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<title>Transfer-matrix method (optics)</title>

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Transfer-matrix method (optics)

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</style><div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="Transfer-matrix_method_(disambiguation)" class="mw-redirect mw-disambig" title="Transfer-matrix method (disambiguation)">Transfer-matrix method</a>.</div>

The transfer-matrix method is a method used in <a href="Optics" title="Optics">optics</a> and <a href="Acoustics" title="Acoustics">acoustics</a> to analyze the propagation of <a href="Electromagnetic_wave" class="mw-redirect" title="Electromagnetic wave">electromagnetic</a> or <a href="Acoustic_wave" title="Acoustic wave">acoustic waves</a> through a <a href="Stratified_medium" class="mw-redirect" title="Stratified medium">stratified medium</a>; a stack of <a href="Thin_film" title="Thin film">thin films</a>.<a href="#cite_note-1">[1]</a><a href="#cite_note-2">[2]</a> This is, for example, relevant for the design of <a href="Anti-reflective_coating" title="Anti-reflective coating">anti-reflective coatings</a> and <a href="Dielectric_mirror" title="Dielectric mirror">dielectric mirrors</a>.

The <a href="Reflection_(physics)" title="Reflection (physics)">reflection</a> of <a href="Light" title="Light">light</a> from a single interface between two <a href="Medium_(optics)" class="mw-redirect" title="Medium (optics)">media</a> is described by the <a href="Fresnel_equations" title="Fresnel equations">Fresnel equations</a>. However, when there are multiple <a href="https://en.wiktionary.org/wiki/interface" class="extiw external" title="wiktionary:interface">interfaces</a>, such as in the figure, the reflections themselves are also partially transmitted and then partially reflected. Depending on the exact path length, these reflections can <a href="Interference_(wave_propagation)" class="mw-redirect" title="Interference (wave propagation)">interfere</a> destructively or constructively. The overall reflection of a layer structure is the sum of an infinite number of reflections.

The transfer-matrix method is based on the fact that, according to <a href="Maxwell's_equations" title="Maxwell's equations">Maxwell's equations</a>, there are simple continuity conditions for the <a href="Electric_field" title="Electric field">electric field</a> across boundaries from one medium to the next. If the field is known at the beginning of a layer, the field at the end of the layer can be derived from a simple <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> operation. A stack of layers can then be represented as a system matrix, which is the product of the individual layer matrices. The final step of the method involves converting the system matrix back into reflection and <a href="Transmission_coefficient" title="Transmission coefficient">transmission coefficients</a>.

<div class="mw-heading mw-heading2"><h2 id="Formalism_for_electromagnetic_waves">Formalism for electromagnetic waves</h2></div>

Below is described how the transfer matrix is applied to <a href="Electromagnetic_waves" class="mw-redirect" title="Electromagnetic waves">electromagnetic waves</a> (for example light) of a given <a href="Frequency" title="Frequency">frequency</a> propagating through a stack of layers at <a href="Surface_normal" class="mw-redirect" title="Surface normal">normal incidence</a>. It can be generalized to deal with incidence at an angle, <a href="Absorption_(electromagnetic_radiation)" title="Absorption (electromagnetic radiation)">absorbing media</a>, and media with <a href="Permeability_(electromagnetism)" title="Permeability (electromagnetism)">magnetic properties</a>. We assume that the stack layers are normal to the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\,}">

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

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</math><img src="./624faa61961bd63f364dee3e97dec7dd48694600.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.475ex; height:1.676ex;" alt="{\displaystyle z\,}" loading="lazy"> axis and that the field within one layer can be represented as the superposition of a left- and right-traveling wave with <a href="Wave_number" class="mw-redirect" title="Wave number">wave number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\,}">

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

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</math><img src="./a5665cb00a844c1c6a671c119c6e4984d28fb851.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.598ex; height:2.176ex;" alt="{\displaystyle k\,}" loading="lazy">,

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<annotation encoding="application/x-tex">{\displaystyle E(z)=E_{r}e^{ikz}+E_{l}e^{-ikz}\,}</annotation>

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</math><img src="./2ffd7b2055a450c4b7ba21f926d8003235847705.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.422ex; height:3.176ex;" alt="{\displaystyle E(z)=E_{r}e^{ikz}+E_{l}e^{-ikz}\,}" loading="lazy">.</dd></dl>

Because it follows from <a href="Maxwell's_equation" class="mw-redirect" title="Maxwell's equation">Maxwell's equation</a> that electric field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\,}">

