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<title>Paraxial approximation</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Paraxial approximation</span></span>

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<p>In <a href="Geometric_optics" class="mw-redirect" title="Geometric optics">geometric optics</a>, the <b>paraxial approximation</b> is a <a href="Small-angle_approximation" title="Small-angle approximation">small-angle approximation</a> used in <a href="Gaussian_optics" title="Gaussian optics">Gaussian optics</a> and <a href="Ray_tracing_(physics)" title="Ray tracing (physics)">ray tracing</a> of light through an optical system (such as a <a href="Lens_(optics)" class="mw-redirect" title="Lens (optics)">lens</a>).<sup id="cite_ref-Greivenkamp_1-0" class="reference"><a href="#cite_note-Greivenkamp-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><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>A <b>paraxial ray</b> is a <a href="Ray_(optics)" title="Ray (optics)">ray</a> that makes a small angle (<i>θ</i>) to the <a href="Optical_axis" title="Optical axis">optical axis</a> of the system, and lies close to the axis throughout the system.<sup id="cite_ref-Greivenkamp_1-1" class="reference"><a href="#cite_note-Greivenkamp-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Generally, this allows three important approximations (for <i>θ</i> in <a href="Radian" title="Radian">radians</a>) for calculation of the ray's path, namely:<sup id="cite_ref-Greivenkamp_1-2" class="reference"><a href="#cite_note-Greivenkamp-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

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<annotation encoding="application/x-tex">{\displaystyle \sin \theta \approx \theta ,\quad \tan \theta \approx \theta \quad {\text{and}}\quad \cos \theta \approx 1.}</annotation>

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</math></span><img src="./638801c838f0d580a5bf4bc1a911ffd3fe3f5418.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:39.181ex; height:2.509ex;" alt="{\displaystyle \sin \theta \approx \theta ,\quad \tan \theta \approx \theta \quad {\text{and}}\quad \cos \theta \approx 1.}" loading="lazy"></span></dd></dl>

<p>The paraxial approximation is used in <a href="Gaussian_optics" title="Gaussian optics">Gaussian optics</a> and <i>first-order</i> ray tracing.<sup id="cite_ref-Greivenkamp_1-3" class="reference"><a href="#cite_note-Greivenkamp-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <a href="Ray_transfer_matrix_analysis" title="Ray transfer matrix analysis">Ray transfer matrix analysis</a> is one method that uses the approximation.

</p><p>In some cases, the second-order approximation is also called "paraxial". The approximations above for sine and tangent do not change for the "second-order" paraxial approximation (the second term in their <a href="Taylor_series" title="Taylor series">Taylor series</a> expansion is zero), while for cosine the second order approximation is

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<annotation encoding="application/x-tex">{\displaystyle \cos \theta \approx 1-{\theta ^{2} \over 2}\ .}</annotation>

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</math></span><img src="./06e4b50ef369c3a0f07a2dc196c43a5d09cfad1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.898ex; height:5.676ex;" alt="{\displaystyle \cos \theta \approx 1-{\theta ^{2} \over 2}\ .}" loading="lazy"></span></dd></dl>

<p>The second-order approximation is accurate within 0.5% for angles under about 10°, but its inaccuracy grows significantly for larger angles.<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>

</p><p>For larger angles it is often necessary to distinguish between <a href="Meridional_ray" class="mw-redirect" title="Meridional ray">meridional rays</a>, which lie in a plane containing the <a href="Optical_axis" title="Optical axis">optical axis</a>, and <a href="Sagittal_ray" class="mw-redirect" title="Sagittal ray">sagittal rays</a>, which do not.

</p><p>Use of the small angle approximations replaces dimensionless trigonometric functions with angles in radians. In <a href="Dimensional_analysis" title="Dimensional analysis">dimensional analysis</a> on optics equations radians are dimensionless and therefore can be ignored.

</p><p>A paraxial approximation is also commonly used in <a href="Physical_optics" title="Physical optics">physical optics</a>. It is used in the derivation of the paraxial wave equation from the homogeneous <a href="Maxwell's_equations" title="Maxwell's equations">Maxwell's equations</a> and, consequently, <a href="Gaussian_beam" title="Gaussian beam">Gaussian beam</a> optics.

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

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</style><cite id="CITEREFGreivenkamp2004" class="citation book cs1">Greivenkamp, John E. (2004). <i>Field Guide to Geometrical Optics</i>. SPIE Field Guides. Vol. 1. <a href="SPIE" title="SPIE">SPIE</a>. pp. <span class="nowrap">19–</span>20. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8194-5294-7</bdi>.</cite></span>

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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFWeisstein2007" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> (2007). <a rel="nofollow" class="external text" href="http://scienceworld.wolfram.com/physics/ParaxialApproximation.html">"Paraxial Approximation"</a>. <i><a href="ScienceWorld" class="mw-redirect" title="ScienceWorld">ScienceWorld</a></i>. <a href="Wolfram_Research" title="Wolfram Research">Wolfram Research</a><span class="reference-accessdate">. Retrieved <span class="nowrap">15 January</span> 2014</span>.</cite></span>

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<cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.wolframalpha.com/input/?i=Plot%5B{%28x+Deg+-+Sin%5Bx+Deg%5D%29%2FSin%5Bx+Deg%5D%2C+%28Tan%5Bx+Deg%5D+-+x+Deg%29%2FTan%5Bx+Deg%5D%2C+%281+-+Cos%5Bx+Deg%5D%29%2FCos%5Bx+Deg%5D}%2C+{x%2C+0%2C+15}%5D">"Paraxial approximation error plot"</a>. <i><a href="Wolfram_Alpha" class="mw-redirect" title="Wolfram Alpha">Wolfram Alpha</a></i>. <a href="Wolfram_Research" title="Wolfram Research">Wolfram Research</a><span class="reference-accessdate">. Retrieved <span class="nowrap">26 August</span> 2014</span>.</cite></span>

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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com/ParaxialApproximationAndTheMirror/">Paraxial Approximation and the Mirror</a> by David Schurig, <a href="The_Wolfram_Demonstrations_Project" class="mw-redirect" title="The Wolfram Demonstrations Project">The Wolfram Demonstrations Project</a>.</li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-04-13" href="https://en.wikipedia.org/wiki/?title=Paraxial_approximation&oldid=1285444143">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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