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Jean-Sébastien
2022-03-07 20:40:36 +01:00
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@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-02 Wed 15:45 -->
<!-- 2022-03-07 Mon 20:38 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1098,14 +1098,6 @@ Table of contents
<li class="toc-currentpage">
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li>
</ul>
@@ -1615,7 +1607,7 @@ Table of contents
</ul>
</details>
</nav>
<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emdm.html">Electromagnetodynamics in Matter</a></li><li><a class="breadcrumb-link"href="emdm_emwm.html">Electromagnetic Waves in Matter</a></li><li><a class="breadcrumb-link"href="emdm_emwm_refl.html">Reflection and Transmission</a></li><li>Oblique Incidence</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refl_ni.html">Normal Incidence&emsp;<small>[emdm.emwm.refl.ni]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_refl_Fe.html">Fresnel's Equations&emsp;<small>[emdm.emwm.refl.Fe]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm_refl.html">Reflection and Transmission&emsp;<small>[emdm.emwm.refl]</small></a></li></ul><div id="outline-container-emdm_emwm_refl_oi" class="outline-5">
<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emdm.html">Electromagnetodynamics in Matter</a></li><li><a class="breadcrumb-link"href="emdm_emwm.html">Electromagnetic Waves in Matter</a></li><li><a class="breadcrumb-link"href="emdm_emwm_refl.html">Reflection and Transmission</a></li><li>Oblique Incidence</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refl_ni.html">Normal Incidence&emsp;<small>[emdm.emwm.refl.ni]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_ad.html">Absorption and Dispersion&emsp;<small>[emdm.emwm.ad]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm_refl.html">Reflection and Transmission&emsp;<small>[emdm.emwm.refl]</small></a></li></ul><div id="outline-container-emdm_emwm_refl_oi" class="outline-5">
<h5 id="emdm_emwm_refl_oi">Oblique Incidence<a class="headline-permalink" href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<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"/>
<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"/>
@@ -1642,14 +1634,28 @@ Transmitted wave:
{\boldsymbol B}_T ({\boldsymbol r},t) = \frac{1}{v_2} \hat{\boldsymbol k}_T \times {\boldsymbol E}_{T} ({\boldsymbol r}, t).
\]
All waves have the same frequency \(\omega\). Since \(\omega = k v\), the three wavevectors are related by
</p>
<div class="eqlabel" id="org862531f">
<p>
<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">
<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"/>
<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"/>
</svg></a>
</p>
<div class="alteqlabels" id="org4435847">
</div>
</div>
<p>
\[
k_I v_1 = k_R v_1 = k_T v_2 ~~\longrightarrow~~ k_I = k_R = \frac{v_2}{v_1} k_T = \frac{n_1}{n_2} k_T
\label{eq:RTObliquek}
\tag{RTobliquek}\label{RTobliquek}
\]
</p>
<p>
These forms for incident, reflected and transmitted wave can be substituted in the boundary conditions (\ref{eq:EMBdryCondAtMediumInterface}). Since these must be valid for any \(x\) and \(y\) (on the interface at \(z=0\)), we must have that the \(x\) and \(y\) components of the wavevectors coincide for all the waves:
These forms for incident, reflected and transmitted wave can be substituted in the boundary conditions <a href="./emdm_Me_bc.html#disc_nfc">disc_nfc</a>. Since these must be valid for any \(x\) and \(y\) (on the interface at \(z=0\)), we must have that the \(x\) and \(y\) components of the wavevectors coincide for all the waves:
\[
k_{I_x} = k_{R_x} = k_{T_x}, \hspace{10mm}
k_{I_y} = k_{R_y} = k_{T_y}
@@ -1659,29 +1665,29 @@ These forms for incident, reflected and transmitted wave can be substituted in t
<p>
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
</p>
<div class="core div" id="orge681e56">
<div class="core div" id="org6f173e3">
<p>
{\bf First law of reflection:}
the incident, reflected and transmitted wave vectors form a plane (called the plane of incidence) which also includes the normal to the surface.
<b>First law of reflection:</b>
the incident, reflected and transmitted wave vectors form a plane (called the plane of incidence) which also includes the normal to the surface.
