Geometrical optics

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menu__item--page-actions-watch"> Watch </a> </li> <li id="page-actions-edit" class="page-actions-menu__list-item"> <a role="button" id="ca-edit" href="/w/index.php?title=Geometrical_optics&action=edit" data-event-name="menu.edit" data-mw="interface" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet edit-page menu__item--page-actions-edit"> Edit </a> </li> </ul> </nav>  <div id="mw-content-subtitle"></div> </div> <div id="bodyContent" class="content"> <div id="mw-content-text" class="mw-body-content"><script>function mfTempOpenSection(id){var block=document.getElementById("mf-section-"+id);block.className+=" open-block";block.previousSibling.className+=" open-block";}</script><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><section class="mf-section-0" id="mf-section-0"> Geometrical optics, or ray optics, is a model of <a href="/wiki/Optics" title="Optics">optics</a> that describes <a href="/wiki/Light" title="Light">light</a> <a href="/wiki/Wave_propagation" class="mw-redirect" title="Wave propagation">propagation</a> in terms of <a href="/wiki/Ray_(optics)" title="Ray (optics)">rays</a>. The ray in geometrical optics is an <a href="/wiki/Abstract_object" class="mw-redirect" title="Abstract object">abstraction</a> useful for approximating the paths along which light propagates under certain circumstances. The simplifying assumptions of geometrical optics include that light rays: <ul><li>propagate in straight-line paths as they travel in a <a href="/wiki/Homogeneity_(physics)" title="Homogeneity (physics)">homogeneous</a> medium</li> <li>bend, and in particular circumstances may split in two, at the <a href="/wiki/Interface_(matter)" title="Interface (matter)">interface</a> between two dissimilar <a href="/wiki/Optical_medium" title="Optical medium">media</a></li> <li>follow curved paths in a medium in which the <a href="/wiki/Refractive_index" title="Refractive index">refractive index</a> changes</li> <li>may be absorbed or reflected.</li></ul> Geometrical optics does not account for certain optical effects such as <a href="/wiki/Diffraction" title="Diffraction">diffraction</a> and <a href="/wiki/Interference_(wave_propagation)" class="mw-redirect" title="Interference (wave propagation)">interference</a>, which are considered in <a href="/wiki/Physical_optics" title="Physical optics">physical optics</a>. This simplification is useful in practice; it is an excellent approximation when the wavelength is small compared to the size of structures with which the light interacts. The techniques are particularly useful in describing geometrical aspects of <a href="/wiki/Image" title="Image">imaging</a>, including <a href="/wiki/Optical_aberration" title="Optical aberration">optical aberrations</a>. <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"><div class="toctitle" lang="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><label class="toctogglelabel" for="toctogglecheckbox"></label></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Explanation">1 Explanation</a></li> <li class="toclevel-1 tocsection-2"><a href="#Reflection">2 Reflection</a></li> <li class="toclevel-1 tocsection-3"><a href="#Refraction">3 Refraction</a></li> <li class="toclevel-1 tocsection-4"><a href="#Underlying_mathematics">4 Underlying mathematics</a> <ul> <li class="toclevel-2 tocsection-5"><a href="#Sommerfeld%E2%80%93Runge_method">4.1 Sommerfeld–Runge method</a></li> <li class="toclevel-2 tocsection-6"><a href="#Luneburg_method">4.2 Luneburg method</a></li> <li class="toclevel-2 tocsection-7"><a href="#General_equation_using_four-vector_notation">4.3 General equation using four-vector notation</a></li> </ul> </li> <li class="toclevel-1 tocsection-8"><a href="#See_also">5 See also</a></li> <li class="toclevel-1 tocsection-9"><a href="#References">6 References</a></li> <li class="toclevel-1 tocsection-10"><a href="#Further_reading">7 Further reading</a> <ul> <li class="toclevel-2 tocsection-11"><a href="#English_translations_of_some_early_books_and_papers">7.1 English translations of some early books and papers</a></li> </ul> </li> <li class="toclevel-1 tocsection-12"><a href="#External_links">8 External links</a></li> </ul> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"><h2 id="Explanation">Explanation</h2> <a role="button" href="/w/index.php?title=Geometrical_optics&action=edit&section=1" title="Edit section: Explanation" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-1 collapsible-block" id="mf-section-1"> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Ray_(optics)" title="Ray (optics)">Ray (optics)</a> and <a href="/wiki/Ray_tracing_(physics)" title="Ray tracing (physics)">Ray tracing (physics)</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Plane_wave_wavefronts_3D.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/Plane_wave_wavefronts_3D.svg/220px-Plane_wave_wavefronts_3D.svg.png" decoding="async" width="220" height="135" class="mw-file-element" data-file-width="327" data-file-height="201"></noscript><span class="lazy-image-placeholder" style="width: 220px;height: 135px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/Plane_wave_wavefronts_3D.svg/220px-Plane_wave_wavefronts_3D.svg.png" data-width="220" data-height="135" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/Plane_wave_wavefronts_3D.svg/330px-Plane_wave_wavefronts_3D.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/Plane_wave_wavefronts_3D.svg/440px-Plane_wave_wavefronts_3D.svg.png 2x" data-class="mw-file-element"> </a><figcaption>As light travels through space, it <a href="/wiki/Oscillation" title="Oscillation">oscillates</a> in <a href="/wiki/Amplitude" title="Amplitude">amplitude</a>. In this image, each maximum amplitude <a href="/wiki/Crest_(physics)" class="mw-redirect" title="Crest (physics)">crest</a> is marked with a <a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a> to illustrate the <a href="/wiki/Wavefront" title="Wavefront">wavefront</a>. The <a href="/wiki/Ray_(optics)" title="Ray (optics)">ray</a> is the arrow <a href="/wiki/Perpendicular" title="Perpendicular">perpendicular</a> to these <a href="/wiki/Parallel_(geometry)" title="Parallel (geometry)">parallel</a> surfaces.</figcaption></figure> A light ray is a <a href="/wiki/Line_(geometry)" title="Line (geometry)">line</a> or <a href="/wiki/Curve" title="Curve">curve</a> that is <a href="/wiki/Perpendicular" title="Perpendicular">perpendicular</a> to the light's <a href="/wiki/Wavefront" title="Wavefront">wavefronts</a> (and is therefore <a href="https://en.wiktionary.org/wiki/collinear" class="extiw" title="wiktionary:collinear">collinear</a> with the <a href="/wiki/Wave_vector" title="Wave vector">wave vector</a>). A slightly more rigorous definition of a light ray follows from <a href="/wiki/Fermat%27s_principle" title="Fermat's principle">Fermat's principle</a>, which states that the path taken between two points by a ray of light is the path that can be traversed in the least time.<a href="#cite_note-1">[1]</a> Geometrical optics is often simplified by making the <a href="/wiki/Paraxial_approximation" title="Paraxial approximation">paraxial approximation</a>, or "small angle approximation". The mathematical behavior then becomes <a href="/wiki/Linear_system" title="Linear system">linear</a>, allowing optical components and systems to be described by simple matrices. This leads to the techniques of <a href="/wiki/Gaussian_optics" title="Gaussian optics">Gaussian optics</a> and paraxial <a href="/wiki/Ray_tracing_(physics)" title="Ray tracing (physics)">ray tracing</a>, which are used to find basic properties of optical systems, such as approximate <a href="/wiki/Image" title="Image">image</a> and object positions and <a href="/wiki/Magnification" title="Magnification">magnifications</a>.<a href="#cite_note-Greivenkamp-2">[2]</a> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"><h2 id="Reflection">Reflection</h2> <a role="button" href="/w/index.php?title=Geometrical_optics&action=edit&section=2" title="Edit section: Reflection" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-2 collapsible-block" id="mf-section-2"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Reflection_(physics)" title="Reflection (physics)">Reflection (physics)</a></div> <figure typeof="mw:File/Frame"><a href="/wiki/File:Reflection_angles.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Reflection_angles.svg/170px-Reflection_angles.svg.png" decoding="async" width="170" height="204" class="mw-file-element" data-file-width="170" data-file-height="204"></noscript><span class="lazy-image-placeholder" style="width: 170px;height: 204px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Reflection_angles.svg/170px-Reflection_angles.svg.png" data-width="170" data-height="204" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Reflection_angles.svg/255px-Reflection_angles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/10/Reflection_angles.svg/340px-Reflection_angles.svg.png 2x" data-class="mw-file-element"> </a><figcaption>Diagram of <a href="/wiki/Specular_reflection" title="Specular reflection">specular reflection</a></figcaption></figure> Glossy surfaces such as <a href="/wiki/Mirror" title="Mirror">mirrors</a> reflect light in a simple, predictable way. This allows for production of reflected images that can be associated with an actual (<a href="/wiki/Real_image" title="Real image">real</a>) or extrapolated (<a href="/wiki/Virtual_image" title="Virtual image">virtual</a>) location in space. With such surfaces, the direction of the reflected ray is determined by the angle the incident ray makes with the <a href="/wiki/Surface_normal" class="mw-redirect" title="Surface normal">surface normal</a>, a line perpendicular to the surface at the point where the ray hits. The incident and reflected rays lie in a single plane, and the angle between the reflected ray and the surface normal is the same as that between the incident ray and the normal.<a href="#cite_note-Geoptics-3">[3]</a> This is known as the <a href="/wiki/Law_of_Reflection" class="mw-redirect" title="Law of Reflection">Law of Reflection</a>. For <a href="/wiki/Plane_mirror" title="Plane mirror">flat mirrors</a>, the law of reflection implies that images of objects are upright and the same distance behind the mirror as the objects are in front of the mirror. The image size is the same as the object size. (The <a href="/wiki/Magnification" title="Magnification">magnification</a> of a flat mirror is equal to one.) The law also implies that <a href="/wiki/Mirror_image" title="Mirror image">mirror images</a> are <a href="/wiki/Parity_(physics)" title="Parity (physics)">parity inverted</a>, which is perceived as a left-right inversion. <a href="/wiki/Curved_mirror" title="Curved mirror">Mirrors with curved surfaces</a> can be modeled by <a href="/wiki/Ray_tracing_(physics)" title="Ray tracing (physics)">ray tracing</a> and using the law of reflection at each point on the surface. For <a href="/wiki/Parabolic_reflector" title="Parabolic reflector">mirrors with parabolic surfaces</a>, parallel rays incident on the mirror produce reflected rays that converge at a common <a href="/wiki/Focus_(optics)" title="Focus (optics)">focus</a>. Other curved surfaces may also focus light, but with aberrations due to the diverging shape causing the focus to be smeared out in space. In particular, spherical mirrors exhibit <a href="/wiki/Spherical_aberration" title="Spherical aberration">spherical aberration</a>. Curved mirrors can form images with magnification greater than or less than one, and the image can be upright or inverted. An upright image formed by reflection in a mirror is always virtual, while an inverted image is real and can be projected onto a screen.<a href="#cite_note-Geoptics-3">[3]</a> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"><h2 id="Refraction">Refraction</h2> <a role="button" href="/w/index.php?title=Geometrical_optics&action=edit&section=3" title="Edit section: Refraction" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-3 collapsible-block" id="mf-section-3"> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Summarize plainlinks metadata ambox ambox-style" role="presentation"><tbody><tr><td class="mbox-text"><div class="mbox-text-span">This section should include a summary of, or be summarized in, another article. See <a href="/wiki/Wikipedia:Summary_style" title="Wikipedia:Summary style">Wikipedia:Summary style</a> for information on how to incorporate it into this article's main text, or the main text of another article. (June 2009)</div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Refraction" title="Refraction">Refraction</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Snells_law.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Snells_law.svg/300px-Snells_law.svg.png" decoding="async" width="300" height="166" class="mw-file-element" data-file-width="641" data-file-height="355"></noscript><span class="lazy-image-placeholder" style="width: 300px;height: 166px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Snells_law.svg/300px-Snells_law.svg.png" data-width="300" data-height="166" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Snells_law.svg/450px-Snells_law.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Snells_law.svg/600px-Snells_law.svg.png 2x" data-class="mw-file-element"> </a><figcaption>Illustration of Snell's Law</figcaption></figure> Refraction occurs when light travels through an area of space that has a changing index of refraction. The simplest case of refraction occurs when there is an interface between a uniform medium with index of refraction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{1}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee784b70e772f55ede5e6e0bdc929994bff63413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.449ex; height:2.009ex;" alt="{\displaystyle n_{1}}"></noscript><span class="lazy-image-placeholder" style="width: 2.449ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee784b70e772f55ede5e6e0bdc929994bff63413" data-alt="{\displaystyle n_{1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and another medium with index of refraction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{2}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/840e456e3058bc0be28e5cf653b170cdbfcc3be4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.449ex; height:2.009ex;" alt="{\displaystyle n_{2}}"></noscript><span class="lazy-image-placeholder" style="width: 2.449ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/840e456e3058bc0be28e5cf653b170cdbfcc3be4" data-alt="{\displaystyle n_{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . In such situations, <a href="/wiki/Snell%27s_Law" class="mw-redirect" title="Snell's Law">Snell's Law</a> describes the resulting deflection of the light ray: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{1}\sin \theta _{1}=n_{2}\sin \theta _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{1}\sin \theta _{1}=n_{2}\sin \theta _{2}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6b2a5ea2d5d1fdf92e3b13e58bd3f91a48236ff" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.545ex; height:2.509ex;" alt="{\displaystyle n_{1}\sin \theta _{1}=n_{2}\sin \theta _{2}}"></noscript><span class="lazy-image-placeholder" style="width: 19.545ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6b2a5ea2d5d1fdf92e3b13e58bd3f91a48236ff" data-alt="{\displaystyle n_{1}\sin \theta _{1}=n_{2}\sin \theta _{2}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{1}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f84b9443d095623e02fd287cd095123d70b0278" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.145ex; height:2.509ex;" alt="{\displaystyle \theta _{1}}"></noscript><span class="lazy-image-placeholder" style="width: 2.145ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f84b9443d095623e02fd287cd095123d70b0278" data-alt="{\displaystyle \theta _{1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{2}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ed6ea624b20b153403979ffaf5434fc36de2990" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.145ex; height:2.509ex;" alt="{\displaystyle \theta _{2}}"></noscript><span class="lazy-image-placeholder" style="width: 2.145ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ed6ea624b20b153403979ffaf5434fc36de2990" data-alt="{\displaystyle \theta _{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  are the angles between the normal (to the interface) and the incident and refracted waves, respectively. This phenomenon is also associated with a changing speed of light as seen from the definition of index of refraction provided above which implies: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{1}\sin \theta _{2}\ =v_{2}\sin \theta _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mtext> </mtext> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{1}\sin \theta _{2}\ =v_{2}\sin \theta _{1}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67f709b46fbb8d6fd025424b4c93af66f9e2b2ef" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.592ex; height:2.509ex;" alt="{\displaystyle v_{1}\sin \theta _{2}\ =v_{2}\sin \theta _{1}}"></noscript><span class="lazy-image-placeholder" style="width: 19.592ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67f709b46fbb8d6fd025424b4c93af66f9e2b2ef" data-alt="{\displaystyle v_{1}\sin \theta _{2}\ =v_{2}\sin \theta _{1}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{1}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98d33f5d498d528bd8c10edc8ac8c34347f32b3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.182ex; height:2.009ex;" alt="{\displaystyle v_{1}}"></noscript><span class="lazy-image-placeholder" style="width: 2.182ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98d33f5d498d528bd8c10edc8ac8c34347f32b3a" data-alt="{\displaystyle v_{1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{2}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb04c423c2cb809c30cac725befa14ffbf4c85f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.182ex; height:2.009ex;" alt="{\displaystyle v_{2}}"></noscript><span class="lazy-image-placeholder" style="width: 2.182ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb04c423c2cb809c30cac725befa14ffbf4c85f3" data-alt="{\displaystyle v_{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  are the wave velocities through the respective media.