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<h2 class="hd hd-2 unit-title">Introduction to Polarization and Polarizers</h2>
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The next property of light that we discuss is polarization—the orientation of the electric field vector. Although, in principle, we could also use the orientation of the magnetic field, the word "polarization" is defined to denote the electric field direction. </p><p>
We begin by writing the most general possible form for a propagating electric field, and explore the properties related to the direction of the total field vector. We see that it can change its orientation in time and space! </p>
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<h2 class="hd hd-2 unit-title">L30v1: Electric Field in EM wave</h2>
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<h3 class="hd hd-2">L30v1: Electric Field in EM wave</h3>
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<h2 class="hd hd-2 unit-title">L30Q1: Formalism</h2>
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Formalism - part a
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<p>
We have introduced notation to express the general form of a plane wave as a set of vectors&#8212;we have seen this concise notation before! </p>
<p>
The general solution for a wave moving in the [mathjaxinline]+\hat{z}[/mathjaxinline] direction is: </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\vec{E}(z,t) = \mathrm{Re}\left[\left(\psi _{1} \hat{x} + \psi _{2} \hat{y}\right) e^{i(kz - \omega t)}\right][/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
with [mathjaxinline]\psi _{1}=A_{1}e^{i\phi _{1}}[/mathjaxinline] and [mathjaxinline]\psi _{2}=A_{2}e^{i\phi _{2}}[/mathjaxinline]. </p>
<p>
The concise form is given by: </p>
<table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\mathrm{\textbf{E}} = \mathrm{Re}\left[\mathrm{\textbf{Z}}e^{i(kz - \omega t)}\right][/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
where both [mathjaxinline]\mathrm{\textbf{E}}[/mathjaxinline] and [mathjaxinline]\mathrm{\textbf{Z}}[/mathjaxinline] are vectors with [mathjaxinline]x[/mathjaxinline] and [mathjaxinline]y[/mathjaxinline] components (which could be imaginary in the case of [mathjaxinline]\mathrm{\textbf{Z}}[/mathjaxinline]). </p>
<p><b class="bfseries">(Part a)</b> Consider the following function: </p>
<table id="a0000000004" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\vec{E}(z,t) = E_{1}\cos (kz - \omega t)\hat{x}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
Determine [mathjaxinline]\mathrm{\textbf{Z}}[/mathjaxinline]. Express your answer as a vector whose elements depend on the variables defined by [mathjaxinline]\vec{E}(z,t)[/mathjaxinline], which could include <code>E_1</code> for [mathjaxinline]E_{1}[/mathjaxinline], <code>k</code>, <code>z</code>, <code>t</code>, or <code>omega</code> for [mathjaxinline]\omega[/mathjaxinline]. Use relevant numerical variables including <code>i</code> or <code>pi</code>. </p>
<p>
Recall, we are using the input convention <code>[[A],[B]]</code> for vectors of the form: </p>
<table id="a0000000005" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\begin{pmatrix} A\\ B\\ \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<p style="display:inline">[mathjaxinline]\textbf{Z}=[/mathjaxinline]</p>
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<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
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<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
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<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
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<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
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<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
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<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
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<code>abs, ln, sqrt</code>
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<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
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<code>sin, cos, tan, sec, csc, cot</code>
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<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
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<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
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<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
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<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
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<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
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<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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Formalism - part b
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<p><b class="bfseries">(Part b)</b> Consider the following function: </p>
<table id="a0000000007" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\vec{E}(z,t) = E_{2}\sin (kz - \omega t)\hat{y}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
Determine [mathjaxinline]\mathrm{\textbf{Z}}[/mathjaxinline]. Express your answer as a vector whose elements depend on the variables defined by [mathjaxinline]\vec{E}(z,t)[/mathjaxinline], which could include <code>E_2</code> for [mathjaxinline]E_{2}[/mathjaxinline], <code>k</code>, <code>z</code>, <code>t</code>, or <code>omega</code> for [mathjaxinline]\omega[/mathjaxinline]. Use relevant numerical variables including <code>i</code> or <code>pi</code>. </p>
<p style="display:inline">[mathjaxinline]\textbf{Z}=[/mathjaxinline]</p>