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

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</math><img src="./9123abddc2ec35f72035ec59f443c79ee052c9ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.163ex; height:2.176ex;" alt="{\displaystyle E\,}" loading="lazy"> and magnetic field (its normalized derivative) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle H={\frac {1}{ik}}Z_{c}{\frac {dE}{dz}}\,}">

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<annotation encoding="application/x-tex">{\textstyle H={\frac {1}{ik}}Z_{c}{\frac {dE}{dz}}\,}</annotation>

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</math><img src="./d60710e979b3c208f7716111eb5d743ac271debd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:13.292ex; height:3.843ex;" alt="{\textstyle H={\frac {1}{ik}}Z_{c}{\frac {dE}{dz}}\,}" loading="lazy"> must be continuous across a boundary, it is convenient to represent the field as the vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (E(z),H(z))\,}">

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<annotation encoding="application/x-tex">{\textstyle (E(z),H(z))\,}</annotation>

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</math><img src="./3013a58f5b7a37b8ed8dc1eceb5902af1d92d83c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.864ex; height:2.843ex;" alt="{\textstyle (E(z),H(z))\,}" loading="lazy">, where

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<annotation encoding="application/x-tex">{\displaystyle H(z)={\frac {1}{Z_{c}}}E_{r}e^{ikz}-{\frac {1}{Z_{c}}}E_{l}e^{-ikz}\,}</annotation>

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</math><img src="./68cdaa1e60c88a3ee2c72cb0de5e30d833df50a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.446ex; height:5.676ex;" alt="{\displaystyle H(z)={\frac {1}{Z_{c}}}E_{r}e^{ikz}-{\frac {1}{Z_{c}}}E_{l}e^{-ikz}\,}" loading="lazy">.</dd></dl>

Since there are two equations relating <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\,}">

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

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</math><img src="./9123abddc2ec35f72035ec59f443c79ee052c9ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.163ex; height:2.176ex;" alt="{\displaystyle E\,}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\,}">

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

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</math><img src="./33810a0db43048d22f06d52f257b8e49e4e977a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.451ex; height:2.176ex;" alt="{\displaystyle H\,}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{r}\,}">

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<annotation encoding="application/x-tex">{\displaystyle E_{r}\,}</annotation>

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</math><img src="./f407a752b38b3c62cd6e61f157d29e8e0825e03e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.076ex; height:2.509ex;" alt="{\displaystyle E_{r}\,}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{l}\,}">

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<annotation encoding="application/x-tex">{\displaystyle E_{l}\,}</annotation>

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</math><img src="./bca1fc3b70b97b402276f72bfdb9c608a9e2738d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.825ex; height:2.509ex;" alt="{\displaystyle E_{l}\,}" loading="lazy">, these two representations are equivalent. In the new representation, propagation over a distance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L\,}">

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

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</math><img src="./d330bc0cd693cc87e3943137dc591038a89f77e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.97ex; height:2.176ex;" alt="{\displaystyle L\,}" loading="lazy"> into the positive direction of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\,}">

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

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</math><img src="./624faa61961bd63f364dee3e97dec7dd48694600.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.475ex; height:1.676ex;" alt="{\displaystyle z\,}" loading="lazy"> is described by the matrix belonging to the <a href="Special_linear_group" title="Special linear group">special linear group</a> SL(2, C)

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<annotation encoding="application/x-tex">{\displaystyle M=\left({\begin{array}{cc}\cos kL&iZ_{c}\sin kL\\{\frac {i}{Z_{c}}}\sin kL&\cos kL\end{array}}\right),}</annotation>

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</math><img src="./7754d60616213abb71c17398dbe16fe8317667df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:32.213ex; height:7.509ex;" alt="{\displaystyle M=\left({\begin{array}{cc}\cos kL&iZ_{c}\sin kL\\{\frac {i}{Z_{c}}}\sin kL&\cos kL\end{array}}\right),}" loading="lazy"></dd></dl>

and

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<annotation encoding="application/x-tex">{\displaystyle \left({\begin{array}{c}E(z+L)\\H(z+L)\end{array}}\right)=M\cdot \left({\begin{array}{c}E(z)\\H(z)\end{array}}\right)}</annotation>

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</math><img src="./59aaa017a46a79c61da9dcef2850ff6d252aab47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.91ex; height:6.176ex;" alt="{\displaystyle \left({\begin{array}{c}E(z+L)\\H(z+L)\end{array}}\right)=M\cdot \left({\begin{array}{c}E(z)\\H(z)\end{array}}\right)}" loading="lazy"></dd></dl>

