Retrieved <span class="nowrap">10 November</span> 2013</span>.</cite></span>

</li>

<li id="cite_note-Goodman2005-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Goodman2005_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Goodman2005_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFGoodman2005" class="citation book cs1">Goodman, Joseph (2005). <i>Introduction to Fourier Optics</i> (3rd ed.). Roberts & Co Publishers. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-9747077-2-4</bdi>.</cite></span>

</li>

<li id="cite_note-Fisher2008-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Fisher2008_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Fisher2008_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFFisher2008" class="citation book cs1">Fisher, Robert (2008). <i>Optical System Design</i> (2nd ed.). The McGraw-Hill Companies, Inc. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780071472487</bdi>.</cite></span>

</li>

<li id="cite_note-Pawley2006-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Pawley2006_4-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFPawley2006" class="citation book cs1">Pawley, James B. (2006). <i>Handbook of confocal microscopy</i> (3rd ed.). Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-25921-X</bdi>.</cite></span>

</li>

<li id="cite_note-OpticsCourseNotesOnOTF-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-OpticsCourseNotesOnOTF_5-0">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://wp.optics.arizona.edu/jcwyant/wp-content/uploads/sites/13/2016/08/OpticalTransferFunction.nb_.pdf">"Optics Course Notes on the calculation of the OTF from the Pupil function"</a> <span class="cs1-format">(PDF)</span><span class="reference-accessdate">. Retrieved <span class="nowrap">2 February</span> 2022</span>.</cite></span>

</li>

</ol></div></div></div><div><div class="zim-footer">

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