<a href="#cite_note-Geoptics-3">[3]</a> Various consequences of Snell's Law include the fact that for light rays traveling from a material with a high index of refraction to a material with a low index of refraction, it is possible for the interaction with the interface to result in zero transmission. This phenomenon is called <a href="/wiki/Total_internal_reflection" title="Total internal reflection">total internal reflection</a> and allows for <a href="/wiki/Fiber_optics" class="mw-redirect" title="Fiber optics">fiber optics</a> technology. As light signals travel down a fiber optic cable, they undergo total internal reflection allowing for essentially no light lost over the length of the cable. It is also possible to produce <a href="/wiki/Plane_polarization" class="mw-redirect" title="Plane polarization">polarized light rays</a> using a combination of reflection and refraction: When a refracted ray and the reflected ray form a <a href="/wiki/Right_angle" title="Right angle">right angle</a>, the reflected ray has the property of "plane polarization". The angle of incidence required for such a scenario is known as <a href="/wiki/Brewster%27s_angle" title="Brewster's angle">Brewster's angle</a>.<a href="#cite_note-Geoptics-3">[3]</a> Snell's Law can be used to predict the deflection of light rays as they pass through "linear media" as long as the indexes of refraction and the geometry of the media are known. For example, the propagation of light through a <a href="/wiki/Prism_(optics)" title="Prism (optics)">prism</a> results in the light ray being deflected depending on the shape and orientation of the prism. Additionally, since different frequencies of light have slightly different indexes of refraction in most materials, refraction can be used to produce <a href="/wiki/Dispersion_(optics)" title="Dispersion (optics)">dispersion</a> <a href="/wiki/Spectrum" title="Spectrum">spectra</a> that appear as rainbows. The discovery of this phenomenon when passing light through a prism is famously attributed to <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a>.<a href="#cite_note-Geoptics-3">[3]</a> Some media have an index of refraction which varies gradually with position and, thus, light rays curve through the medium rather than travel in straight lines. This effect is what is responsible for <a href="/wiki/Mirage" title="Mirage">mirages</a> seen on hot days where the changing index of refraction of the air causes the light rays to bend creating the appearance of specular reflections in the distance (as if on the surface of a pool of water). Material that has a varying index of refraction is called a gradient-index (GRIN) material and has many useful properties used in modern optical scanning technologies including <a href="/wiki/Photocopiers" class="mw-redirect" title="Photocopiers">photocopiers</a> and <a href="/wiki/Image_scanner" title="Image scanner">scanners</a>. The phenomenon is studied in the field of <a href="/wiki/Gradient-index_optics" title="Gradient-index optics">gradient-index optics</a>.<a href="#cite_note-4">[4]</a> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Lens3b.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Lens3b.svg/360px-Lens3b.svg.png" decoding="async" width="360" height="257" class="mw-file-element" data-file-width="362" data-file-height="258"></noscript><span class="lazy-image-placeholder" style="width: 360px;height: 257px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Lens3b.svg/360px-Lens3b.svg.png" data-width="360" data-height="257" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Lens3b.svg/540px-Lens3b.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/97/Lens3b.svg/720px-Lens3b.svg.png 2x" data-class="mw-file-element"> </a><figcaption>A ray tracing diagram for a simple converging lens.</figcaption></figure> A device which produces converging or diverging light rays due to refraction is known as a <a href="/wiki/Lens_(optics)" class="mw-redirect" title="Lens (optics)">lens</a>. Thin lenses produce focal points on either side that can be modeled using the <a href="/wiki/Lensmaker%27s_equation" class="mw-redirect" title="Lensmaker's equation">lensmaker's equation</a>.<a href="#cite_note-hecht-5">[5]</a> In general, two types of lenses exist: <a href="/wiki/Convex_lens" class="mw-redirect" title="Convex lens">convex lenses</a>, which cause parallel light rays to converge, and <a href="/wiki/Concave_lens" class="mw-redirect" title="Concave lens">concave lenses</a>, which cause parallel light rays to diverge. The detailed prediction of how images are produced by these lenses can be made using ray-tracing similar to curved mirrors. Similarly to curved mirrors, thin lenses follow a simple equation that determines the location of the images given a particular focal length (<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><noscript><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}"></noscript><span class="lazy-image-placeholder" style="width: 1.279ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" data-alt="{\displaystyle f}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ) and object distance (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{1}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf84e7fd4fb8259a9b37f956afdf83ee2a020f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{1}}"></noscript><span class="lazy-image-placeholder" style="width: 2.479ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf84e7fd4fb8259a9b37f956afdf83ee2a020f9" data-alt="{\displaystyle S_{1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ): <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{S_{1}}}+{\frac {1}{S_{2}}}={\frac {1}{f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>f</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{S_{1}}}+{\frac {1}{S_{2}}}={\frac {1}{f}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d5471b983b47c062376f96c793d2a5fb8b04761" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.684ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{S_{1}}}+{\frac {1}{S_{2}}}={\frac {1}{f}}}"></noscript><span class="lazy-image-placeholder" style="width: 14.684ex;height: 5.676ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d5471b983b47c062376f96c793d2a5fb8b04761" data-alt="{\displaystyle {\frac {1}{S_{1}}}+{\frac {1}{S_{2}}}={\frac {1}{f}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{2}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1143e284d5f25cef778ab482edf6617a523ddd9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{2}}"></noscript><span class="lazy-image-placeholder" style="width: 2.479ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1143e284d5f25cef778ab482edf6617a523ddd9f" data-alt="{\displaystyle S_{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is the distance associated with the image and is considered by convention to be negative if on the same side of the lens as the object and positive if on the opposite side of the lens.<a href="#cite_note-hecht-5">[5]</a> The focal length f is considered negative for concave lenses. Incoming parallel rays are focused by a convex lens into an inverted real image one focal length from the lens, on the far side of the lens. <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:2015-05-25_0820Incoming_parallel_rays_are_focused_by_a_convex_lens_into_an_inverted_real_image_one_focal_length_from_the_lens,_on_the_far_side_of_the.png" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/2015-05-25_0820Incoming_parallel_rays_are_focused_by_a_convex_lens_into_an_inverted_real_image_one_focal_length_from_the_lens%2C_on_the_far_side_of_the.png/220px-2015-05-25_0820Incoming_parallel_rays_are_focused_by_a_convex_lens_into_an_inverted_real_image_one_focal_length_from_the_lens%2C_on_the_far_side_of_the.png" decoding="async" width="220" height="123" class="mw-file-element" data-file-width="948" data-file-height="531"></noscript><span class="lazy-image-placeholder" style="width: 220px;height: 123px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/2015-05-25_0820Incoming_parallel_rays_are_focused_by_a_convex_lens_into_an_inverted_real_image_one_focal_length_from_the_lens%2C_on_the_far_side_of_the.png/220px-2015-05-25_0820Incoming_parallel_rays_are_focused_by_a_convex_lens_into_an_inverted_real_image_one_focal_length_from_the_lens%2C_on_the_far_side_of_the.png" data-width="220" data-height="123" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/2015-05-25_0820Incoming_parallel_rays_are_focused_by_a_convex_lens_into_an_inverted_real_image_one_focal_length_from_the_lens%2C_on_the_far_side_of_the.png/330px-2015-05-25_0820Incoming_parallel_rays_are_focused_by_a_convex_lens_into_an_inverted_real_image_one_focal_length_from_the_lens%2C_on_the_far_side_of_the.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1e/2015-05-25_0820Incoming_parallel_rays_are_focused_by_a_convex_lens_into_an_inverted_real_image_one_focal_length_from_the_lens%2C_on_the_far_side_of_the.png/440px-2015-05-25_0820Incoming_parallel_rays_are_focused_by_a_convex_lens_into_an_inverted_real_image_one_focal_length_from_the_lens%2C_on_the_far_side_of_the.png 2x" data-class="mw-file-element"> </a><figcaption>Incoming parallel rays are focused by a convex lens into an inverted real image one focal length from the lens, on the far side of the lens</figcaption></figure> Rays from an object at finite distance are focused further from the lens than the focal distance; the closer the object is to the lens, the further the image is from the lens. With concave lenses, incoming parallel rays diverge after going through the lens, in such a way that they seem to have originated at an upright virtual image one focal length from the lens, on the same side of the lens that the parallel rays are approaching on. <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:2015-05-25_0836With_concave_lenses,_incoming_parallel_rays_diverge_after_going_through_the_lens,_in_such_a_way_that_they_seem_to_have_originated_at_an.png" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/2015-05-25_0836With_concave_lenses%2C_incoming_parallel_rays_diverge_after_going_through_the_lens%2C_in_such_a_way_that_they_seem_to_have_originated_at_an.png/220px-2015-05-25_0836With_concave_lenses%2C_incoming_parallel_rays_diverge_after_going_through_the_lens%2C_in_such_a_way_that_they_seem_to_have_originated_at_an.png" decoding="async" width="220" height="189" class="mw-file-element" data-file-width="625" data-file-height="536"></noscript><span class="lazy-image-placeholder" style="width: 220px;height: 189px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/2015-05-25_0836With_concave_lenses%2C_incoming_parallel_rays_diverge_after_going_through_the_lens%2C_in_such_a_way_that_they_seem_to_have_originated_at_an.png/220px-2015-05-25_0836With_concave_lenses%2C_incoming_parallel_rays_diverge_after_going_through_the_lens%2C_in_such_a_way_that_they_seem_to_have_originated_at_an.png" data-width="220" data-height="189" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/2015-05-25_0836With_concave_lenses%2C_incoming_parallel_rays_diverge_after_going_through_the_lens%2C_in_such_a_way_that_they_seem_to_have_originated_at_an.png/330px-2015-05-25_0836With_concave_lenses%2C_incoming_parallel_rays_diverge_after_going_through_the_lens%2C_in_such_a_way_that_they_seem_to_have_originated_at_an.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3d/2015-05-25_0836With_concave_lenses%2C_incoming_parallel_rays_diverge_after_going_through_the_lens%2C_in_such_a_way_that_they_seem_to_have_originated_at_an.png/440px-2015-05-25_0836With_concave_lenses%2C_incoming_parallel_rays_diverge_after_going_through_the_lens%2C_in_such_a_way_that_they_seem_to_have_originated_at_an.png 2x" data-class="mw-file-element"> </a><figcaption>With concave lenses, incoming parallel rays diverge after going through the lens, in such a way that they seem to have originated at an upright virtual image one focal length from the lens, on the same side of the lens that the parallel rays are approaching on.</figcaption></figure> Rays from an object at finite distance are associated with a virtual image that is closer to the lens than the focal length, and on the same side of the lens as the object. The closer the object is to the lens, the closer the virtual image is to the lens. <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Virtualimageframerate1.gif" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Virtualimageframerate1.gif/220px-Virtualimageframerate1.gif" decoding="async" width="220" height="231" class="mw-file-element" data-file-width="597" data-file-height="626"></noscript><span class="lazy-image-placeholder" style="width: 220px;height: 231px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Virtualimageframerate1.gif/220px-Virtualimageframerate1.gif" data-width="220" data-height="231" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Virtualimageframerate1.gif/330px-Virtualimageframerate1.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Virtualimageframerate1.gif/440px-Virtualimageframerate1.gif 2x" data-class="mw-file-element"> </a><figcaption>Rays from an object at finite distance are associated with a virtual image that is closer to the lens than the focal length, and on the same side of the lens as the object.</figcaption></figure> Likewise, the magnification of a lens is given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=-{\frac {S_{2}}{S_{1}}}={\frac {f}{f-S_{1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>f</mi> <mrow> <mi>f</mi> <mo>−</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=-{\frac {S_{2}}{S_{1}}}={\frac {f}{f-S_{1}}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ba81b22aed933e15e889507aa5b4a3f5d6bc0fd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.197ex; height:5.843ex;" alt="{\displaystyle M=-{\frac {S_{2}}{S_{1}}}={\frac {f}{f-S_{1}}}}"></noscript><span class="lazy-image-placeholder" style="width: 21.197ex;height: 5.843ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ba81b22aed933e15e889507aa5b4a3f5d6bc0fd" data-alt="{\displaystyle M=-{\frac {S_{2}}{S_{1}}}={\frac {f}{f-S_{1}}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  where the negative sign is given, by convention, to indicate an upright object for positive values and an inverted object for negative values. Similar to mirrors, upright images produced by single lenses are virtual while inverted images are real.<a href="#cite_note-Geoptics-3">[3]</a> Lenses suffer from <a href="/wiki/Optical_aberration" title="Optical aberration">aberrations</a> that distort images and focal points. These are due to both to geometrical imperfections and due to the changing index of refraction for different wavelengths of light (<a href="/wiki/Chromatic_aberration" title="Chromatic aberration">chromatic aberration</a>).