Such a matrix can represent propagation through a layer if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\,}">

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

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</math><img src="./a5665cb00a844c1c6a671c119c6e4984d28fb851.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.598ex; height:2.176ex;" alt="{\displaystyle k\,}" loading="lazy"> is the wave number in the medium and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L\,}">

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

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</math><img src="./d330bc0cd693cc87e3943137dc591038a89f77e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.97ex; height:2.176ex;" alt="{\displaystyle L\,}" loading="lazy"> the thickness of the layer:

For a system with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\,}">

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

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</math><img src="./ed4f97d800eaacd982789661ac3896e1181d6137.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.451ex; height:2.176ex;" alt="{\displaystyle N\,}" loading="lazy"> layers, each layer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j\,}">

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

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</math><img src="./30f268d4be31700461b9f20cabb0724899ad5d27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:1.372ex; height:2.509ex;" alt="{\displaystyle j\,}" loading="lazy"> has a transfer matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{j}\,}">

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<annotation encoding="application/x-tex">{\displaystyle M_{j}\,}</annotation>

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</math><img src="./6c9e96f930013288cd587ad470f97b404f66e38a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.551ex; height:2.843ex;" alt="{\displaystyle M_{j}\,}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j\,}">

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

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</math><img src="./30f268d4be31700461b9f20cabb0724899ad5d27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:1.372ex; height:2.509ex;" alt="{\displaystyle j\,}" loading="lazy"> increases towards higher <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\,}">

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

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</math><img src="./624faa61961bd63f364dee3e97dec7dd48694600.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.475ex; height:1.676ex;" alt="{\displaystyle z\,}" loading="lazy"> values. The system transfer matrix is then

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<annotation encoding="application/x-tex">{\displaystyle M_{s}=M_{N}\cdot \ldots \cdot M_{2}\cdot M_{1}.}</annotation>

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</math><img src="./75eef74b7967aa2c022646f740e99aceafa610d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.325ex; height:2.509ex;" alt="{\displaystyle M_{s}=M_{N}\cdot \ldots \cdot M_{2}\cdot M_{1}.}" loading="lazy"></dd></dl>

Typically, one would like to know the <a href="Reflectance" title="Reflectance">reflectance</a> and <a href="Transmittance" title="Transmittance">transmittance</a> of the layer structure. If the layer stack starts at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=0\,}">

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<annotation encoding="application/x-tex">{\displaystyle z=0\,}</annotation>

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</math><img src="./2643b34ec816883ada7a536c1e4e0fcc23e78464.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.736ex; height:2.176ex;" alt="{\displaystyle z=0\,}" loading="lazy">, then for negative <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\,}">

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

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</math><img src="./624faa61961bd63f364dee3e97dec7dd48694600.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.475ex; height:1.676ex;" alt="{\displaystyle z\,}" loading="lazy">, the field is described as

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<annotation encoding="application/x-tex">{\displaystyle E_{L}(z)=E_{0}e^{ik_{L}z}+rE_{0}e^{-ik_{L}z},\qquad z<0,}</annotation>

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</math><img src="./003ffae6ae35c5f67decbe156bfd59f86b6bcf0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.608ex; height:3.176ex;" alt="{\displaystyle E_{L}(z)=E_{0}e^{ik_{L}z}+rE_{0}e^{-ik_{L}z},\qquad z<0,}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{0}\,}">

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<annotation encoding="application/x-tex">{\displaystyle E_{0}\,}</annotation>

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</math><img src="./4ee33d096c508161625d49c0653114749942919f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.157ex; height:2.509ex;" alt="{\displaystyle E_{0}\,}" loading="lazy"> is the amplitude of the incoming wave, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{L}\,}">

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<annotation encoding="application/x-tex">{\displaystyle k_{L}\,}</annotation>

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</math><img src="./95132349e1d31b87b3530f7e374fad32e1c2d5ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.95ex; height:2.509ex;" alt="{\displaystyle k_{L}\,}" loading="lazy"> the wave number in the left medium, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\,}">

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

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</math><img src="./f08ce4d4c86c5b43f36c8435fb598da6471047c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.436ex; height:1.676ex;" alt="{\displaystyle r\,}" loading="lazy"> is the amplitude (not intensity!) reflectance coefficient of the layer structure. On the other side of the layer structure, the field consists of a right-propagating transmitted field