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<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
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<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
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<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
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<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
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enter <code> 2+3*2 </code> for 8 </td>
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<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
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enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
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<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
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<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
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<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
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<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
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<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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<h2 class="hd hd-2 unit-title">L30Q2: Polarization States</h2>
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Polarization States - part a
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<p>
A wave is linearly polarized if its complex amplitude can be expressed in the following form: </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\mathrm{\textbf{Z}}_{0} = E_{0} \begin{pmatrix} 1\\ 0\\ \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
where we are using the subscript [mathjaxinline]0[/mathjaxinline] to denote this specific state, which is polarized in the x-direction. </p>
<p>
Note, that the rotation matrix [mathjaxinline]\mathrm{\textbf{R}}[/mathjaxinline] can rotate a linearly polarized state through an angle [mathjaxinline]\theta[/mathjaxinline] and the wave is <i class="itshape">still</i> linearly polarized (this corresponds to a change of coordinates): </p>
<table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\mathrm{\textbf{R}} = \begin{pmatrix} \cos \theta &amp; &amp; -\sin \theta \\ \sin \theta &amp; &amp; \cos \theta \\ \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<table id="a0000000004" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
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<td class="equation" style="width:80%; border:none">[mathjax]\mathrm{\textbf{R}}\cdot \mathrm{\textbf{Z}} = E_{0} \begin{pmatrix} \cos \theta \\ \sin \theta \\ \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<p>
The angle that the vector [mathjaxinline]\mathrm{\textbf{Z}}[/mathjaxinline] makes with respect to the x-axis is the angle of polarization. This angle can be determined through the atan formula (where [mathjaxinline]y[/mathjaxinline] and [mathjaxinline]x[/mathjaxinline] are the components of the vector): </p>
<table id="a0000000005" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
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<td class="equation" style="width:80%; border:none">[mathjax]\mathrm{atan}(x,y)= \begin{cases} \arctan (\frac{y}{x}) &amp; \mathrm{if}\, x\gt 0\\ \arctan (\frac{y}{x}) + \pi &amp; \mathrm{if}\, x\lt 0\, \mathrm{and}\, y\geq 0\\ \arctan (\frac{y}{x}) - \pi &amp; \mathrm{if}\, x\lt 0\, \mathrm{and}\, y\lt 0\\ +\frac{\pi }{2} &amp; \mathrm{if}\, x=0\, \mathrm{and}\, y\gt 0\\ -\frac{\pi }{2} &amp; \mathrm{if}\, x=0\, \mathrm{and}\, y\lt 0\\ \mathrm{undefined} &amp; \mathrm{if}\, x=0\, \mathrm{and}\, y=0 \\ \end{cases}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<p>
Indicate whether each of the following polarization states represents a linearly polarized wave. If so, indicate the angle of polarization in degrees. If not, give <code>NA</code> for [mathjaxinline]\theta[/mathjaxinline]. </p>
<p><b class="bfseries">(Part a)</b> The state given by: </p>
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<td class="equation" style="width:80%; border:none">[mathjax]\mathrm{\textbf{Z}} = E_{0} \begin{pmatrix} 1\\ 1\\ \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<p style="display:inline">Is it linearly polarized?</p>
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<p style="display:inline">[mathjaxinline]\theta =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\mathrm{deg}[/mathjaxinline]</p>
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<code>2/3</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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Polarization States - part b
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<p><b class="bfseries">(Part b)</b> The state given by: </p>
<table id="a0000000007" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\mathrm{\textbf{Z}} = E_{0} \begin{pmatrix} 1\\ \dfrac {1}{2}\\ \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
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<p>
<p style="display:inline">Is it linearly polarized?</p>
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<p style="display:inline">[mathjaxinline]\theta =[/mathjaxinline] </p>
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<div id="display_lect_19_03_1b_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_lect_19_03_1b_3_1_dynamath" name="input_lect_19_03_1b_3_1_dynamath"/>
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</div></div>
<p style="display:inline">[mathjaxinline]\mathrm{deg}[/mathjaxinline]</p>
</p>
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<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