<a href="#cite_note-Geoptics-3">[3]</a> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"><h2 id="Underlying_mathematics">Underlying mathematics</h2> <a role="button" href="/w/index.php?title=Geometrical_optics&action=edit&section=4" title="Edit section: Underlying mathematics" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-4 collapsible-block" id="mf-section-4"> As a mathematical study, geometrical optics emerges as a short-<a href="/wiki/Wavelength" title="Wavelength">wavelength</a> limit for solutions to <a href="/wiki/Hyperbolic_partial_differential_equation" title="Hyperbolic partial differential equation">hyperbolic partial differential equations</a> (Sommerfeld–Runge method) or as a property of propagation of field discontinuities according to Maxwell's equations (Luneburg method). In this short-wavelength limit, it is possible to approximate the solution locally by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(t,x)\approx a(t,x)e^{i(k\cdot x-\omega t)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>⋅</mo> <mi>x</mi> <mo>−</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(t,x)\approx a(t,x)e^{i(k\cdot x-\omega t)}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73ecac4272cb6730876ac017be47c590d457cd30" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.994ex; height:3.343ex;" alt="{\displaystyle u(t,x)\approx a(t,x)e^{i(k\cdot x-\omega t)}}"></noscript><span class="lazy-image-placeholder" style="width: 23.994ex;height: 3.343ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73ecac4272cb6730876ac017be47c590d457cd30" data-alt="{\displaystyle u(t,x)\approx a(t,x)e^{i(k\cdot x-\omega t)}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k,\omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>,</mo> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k,\omega }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc8fcd6413d09acb97893ca8cc6be3ed6ba6cedb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.691ex; height:2.509ex;" alt="{\displaystyle k,\omega }"></noscript><span class="lazy-image-placeholder" style="width: 3.691ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc8fcd6413d09acb97893ca8cc6be3ed6ba6cedb" data-alt="{\displaystyle k,\omega }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  satisfy a <a href="/wiki/Dispersion_relation" title="Dispersion relation">dispersion relation</a>, and the amplitude <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(t,x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(t,x)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e1d4ef4e319ce54aea3d49a712ac36a24111e3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.242ex; height:2.843ex;" alt="{\displaystyle a(t,x)}"></noscript><span class="lazy-image-placeholder" style="width: 6.242ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e1d4ef4e319ce54aea3d49a712ac36a24111e3a" data-alt="{\displaystyle a(t,x)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  varies slowly. More precisely, the <a href="/wiki/Leading-order_term" title="Leading-order term">leading order</a> solution takes the form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}(t,x)e^{i\varphi (t,x)/\varepsilon }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ε</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}(t,x)e^{i\varphi (t,x)/\varepsilon }.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ec2990d7efefa744f40f115bb420112fbab2178" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.76ex; height:3.343ex;" alt="{\displaystyle a_{0}(t,x)e^{i\varphi (t,x)/\varepsilon }.}"></noscript><span class="lazy-image-placeholder" style="width: 15.76ex;height: 3.343ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ec2990d7efefa744f40f115bb420112fbab2178" data-alt="{\displaystyle a_{0}(t,x)e^{i\varphi (t,x)/\varepsilon }.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  The phase <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (t,x)/\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (t,x)/\varepsilon }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5fe3431316bde60669622137f4fe2cc2323e4df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.779ex; height:2.843ex;" alt="{\displaystyle \varphi (t,x)/\varepsilon }"></noscript><span class="lazy-image-placeholder" style="width: 8.779ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5fe3431316bde60669622137f4fe2cc2323e4df" data-alt="{\displaystyle \varphi (t,x)/\varepsilon }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  can be linearized to recover large wavenumber <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k:=\nabla _{x}\varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>:=</mo> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k:=\nabla _{x}\varphi }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81b5f76496990486d559e52059a183c047c118a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.585ex; height:2.676ex;" alt="{\displaystyle k:=\nabla _{x}\varphi }"></noscript><span class="lazy-image-placeholder" style="width: 9.585ex;height: 2.676ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81b5f76496990486d559e52059a183c047c118a4" data-alt="{\displaystyle k:=\nabla _{x}\varphi }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , and frequency <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega :=-\partial _{t}\varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>:=</mo> <mo>−</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega :=-\partial _{t}\varphi }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e5ab7b7d8d64d26d3d32cc0f9b17de13f4a563e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.58ex; height:2.676ex;" alt="{\displaystyle \omega :=-\partial _{t}\varphi }"></noscript><span class="lazy-image-placeholder" style="width: 10.58ex;height: 2.676ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e5ab7b7d8d64d26d3d32cc0f9b17de13f4a563e" data-alt="{\displaystyle \omega :=-\partial _{t}\varphi }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . The amplitude <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/693ad9f934775838bd72406b41ada4a59785d7ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{0}}"></noscript><span class="lazy-image-placeholder" style="width: 2.284ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/693ad9f934775838bd72406b41ada4a59785d7ba" data-alt="{\displaystyle a_{0}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  satisfies a <a href="/wiki/Transport_equation" class="mw-redirect" title="Transport equation">transport equation</a>. The small parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon \,}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/137e3f3e97dfca591286d3815815dd6470bdf77b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:1.676ex;" alt="{\displaystyle \varepsilon \,}"></noscript><span class="lazy-image-placeholder" style="width: 1.471ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/137e3f3e97dfca591286d3815815dd6470bdf77b" data-alt="{\displaystyle \varepsilon \,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  enters the scene due to highly oscillatory initial conditions. Thus, when initial conditions oscillate much faster than the coefficients of the differential equation, solutions will be highly oscillatory, and transported along rays. Assuming coefficients in the differential equation are smooth, the rays will be too. In other words, <a href="/wiki/Refraction" title="Refraction">refraction</a> does not take place. The motivation for this technique comes from studying the typical scenario of light propagation where short wavelength light travels along rays that minimize (more or less) its travel time. Its full application requires tools from <a href="/wiki/Microlocal_analysis" title="Microlocal analysis">microlocal analysis</a>. <div class="mw-heading mw-heading3"><h3 id="Sommerfeld–Runge_method">Sommerfeld–Runge method</h3> <a role="button" href="/w/index.php?title=Geometrical_optics&action=edit&section=5" title="Edit section: Sommerfeld–Runge method" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> The method of obtaining equations of geometrical optics by taking the limit of zero wavelength was first described by <a href="/wiki/Arnold_Sommerfeld" title="Arnold Sommerfeld">Arnold Sommerfeld</a> and J. Runge in 1911.<a href="#cite_note-6">[6]</a> Their derivation was based on an oral remark by <a href="/wiki/Peter_Debye" title="Peter Debye">Peter Debye</a>.<a href="#cite_note-7">[7]</a><a href="#cite_note-8">[8]</a> Consider a monochromatic scalar field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (\mathbf {r} ,t)=\phi (\mathbf {r} )e^{i\omega t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (\mathbf {r} ,t)=\phi (\mathbf {r} )e^{i\omega t}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c70010bd4431aa79c3be8725f315fedef4bc30c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.193ex; height:3.176ex;" alt="{\displaystyle \psi (\mathbf {r} ,t)=\phi (\mathbf {r} )e^{i\omega t}}"></noscript><span class="lazy-image-placeholder" style="width: 17.193ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c70010bd4431aa79c3be8725f315fedef4bc30c7" data-alt="{\displaystyle \psi (\mathbf {r} ,t)=\phi (\mathbf {r} )e^{i\omega t}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></noscript><span class="lazy-image-placeholder" style="width: 1.513ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" data-alt="{\displaystyle \psi }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  could be any of the components of <a href="/wiki/Electric_field" title="Electric field">electric</a> or <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a> and hence the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></noscript><span class="lazy-image-placeholder" style="width: 1.385ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" data-alt="{\displaystyle \phi }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  satisfy the wave equation <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla ^{2}\phi +k_{o}^{2}n(\mathbf {r} )^{2}\phi =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ϕ</mi> <mo>+</mo> <msubsup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>n</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ϕ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}\phi +k_{o}^{2}n(\mathbf {r} )^{2}\phi =0}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0535f156611a553911a6267e55a12f63ce4756f8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.488ex; height:3.176ex;" alt="{\displaystyle \nabla ^{2}\phi +k_{o}^{2}n(\mathbf {r} )^{2}\phi =0}"></noscript><span class="lazy-image-placeholder" style="width: 20.488ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0535f156611a553911a6267e55a12f63ce4756f8" data-alt="{\displaystyle \nabla ^{2}\phi +k_{o}^{2}n(\mathbf {r} )^{2}\phi =0}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{o}=\omega /c=2\pi /\lambda _{o}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mo>=</mo> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{o}=\omega /c=2\pi /\lambda _{o}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dcd39037736a469d6079b925e28468c696d0f05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.095ex; height:2.843ex;" alt="{\displaystyle k_{o}=\omega /c=2\pi /\lambda _{o}}"></noscript><span class="lazy-image-placeholder" style="width: 18.095ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dcd39037736a469d6079b925e28468c696d0f05" data-alt="{\displaystyle k_{o}=\omega /c=2\pi /\lambda _{o}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></noscript><span class="lazy-image-placeholder" style="width: 1.007ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" data-alt="{\displaystyle c}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  being the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a> in vacuum. Here, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n(\mathbf {r} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n(\mathbf {r} )}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d809ed046365b51548655ff862ddef471e75c837" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.306ex; height:2.843ex;" alt="{\displaystyle n(\mathbf {r} )}"></noscript><span class="lazy-image-placeholder" style="width: 4.306ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d809ed046365b51548655ff862ddef471e75c837" data-alt="{\displaystyle n(\mathbf {r} )}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is the <a href="/wiki/Refractive_index" title="Refractive index">refractive index</a> of the medium. Without loss of generality, let us introduce <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi =A(k_{o},\mathbf {r} )e^{ik_{o}S(\mathbf {r} )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>=</mo> <mi>A</mi> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mi>S</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi =A(k_{o},\mathbf {r} )e^{ik_{o}S(\mathbf {r} )}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b51ff8ac81d67cc520f8654bb85c1a5031129e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.083ex; height:3.343ex;" alt="{\displaystyle \phi =A(k_{o},\mathbf {r} )e^{ik_{o}S(\mathbf {r} )}}"></noscript><span class="lazy-image-placeholder" style="width: 19.083ex;height: 3.343ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b51ff8ac81d67cc520f8654bb85c1a5031129e4" data-alt="{\displaystyle \phi =A(k_{o},\mathbf {r} )e^{ik_{o}S(\mathbf {r} )}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  to convert the equation to <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -k_{o}^{2}A[(\nabla S)^{2}-n^{2}]+2ik_{o}(\nabla S\cdot \nabla A)+ik_{o}A\nabla ^{2}S+\nabla ^{2}A=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msubsup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>A</mi> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mi>S</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">]</mo> <mo>+</mo> <mn>2</mn> <mi>i</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mi>S</mi> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mi>A</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mi>A</mi> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>S</mi> <mo>+</mo> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>A</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -k_{o}^{2}A[(\nabla S)^{2}-n^{2}]+2ik_{o}(\nabla S\cdot \nabla A)+ik_{o}A\nabla ^{2}S+\nabla ^{2}A=0.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b472952e8003d9916152f34e2369eee0f63a95" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:60.945ex; height:3.176ex;" alt="{\displaystyle -k_{o}^{2}A[(\nabla S)^{2}-n^{2}]+2ik_{o}(\nabla S\cdot \nabla A)+ik_{o}A\nabla ^{2}S+\nabla ^{2}A=0.}"></noscript><span class="lazy-image-placeholder" style="width: 60.945ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b472952e8003d9916152f34e2369eee0f63a95" data-alt="{\displaystyle -k_{o}^{2}A[(\nabla S)^{2}-n^{2}]+2ik_{o}(\nabla S\cdot \nabla A)+ik_{o}A\nabla ^{2}S+\nabla ^{2}A=0.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  Since the underlying principle of geometrical optics lies in the limit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{o}\sim k_{o}^{-1}\rightarrow 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mo>∼</mo> <msubsup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{o}\sim k_{o}^{-1}\rightarrow 0}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aafd2c6490f745651d70b4641170c4da814e16a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.804ex; height:3.009ex;" alt="{\displaystyle \lambda _{o}\sim k_{o}^{-1}\rightarrow 0}"></noscript><span class="lazy-image-placeholder" style="width: 13.804ex;height: 3.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aafd2c6490f745651d70b4641170c4da814e16a7" data-alt="{\displaystyle \lambda _{o}\sim k_{o}^{-1}\rightarrow 0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , the following asymptotic series is assumed, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(k_{o},\mathbf {r} )=\sum _{m=0}^{\infty }{\frac {A_{m}(\mathbf {r} )}{(ik_{o})^{m}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(k_{o},\mathbf {r} )=\sum _{m=0}^{\infty }{\frac {A_{m}(\mathbf {r} )}{(ik_{o})^{m}}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f720666cf0f29800ccac5ac13ef8aa85734e01ee" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.322ex; height:6.843ex;" alt="{\displaystyle A(k_{o},\mathbf {r} )=\sum _{m=0}^{\infty }{\frac {A_{m}(\mathbf {r} )}{(ik_{o})^{m}}}}"></noscript><span class="lazy-image-placeholder" style="width: 22.322ex;height: 6.843ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f720666cf0f29800ccac5ac13ef8aa85734e01ee" data-alt="{\displaystyle A(k_{o},\mathbf {r} )=\sum _{m=0}^{\infty }{\frac {A_{m}(\mathbf {r} )}{(ik_{o})^{m}}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  For large but finite value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{o}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{o}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c06248902993a3b78e87a0c2c7aa508312870f30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.241ex; height:2.509ex;" alt="{\displaystyle k_{o}}"></noscript><span class="lazy-image-placeholder" style="width: 2.241ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c06248902993a3b78e87a0c2c7aa508312870f30" data-alt="{\displaystyle k_{o}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , the series diverges, and one has to be careful in keeping only appropriate first few terms. For each value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{o}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{o}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c06248902993a3b78e87a0c2c7aa508312870f30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.241ex; height:2.509ex;" alt="{\displaystyle k_{o}}"></noscript><span class="lazy-image-placeholder" style="width: 2.241ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c06248902993a3b78e87a0c2c7aa508312870f30" data-alt="{\displaystyle k_{o}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , one can find an optimum number of terms to be kept and adding more terms than the optimum number might result in a poorer approximation.