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<annotation encoding="application/x-tex">{\displaystyle E_{R}(z)=tE_{0}e^{ik_{R}z},\qquad z>L',}</annotation>

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</math><img src="./cd992ea5497284ecf74bfc2f00e293027ce868fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.266ex; height:3.176ex;" alt="{\displaystyle E_{R}(z)=tE_{0}e^{ik_{R}z},\qquad z>L',}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\,}">

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

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</math><img src="./946383a7c6d1876177c662a95b369ced2ad99cd9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.227ex; height:2.009ex;" alt="{\displaystyle t\,}" loading="lazy"> is the amplitude transmittance, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{R}\,}">

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

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</math><img src="./1ec38df47b311477d24dd891eb319d7b960df201.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.078ex; height:2.509ex;" alt="{\displaystyle k_{R}\,}" loading="lazy"> is the wave number in the rightmost medium, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L'}">

<msup>

</msup>

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

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</math><img src="./7a5bf7164faabe6b737c3d5269366b5e17f2ac7c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.268ex; height:2.509ex;" alt="{\displaystyle L'}" loading="lazy"> is the total thickness. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle H_{L}={\frac {1}{ik}}Z_{c}{\frac {dE_{L}}{dz}}\,}">

<msub>

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<annotation encoding="application/x-tex">{\textstyle H_{L}={\frac {1}{ik}}Z_{c}{\frac {dE_{L}}{dz}}\,}</annotation>

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</math><img src="./c6db4c4c89255b277e2e55ee5a0bef865689adcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:15.542ex; height:4.176ex;" alt="{\textstyle H_{L}={\frac {1}{ik}}Z_{c}{\frac {dE_{L}}{dz}}\,}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle H_{R}={\frac {1}{ik}}Z_{c}{\frac {dE_{R}}{dz}}\,}">

<msub>

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</msub>

<mfrac>

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</mfrac>

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<msub>

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</msub>

<mfrac>

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<msub>

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<annotation encoding="application/x-tex">{\textstyle H_{R}={\frac {1}{ik}}Z_{c}{\frac {dE_{R}}{dz}}\,}</annotation>

</semantics>

</math><img src="./c4d64144f19d4510cc9fe84f95c1f1425b86dba5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:15.774ex; height:4.176ex;" alt="{\textstyle H_{R}={\frac {1}{ik}}Z_{c}{\frac {dE_{R}}{dz}}\,}" loading="lazy">, then one can solve

<mrow>

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<annotation encoding="application/x-tex">{\displaystyle \left({\begin{array}{c}E(z_{R})\\H(z_{R})\end{array}}\right)=M\cdot \left({\begin{array}{c}E(0)\\H(0)\end{array}}\right)}</annotation>

</semantics>

</math><img src="./e184a389984d1705bffcd725ddd80f6df21f6ac9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.034ex; height:6.176ex;" alt="{\displaystyle \left({\begin{array}{c}E(z_{R})\\H(z_{R})\end{array}}\right)=M\cdot \left({\begin{array}{c}E(0)\\H(0)\end{array}}\right)}" loading="lazy"></dd></dl>

in terms of the matrix elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{mn}\,}">

<msub>

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</msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{mn}\,}</annotation>

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</math><img src="./3e9a665b60fed6255a8c0708b37ba56b526b6f04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.302ex; height:2.509ex;" alt="{\displaystyle M_{mn}\,}" loading="lazy"> of the system matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{s}\,}">

<msub>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{s}\,}</annotation>

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</math><img src="./befebd67b63fe4010fa26aebfc304094639ca74d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.644ex; height:2.509ex;" alt="{\displaystyle M_{s}\,}" loading="lazy"> and obtain

<msub>

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<msup>

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</msup>

<mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle t=2ik_{L}e^{-ik_{R}L}\left[{\frac {1}{-M_{21}+k_{L}k_{R}M_{12}+i(k_{R}M_{11}+k_{L}M_{22})}}\right]}</annotation>

</semantics>

</math><img src="./dfa4d3375427d1301ca9ca5267f4b2930e991054.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:58.428ex; height:6.343ex;" alt="{\displaystyle t=2ik_{L}e^{-ik_{R}L}\left[{\frac {1}{-M_{21}+k_{L}k_{R}M_{12}+i(k_{R}M_{11}+k_{L}M_{22})}}\right]}" loading="lazy"></dd></dl>

and

<mrow>

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<msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle r=\left[{\frac {(M_{21}+k_{L}k_{R}M_{12})+i(k_{L}M_{22}-k_{R}M_{11})}{(-M_{21}+k_{L}k_{R}M_{12})+i(k_{L}M_{22}+k_{R}M_{11})}}\right]}</annotation>