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<code>2520</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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<h2 class="hd hd-2 unit-title">L30Q3: Circularly Polarized Waves</h2>
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Circularly polarized waves
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For circularly polarized light, we stated that the total electric field vector will "precess" in the [mathjaxinline]xy[/mathjaxinline]-plane as a function of time. </p>
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Which of the following will INCREASE the rate of precession? Select ALL that apply. </p>
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<text>c) increase the frequency of the wave</text>
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<text>d) decrease the frequency of the wave</text>
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<text>e) increase the amplitude of each component of the wave equally</text>
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<text>f) decrease the amplitude of each component of the wave equally</text>
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<h2 class="hd hd-2 unit-title">L30v4: Eliptically Polarized Light</h2>
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<h2 class="hd hd-2 unit-title">L30Q4: Creation of Elliptically Polarized Waves</h2>
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Creation of elliptically polarized waves
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We have seen that a circularly polarized wave is created by two orthogonal plane waves with equal amplitude, but a phase shift of [mathjaxinline]\pi /2[/mathjaxinline]. The polarization state of the system can be written as: </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
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<td class="equation" style="width:80%; border:none">[mathjax]\mathrm{\textbf{Z}} = E_{0} \begin{pmatrix} 1\\ \pm i\\ \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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Beginning with a circularly polarized wave, which of the following would generate an elliptically polarized wave? Select ALL that apply. </p>
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<text>a) change the frequency of the wave</text>
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<text>b) change the amplitude of each field component by the same magnitude</text>
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<text>c) change amplitude of only one field component</text>
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<text>d) change the phase between the field components by an amount [mathjaxinline]\delta \lt \pi /2[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">L30Q5: Different Polarization States [WITH SIMULATION]</h2>
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Different Polarization States
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Each of the following plots shows the normalized magnitude of the [mathjaxinline]x[/mathjaxinline] and [mathjaxinline]y[/mathjaxinline] components of the electric field vector over one complete period of oscillation. </p>
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In the questions below, indicate which of the diagrams above corresponds to the polarization state that is given and state the "type" of polarization: linear, circular, or elliptical. Let [mathjaxinline]E_{0}=1[/mathjaxinline]. </p>
<p><b class="bfseries">(Part a)</b> Polarization State </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\mathrm{\textbf{Z}}_{1} = \frac{1}{\sqrt {2}} \begin{pmatrix} 1\\ -1\\ \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<p><b class="bfseries">(Part i)</b> Which diagram depicts this polarization state? <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="inputtype option-input ">
<select name="input_lect_19_05_2_1" id="input_lect_19_05_2_1" aria-describedby="status_lect_19_05_2_1">
<option value="option_lect_19_05_2_1_dummy_default">Select an option</option>
<option value="a)"> a)</option>
<option value="b)"> b)</option>
<option value="c)"> c)</option>
<option value="d)"> d)</option>
<option value="e)"> e)</option>
<option value="f)"> f)</option>
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<span class="status unanswered" id="status_lect_19_05_2_1" data-tooltip="Not yet answered.">
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<p class="answer" id="answer_lect_19_05_2_1"/>
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<p><b class="bfseries">(Part ii)</b> What is the type of polarization? <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="inputtype option-input ">
<select name="input_lect_19_05_3_1" id="input_lect_19_05_3_1" aria-describedby="status_lect_19_05_3_1">
<option value="option_lect_19_05_3_1_dummy_default">Select an option</option>
<option value="a) linear"> a) linear</option>
<option value="b) circular"> b) circular</option>
<option value="c) elliptical"> c) elliptical</option>
</select>
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<span class="status unanswered" id="status_lect_19_05_3_1" data-tooltip="Not yet answered.">
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<p class="answer" id="answer_lect_19_05_3_1"/>
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<p><b class="bfseries">(Part b)</b> Polarization State </p>
<table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\mathrm{\textbf{Z}}_{2} = \frac{1}{\sqrt {2}} \begin{pmatrix} i\\ 1\\ \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p><b class="bfseries">(Part i)</b> Which diagram depicts this polarization state? <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="inputtype option-input ">
<select name="input_lect_19_05_4_1" id="input_lect_19_05_4_1" aria-describedby="status_lect_19_05_4_1">
<option value="option_lect_19_05_4_1_dummy_default">Select an option</option>