<a href="#cite_note-9">[9]</a> Substituting the series into the equation and collecting terms of different orders, one finds <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}O(k_{o}^{2}):&\quad (\nabla S)^{2}=n^{2},\\[1ex]O(k_{o}):&\quad 2\nabla S\cdot \nabla A_{0}+A_{0}\nabla ^{2}S=0,\\[1ex]O(1):&\quad 2\nabla S\cdot \nabla A_{1}+A_{1}\nabla ^{2}S=-\nabla ^{2}A_{0},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.73em 0.73em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>O</mi> <mo stretchy="false">(</mo> <msubsup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>:</mo> </mtd> <mtd> <mi></mi> <mspace width="1em"></mspace> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mi>S</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>O</mi> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>:</mo> </mtd> <mtd> <mspace width="1em"></mspace> <mn>2</mn> <mi mathvariant="normal">∇</mi> <mi>S</mi> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>S</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>:</mo> </mtd> <mtd> <mspace width="1em"></mspace> <mn>2</mn> <mi mathvariant="normal">∇</mi> <mi>S</mi> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>S</mi> <mo>=</mo> <mo>−</mo> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}O(k_{o}^{2}):&\quad (\nabla S)^{2}=n^{2},\\[1ex]O(k_{o}):&\quad 2\nabla S\cdot \nabla A_{0}+A_{0}\nabla ^{2}S=0,\\[1ex]O(1):&\quad 2\nabla S\cdot \nabla A_{1}+A_{1}\nabla ^{2}S=-\nabla ^{2}A_{0},\end{aligned}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb3606c5d5e65d16712ad728c4dae20cec772bc4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:42.692ex; height:11.843ex;" alt="{\displaystyle {\begin{aligned}O(k_{o}^{2}):&\quad (\nabla S)^{2}=n^{2},\\[1ex]O(k_{o}):&\quad 2\nabla S\cdot \nabla A_{0}+A_{0}\nabla ^{2}S=0,\\[1ex]O(1):&\quad 2\nabla S\cdot \nabla A_{1}+A_{1}\nabla ^{2}S=-\nabla ^{2}A_{0},\end{aligned}}}"></noscript><span class="lazy-image-placeholder" style="width: 42.692ex;height: 11.843ex;vertical-align: -5.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb3606c5d5e65d16712ad728c4dae20cec772bc4" data-alt="{\displaystyle {\begin{aligned}O(k_{o}^{2}):&\quad (\nabla S)^{2}=n^{2},\\[1ex]O(k_{o}):&\quad 2\nabla S\cdot \nabla A_{0}+A_{0}\nabla ^{2}S=0,\\[1ex]O(1):&\quad 2\nabla S\cdot \nabla A_{1}+A_{1}\nabla ^{2}S=-\nabla ^{2}A_{0},\end{aligned}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  in general, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(k_{o}^{1-m}):\quad 2\nabla S\cdot \nabla A_{m}+A_{m}\nabla ^{2}S=-\nabla ^{2}A_{m-1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <msubsup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>m</mi> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>:</mo> <mspace width="1em"></mspace> <mn>2</mn> <mi mathvariant="normal">∇</mi> <mi>S</mi> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>S</mi> <mo>=</mo> <mo>−</mo> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(k_{o}^{1-m}):\quad 2\nabla S\cdot \nabla A_{m}+A_{m}\nabla ^{2}S=-\nabla ^{2}A_{m-1}.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/625b16cf514922cb1c67cc14b91b3651d956be81" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.27ex; height:3.176ex;" alt="{\displaystyle O(k_{o}^{1-m}):\quad 2\nabla S\cdot \nabla A_{m}+A_{m}\nabla ^{2}S=-\nabla ^{2}A_{m-1}.}"></noscript><span class="lazy-image-placeholder" style="width: 49.27ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/625b16cf514922cb1c67cc14b91b3651d956be81" data-alt="{\displaystyle O(k_{o}^{1-m}):\quad 2\nabla S\cdot \nabla A_{m}+A_{m}\nabla ^{2}S=-\nabla ^{2}A_{m-1}.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  The first equation is known as the <a href="/wiki/Eikonal_equation" title="Eikonal equation">eikonal equation</a>, which determines the eikonal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(\mathbf {r} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(\mathbf {r} )}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/257bcdb41aa0e6d1d0772f4c92a82a4baa4a9116" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.411ex; height:2.843ex;" alt="{\displaystyle S(\mathbf {r} )}"></noscript><span class="lazy-image-placeholder" style="width: 4.411ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/257bcdb41aa0e6d1d0772f4c92a82a4baa4a9116" data-alt="{\displaystyle S(\mathbf {r} )}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is a <a href="/wiki/Hamilton%E2%80%93Jacobi_equation" title="Hamilton–Jacobi equation">Hamilton–Jacobi equation</a>, written for example in Cartesian coordinates becomes <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {\partial S}{\partial x}}\right)^{2}+\left({\frac {\partial S}{\partial y}}\right)^{2}+\left({\frac {\partial S}{\partial z}}\right)^{2}=n^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial S}{\partial x}}\right)^{2}+\left({\frac {\partial S}{\partial y}}\right)^{2}+\left({\frac {\partial S}{\partial z}}\right)^{2}=n^{2}.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef5c8347fc6d6c8409afd70bff7363846afbb02a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.262ex; height:6.509ex;" alt="{\displaystyle \left({\frac {\partial S}{\partial x}}\right)^{2}+\left({\frac {\partial S}{\partial y}}\right)^{2}+\left({\frac {\partial S}{\partial z}}\right)^{2}=n^{2}.}"></noscript><span class="lazy-image-placeholder" style="width: 36.262ex;height: 6.509ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef5c8347fc6d6c8409afd70bff7363846afbb02a" data-alt="{\displaystyle \left({\frac {\partial S}{\partial x}}\right)^{2}+\left({\frac {\partial S}{\partial y}}\right)^{2}+\left({\frac {\partial S}{\partial z}}\right)^{2}=n^{2}.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  The remaining equations determine the functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{m}(\mathbf {r} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{m}(\mathbf {r} )}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aa5d805ebf604bddf726e5e6f6c2485d3e488f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.33ex; height:2.843ex;" alt="{\displaystyle A_{m}(\mathbf {r} )}"></noscript><span class="lazy-image-placeholder" style="width: 6.33ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aa5d805ebf604bddf726e5e6f6c2485d3e488f1" data-alt="{\displaystyle A_{m}(\mathbf {r} )}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . <div class="mw-heading mw-heading3"><h3 id="Luneburg_method">Luneburg method</h3> <a role="button" href="/w/index.php?title=Geometrical_optics&action=edit&section=6" title="Edit section: Luneburg method" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> The method of obtaining equations of geometrical optics by analysing surfaces of discontinuities of solutions to Maxwell's equations was first described by <a href="/wiki/Rudolf_Luneburg" title="Rudolf Luneburg">Rudolf Karl Luneburg</a> in 1944.<a href="#cite_note-10">[10]</a> It does not restrict the electromagnetic field to have a special form required by the Sommerfeld-Runge method which assumes the amplitude <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(k_{o},\mathbf {r} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(k_{o},\mathbf {r} )}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa566b637906e1e668837bcc111b0d8b2f550c86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.929ex; height:2.843ex;" alt="{\displaystyle A(k_{o},\mathbf {r} )}"></noscript><span class="lazy-image-placeholder" style="width: 7.929ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa566b637906e1e668837bcc111b0d8b2f550c86" data-alt="{\displaystyle A(k_{o},\mathbf {r} )}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and phase <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(\mathbf {r} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(\mathbf {r} )}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/257bcdb41aa0e6d1d0772f4c92a82a4baa4a9116" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.411ex; height:2.843ex;" alt="{\displaystyle S(\mathbf {r} )}"></noscript><span class="lazy-image-placeholder" style="width: 4.411ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/257bcdb41aa0e6d1d0772f4c92a82a4baa4a9116" data-alt="{\displaystyle S(\mathbf {r} )}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  satisfy the equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \lim _{k_{0}\to \infty }{\frac {1}{k_{0}}}\left({\frac {1}{A}}\,\nabla S\cdot \nabla A+{\frac {1}{2}}\nabla ^{2}S\right)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>A</mi> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal">∇</mi> <mi>S</mi> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mi>A</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>S</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \lim _{k_{0}\to \infty }{\frac {1}{k_{0}}}\left({\frac {1}{A}}\,\nabla S\cdot \nabla A+{\frac {1}{2}}\nabla ^{2}S\right)=0}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b439cd05b7b46408ac194e904b623fc6c0176cfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:39.008ex; height:4.843ex;" alt="{\textstyle \lim _{k_{0}\to \infty }{\frac {1}{k_{0}}}\left({\frac {1}{A}}\,\nabla S\cdot \nabla A+{\frac {1}{2}}\nabla ^{2}S\right)=0}"></noscript><span class="lazy-image-placeholder" style="width: 39.008ex;height: 4.843ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b439cd05b7b46408ac194e904b623fc6c0176cfd" data-alt="{\textstyle \lim _{k_{0}\to \infty }{\frac {1}{k_{0}}}\left({\frac {1}{A}}\,\nabla S\cdot \nabla A+{\frac {1}{2}}\nabla ^{2}S\right)=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . This condition is satisfied by e.g. plane waves but is not additive. The main conclusion of Luneburg's approach is the following: Theorem. Suppose the fields <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} (x,y,z,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} (x,y,z,t)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69562d727b73352500048436d0e82d2ee1b1a765" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.081ex; height:2.843ex;" alt="{\displaystyle \mathbf {E} (x,y,z,t)}"></noscript><span class="lazy-image-placeholder" style="width: 11.081ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69562d727b73352500048436d0e82d2ee1b1a765" data-alt="{\displaystyle \mathbf {E} (x,y,z,t)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {H} (x,y,z,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {H} (x,y,z,t)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19d41f4e271e71ec921605d8ea13b383b35f51f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.416ex; height:2.843ex;" alt="{\displaystyle \mathbf {H} (x,y,z,t)}"></noscript><span class="lazy-image-placeholder" style="width: 11.416ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19d41f4e271e71ec921605d8ea13b383b35f51f9" data-alt="{\displaystyle \mathbf {H} (x,y,z,t)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  (in a linear isotropic medium described by dielectric constants <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon (x,y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon (x,y,z)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f012f6b103fc2b3ec95a071ca76a8c899427142" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.534ex; height:2.843ex;" alt="{\displaystyle \varepsilon (x,y,z)}"></noscript><span class="lazy-image-placeholder" style="width: 8.534ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f012f6b103fc2b3ec95a071ca76a8c899427142" data-alt="{\displaystyle \varepsilon (x,y,z)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (x,y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (x,y,z)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56ea7df220cc40d21122e2563917cfd110ddad97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.852ex; height:2.843ex;" alt="{\displaystyle \mu (x,y,z)}"></noscript><span class="lazy-image-placeholder" style="width: 8.852ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56ea7df220cc40d21122e2563917cfd110ddad97" data-alt="{\displaystyle \mu (x,y,z)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ) have finite discontinuities along a (moving) surface in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{3}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42ee047387e551a89e8481e1a9e974dcc5fd5acc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.057ex; height:2.676ex;" alt="{\displaystyle \mathbf {R} ^{3}}"></noscript><span class="lazy-image-placeholder" style="width: 3.057ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42ee047387e551a89e8481e1a9e974dcc5fd5acc" data-alt="{\displaystyle \mathbf {R} ^{3}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  described by the equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x,y,z)-ct=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>c</mi> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x,y,z)-ct=0}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a58f0afeadfcf464e3c6ef5466b13297286161ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.911ex; height:2.843ex;" alt="{\displaystyle \psi (x,y,z)-ct=0}"></noscript><span class="lazy-image-placeholder" style="width: 17.911ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a58f0afeadfcf464e3c6ef5466b13297286161ef" data-alt="{\displaystyle \psi (x,y,z)-ct=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Then Maxwell's equations in the integral form imply that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></noscript><span class="lazy-image-placeholder" style="width: 1.513ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" data-alt="{\displaystyle \psi }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  satisfies the <a href="/wiki/Eikonal_equation" title="Eikonal equation">eikonal equation</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{x}^{2}+\psi _{y}^{2}+\psi _{z}^{2}=\varepsilon \mu =n^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mi>ε</mi> <mi>μ</mi> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{x}^{2}+\psi _{y}^{2}+\psi _{z}^{2}=\varepsilon \mu =n^{2},}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd589b013dcba6e820a77a6f89644d02cda6faf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.279ex; height:3.343ex;" alt="{\displaystyle \psi _{x}^{2}+\psi _{y}^{2}+\psi _{z}^{2}=\varepsilon \mu =n^{2},}"></noscript><span class="lazy-image-placeholder" style="width: 25.279ex;height: 3.343ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd589b013dcba6e820a77a6f89644d02cda6faf" data-alt="{\displaystyle \psi _{x}^{2}+\psi _{y}^{2}+\psi _{z}^{2}=\varepsilon \mu =n^{2},}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  where <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><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" data-alt="{\displaystyle n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is the index of refraction of the medium (Gaussian units). An example of such surface of discontinuity is the initial wave front emanating from a source that starts radiating at a certain instant of time. The surfaces of field discontinuity thus become geometrical optics wave fronts with the corresponding geometrical optics fields defined as: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {E} ^{*}(x,y,z)&=\mathbf {E} (x,y,z,\psi (x,y,z)/c)\\[1ex]\mathbf {H} ^{*}(x,y,z)&=\mathbf {H} (x,y,z,\psi (x,y,z)/c)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.73em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {E} ^{*}(x,y,z)&=\mathbf {E} (x,y,z,\psi (x,y,z)/c)\\[1ex]\mathbf {H} ^{*}(x,y,z)&=\mathbf {H} (x,y,z,\psi (x,y,z)/c)\end{aligned}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5874fbdbc2c4ebc3b91ea38745cae8d08c140e68" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:36.155ex; height:7.176ex;" alt="{\displaystyle {\begin{aligned}\mathbf {E} ^{*}(x,y,z)&=\mathbf {E} (x,y,z,\psi (x,y,z)/c)\\[1ex]\mathbf {H} ^{*}(x,y,z)&=\mathbf {H} (x,y,z,\psi (x,y,z)/c)\end{aligned}}}"></noscript><span class="lazy-image-placeholder" style="width: 36.155ex;height: 7.176ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5874fbdbc2c4ebc3b91ea38745cae8d08c140e68" data-alt="{\displaystyle {\begin{aligned}\mathbf {E} ^{*}(x,y,z)&=\mathbf {E} (x,y,z,\psi (x,y,z)/c)\\[1ex]\mathbf {H} ^{*}(x,y,z)&=\mathbf {H} (x,y,z,\psi (x,y,z)/c)\end{aligned}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  Those fields obey transport equations consistent with the transport equations of the Sommerfeld-Runge approach. Light rays in Luneburg's theory are defined as trajectories orthogonal to the discontinuity surfaces and can be shown to obey <a href="/wiki/Fermat%27s_principle" title="Fermat's principle">Fermat's principle</a> of least time thus establishing the identity of those rays with light rays of standard optics. The above developments can be generalised to anisotropic media.<a href="#cite_note-11">[11]</a> The proof of Luneburg's theorem is based on investigating how Maxwell's equations govern the propagation of discontinuities of solutions. The basic technical lemma is as follows: A technical lemma. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (x,y,z,t)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (x,y,z,t)=0}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/090de17e6d07b64b1ccbac5af128efef0335daa1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.105ex; height:2.843ex;" alt="{\displaystyle \varphi (x,y,z,t)=0}"></noscript><span class="lazy-image-placeholder" style="width: 15.105ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/090de17e6d07b64b1ccbac5af128efef0335daa1" data-alt="{\displaystyle \varphi (x,y,z,t)=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  be a hypersurface (a 3-dimensional manifold) in spacetime <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{4}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd186b8abaeed43c225609f9556d8ee783b6a7fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.057ex; height:2.676ex;" alt="{\displaystyle \mathbf {R} ^{4}}"></noscript><span class="lazy-image-placeholder" style="width: 3.057ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd186b8abaeed43c225609f9556d8ee783b6a7fd" data-alt="{\displaystyle \mathbf {R} ^{4}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  on which one or more of: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} (x,y,z,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} (x,y,z,t)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69562d727b73352500048436d0e82d2ee1b1a765" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.081ex; height:2.843ex;" alt="{\displaystyle \mathbf {E} (x,y,z,t)}"></noscript><span class="lazy-image-placeholder" style="width: 11.081ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69562d727b73352500048436d0e82d2ee1b1a765" data-alt="{\displaystyle \mathbf {E} (x,y,z,t)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {H} (x,y,z,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {H} (x,y,z,t)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19d41f4e271e71ec921605d8ea13b383b35f51f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.416ex; height:2.843ex;" alt="{\displaystyle \mathbf {H} (x,y,z,t)}"></noscript><span class="lazy-image-placeholder" style="width: 11.416ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19d41f4e271e71ec921605d8ea13b383b35f51f9" data-alt="{\displaystyle \mathbf {H} (x,y,z,t)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon (x,y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon (x,y,z)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f012f6b103fc2b3ec95a071ca76a8c899427142" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.534ex; height:2.843ex;" alt="{\displaystyle \varepsilon (x,y,z)}"></noscript><span class="lazy-image-placeholder" style="width: 8.534ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f012f6b103fc2b3ec95a071ca76a8c899427142" data-alt="{\displaystyle \varepsilon (x,y,z)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (x,y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (x,y,z)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56ea7df220cc40d21122e2563917cfd110ddad97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.852ex; height:2.843ex;" alt="{\displaystyle \mu (x,y,z)}"></noscript><span class="lazy-image-placeholder" style="width: 8.852ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56ea7df220cc40d21122e2563917cfd110ddad97" data-alt="{\displaystyle \mu (x,y,z)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , have a finite discontinuity. Then at each point of the hypersurface the following formulas hold: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\nabla \varphi \cdot [\varepsilon \mathbf {E} ]&=0\\[1ex]\nabla \varphi \cdot [\mu \mathbf {H} ]&=0\\[1ex]\nabla \varphi \times [\mathbf {E} ]+{\frac {1}{c}}\,\varphi _{t}\,[\mu \mathbf {H} ]&=0\\[1ex]\nabla \varphi \times [\mathbf {H} ]-{\frac {1}{c}}\,\varphi _{t}\,[\varepsilon \mathbf {E} ]&=0\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.73em 0.73em 0.73em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo>⋅</mo> <mo stretchy="false">[</mo> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo>⋅</mo> <mo stretchy="false">[</mo> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo>×</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">]</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>c</mi> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mo stretchy="false">[</mo> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo>×</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo stretchy="false">]</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>c</mi> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mo stretchy="false">[</mo> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\nabla \varphi \cdot [\varepsilon \mathbf {E} ]&=0\\[1ex]\nabla \varphi \cdot [\mu \mathbf {H} ]&=0\\[1ex]\nabla \varphi \times [\mathbf {E} ]+{\frac {1}{c}}\,\varphi _{t}\,[\mu \mathbf {H} ]&=0\\[1ex]\nabla \varphi \times [\mathbf {H} ]-{\frac {1}{c}}\,\varphi _{t}\,[\varepsilon \mathbf {E} ]&=0\end{aligned}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa8b556c2167374be68145158bc6612579712a50" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.338ex; width:27.106ex; height:19.843ex;" alt="{\displaystyle {\begin{aligned}\nabla \varphi \cdot [\varepsilon \mathbf {E} ]&=0\\[1ex]\nabla \varphi \cdot [\mu \mathbf {H} ]&=0\\[1ex]\nabla \varphi \times [\mathbf {E} ]+{\frac {1}{c}}\,\varphi _{t}\,[\mu \mathbf {H} ]&=0\\[1ex]\nabla \varphi \times [\mathbf {H} ]-{\frac {1}{c}}\,\varphi _{t}\,[\varepsilon \mathbf {E} ]&=0\end{aligned}}}"></noscript><span class="lazy-image-placeholder" style="width: 27.106ex;height: 19.843ex;vertical-align: -9.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa8b556c2167374be68145158bc6612579712a50" data-alt="{\displaystyle {\begin{aligned}\nabla \varphi \cdot [\varepsilon \mathbf {E} ]&=0\\[1ex]\nabla \varphi \cdot [\mu \mathbf {H} ]&=0\\[1ex]\nabla \varphi \times [\mathbf {E} ]+{\frac {1}{c}}\,\varphi _{t}\,[\mu \mathbf {H} ]&=0\\[1ex]\nabla \varphi \times [\mathbf {H} ]-{\frac {1}{c}}\,\varphi _{t}\,[\varepsilon \mathbf {E} ]&=0\end{aligned}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  where the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3d0e93b78c50237f9ea83d027e4ebbdaef354b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \nabla }"></noscript><span class="lazy-image-placeholder" style="width: 1.936ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3d0e93b78c50237f9ea83d027e4ebbdaef354b2" data-alt="{\displaystyle \nabla }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  operator acts in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xyz}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xyz}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d3957b3fbfe29dfdb2f7cdad4e3d7fc2ea7be84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.573ex; height:2.009ex;" alt="{\displaystyle xyz}"></noscript><span class="lazy-image-placeholder" style="width: 3.573ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d3957b3fbfe29dfdb2f7cdad4e3d7fc2ea7be84" data-alt="{\displaystyle xyz}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> -space (for every fixed <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></noscript><span class="lazy-image-placeholder" style="width: 0.84ex;height: 2.009ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" data-alt="{\displaystyle t}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ) and the square brackets denote the difference in values on both sides of the discontinuity surface (set up according to an arbitrary but fixed convention, e.g. the gradient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \varphi }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d135f308e43463a63104ad85008b3b072c3e938" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.456ex; height:2.676ex;" alt="{\displaystyle \nabla \varphi }"></noscript><span class="lazy-image-placeholder" style="width: 3.456ex;height: 2.676ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d135f308e43463a63104ad85008b3b072c3e938" data-alt="{\displaystyle \nabla \varphi }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  pointing in the direction of the quantities being subtracted from). Sketch of proof. Start with Maxwell's equations away from the sources (Gaussian units): <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\nabla \cdot \varepsilon \mathbf {E} =0\\[1ex]\nabla \cdot \mu \mathbf {H} =0\\[1ex]\nabla \times \mathbf {E} +{\tfrac {\mu }{c}}\,\mathbf {H} _{t}=0\\[1ex]\nabla \times \mathbf {H} -{\tfrac {\varepsilon }{c}}\,\mathbf {E} _{t}=0\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.73em 0.73em 0.73em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>μ</mi> <mi>c</mi> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace"></mspace> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>ε</mi> <mi>c</mi> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace"></mspace> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\nabla \cdot \varepsilon \mathbf {E} =0\\[1ex]\nabla \cdot \mu \mathbf {H} =0\\[1ex]\nabla \times \mathbf {E} +{\tfrac {\mu }{c}}\,\mathbf {H} _{t}=0\\[1ex]\nabla \times \mathbf {H} -{\tfrac {\varepsilon }{c}}\,\mathbf {E} _{t}=0\end{aligned}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03bda406cbd94f13a917d47352138314b2cf80ef" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:19.518ex; height:15.843ex;" alt="{\displaystyle {\begin{aligned}\nabla \cdot \varepsilon \mathbf {E} =0\\[1ex]\nabla \cdot \mu \mathbf {H} =0\\[1ex]\nabla \times \mathbf {E} +{\tfrac {\mu }{c}}\,\mathbf {H} _{t}=0\\[1ex]\nabla \times \mathbf {H} -{\tfrac {\varepsilon }{c}}\,\mathbf {E} _{t}=0\end{aligned}}}"></noscript><span class="lazy-image-placeholder" style="width: 19.518ex;height: 15.843ex;vertical-align: -7.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03bda406cbd94f13a917d47352138314b2cf80ef" data-alt="{\displaystyle {\begin{aligned}\nabla \cdot \varepsilon \mathbf {E} =0\\[1ex]\nabla \cdot \mu \mathbf {H} =0\\[1ex]\nabla \times \mathbf {E} +{\tfrac {\mu }{c}}\,\mathbf {H} _{t}=0\\[1ex]\nabla \times \mathbf {H} -{\tfrac {\varepsilon }{c}}\,\mathbf {E} _{t}=0\end{aligned}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  Using Stokes' theorem in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{4}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd186b8abaeed43c225609f9556d8ee783b6a7fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.057ex; height:2.676ex;" alt="{\displaystyle \mathbf {R} ^{4}}"></noscript><span class="lazy-image-placeholder" style="width: 3.057ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd186b8abaeed43c225609f9556d8ee783b6a7fd" data-alt="{\displaystyle \mathbf {R} ^{4}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  one can conclude from the first of the above equations that for any domain <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><noscript><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}"></noscript><span class="lazy-image-placeholder" style="width: 1.924ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" data-alt="{\displaystyle D}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{4}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd186b8abaeed43c225609f9556d8ee783b6a7fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.057ex; height:2.676ex;" alt="{\displaystyle \mathbf {R} ^{4}}"></noscript><span class="lazy-image-placeholder" style="width: 3.057ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd186b8abaeed43c225609f9556d8ee783b6a7fd" data-alt="{\displaystyle \mathbf {R} ^{4}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  with a piecewise smooth (3-dimensional) boundary <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"></noscript><span class="lazy-image-placeholder" style="width: 1.453ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" data-alt="{\displaystyle \Gamma }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  the following is true: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{\Gamma }(\mathbf {M} \cdot \varepsilon \mathbf {E} )\,dS=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mo>⋅</mo> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>S</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{\Gamma }(\mathbf {M} \cdot \varepsilon \mathbf {E} )\,dS=0}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/520bccddf30a9746dbcbad51ee62be82b11f767c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.782ex; height:5.676ex;" alt="{\displaystyle \oint _{\Gamma }(\mathbf {M} \cdot \varepsilon \mathbf {E} )\,dS=0}"></noscript><span class="lazy-image-placeholder" style="width: 18.782ex;height: 5.676ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/520bccddf30a9746dbcbad51ee62be82b11f767c" data-alt="{\displaystyle \oint _{\Gamma }(\mathbf {M} \cdot \varepsilon \mathbf {E} )\,dS=0}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M} =(x_{N},y_{N},z_{N})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M} =(x_{N},y_{N},z_{N})}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8630cc5b4d1a7c94d933c4198d9d0386b6769e9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.138ex; height:2.843ex;" alt="{\displaystyle \mathbf {M} =(x_{N},y_{N},z_{N})}"></noscript><span class="lazy-image-placeholder" style="width: 18.138ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8630cc5b4d1a7c94d933c4198d9d0386b6769e9e" data-alt="{\displaystyle \mathbf {M} =(x_{N},y_{N},z_{N})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is the projection of the outward unit normal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{N},y_{N},z_{N},t_{N})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{N},y_{N},z_{N},t_{N})}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/712a60a503dea57be72e3ce2f54fc4fadec6533b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.067ex; height:2.843ex;" alt="{\displaystyle (x_{N},y_{N},z_{N},t_{N})}"></noscript><span class="lazy-image-placeholder" style="width: 16.067ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/712a60a503dea57be72e3ce2f54fc4fadec6533b" data-alt="{\displaystyle (x_{N},y_{N},z_{N},t_{N})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"></noscript><span class="lazy-image-placeholder" style="width: 1.453ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" data-alt="{\displaystyle \Gamma }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  onto the 3D slice <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t={\rm {const}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t={\rm {const}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3d172decd4c60b544d5974710f3039d7d2a57d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.246ex; height:2.009ex;" alt="{\displaystyle t={\rm {const}}}"></noscript><span class="lazy-image-placeholder" style="width: 9.246ex;height: 2.009ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3d172decd4c60b544d5974710f3039d7d2a57d7" data-alt="{\displaystyle t={\rm {const}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dS}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dS}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66f6ab8a2a8b85df78efbea896fcfeb3d4ea39e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.715ex; height:2.176ex;" alt="{\displaystyle dS}"></noscript><span class="lazy-image-placeholder" style="width: 2.715ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66f6ab8a2a8b85df78efbea896fcfeb3d4ea39e4" data-alt="{\displaystyle dS}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is the volume 3-form on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"></noscript><span class="lazy-image-placeholder" style="width: 1.453ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" data-alt="{\displaystyle \Gamma }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Similarly, one establishes the following from the remaining Maxwell's equations: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\oint _{\Gamma }\left(\mathbf {M} \cdot \mu \mathbf {H} \right)dS&=0\\[1.55ex]\oint _{\Gamma }\left(\mathbf {M} \times \mathbf {E} +{\frac {\mu }{c}}\,t_{N}\,\mathbf {H} \right)dS&=0\\[1.55ex]\oint _{\Gamma }\left(\mathbf {M} \times \mathbf {H} -{\frac {\varepsilon }{c}}\,t_{N}\,\mathbf {E} \right)dS&=0\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.966em 0.966em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mo>⋅</mo> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>S</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>μ</mi> <mi>c</mi> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>S</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ε</mi> <mi>c</mi> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>S</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\oint _{\Gamma }\left(\mathbf {M} \cdot \mu \mathbf {H} \right)dS&=0\\[1.55ex]\oint _{\Gamma }\left(\mathbf {M} \times \mathbf {E} +{\frac {\mu }{c}}\,t_{N}\,\mathbf {H} \right)dS&=0\\[1.55ex]\oint _{\Gamma }\left(\mathbf {M} \times \mathbf {H} -{\frac {\varepsilon }{c}}\,t_{N}\,\mathbf {E} \right)dS&=0\end{aligned}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be8704c5e24fd980e49c510d1418ff6c1f39af3c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.838ex; width:31.439ex; height:20.843ex;" alt="{\displaystyle {\begin{aligned}\oint _{\Gamma }\left(\mathbf {M} \cdot \mu \mathbf {H} \right)dS&=0\\[1.55ex]\oint _{\Gamma }\left(\mathbf {M} \times \mathbf {E} +{\frac {\mu }{c}}\,t_{N}\,\mathbf {H} \right)dS&=0\\[1.55ex]\oint _{\Gamma }\left(\mathbf {M} \times \mathbf {H} -{\frac {\varepsilon }{c}}\,t_{N}\,\mathbf {E} \right)dS&=0\end{aligned}}}"></noscript><span class="lazy-image-placeholder" style="width: 31.439ex;height: 20.843ex;vertical-align: -9.