</p>
</div>
<p>
Specializing (\ref{eq:RTObliquek}) to our notations, we have
Specializing <a href="./emdm_emwm_refl_oi.html#RTobliquek">RTobliquek</a> to our notations, we have
\[
k_I \sin \theta_I = k_R \sin \theta_R = k_T \sin \theta_T
\]
with the incidence (\(\theta_I\)) and reflection (\(\theta_R\)) angles
and the angle of refraction (\(\theta_T\)) obey the following laws:
</p>
<div class="core div" id="orgfbf5032">
<div class="core div" id="orgbf9459c">
<p>
{\bf Law of reflection}
\[
<b>Law of reflection</b>
\[
\theta_I = \theta_R
\]
{\bf Law of refraction (Snell's law)}
\[
<b>Law of refraction (Snell's law)</b>
\[
n_1 \sin \theta_I = n_2 \sin \theta_T
\]
</p>
@@ -1691,13 +1697,25 @@ and the angle of refraction (\(\theta_T\)) obey the following laws:
<p>
This takes care of the spatially-dependent exponential factors in the boundary conditions. The coefficients must further obey
</p>
<div class="eqlabel" id="org8d76e61">
<p>
<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">
<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"/>
<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"/>
</svg></a>
</p>
<div class="alteqlabels" id="org2c4d03c">
</div>
</div>
\begin{align}
\varepsilon_1 \left({\boldsymbol E}_{0_I} + {\boldsymbol E}_{0_R} \right)_z &amp;= \varepsilon_2 \left({\boldsymbol E}_{0_T} \right)_z,
&amp; \hspace{10mm}
\left({\boldsymbol B}_{0_I} +{\boldsymbol B}_{0_R} \right)_z &amp;= \left({\boldsymbol B}_{0_T}\right)_z, \nonumber \\
\left( {\boldsymbol E}_{0_I} + {\boldsymbol E}_{0_R} \right)_{x,y} &amp;= \left({\boldsymbol E}_{0_T}\right)_{x,y},
&amp; \frac{1}{\mu_1} \left({\boldsymbol B}_{0_I} + {\boldsymbol B}_{0_R} \right)_{x,y} &amp;= \frac{1}{\mu_2} \left({\boldsymbol B}_{0_T}\right)_{x,y}.
\label{eq:EMBdryCondAtMediumInterface:amp}
\tag{EBRT}\label{EBRT}
\end{align}
<p>
@@ -1709,8 +1727,8 @@ The two cases of polarization parallel and perpendicular to the plane of inciden
</p>
<p>
\paragraph{Polarization in plane of incidence:}
in this case the first equation of (\ref{eq:EMBdryCondAtMediumInterface:amp}) gives
<b>Polarization in plane of incidence</b>:
in this case the first equation of <a href="./emdm_emwm_refl_oi.html#EBRT">EBRT</a> gives
\[
\varepsilon_1 \left(-E_{0_I} \sin \theta_I + E_{0_R} \sin \theta_R \right) = -\varepsilon_2 E_{0_T} \sin \theta_T.
\]
@@ -1732,13 +1750,28 @@ while the third equation becomes
\]
Writing everything in terms of the incident amplitude, we get
</p>
<div class="main div" id="orgcf2803c">
<div class="main div" id="org3866053">
<p>
<b>Fresnel's equations for reflection and transmission amplitudes (parallel case)</b>
</p>
<div class="eqlabel" id="org31e8ee3">
<p>
<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">
<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"/>
<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"/>
</svg></a>
</p>
<div class="alteqlabels" id="org1f91c78">
</div>
</div>
<p>
{\bf Fresnel's equations for reflection and transmission amplitudes (parallel case)}
\[
E_{0_R} = \frac{\alpha - \beta}{\alpha + \beta} E_{0_I},
\hspace{10mm}
E_{0_T} = \frac{2}{\alpha + \beta} E_{0_I}
\tag{Fresnel}\label{Fresnel}
\]
</p>
@@ -1749,15 +1782,28 @@ Amplitudes for transmitted and reflected wave: depend on angle of incidence:
\[
\alpha = \frac{\sqrt{1 - \sin^2 \theta_T}}{\cos \theta_I} = \frac{\left[1 - \left(\frac{n_1}{n_2}\right)^2 \sin^2 \theta_I\right]^{1/2}}{\cos \theta_I}
\]
Behaviour: for \(\theta_I = 0\) we recover (\ref{Gr(9.82)}).
Behaviour: for \(\theta_I = 0\) we recover <a href="./emdm_emwm_refl_ni.html#ERT">ERT</a>.
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
</p>
<div class="main div" id="org7503b43">
<div class="main div" id="org328f2a5">
<div class="eqlabel" id="org3b9cec2">
<p>
{\bf Brewster's angle {\it (at which the reflected wave amplitude vanishes)}}
\[
\theta_B = \arcsin \left[ \frac{1 - \beta^2}{(n_1/n_2)^2 - \beta^2} \right]^{1/2}
\]
<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">
<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"/>
<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"/>
</svg></a>
</p>
<div class="alteqlabels" id="orgf448df4">
</div>
</div>
<p>
<b>Brewster's angle</b> <i>(at which the reflected wave amplitude vanishes)</i>
\[
\theta_B = \arcsin \left[ \frac{1 - \beta^2}{(n_1/n_2)^2 - \beta^2} \right]^{1/2}
\tag{Brewster}\label{Brewster}
\]
</p>
</div>
@@ -1785,9 +1831,7 @@ Of course, we get \(R + T = 1\) as expected.
</div>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refl_ni.html">Normal Incidence&emsp;<small>[emdm.emwm.refl.ni]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_refl_Fe.html">Fresnel's Equations&emsp;<small>[emdm.emwm.refl.Fe]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm_refl.html">Reflection and Transmission&emsp;<small>[emdm.emwm.refl]</small></a></li></ul>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refl_ni.html">Normal Incidence&emsp;<small>[emdm.emwm.refl.ni]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_ad.html">Absorption and Dispersion&emsp;<small>[emdm.emwm.ad]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm_refl.html">Reflection and Transmission&emsp;<small>[emdm.emwm.refl]</small></a></li></ul>
<br>
<hr>
<div class="license">
@@ -1802,7 +1846,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p>
</div>