</semantics>

</math><img src="./92d459d66dc234881f54ddac51038a7cb2bcd2c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:49.217ex; height:6.509ex;" alt="{\displaystyle r=\left[{\frac {(M_{21}+k_{L}k_{R}M_{12})+i(k_{L}M_{22}-k_{R}M_{11})}{(-M_{21}+k_{L}k_{R}M_{12})+i(k_{L}M_{22}+k_{R}M_{11})}}\right]}" loading="lazy">.</dd></dl>

The transmittance and reflectance (i.e., the fractions of the incident intensity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left|E_{0}\right|^{2}}">

<msup>

<mrow>

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\textstyle \left|E_{0}\right|^{2}}</annotation>

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</math><img src="./938586808e252a67afc8547aba8457abe6c36898.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.117ex; height:3.343ex;" alt="{\textstyle \left|E_{0}\right|^{2}}" loading="lazy"> transmitted and reflected by the layer) are often of more practical use and are given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle T={\frac {k_{R}}{k_{L}}}|t|^{2}\,}">

<mfrac>

<msub>

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</msub>

<msub>

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<msup>

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<annotation encoding="application/x-tex">{\textstyle T={\frac {k_{R}}{k_{L}}}|t|^{2}\,}</annotation>

</semantics>

</math><img src="./014ef3ecf9b6b2c9bfb6ea2fcc7d7c7e8ed2f9a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:11.179ex; height:4.509ex;" alt="{\textstyle T={\frac {k_{R}}{k_{L}}}|t|^{2}\,}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=|r|^{2}\,}">

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle R=|r|^{2}\,}</annotation>

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</math><img src="./eafa05eb832e2d4cacb9c19dcb1c4f81c60b5d11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.646ex; height:3.343ex;" alt="{\displaystyle R=|r|^{2}\,}" loading="lazy">, respectively (at normal incidence).

<div class="mw-heading mw-heading3"><h3 id="Example">Example</h3></div>

As an illustration, consider a single layer of glass with a refractive index n and thickness d suspended in air at a wave number k (in air). In glass, the wave number is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k'=nk\,}">

<msup>

</msup>

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<annotation encoding="application/x-tex">{\displaystyle k'=nk\,}</annotation>

</semantics>

</math><img src="./57f0904bb78824a0edfc99f41c7c5399a7be96f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.987ex; height:2.509ex;" alt="{\displaystyle k'=nk\,}" loading="lazy">. The transfer matrix is

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<annotation encoding="application/x-tex">{\displaystyle M=\left({\begin{array}{cc}\cos k'd&\sin(k'd)/k'\\-k'\sin k'd&\cos k'd\end{array}}\right)}</annotation>

</semantics>

</math><img src="./893b6b706c4d4e25b8ea66355872dfac5e195d18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.317ex; height:6.176ex;" alt="{\displaystyle M=\left({\begin{array}{cc}\cos k'd&\sin(k'd)/k'\\-k'\sin k'd&\cos k'd\end{array}}\right)}" loading="lazy">.</dd></dl>

The amplitude reflection coefficient can be simplified to

<mfrac>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle r={\frac {(1/n-n)\sin(k'd)}{(n+1/n)\sin(k'd)+2i\cos(k'd)}}}</annotation>

</semantics>

</math><img src="./1d02151d26c8d0b26038c759537599368731d9e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.136ex; height:6.509ex;" alt="{\displaystyle r={\frac {(1/n-n)\sin(k'd)}{(n+1/n)\sin(k'd)+2i\cos(k'd)}}}" loading="lazy">.</dd></dl>

This configuration effectively describes a <a href="Fabry%E2%80%93P%C3%A9rot_interferometer" title="Fabry–Pérot interferometer">Fabry–Pérot interferometer</a> or etalon: for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle k'd=0,\pi ,2\pi ,\cdots \,}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle k'd=0,\pi ,2\pi ,\cdots \,}</annotation>

</semantics>

</math><img src="./2ef14fed6900ed715fd1c3775ff9caf39843c597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.411ex; height:2.676ex;" alt="{\textstyle k'd=0,\pi ,2\pi ,\cdots \,}" loading="lazy">, the reflection vanishes.