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be8704c5e24fd980e49c510d1418ff6c1f39af3c" data-alt="{\displaystyle {\begin{aligned}\oint _{\Gamma }\left(\mathbf {M} \cdot \mu \mathbf {H} \right)dS&=0\\[1.55ex]\oint _{\Gamma }\left(\mathbf {M} \times \mathbf {E} +{\frac {\mu }{c}}\,t_{N}\,\mathbf {H} \right)dS&=0\\[1.55ex]\oint _{\Gamma }\left(\mathbf {M} \times \mathbf {H} -{\frac {\varepsilon }{c}}\,t_{N}\,\mathbf {E} \right)dS&=0\end{aligned}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  Now by considering arbitrary small sub-surfaces <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{0}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44f27c3fa0660b68ef8fa747442140655cff65cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.507ex; height:2.509ex;" alt="{\displaystyle \Gamma _{0}}"></noscript><span class="lazy-image-placeholder" style="width: 2.507ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44f27c3fa0660b68ef8fa747442140655cff65cd" data-alt="{\displaystyle \Gamma _{0}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"></noscript><span class="lazy-image-placeholder" style="width: 1.453ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" data-alt="{\displaystyle \Gamma }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and setting up small neighbourhoods surrounding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{0}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44f27c3fa0660b68ef8fa747442140655cff65cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.507ex; height:2.509ex;" alt="{\displaystyle \Gamma _{0}}"></noscript><span class="lazy-image-placeholder" style="width: 2.507ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44f27c3fa0660b68ef8fa747442140655cff65cd" data-alt="{\displaystyle \Gamma _{0}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{4}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd186b8abaeed43c225609f9556d8ee783b6a7fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.057ex; height:2.676ex;" alt="{\displaystyle \mathbf {R} ^{4}}"></noscript><span class="lazy-image-placeholder" style="width: 3.057ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd186b8abaeed43c225609f9556d8ee783b6a7fd" data-alt="{\displaystyle \mathbf {R} ^{4}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , and subtracting the above integrals accordingly, one obtains: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\int _{\Gamma _{0}}(\nabla \varphi \cdot [\varepsilon \mathbf {E} ])\,{dS \over \|\nabla ^{4D}\varphi \|}&=0\\[1ex]\int _{\Gamma _{0}}(\nabla \varphi \cdot [\mu \mathbf {H} ])\,{dS \over \|\nabla ^{4D}\varphi \|}&=0\\[1ex]\int _{\Gamma _{0}}\left(\nabla \varphi \times [\mathbf {E} ]+{1 \over c}\,\varphi _{t}\,[\mu \mathbf {H} ]\right)\,{\frac {dS}{\|\nabla ^{4D}\varphi \|}}&=0\\[1ex]\int _{\Gamma _{0}}\left(\nabla \varphi \times [\mathbf {H} ]-{1 \over c}\,\varphi _{t}\,[\varepsilon \mathbf {E} ]\right)\,{\frac {dS}{\|\nabla ^{4D}\varphi \|}}&=0\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.73em 0.73em 0.73em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo>⋅</mo> <mo stretchy="false">[</mo> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>S</mi> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mi>D</mi> </mrow> </msup> <mi>φ</mi> <mo fence="false" stretchy="false">‖</mo> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo>⋅</mo> <mo stretchy="false">[</mo> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>S</mi> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mi>D</mi> </mrow> </msup> <mi>φ</mi> <mo fence="false" stretchy="false">‖</mo> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo>×</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">]</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>c</mi> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mo stretchy="false">[</mo> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo stretchy="false">]</mo> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>S</mi> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mi>D</mi> </mrow> </msup> <mi>φ</mi> <mo fence="false" stretchy="false">‖</mo> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo>×</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo stretchy="false">]</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>c</mi> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mo stretchy="false">[</mo> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">]</mo> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>S</mi> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mi>D</mi> </mrow> </msup> <mi>φ</mi> <mo fence="false" stretchy="false">‖</mo> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int _{\Gamma _{0}}(\nabla \varphi \cdot [\varepsilon \mathbf {E} ])\,{dS \over \|\nabla ^{4D}\varphi \|}&=0\\[1ex]\int _{\Gamma _{0}}(\nabla \varphi \cdot [\mu \mathbf {H} ])\,{dS \over \|\nabla ^{4D}\varphi \|}&=0\\[1ex]\int _{\Gamma _{0}}\left(\nabla \varphi \times [\mathbf {E} ]+{1 \over c}\,\varphi _{t}\,[\mu \mathbf {H} ]\right)\,{\frac {dS}{\|\nabla ^{4D}\varphi \|}}&=0\\[1ex]\int _{\Gamma _{0}}\left(\nabla \varphi \times [\mathbf {H} ]-{1 \over c}\,\varphi _{t}\,[\varepsilon \mathbf {E} ]\right)\,{\frac {dS}{\|\nabla ^{4D}\varphi \|}}&=0\end{aligned}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36b6c0263cbfbc789c99ad82840379983c668293" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.671ex; width:44.104ex; height:28.509ex;" alt="{\displaystyle {\begin{aligned}\int _{\Gamma _{0}}(\nabla \varphi \cdot [\varepsilon \mathbf {E} ])\,{dS \over \|\nabla ^{4D}\varphi \|}&=0\\[1ex]\int _{\Gamma _{0}}(\nabla \varphi \cdot [\mu \mathbf {H} ])\,{dS \over \|\nabla ^{4D}\varphi \|}&=0\\[1ex]\int _{\Gamma _{0}}\left(\nabla \varphi \times [\mathbf {E} ]+{1 \over c}\,\varphi _{t}\,[\mu \mathbf {H} ]\right)\,{\frac {dS}{\|\nabla ^{4D}\varphi \|}}&=0\\[1ex]\int _{\Gamma _{0}}\left(\nabla \varphi \times [\mathbf {H} ]-{1 \over c}\,\varphi _{t}\,[\varepsilon \mathbf {E} ]\right)\,{\frac {dS}{\|\nabla ^{4D}\varphi \|}}&=0\end{aligned}}}"></noscript><span class="lazy-image-placeholder" style="width: 44.104ex;height: 28.509ex;vertical-align: -13.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36b6c0263cbfbc789c99ad82840379983c668293" data-alt="{\displaystyle {\begin{aligned}\int _{\Gamma _{0}}(\nabla \varphi \cdot [\varepsilon \mathbf {E} ])\,{dS \over \|\nabla ^{4D}\varphi \|}&=0\\[1ex]\int _{\Gamma _{0}}(\nabla \varphi \cdot [\mu \mathbf {H} ])\,{dS \over \|\nabla ^{4D}\varphi \|}&=0\\[1ex]\int _{\Gamma _{0}}\left(\nabla \varphi \times [\mathbf {E} ]+{1 \over c}\,\varphi _{t}\,[\mu \mathbf {H} ]\right)\,{\frac {dS}{\|\nabla ^{4D}\varphi \|}}&=0\\[1ex]\int _{\Gamma _{0}}\left(\nabla \varphi \times [\mathbf {H} ]-{1 \over c}\,\varphi _{t}\,[\varepsilon \mathbf {E} ]\right)\,{\frac {dS}{\|\nabla ^{4D}\varphi \|}}&=0\end{aligned}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla ^{4D}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mi>D</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla ^{4D}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b40650f6e9aee5914af84d1e94b3e483f91e88c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.351ex; height:2.676ex;" alt="{\displaystyle \nabla ^{4D}}"></noscript><span class="lazy-image-placeholder" style="width: 4.351ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b40650f6e9aee5914af84d1e94b3e483f91e88c" data-alt="{\displaystyle \nabla ^{4D}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  denotes the gradient in the 4D <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xyzt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <mi>z</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xyzt}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18208e4087d68de97248f1b32cd4c5f963d412f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.413ex; height:2.343ex;" alt="{\displaystyle xyzt}"></noscript><span class="lazy-image-placeholder" style="width: 4.413ex;height: 2.343ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18208e4087d68de97248f1b32cd4c5f963d412f4" data-alt="{\displaystyle xyzt}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> -space. And since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{0}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44f27c3fa0660b68ef8fa747442140655cff65cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.507ex; height:2.509ex;" alt="{\displaystyle \Gamma _{0}}"></noscript><span class="lazy-image-placeholder" style="width: 2.507ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44f27c3fa0660b68ef8fa747442140655cff65cd" data-alt="{\displaystyle \Gamma _{0}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is arbitrary, the integrands must be equal to 0 which proves the lemma. It's now easy to show that as they propagate through a continuous medium, the discontinuity surfaces obey the eikonal equation. Specifically, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"></noscript><span class="lazy-image-placeholder" style="width: 1.083ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" data-alt="{\displaystyle \varepsilon }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></noscript><span class="lazy-image-placeholder" style="width: 1.402ex;height: 2.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" data-alt="{\displaystyle \mu }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  are continuous, then the discontinuities of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d7f22b39d51f780fc02859059c1757c606b9de2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.757ex; height:2.176ex;" alt="{\displaystyle \mathbf {E} }"></noscript><span class="lazy-image-placeholder" style="width: 1.757ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d7f22b39d51f780fc02859059c1757c606b9de2" data-alt="{\displaystyle \mathbf {E} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {H} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f017b876ed763037d8818ec5dfbbdc6703e0f683" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.091ex; height:2.176ex;" alt="{\displaystyle \mathbf {H} }"></noscript><span class="lazy-image-placeholder" style="width: 2.091ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f017b876ed763037d8818ec5dfbbdc6703e0f683" data-alt="{\displaystyle \mathbf {H} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  satisfy: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\varepsilon \mathbf {E} ]=\varepsilon [\mathbf {E} ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">]</mo> <mo>=</mo> <mi>ε</mi> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [\varepsilon \mathbf {E} ]=\varepsilon [\mathbf {E} ]}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e2919510029905936e2b8d8149e1f0a7c4cf97a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.367ex; height:2.843ex;" alt="{\displaystyle [\varepsilon \mathbf {E} ]=\varepsilon [\mathbf {E} ]}"></noscript><span class="lazy-image-placeholder" style="width: 11.367ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e2919510029905936e2b8d8149e1f0a7c4cf97a" data-alt="{\displaystyle [\varepsilon \mathbf {E} ]=\varepsilon [\mathbf {E} ]}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\mu \mathbf {H} ]=\mu [\mathbf {H} ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo stretchy="false">]</mo> <mo>=</mo> <mi>μ</mi> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [\mu \mathbf {H} ]=\mu [\mathbf {H} ]}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70b7d98f0ed5b58d0201a273be52bbcd79b288b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.672ex; height:2.843ex;" alt="{\displaystyle [\mu \mathbf {H} ]=\mu [\mathbf {H} ]}"></noscript><span class="lazy-image-placeholder" style="width: 12.672ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70b7d98f0ed5b58d0201a273be52bbcd79b288b6" data-alt="{\displaystyle [\mu \mathbf {H} ]=\mu [\mathbf {H} ]}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . In this case the last two equations of the lemma can be written as: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\nabla \varphi \times [\mathbf {E} ]+{\mu \over c}\,\varphi _{t}\,[\mathbf {H} ]&=0\\[1ex]\nabla \varphi \times [\mathbf {H} ]-{\varepsilon \over c}\,\varphi _{t}\,[\mathbf {E} ]&=0\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.73em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo>×</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">]</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>μ</mi> <mi>c</mi> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo>×</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo stretchy="false">]</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ε</mi> <mi>c</mi> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\nabla \varphi \times [\mathbf {E} ]+{\mu \over c}\,\varphi _{t}\,[\mathbf {H} ]&=0\\[1ex]\nabla \varphi \times [\mathbf {H} ]-{\varepsilon \over c}\,\varphi _{t}\,[\mathbf {E} ]&=0\end{aligned}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d70f582d3447fcf917d75931ce5a535dec17cc9d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:25.943ex; height:10.843ex;" alt="{\displaystyle {\begin{aligned}\nabla \varphi \times [\mathbf {E} ]+{\mu \over c}\,\varphi _{t}\,[\mathbf {H} ]&=0\\[1ex]\nabla \varphi \times [\mathbf {H} ]-{\varepsilon \over c}\,\varphi _{t}\,[\mathbf {E} ]&=0\end{aligned}}}"></noscript><span class="lazy-image-placeholder" style="width: 25.943ex;height: 10.843ex;vertical-align: -4.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d70f582d3447fcf917d75931ce5a535dec17cc9d" data-alt="{\displaystyle {\begin{aligned}\nabla \varphi \times [\mathbf {E} ]+{\mu \over c}\,\varphi _{t}\,[\mathbf {H} ]&=0\\[1ex]\nabla \varphi \times [\mathbf {H} ]-{\varepsilon \over c}\,\varphi _{t}\,[\mathbf {E} ]&=0\end{aligned}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  Taking the cross product of the second equation with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \varphi }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d135f308e43463a63104ad85008b3b072c3e938" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.456ex; height:2.676ex;" alt="{\displaystyle \nabla \varphi }"></noscript><span class="lazy-image-placeholder" style="width: 3.456ex;height: 2.676ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d135f308e43463a63104ad85008b3b072c3e938" data-alt="{\displaystyle \nabla \varphi }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and substituting the first yields: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \varphi \times (\nabla \varphi \times [\mathbf {H} ])-{\varepsilon \over c}\,\varphi _{t}\,(\nabla \varphi \times [\mathbf {E} ])=(\nabla \varphi \cdot [\mathbf {H} ])\,\nabla \varphi -\|\nabla \varphi \|^{2}\,[\mathbf {H} ]+{\varepsilon \mu \over c^{2}}\varphi _{t}^{2}\,[\mathbf {H} ]=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo>×</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ε</mi> <mi>c</mi> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo>×</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo>⋅</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo>−</mo> <mo fence="false" stretchy="false">‖</mo> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo stretchy="false">]</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ε</mi> <mi>μ</mi> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mspace width="thinmathspace"></mspace> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo stretchy="false">]</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \varphi \times (\nabla \varphi \times [\mathbf {H} ])-{\varepsilon \over c}\,\varphi _{t}\,(\nabla \varphi \times [\mathbf {E} ])=(\nabla \varphi \cdot [\mathbf {H} ])\,\nabla \varphi -\|\nabla \varphi \|^{2}\,[\mathbf {H} ]+{\varepsilon \mu \over c^{2}}\varphi _{t}^{2}\,[\mathbf {H} ]=0}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5b5c97784e35d3caadde1e799d60b9e70aaeb90" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:84.312ex; height:5.176ex;" alt="{\displaystyle \nabla \varphi \times (\nabla \varphi \times [\mathbf {H} ])-{\varepsilon \over c}\,\varphi _{t}\,(\nabla \varphi \times [\mathbf {E} ])=(\nabla \varphi \cdot [\mathbf {H} ])\,\nabla \varphi -\|\nabla \varphi \|^{2}\,[\mathbf {H} ]+{\varepsilon \mu \over c^{2}}\varphi _{t}^{2}\,[\mathbf {H} ]=0}"></noscript><span class="lazy-image-placeholder" style="width: 84.312ex;height: 5.176ex;vertical-align: -2.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5b5c97784e35d3caadde1e799d60b9e70aaeb90" data-alt="{\displaystyle \nabla \varphi \times (\nabla \varphi \times [\mathbf {H} ])-{\varepsilon \over c}\,\varphi _{t}\,(\nabla \varphi \times [\mathbf {E} ])=(\nabla \varphi \cdot [\mathbf {H} ])\,\nabla \varphi -\|\nabla \varphi \|^{2}\,[\mathbf {H} ]+{\varepsilon \mu \over c^{2}}\varphi _{t}^{2}\,[\mathbf {H} ]=0}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  The continuity of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></noscript><span class="lazy-image-placeholder" style="width: 1.402ex;height: 2.