Update 2022-03-15 10:07
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Jean-Sébastien
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<!DOCTYPE html>
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<html lang="en">
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<head>
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<!-- 2022-03-07 Mon 20:38 -->
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<!-- 2022-03-15 Tue 08:10 -->
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<meta charset="utf-8">
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<meta name="viewport" content="width=device-width, initial-scale=1">
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<title>Pre-Quantum Electrodynamics</title>
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@@ -1310,10 +1310,6 @@ Table of contents
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</summary>
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<ul>
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<li>
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<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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</li>
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<li>
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<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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</li>
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@@ -1352,6 +1348,10 @@ Table of contents
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<li>
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<a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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</li>
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<li>
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<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
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</li>
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</ul>
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@@ -1635,14 +1635,14 @@ Transmitted wave:
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\]
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All waves have the same frequency \(\omega\). Since \(\omega = k v\), the three wavevectors are related by
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</p>
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<div class="eqlabel" id="org862531f">
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<div class="eqlabel" id="org792ed2b">
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<p>
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<a id="RTobliquek"></a><a href="./emdm_emwm_refl_oi.html#RTobliquek"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
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<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
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</svg></a>
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</p>
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<div class="alteqlabels" id="org4435847">
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<div class="alteqlabels" id="org1997374">
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</div>
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@@ -1665,7 +1665,7 @@ These forms for incident, reflected and transmitted wave can be substituted in t
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<p>
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From now on we will orient the axes so that \({\boldsymbol k}_I\) lies in the \(xz\) plane. This means that \({\boldsymbol k}_R\) and \({\boldsymbol k}_T\) also lie in that plane. This is the
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</p>
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<div class="core div" id="org6f173e3">
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<div class="core div" id="orgbf226b1">
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<p>
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<b>First law of reflection:</b>
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the incident, reflected and transmitted wave vectors form a plane (called the plane of incidence) which also includes the normal to the surface.
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@@ -1680,7 +1680,7 @@ Specializing <a href="./emdm_emwm_refl_oi.html#RTobliquek">RTobliquek</a> to our
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with the incidence (\(\theta_I\)) and reflection (\(\theta_R\)) angles
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and the angle of refraction (\(\theta_T\)) obey the following laws:
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</p>
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<div class="core div" id="orgbf9459c">
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<div class="core div" id="org000f37c">
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<p>
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<b>Law of reflection</b>
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\[
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@@ -1697,14 +1697,14 @@ and the angle of refraction (\(\theta_T\)) obey the following laws:
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<p>
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This takes care of the spatially-dependent exponential factors in the boundary conditions. The coefficients must further obey
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</p>
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<div class="eqlabel" id="org8d76e61">
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<div class="eqlabel" id="orgdc1db4c">
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<p>
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<a id="EBRT"></a><a href="./emdm_emwm_refl_oi.html#EBRT"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
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<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
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</svg></a>
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</p>
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<div class="alteqlabels" id="org2c4d03c">
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<div class="alteqlabels" id="orgf69a0d1">
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</div>
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@@ -1750,18 +1750,18 @@ while the third equation becomes
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\]
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Writing everything in terms of the incident amplitude, we get
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</p>
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<div class="main div" id="org3866053">
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<div class="main div" id="org7e347a7">
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<p>
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<b>Fresnel's equations for reflection and transmission amplitudes (parallel case)</b>
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</p>
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<div class="eqlabel" id="org31e8ee3">
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<div class="eqlabel" id="org5c049e2">
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<p>
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<a id="Fresnel"></a><a href="./emdm_emwm_refl_oi.html#Fresnel"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
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<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
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</svg></a>
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</p>
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<div class="alteqlabels" id="org1f91c78">
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<div class="alteqlabels" id="org50e3086">
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</div>
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@@ -1785,15 +1785,15 @@ Amplitudes for transmitted and reflected wave: depend on angle of incidence:
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Behaviour: for \(\theta_I = 0\) we recover <a href="./emdm_emwm_refl_ni.html#ERT">ERT</a>.
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For grazing waves \(\theta_I \rightarrow \pi/2\) we have that \(\alpha \rightarrow \infty\) and the wave is totally reflected. The most interesting angle is the one at which \(\alpha = \beta\) and the reflected wave has zero amplitude. This is known as
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</p>
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<div class="main div" id="org328f2a5">
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<div class="eqlabel" id="org3b9cec2">
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<div class="main div" id="org73df67c">
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<div class="eqlabel" id="org8295eee">
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<p>
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<a id="Brewster"></a><a href="./emdm_emwm_refl_oi.html#Brewster"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
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<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
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</svg></a>
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</p>
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<div class="alteqlabels" id="orgf448df4">
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<div class="alteqlabels" id="org2f65b27">
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</div>
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@@ -1846,7 +1846,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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</div>
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<div id="postamble" class="status">
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<p class="author">Author: Jean-Sébastien Caux</p>
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<p class="date">Created: 2022-03-07 Mon 20:38</p>
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<p class="date">Created: 2022-03-15 Tue 08:10</p>
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<p class="validation"></p>
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</div>
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