<div class="mw-heading mw-heading2"><h2 id="Acoustic_waves">Acoustic waves</h2></div>

It is possible to apply the transfer-matrix method to sound waves. Instead of the electric field E and its derivative H, the displacement u and the <a href="Stress_(physics)" class="mw-redirect" title="Stress (physics)">stress</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma =Cdu/dz}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma =Cdu/dz}</annotation>

</semantics>

</math><img src="./b142003e4e4e1f09c7e05c23d51d2e9c51165b9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.206ex; height:2.843ex;" alt="{\displaystyle \sigma =Cdu/dz}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> is the <a href="P-wave_modulus" title="P-wave modulus">p-wave modulus</a>, should be used.

<div class="mw-heading mw-heading2"><h2 id="Abeles_matrix_formalism">Abeles matrix formalism</h2></div>

The Abeles matrix method<a href="#cite_note-3">[3]</a><a href="#cite_note-4">[4]</a><a href="#cite_note-5">[5]</a> is a computationally fast and easy way to calculate the specular reflectivity from a stratified interface, as a function of the perpendicular <a href="Momentum_transfer" title="Momentum transfer">momentum transfer</a>, Qz:

<msub>

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<annotation encoding="application/x-tex">{\displaystyle Q_{z}={\frac {4\pi }{\lambda }}\sin \theta =2k_{z}}</annotation>

</semantics>

</math><img src="./999f9e6677c23bbbfa2be5e2d5668f642053289d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.463ex; height:5.343ex;" alt="{\displaystyle Q_{z}={\frac {4\pi }{\lambda }}\sin \theta =2k_{z}}" loading="lazy"></dd></dl>

where θ is the angle of incidence/reflection of the incident <a href="Radiation" title="Radiation">radiation</a> and λ is the wavelength of the radiation.

The measured reflectivity depends on the variation in the scattering length density (<dfn>SLD</dfn>)

profile, ρ(z), perpendicular to the interface. Although the scattering length density profile

is normally a continuously varying function, the interfacial structure can often be well approximated

by a slab model in which layers of thickness (dn), scattering length density (ρn) and roughness (σn,n+1) are sandwiched between the super- and sub-phases. One then uses a refinement procedure to minimise the differences between the theoretical and measured reflectivity curves, by changing the parameters that describe each layer.

In this description the interface is split into n layers. Since the incident neutron beam

is refracted by each of the layers the wavevector k, in layer n, is given by:

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<annotation encoding="application/x-tex">{\displaystyle k_{n}={\sqrt {{k_{z}}^{2}-4\pi ({\rho }_{n}-{\rho }_{0})}}}</annotation>

</semantics>

</math><img src="./167b68b0c772693ac56c18695dada535837e96d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:25.78ex; height:4.843ex;" alt="{\displaystyle k_{n}={\sqrt {{k_{z}}^{2}-4\pi ({\rho }_{n}-{\rho }_{0})}}}" loading="lazy"></dd></dl>

The <a href="Fresnel_equations" title="Fresnel equations">Fresnel reflection</a> coefficient between layer n and n+1 is then given by:

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<annotation encoding="application/x-tex">{\displaystyle r_{n,n+1}={\frac {k_{n}-k_{n+1}}{k_{n}+k_{n+1}}}}</annotation>

</semantics>

</math><img src="./2459e92f1ee9f966c916ea68053f915f8f279832.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.546ex; height:5.843ex;" alt="{\displaystyle r_{n,n+1}={\frac {k_{n}-k_{n+1}}{k_{n}+k_{n+1}}}}" loading="lazy"></dd></dl>

Because the interface between each layer is unlikely to be perfectly smooth the roughness/diffuseness of each interface modifies the Fresnel coefficient and is accounted for by an <a href="Error_function" title="Error function">error function</a>,<a href="#cite_note-FOOTNOTENévotCroce1980-6">[6]</a>

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<annotation encoding="application/x-tex">{\displaystyle r_{n,n+1}={\frac {k_{n}-k_{n+1}}{k_{n}+k_{n+1}}}\exp(-2k_{n}k_{n+1}{\sigma _{n,n+1}}^{2}).}</annotation>

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</math><img src="./f666f5a8b15e1c0fabee9a41c38a63c616ab740d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:43.017ex; height:5.843ex;" alt="{\displaystyle r_{n,n+1}={\frac {k_{n}-k_{n+1}}{k_{n}+k_{n+1}}}\exp(-2k_{n}k_{n+1}{\sigma _{n,n+1}}^{2}).}" loading="lazy"></dd></dl>

A phase factor, β, is introduced, which accounts for the thickness of each layer.