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" data-alt="{\displaystyle \mu }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and the second equation of the lemma imply: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \varphi \cdot [\mathbf {H} ]=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo>⋅</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo stretchy="false">]</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \varphi \cdot [\mathbf {H} ]=0}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b304b5ef3a9a1871057c81aca26b4a39dfd4f0f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.781ex; height:2.843ex;" alt="{\displaystyle \nabla \varphi \cdot [\mathbf {H} ]=0}"></noscript><span class="lazy-image-placeholder" style="width: 12.781ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b304b5ef3a9a1871057c81aca26b4a39dfd4f0f5" data-alt="{\displaystyle \nabla \varphi \cdot [\mathbf {H} ]=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , hence, for points lying on the surface <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =0}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/192287b02f5764a18fe39f37b8199d72000aa220" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.781ex; height:2.676ex;" alt="{\displaystyle \varphi =0}"></noscript><span class="lazy-image-placeholder" style="width: 5.781ex;height: 2.676ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/192287b02f5764a18fe39f37b8199d72000aa220" data-alt="{\displaystyle \varphi =0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  only: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\nabla \varphi \|^{2}={\varepsilon \mu \over c^{2}}\varphi _{t}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ε</mi> <mi>μ</mi> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\nabla \varphi \|^{2}={\varepsilon \mu \over c^{2}}\varphi _{t}^{2}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e05ee647c9c658987e241b787c1123942bc10b7f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.829ex; height:5.176ex;" alt="{\displaystyle \|\nabla \varphi \|^{2}={\varepsilon \mu \over c^{2}}\varphi _{t}^{2}}"></noscript><span class="lazy-image-placeholder" style="width: 15.829ex;height: 5.176ex;vertical-align: -2.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e05ee647c9c658987e241b787c1123942bc10b7f" data-alt="{\displaystyle \|\nabla \varphi \|^{2}={\varepsilon \mu \over c^{2}}\varphi _{t}^{2}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  (Notice the presence of the discontinuity is essential in this step as we'd be dividing by zero otherwise.) Because of the physical considerations one can assume without loss of generality that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></noscript><span class="lazy-image-placeholder" style="width: 1.52ex;height: 2.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" data-alt="{\displaystyle \varphi }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is of the following form: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (x,y,z,t)=\psi (x,y,z)-ct}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>c</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (x,y,z,t)=\psi (x,y,z)-ct}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/104d591df6962ae238ee14af4c038a2f97316e5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.593ex; height:2.843ex;" alt="{\displaystyle \varphi (x,y,z,t)=\psi (x,y,z)-ct}"></noscript><span class="lazy-image-placeholder" style="width: 27.593ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/104d591df6962ae238ee14af4c038a2f97316e5b" data-alt="{\displaystyle \varphi (x,y,z,t)=\psi (x,y,z)-ct}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , i.e. a 2D surface moving through space, modelled as level surfaces of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></noscript><span class="lazy-image-placeholder" style="width: 1.513ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" data-alt="{\displaystyle \psi }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . (Mathematically <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></noscript><span class="lazy-image-placeholder" style="width: 1.513ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" data-alt="{\displaystyle \psi }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  exists if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{t}\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{t}\neq 0}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf3a90f8dea29439d414ecc6df498ee26035f0e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.607ex; height:2.676ex;" alt="{\displaystyle \varphi _{t}\neq 0}"></noscript><span class="lazy-image-placeholder" style="width: 6.607ex;height: 2.676ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf3a90f8dea29439d414ecc6df498ee26035f0e3" data-alt="{\displaystyle \varphi _{t}\neq 0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  by the <a href="/wiki/Implicit_function_theorem" title="Implicit function theorem">implicit function theorem</a>.) The above equation written in terms of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></noscript><span class="lazy-image-placeholder" style="width: 1.513ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" data-alt="{\displaystyle \psi }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  becomes: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\nabla \psi \|^{2}={\varepsilon \mu \over c^{2}}\,(-c)^{2}=\varepsilon \mu =n^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi mathvariant="normal">∇</mi> <mi>ψ</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ε</mi> <mi>μ</mi> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mo stretchy="false">(</mo> <mo>−</mo> <mi>c</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>ε</mi> <mi>μ</mi> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\nabla \psi \|^{2}={\varepsilon \mu \over c^{2}}\,(-c)^{2}=\varepsilon \mu =n^{2}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13c986fe266371dd3bd6a0560b52f43963836675" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:30.445ex; height:5.176ex;" alt="{\displaystyle \|\nabla \psi \|^{2}={\varepsilon \mu \over c^{2}}\,(-c)^{2}=\varepsilon \mu =n^{2}}"></noscript><span class="lazy-image-placeholder" style="width: 30.445ex;height: 5.176ex;vertical-align: -2.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13c986fe266371dd3bd6a0560b52f43963836675" data-alt="{\displaystyle \|\nabla \psi \|^{2}={\varepsilon \mu \over c^{2}}\,(-c)^{2}=\varepsilon \mu =n^{2}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  i.e., <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{x}^{2}+\psi _{y}^{2}+\psi _{z}^{2}=n^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{x}^{2}+\psi _{y}^{2}+\psi _{z}^{2}=n^{2}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/481d1a2c0efc7a5b1e39b328ac544236b8bb3a89" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.049ex; height:3.343ex;" alt="{\displaystyle \psi _{x}^{2}+\psi _{y}^{2}+\psi _{z}^{2}=n^{2}}"></noscript><span class="lazy-image-placeholder" style="width: 19.049ex;height: 3.343ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/481d1a2c0efc7a5b1e39b328ac544236b8bb3a89" data-alt="{\displaystyle \psi _{x}^{2}+\psi _{y}^{2}+\psi _{z}^{2}=n^{2}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  which is the eikonal equation and it holds for all <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><noscript><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}"></noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" data-alt="{\displaystyle x}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></noscript><span class="lazy-image-placeholder" style="width: 1.155ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" data-alt="{\displaystyle y}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></noscript><span class="lazy-image-placeholder" style="width: 1.088ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" data-alt="{\displaystyle z}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , since the variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></noscript><span class="lazy-image-placeholder" style="width: 0.84ex;height: 2.009ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" data-alt="{\displaystyle t}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is absent. Other laws of optics like <a href="/wiki/Snell%27s_law" title="Snell's law">Snell's law</a> and <a href="/wiki/Fresnel_equations#Complex_amplitude_reflection_and_transmission_coefficients" title="Fresnel equations">Fresnel formulae</a> can be similarly obtained by considering discontinuities in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"></noscript><span class="lazy-image-placeholder" style="width: 1.083ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" data-alt="{\displaystyle \varepsilon }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></noscript><span class="lazy-image-placeholder" style="width: 1.402ex;height: 2.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" data-alt="{\displaystyle \mu }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . <div class="mw-heading mw-heading3"><h3 id="General_equation_using_four-vector_notation">General equation using four-vector notation</h3> <a role="button" href="/w/index.php?title=Geometrical_optics&action=edit&section=7" title="Edit section: General equation using four-vector notation" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> In <a href="/wiki/Four-vector" title="Four-vector">four-vector</a> notation used in <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>, the wave equation can be written as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial ^{2}\psi }{\partial x_{i}\partial x^{i}}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ψ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}\psi }{\partial x_{i}\partial x^{i}}}=0}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ed6ac047e49a2b469aa9ba9b5e0c73acd8585e2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.992ex; height:6.343ex;" alt="{\displaystyle {\frac {\partial ^{2}\psi }{\partial x_{i}\partial x^{i}}}=0}"></noscript><span class="lazy-image-placeholder" style="width: 11.992ex;height: 6.343ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ed6ac047e49a2b469aa9ba9b5e0c73acd8585e2" data-alt="{\displaystyle {\frac {\partial ^{2}\psi }{\partial x_{i}\partial x^{i}}}=0}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  and the substitution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi =Ae^{iS/\varepsilon }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>=</mo> <mi>A</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ε</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi =Ae^{iS/\varepsilon }}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53fbc2afc93384253f4ece9aee62b0ae1e3f4f6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.886ex; height:3.176ex;" alt="{\displaystyle \psi =Ae^{iS/\varepsilon }}"></noscript><span class="lazy-image-placeholder" style="width: 10.886ex;height: 3.176ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53fbc2afc93384253f4ece9aee62b0ae1e3f4f6e" data-alt="{\displaystyle \psi =Ae^{iS/\varepsilon }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  leads to<a href="#cite_note-12">[12]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {A}{\varepsilon ^{2}}}{\frac {\partial S}{\partial x_{i}}}{\frac {\partial S}{\partial x^{i}}}+{\frac {2i}{\varepsilon }}{\frac {\partial A}{\partial x_{i}}}{\frac {\partial S}{\partial x^{i}}}+{\frac {iA}{\varepsilon }}{\frac {\partial ^{2}S}{\partial x_{i}\partial x^{i}}}+{\frac {\partial ^{2}A}{\partial x_{i}\partial x^{i}}}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>i</mi> </mrow> <mi>ε</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>A</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mi>A</mi> </mrow> <mi>ε</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>A</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {A}{\varepsilon ^{2}}}{\frac {\partial S}{\partial x_{i}}}{\frac {\partial S}{\partial x^{i}}}+{\frac {2i}{\varepsilon }}{\frac {\partial A}{\partial x_{i}}}{\frac {\partial S}{\partial x^{i}}}+{\frac {iA}{\varepsilon }}{\frac {\partial ^{2}S}{\partial x_{i}\partial x^{i}}}+{\frac {\partial ^{2}A}{\partial x_{i}\partial x^{i}}}=0.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/234226b235515e1768f833923eeeaa6638498d60" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:56.99ex; height:6.343ex;" alt="{\displaystyle -{\frac {A}{\varepsilon ^{2}}}{\frac {\partial S}{\partial x_{i}}}{\frac {\partial S}{\partial x^{i}}}+{\frac {2i}{\varepsilon }}{\frac {\partial A}{\partial x_{i}}}{\frac {\partial S}{\partial x^{i}}}+{\frac {iA}{\varepsilon }}{\frac {\partial ^{2}S}{\partial x_{i}\partial x^{i}}}+{\frac {\partial ^{2}A}{\partial x_{i}\partial x^{i}}}=0.}"></noscript><span class="lazy-image-placeholder" style="width: 56.99ex;height: 6.343ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/234226b235515e1768f833923eeeaa6638498d60" data-alt="{\displaystyle -{\frac {A}{\varepsilon ^{2}}}{\frac {\partial S}{\partial x_{i}}}{\frac {\partial S}{\partial x^{i}}}+{\frac {2i}{\varepsilon }}{\frac {\partial A}{\partial x_{i}}}{\frac {\partial S}{\partial x^{i}}}+{\frac {iA}{\varepsilon }}{\frac {\partial ^{2}S}{\partial x_{i}\partial x^{i}}}+{\frac {\partial ^{2}A}{\partial x_{i}\partial x^{i}}}=0.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  Therefore, the eikonal equation is given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial S}{\partial x_{i}}}{\frac {\partial S}{\partial x^{i}}}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial S}{\partial x_{i}}}{\frac {\partial S}{\partial x^{i}}}=0.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45371ba47166297386c111caf340ce8be4e19fae" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.475ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial S}{\partial x_{i}}}{\frac {\partial S}{\partial x^{i}}}=0.}"></noscript><span class="lazy-image-placeholder" style="width: 13.475ex;height: 5.843ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45371ba47166297386c111caf340ce8be4e19fae" data-alt="{\displaystyle {\frac {\partial S}{\partial x_{i}}}{\frac {\partial S}{\partial x^{i}}}=0.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  Once eikonal is found by solving the above equation, the wave four-vector can be found from <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{i}=-{\frac {\partial S}{\partial x^{i}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{i}=-{\frac {\partial S}{\partial x^{i}}}.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b8ecd07ef682af3edac405a22faef06af1fd5b7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.848ex; height:5.676ex;" alt="{\displaystyle k_{i}=-{\frac {\partial S}{\partial x^{i}}}.}"></noscript><span class="lazy-image-placeholder" style="width: 11.848ex;height: 5.676ex;vertical-align: -2.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b8ecd07ef682af3edac405a22faef06af1fd5b7" data-alt="{\displaystyle k_{i}=-{\frac {\partial S}{\partial x^{i}}}.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"><h2 id="See_also">See also</h2> <a role="button" href="/w/index.php?title=Geometrical_optics&action=edit&section=8" title="Edit section: See also" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-5 collapsible-block" id="mf-section-5"> <ul><li><a href="/wiki/Hamiltonian_optics" title="Hamiltonian optics">Hamiltonian optics</a></li> <li><a href="/wiki/Geometrical_acoustics" title="Geometrical acoustics">Geometrical acoustics</a></li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"><h2 id="References">References</h2> <a role="button" href="/w/index.php?title=Geometrical_optics&action=edit&section=9" title="Edit section: References" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-6 collapsible-block" id="mf-section-6"> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <a href="/wiki/Arthur_Schuster" title="Arthur Schuster">Arthur Schuster</a>, An Introduction to the Theory of Optics, London: Edward Arnold, 1904 <a rel="nofollow" class="external text" href="https://archive.org/details/anintroductiont02schugoog/page/n62">online</a>. </li> <li id="cite_note-Greivenkamp-2"><a href="#cite_ref-Greivenkamp_2-0">^</a> <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="CITEREFGreivenkamp2004" class="citation book cs1">Greivenkamp, John E. (2004). Field Guide to Geometrical Optics. SPIE Field Guides. Vol. 1. <a href="/wiki/SPIE" title="SPIE">SPIE</a>. pp. 19–20. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8194-5294-7" title="Special:BookSources/0-8194-5294-7"><bdi>0-8194-5294-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Field+Guide+to+Geometrical+Optics&rft.series=SPIE+Field+Guides&rft.pages=19-20&rft.pub=SPIE&rft.date=2004&rft.isbn=0-8194-5294-7&rft.aulast=Greivenkamp&rft.aufirst=John+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometrical+optics" class="Z3988"> </li> <li id="cite_note-Geoptics-3">^ <a href="#cite_ref-Geoptics_3-0">a</a> <a href="#cite_ref-Geoptics_3-1">b</a> <a href="#cite_ref-Geoptics_3-2">c</a> <a href="#cite_ref-Geoptics_3-3">d</a> <a href="#cite_ref-Geoptics_3-4">e</a> <a href="#cite_ref-Geoptics_3-5">f</a> <a href="#cite_ref-Geoptics_3-6">g</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHugh_D._Young1992" class="citation book cs1">Hugh D. Young (1992). <a rel="nofollow" class="external text" href="https://archive.org/details/universityphysic8edyoun">University Physics 8e</a>. Addison-Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-52981-5" title="Special:BookSources/0-201-52981-5"><bdi>0-201-52981-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=University+Physics+8e&rft.pub=Addison-Wesley&rft.date=1992&rft.isbn=0-201-52981-5&rft.au=Hugh+D.