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<annotation encoding="application/x-tex">{\displaystyle \beta _{0}=0}</annotation>

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</math><img src="./b4893ae08e9fbb4806f10525517361b5b149b8b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.631ex; height:2.509ex;" alt="{\displaystyle \beta _{0}=0}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle \beta _{n}=ik_{n}d_{n}}</annotation>

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</math><img src="./cc9bf98c58bd99323105502f64b4f86fb5799400.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.292ex; height:2.509ex;" alt="{\displaystyle \beta _{n}=ik_{n}d_{n}}" loading="lazy"></dd></dl>

where i2 = −1.

A characteristic matrix, cn is then calculated for each layer.

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<annotation encoding="application/x-tex">{\displaystyle c_{n}=\left[{\begin{array}{cc}\exp \left(\beta _{n}\right)&r_{n,n+1}\exp \left(\beta _{n}\right)\\r_{n,n+1}\exp \left(-\beta _{n}\right)&\exp \left(-\beta _{n}\right)\end{array}}\right]}</annotation>

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</math><img src="./9ea5be14ff3d9ecce8889029f1607862be22f163.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:40.849ex; height:6.509ex;" alt="{\displaystyle c_{n}=\left[{\begin{array}{cc}\exp \left(\beta _{n}\right)&r_{n,n+1}\exp \left(\beta _{n}\right)\\r_{n,n+1}\exp \left(-\beta _{n}\right)&\exp \left(-\beta _{n}\right)\end{array}}\right]}" loading="lazy"></dd></dl>

The resultant matrix is defined as the ordered product of these characteristic matrices

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<annotation encoding="application/x-tex">{\displaystyle M=\prod _{n}c_{n}}</annotation>

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</math><img src="./957d949f5417398a8058927c8d2d191a328364f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:11.123ex; height:5.509ex;" alt="{\displaystyle M=\prod _{n}c_{n}}" loading="lazy"></dd></dl>

from which the reflectivity is calculated as:

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<mfrac>

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</mfrac>

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<annotation encoding="application/x-tex">{\displaystyle R=\left|{\frac {M_{10}}{M_{00}}}\right|^{2}}</annotation>

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</math><img src="./29f2ffad118a0747599a9ec82d02e83724fe41c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.177ex; height:6.176ex;" alt="{\displaystyle R=\left|{\frac {M_{10}}{M_{00}}}\right|^{2}}" loading="lazy"></dd></dl>

<ul><li><a href="Neutron_reflectometry" title="Neutron reflectometry">Neutron reflectometry</a></li>

<li><a href="Ellipsometry" title="Ellipsometry">Ellipsometry</a></li>

<li><a href="Jones_calculus" title="Jones calculus">Jones calculus</a></li>

<li><a href="X-ray_reflectivity" title="X-ray reflectivity">X-ray reflectivity</a></li>

<li><a href="Scattering-matrix_method" title="Scattering-matrix method">Scattering-matrix method</a></li></ul>

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

<li id="cite_note-1"><a href="#cite_ref-1">^</a> Born, M.; Wolf, E., <a href="Principles_of_Optics" title="Principles of Optics">Principles of optics: electromagnetic theory of propagation, interference and diffraction of light</a>. Oxford, Pergamon Press, 1964.

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> Mackay, T. G.; Lakhtakia, A., The Transfer-Matrix Method in Electromagnetics and Optics. San Rafael, CA, Morgan and Claypool, 2020. <style data-mw-deduplicate="TemplateStyles:r1238218222">

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</style><a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2200%2FS00993ED1V01Y202002EMA001">10.2200/S00993ED1V01Y202002EMA001</a>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> O. S. Heavens. Optical Properties of Thin Films. Butterworth, London (1955).

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFNévotCroce1980" class="citation journal cs1 cs1-prop-foreign-lang-source">Névot, L.; Croce, P. (1980). <a rel="nofollow" class="external text" href="https://hal.archives-ouvertes.fr/jpa-00244786/file/ajp-rphysap_1980_15_3_761_0.pdf">"Caractérisation des surfaces par réflexion rasante de rayons X. Application à l'étude du polissage de quelques verres silicates"</a> (PDF). Revue de Physique Appliquée (in French). 15 (3). EDP Sciences: 761–779. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1051%2Frphysap%3A01980001503076100">10.1051/rphysap:01980001503076100</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0035-1687">0035-1687</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:128834171">128834171</a>.</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFAbelès1950" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="Florin_Abel%C3%A8s" title="Florin Abelès">Abelès, Florin</a> (1950). <a rel="nofollow" class="external text" href="https://hal.science/jpa-00234261/file/ajp-jphysrad_1950_11_7_307_0.pdf">"La théorie générale des couches minces"</a> [The generalized theory of thin films] (PDF). Journal de Physique et le Radium (in French). 11 (7). EDP Sciences: 307–309. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1051%2Fjphysrad%3A01950001107030700">10.1051/jphysrad:01950001107030700</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0368-3842">0368-3842</a>.</cite>

</li>

<li id="cite_note-FOOTNOTENévotCroce1980-6"><a href="#cite_ref-FOOTNOTENévotCroce1980_6-0">^</a> <a href="#CITEREFNévotCroce1980">Névot & Croce (1980)</a>.

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://www.jedsoft.org/physics/notes/multilayer.pdf">Multilayer Reflectivity</a>: first-principles derivation of the transmission and reflection probabilities from a multilayer with complex indices of refraction.</li>

<li><a rel="nofollow" class="external text" href="https://ocw.mit.edu/courses/materials-science-and-engineering/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/lecture-notes/MIT3_024S13_2012lec23.pdf">Layered Materials and Photonic Band Diagrams</a> (Lecture 23) in MIT Open Course <a rel="nofollow" class="external text" href="https://ocw.mit.edu/courses/materials-science-and-engineering/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/lecture-notes/">Electronic, Optical and Magnetic Properties of Materials</a>.</li>

<li><a rel="nofollow" class="external text" href="https://ocw.mit.edu/courses/mechanical-engineering/2-57-nano-to-macro-transport-processes-spring-2012/video-lectures/lecture-13-em-wave-propagation-through-thin-films-multilayers/">EM Wave Propagation Through Thin Films & Multilayers</a> (Lecture 13) in MIT Open Course <a rel="nofollow" class="external text" href="https://ocw.mit.edu/courses/mechanical-engineering/2-57-nano-to-macro-transport-processes-spring-2012">Nano-to-Macro Transport Processes</a>. Includes short discussion acoustic waves.</li></ul>

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

There are a number of computer programs that implement this calculation:

<ul><li><a rel="nofollow" class="external text" href="http://people.csail.mit.edu/jaffer/FreeSnell/">FreeSnell</a> is a stand-alone computer program that implements the transfer-matrix method, including more advanced aspects such as granular films.</li>

<li><a rel="nofollow" class="external text" href="http://thinfilm.hansteen.net/">Thinfilm</a> is a web interface that implements the transfer-matrix method, outputting reflection and transmission coefficients, and also <a href="Ellipsometer" class="mw-redirect" title="Ellipsometer">ellipsometric</a> parameters Psi and Delta.</li>

<li><a rel="nofollow" class="external text" href="http://www.luxpop.com/#Thin_Film_Stack">Luxpop.com</a> is another web interface that implements the transfer-matrix method.</li>

<li><a rel="nofollow" class="external text" href="http://sjbyrnes.com/?page_id=12">Transfer-matrix calculating programs in Python and in Mathematica</a>.</li>

<li><a rel="nofollow" class="external text" href="https://lbolla.github.io/EMpy/">EMPy ("Electromagnetic Python") software</a>.</li>

<li><a rel="nofollow" class="external text" href="http://motofit.sourceforge.net/">motofit</a> is a program for analysing neutron and X-ray reflectometry data.</li>

<li><a rel="nofollow" class="external text" href="http://larfis.polymtl.ca/index.php/en/links/openfilters">OpenFilters</a> is a program for designing optical filters.</li>

<li><a rel="nofollow" class="external text" href="https://github.com/gevero/py_matrix">Py_matrix</a> is an open source Python code that implements the transfer-matrix method for multilayers with arbitrary dielectric tensors. It was especially created for plasmonic and magnetoplasmonic calculations.</li>

<li><a rel="nofollow" class="external text" href="https://ncnr.nist.gov/instruments/magik/calculators/calcR_d3_dark.html">In-browser calculator and fitter</a> Javascript interactive reflectivity calculator using matrix method and Nevot-Croce roughness approximation (calculation kernel converted from C via <a href="Emscripten" title="Emscripten">Emscripten</a>)</li></ul></div><div><div class="zim-footer">

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