+Young&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Funiversityphysic8edyoun&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometrical+optics" class="Z3988"> Chapter 35. </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> E. W. Marchand, Gradient Index Optics, New York, NY, Academic Press, 1978. </li> <li id="cite_note-hecht-5">^ <a href="#cite_ref-hecht_5-0">a</a> <a href="#cite_ref-hecht_5-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHecht1987" class="citation book cs1">Hecht, Eugene (1987). Optics (2nd ed.). Addison Wesley. <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%3AGeometrical+optics" class="Z3988"> Chapters 5 & 6. </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> Sommerfeld, A., & Runge, J. (1911). Anwendung der Vektorrechnung auf die Grundlagen der geometrischen Optik. Annalen der Physik, 340(7), 277-298. </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> Born, M., & Wolf, E. (2013). <a href="/wiki/Principles_of_Optics" title="Principles of Optics">Principles of optics: electromagnetic theory of propagation, interference and diffraction of light</a>. Elsevier. </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSommerfieldJ." class="citation web cs1">Sommerfield, A.; J., Runge. <a rel="nofollow" class="external text" href="https://www.neo-classical-physics.info/uploads/3/0/6/5/3065888/sommerfeld_-_geometrical_optics.pdf">"The application of vector calculus to the foundations of geometrical optics"</a> (PDF). Neo-classical physics. Translated by D. H. Delphenich. Retrieved 3 November 2023.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Neo-classical+physics&rft.atitle=The+application+of+vector+calculus+to+the+foundations+of+geometrical+optics&rft.aulast=Sommerfield&rft.aufirst=A.&rft.au=J.%2C+Runge&rft_id=https%3A%2F%2Fwww.neo-classical-physics.info%2Fuploads%2F3%2F0%2F6%2F5%2F3065888%2Fsommerfeld_-_geometrical_optics.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeometrical+optics" class="Z3988"> </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> Borowitz, S. (1967). Fundamentals of quantum mechanics, particles, waves, and wave mechanics. </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> Luneburg, R. K., Mathematical Theory of Optics, Brown University Press 1944 [mimeographed notes], University of California Press 1964 </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> Kline, M., Kay, I. W., Electromagnetic Theory and Geometrical Optics, Interscience Publishers 1965 </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> Landau, L. D., & Lifshitz, E. M. (1975). The classical theory of fields. </li> </ol></div></div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(7)"><h2 id="Further_reading">Further reading</h2> <a role="button" href="/w/index.php?title=Geometrical_optics&action=edit&section=10" title="Edit section: Further reading" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-7 collapsible-block" id="mf-section-7"> <ul><li><a href="/wiki/Robert_Alfred_Herman" title="Robert Alfred Herman">Robert Alfred Herman</a> (1900) <a rel="nofollow" class="external text" href="https://archive.org/details/atreatiseongeom00hermgoog">A Treatise on Geometrical optics</a> from <a href="/wiki/Archive.org" class="mw-redirect" title="Archive.org">Archive.org</a>.</li> <li><a rel="nofollow" class="external text" href="http://www.wdl.org/en/item/2852">"The Light of the Eyes and the Enlightened Landscape of Vision"</a> is a manuscript, in Arabic, about geometrical optics, dating from the 16th century.</li> <li><a rel="nofollow" class="external text" href="https://books.google.com/books?id=TpY_AAAAYAAJ&pg=PA69">Theory of Systems of Rays</a> – W.R. Hamilton in Transactions of the Royal Irish Academy, Vol. XV, 1828.</li></ul> <div class="mw-heading mw-heading3"><h3 id="English_translations_of_some_early_books_and_papers">English translations of some early books and papers</h3> <a role="button" href="/w/index.php?title=Geometrical_optics&action=edit&section=11" title="Edit section: English translations of some early books and papers" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <ul><li><a rel="nofollow" class="external text" href="http://neo-classical-physics.info/uploads/3/0/6/5/3065888/bruns_-_the_eikonal.pdf">H. Bruns, "Das Eikonal"</a></li> <li><a rel="nofollow" class="external text" href="http://neo-classical-physics.info/uploads/3/0/6/5/3065888/malus_-_optics.pdf">M. Malus, "Optique"</a></li> <li><a rel="nofollow" class="external text" href="http://neo-classical-physics.info/uploads/3/0/6/5/3065888/plucker_-_general_form_of_light_rays.pdf">J. Plucker, "Discussion of the general form for light waves"</a></li> <li><a rel="nofollow" class="external text" href="http://neo-classical-physics.info/uploads/3/0/6/5/3065888/kummer_-_rectilinear_ray_systems.pdf">E. Kummer, "General theory of rectilinear ray systems"</a></li> <li><a rel="nofollow" class="external text" href="http://neo-classical-physics.info/uploads/3/0/6/5/3065888/kummer_-_rectilinear_systems_of_light_rays.pdf">E. Kummer, presentation on optically-realizable rectilinear ray systems</a></li> <li><a rel="nofollow" class="external text" href="http://neo-classical-physics.info/uploads/3/0/6/5/3065888/meibauer_-_rectilinear_ray_systems.pdf">R. Meibauer, "Theory of rectilinear systems of light rays"</a></li> <li><a rel="nofollow" class="external text" href="http://neo-classical-physics.info/uploads/3/0/6/5/3065888/pasch_-_focal_and_singularity_surfaces.pdf">M. Pasch, "On the focal surfaces of ray systems and the singularity surfaces of complexes"</a></li> <li><a rel="nofollow" class="external text" href="http://neo-classical-physics.info/uploads/3/0/6/5/3065888/levistal_-_geometrical_optics.pdf">A. Levistal, "Research in geometrical optics"</a></li> <li><a rel="nofollow" class="external text" href="http://neo-classical-physics.info/uploads/3/0/6/5/3065888/klein_-_bruns_eikonal.pdf">F. Klein, "On the Bruns eikonal"</a></li> <li><a rel="nofollow" class="external text" href="http://neo-classical-physics.info/uploads/3/0/6/5/3065888/dontot_-_integral_invariants_and_geom._opt..pdf">R. Dontot, "On integral invariants and some points of geometrical optics"</a></li> <li><a rel="nofollow" class="external text" href="http://neo-classical-physics.info/uploads/3/0/6/5/3065888/de_donder_-_int._inv._in_optics.pdf">T. de Donder, "On the integral invariants of optics"</a></li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(8)"><h2 id="External_links">External links</h2> <a role="button" href="/w/index.php?title=Geometrical_optics&action=edit&section=12" title="Edit section: External links" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-8 collapsible-block" id="mf-section-8"> <ul><li><a rel="nofollow" class="external text" href="https://feynmanlectures.caltech.edu/I_27.html">Feynman's lecture on Geometrical Optics</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output 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</div> </a> <div class="post-content footer-content"> <div id='mw-data-after-content'> <div class="read-more-container"></div> </div> <div id="p-lang"> <h4>Languages</h4> <section> <ul id="p-variants" class="minerva-languages"></ul> <ul class="minerva-languages"><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A8%D8%B5%D8%B1%D9%8A%D8%A7%D8%AA_%D9%87%D9%86%D8%AF%D8%B3%D9%8A%D8%A9" title="بصريات هندسية – Arabic" lang="ar" hreflang="ar" data-title="بصريات هندسية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%93%D0%B5%D0%B0%D0%BC%D0%B5%D1%82%D1%80%D1%8B%D1%87%D0%BD%D0%B0%D1%8F_%D0%BE%D0%BF%D1%82%D1%8B%D0%BA%D0%B0" title="Геаметрычная оптыка – Belarusian" lang="be" hreflang="be" data-title="Геаметрычная оптыка" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%BD%D0%B0_%D0%BE%D0%BF%D1%82%D0%B8%D0%BA%D0%B0" title="Геометрична оптика – Bulgarian" lang="bg" hreflang="bg" data-title="Геометрична оптика" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/%C3%92ptica_geom%C3%A8trica" title="Òptica geomètrica – Catalan" lang="ca" hreflang="ca" data-title="Òptica geomètrica" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D0%BB%D0%BB%D0%B5_%D0%BE%D0%BF%D1%82%D0%B8%D0%BA%D0%B0" title="Геометрилле оптика – Chuvash" lang="cv" hreflang="cv" data-title="Геометрилле оптика" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Geometrick%C3%A1_optika" title="Geometrická optika – Czech" lang="cs" hreflang="cs" data-title="Geometrická optika" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Geometrisk_optik" title="Geometrisk optik – Danish" lang="da" hreflang="da" data-title="Geometrisk optik" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Geometrische_Optik" title="Geometrische Optik – German" lang="de" hreflang="de" data-title="Geometrische Optik" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Geomeetriline_optika" title="Geomeetriline optika – Estonian" lang="et" hreflang="et" data-title="Geomeetriline optika" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%93%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%B9%CE%BA%CE%AE_%CE%BF%CF%80%CF%84%CE%B9%CE%BA%CE%AE" title="Γεωμετρική οπτική – Greek" lang="el" hreflang="el" data-title="Γεωμετρική οπτική" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/%C3%93ptica_geom%C3%A9trica" title="Óptica geométrica – Spanish" lang="es" hreflang="es" data-title="Óptica geométrica" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Optika_geometriko" title="Optika geometriko – Basque" lang="eu" hreflang="eu" data-title="Optika geometriko" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%86%D9%88%D8%B1%D8%B4%D9%86%D8%A7%D8%B3%DB%8C_%D9%87%D9%86%D8%AF%D8%B3%DB%8C" title="نورشناسی هندسی – Persian" lang="fa" hreflang="fa" data-title="نورشناسی هندسی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Optique_g%C3%A9om%C3%A9trique" title="Optique géométrique – French" lang="fr" hreflang="fr" data-title="Optique géométrique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Optaic_gheoim%C3%A9adrach" title="Optaic gheoiméadrach – Irish" lang="ga" hreflang="ga" data-title="Optaic gheoiméadrach" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B8%B0%ED%95%98%EA%B4%91%ED%95%99" title="기하광학 – Korean" lang="ko" hreflang="ko" data-title="기하광학" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B5%D6%80%D5%AF%D6%80%D5%A1%D5%B9%D5%A1%D6%83%D5%A1%D5%AF%D5%A1%D5%B6_%D6%85%D5%BA%D5%BF%D5%AB%D5%AF%D5%A1" title="Երկրաչափական օպտիկա – Armenian" lang="hy" hreflang="hy" data-title="Երկրաչափական օպտիկա" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%9C%E0%A5%8D%E0%A4%AF%E0%A4%BE%E0%A4%AE%E0%A4%BF%E0%A4%A4%E0%A5%80%E0%A4%AF_%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%95%E0%A4%BE%E0%A4%B6%E0%A4%BF%E0%A4%95%E0%A5%80" title="ज्यामितीय प्रकाशिकी – Hindi" lang="hi" hreflang="hi" data-title="ज्यामितीय प्रकाशिकी" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Geometrijska_optika" title="Geometrijska optika – Croatian" lang="hr" hreflang="hr" data-title="Geometrijska optika" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Optika_geometris" title="Optika geometris – Indonesian" lang="id" hreflang="id" data-title="Optika geometris" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Ottica_geometrica" title="Ottica geometrica – Italian" lang="it" hreflang="it" data-title="Ottica geometrica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%95%D7%A4%D7%98%D7%99%D7%A7%D7%94_%D7%92%D7%90%D7%95%D7%9E%D7%98%D7%A8%D7%99%D7%AA" title="אופטיקה גאומטרית – Hebrew" lang="he" hreflang="he" data-title="אופטיקה גאומטרית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%9C%E0%B3%8D%E0%B2%AF%E0%B2%BE%E0%B2%AE%E0%B2%BF%E0%B2%A4%E0%B3%80%E0%B2%AF_%E0%B2%A6%E0%B3%8D%E0%B2%AF%E0%B3%81%E0%B2%A4%E0%B2%BF%E0%B2%B6%E0%B2%BE%E0%B2%B8%E0%B3%8D%E0%B2%B0%E0%B3%8D%E0%B2%A4" title="ಜ್ಯಾಮಿತೀಯ ದ್ಯುತಿಶಾಸ್ರ್ತ – Kannada" lang="kn" hreflang="kn" data-title="ಜ್ಯಾಮಿತೀಯ ದ್ಯುತಿಶಾಸ್ರ್ತ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target">ಕನ್ನಡ</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F%D0%BB%D1%8B%D2%9B_%D0%BE%D0%BF%D1%82%D0%B8%D0%BA%D0%B0" title="Геометриялық оптика – Kazakh" lang="kk" hreflang="kk" data-title="Геометриялық оптика" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Geometresch_Optik" title="Geometresch Optik – Luxembourgish" lang="lb" hreflang="lb" data-title="Geometresch Optik" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target">Lëtzebuergesch</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Geometrin%C4%97_optika" title="Geometrinė optika – Lithuanian" lang="lt" hreflang="lt" data-title="Geometrinė optika" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Geometriai_optika" title="Geometriai optika – Hungarian" lang="hu" hreflang="hu" data-title="Geometriai optika" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%81%D0%BA%D0%B0_%D0%BE%D0%BF%D1%82%D0%B8%D0%BA%D0%B0" title="Геометриска оптика – Macedonian" lang="mk" hreflang="mk" data-title="Геометриска оптика" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Optik_geometri" title="Optik geometri – Malay" lang="ms" hreflang="ms" data-title="Optik geometri" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Geometrische_optica" title="Geometrische optica – Dutch" lang="nl" hreflang="nl" data-title="Geometrische optica" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%B9%BE%E4%BD%95%E5%85%89%E5%AD%A6" title="幾何光学 – Japanese" lang="ja" hreflang="ja" data-title="幾何光学" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Geometrisk_optikk" title="Geometrisk optikk – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Geometrisk optikk" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Optica_geometrica" title="Optica geometrica – Occitan" lang="oc" hreflang="oc" data-title="Optica geometrica" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target">Occitan</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Geometrik_optika" title="Geometrik optika – Uzbek" lang="uz" hreflang="uz" data-title="Geometrik optika" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Optyka_geometryczna" title="Optyka geometryczna – Polish" lang="pl" hreflang="pl" data-title="Optyka geometryczna" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/%C3%93ptica_geom%C3%A9trica" title="Óptica geométrica – Portuguese" lang="pt" hreflang="pt" data-title="Óptica geométrica" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Optic%C4%83_geometric%C4%83" title="Optică geometrică – Romanian" lang="ro" hreflang="ro" data-title="Optică geometrică" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D0%BE%D0%BF%D1%82%D0%B8%D0%BA%D0%B0" title="Геометрическая оптика – Russian" lang="ru" hreflang="ru" data-title="Геометрическая оптика" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Geometrijska_optika" title="Geometrijska optika – Slovenian" lang="sl" hreflang="sl" data-title="Geometrijska optika" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Geometrijska_optika" title="Geometrijska optika – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Geometrijska optika" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/S%C3%A4deoptiikka" title="Sädeoptiikka – Finnish" lang="fi" hreflang="fi" data-title="Sädeoptiikka" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Geometrisk_optik" title="Geometrisk optik – Swedish" lang="sv" hreflang="sv" data-title="Geometrisk optik" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%97%E0%B8%B1%E0%B8%A8%E0%B8%99%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C%E0%B9%80%E0%B8%8A%E0%B8%B4%E0%B8%87%E0%B9%80%E0%B8%A3%E0%B8%82%E0%B8%B2%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95" title="ทัศนศาสตร์เชิงเรขาคณิต – Thai" lang="th" hreflang="th" data-title="ทัศนศาสตร์เชิงเรขาคณิต" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Geometrik_optik" title="Geometrik optik – Turkish" lang="tr" hreflang="tr" data-title="Geometrik optik" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%BD%D0%B0_%D0%BE%D0%BF%D1%82%D0%B8%D0%BA%D0%B0" title="Геометрична оптика – Ukrainian" lang="uk" hreflang="uk" data-title="Геометрична оптика" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%DB%81%D9%86%D8%AF%D8%B3%DB%8C_%D8%A8%D8%B5%D8%B1%DB%8C%D8%A7%D8%AA" title="ہندسی بصریات – Urdu" lang="ur" hreflang="ur" data-title="ہندسی بصریات" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%B9%BE%E4%BD%95%E5%85%89%E5%AD%B8" title="幾何光學 – Cantonese" lang="yue" hreflang="yue" data-title="幾何光學" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target">粵語</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%87%A0%E4%BD%95%E5%85%89%E5%AD%A6" title="几何光学 – Chinese" lang="zh" hreflang="zh" data-title="几何光学" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li></ul> </section> </div> <div class="minerva-footer-logo"><img src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" alt="Wikipedia" width="120" height="18" style="width: 7.5em; height: 1.125em;"/> </div> <ul id="footer-info" class="footer-info hlist hlist-separated"> <li id="footer-info-lastmod"> This page was last edited on 11 November 2024, at 18:24 (UTC).</li> <li id="footer-info-copyright">Content is available under <a class="external" rel="nofollow" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">CC BY-SA 4.0</a> unless otherwise noted.</li> </ul> <ul id="footer-places" class="footer-places hlist hlist-separated"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a 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Geometrical optics - Wikipedia