<option value="a)"> a)</option>
<option value="b)"> b)</option>
<option value="c)"> c)</option>
<option value="d)"> d)</option>
<option value="e)"> e)</option>
<option value="f)"> f)</option>
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<span class="status unanswered" id="status_lect_19_05_4_1" data-tooltip="Not yet answered.">
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<p class="answer" id="answer_lect_19_05_4_1"/>
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<p><b class="bfseries">(Part ii)</b> What is the type of polarization? <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><div class="inputtype option-input ">
<select name="input_lect_19_05_5_1" id="input_lect_19_05_5_1" aria-describedby="status_lect_19_05_5_1">
<option value="option_lect_19_05_5_1_dummy_default">Select an option</option>
<option value="a) linear"> a) linear</option>
<option value="b) circular"> b) circular</option>
<option value="c) elliptical"> c) elliptical</option>
</select>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_19_05_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<p class="answer" id="answer_lect_19_05_5_1"/>
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<div class="solution-span">
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<p><b class="bfseries">(Part c)</b> Polarization State </p>
<table id="a0000000004" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
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<td class="equation" style="width:80%; border:none">[mathjax]\mathrm{\textbf{Z}}_{3} = \frac{1}{\sqrt {2}} \begin{pmatrix} 1\\ e^{i\pi /4}\\ \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p><b class="bfseries">(Part i)</b> Which diagram depicts this polarization state? <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 5" role="group"><div class="inputtype option-input ">
<select name="input_lect_19_05_6_1" id="input_lect_19_05_6_1" aria-describedby="status_lect_19_05_6_1">
<option value="option_lect_19_05_6_1_dummy_default">Select an option</option>
<option value="a)"> a)</option>
<option value="b)"> b)</option>
<option value="c)"> c)</option>
<option value="d)"> d)</option>
<option value="e)"> e)</option>
<option value="f)"> f)</option>
</select>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_19_05_6_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<p class="answer" id="answer_lect_19_05_6_1"/>
</div></div> </p>
<p><b class="bfseries">(Part ii)</b> What is the type of polarization? <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 6" role="group"><div class="inputtype option-input ">
<select name="input_lect_19_05_7_1" id="input_lect_19_05_7_1" aria-describedby="status_lect_19_05_7_1">
<option value="option_lect_19_05_7_1_dummy_default">Select an option</option>
<option value="a) linear"> a) linear</option>
<option value="b) circular"> b) circular</option>
<option value="c) elliptical"> c) elliptical</option>
</select>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_19_05_7_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<p class="answer" id="answer_lect_19_05_7_1"/>
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<h2>Run the Interactive Python Visualization that Generated the Plots Above!</h2><p>The widget is run in a Jupyter notebook, accessible through the button below. <b>NOTE: The notebook may take up to 3 mintues to load! Please be patient!</b></p><p><div align="center"><a href="https://mybinder.org/v2/gh/mitx-803/vis/master?filepath=polarization_states_1.ipynb" class="btn btn-primary" target="_blank" style="color:#FFFFFF;">ACCESS JUPYTER NOTEBOOK HERE</a></div></p><p><div class="hideshowbox"><h4 onclick="hideshow(this);" style="margin: 0px">How to Run Jupyter Notebooks (expand this section if you need a reminder!)<span class="icon-caret-down toggleimage"/></h4><div class="hideshowcontent"><p><h3>Running Notebooks on an External Server</h3></p><p>To access a simulation, click the "ACCESS JUPYTER NOTEBOOK HERE" button. This will bring you to a loading page, hosted by <i class="itshape">mybinder.org</i> (the loading time is anywhere from 20 seconds to 3 minutes). The Jupyter notebooks are run externally to the course, on a server which runs an instance of Python. There is no need to install Python or related dependencies!</p><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_06.html" width="720" height="590" scrolling="no" frameborder="0"/></div><p><h3>Initializing the Program</h3></p><p>Once loaded, you will see a Jupyter notebook in your browser! You will have to click a button to initialize the program. The button is indicated in the instructions within the notebook, and also shown below.</p><div align="center"><img width="700" src="/assets/courseware/v1/3f6c044fc06f79d82bb2e8a97f7dd11a/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_binder_initialize_button.png"/></div><p/><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_07.html" width="720" height="602" scrolling="no" frameborder="0"/></div><p><h3>Instructions and Source Code</h3></p><p>Each notebook has self-contained instructions on how to use the Python simulation. Additionally, you may toggle the button at the bottom of the notebook to view/augment the source code.</p><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_08.html" width="720" height="608" scrolling="no" frameborder="0"/></div><p><h3>Saving/Running Notebooks Locally</h3></p><p>Finally, you can dowload each notebook to run locally. Additionally, you can visit the git repository to download all notebooks in the course. In order to run notebooks locally, you must install Python and its dependencies. We cannot help with this process, but we encourage you to look at the resources below, if you are interested.</p><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_09.html" width="720" height="609" scrolling="no" frameborder="0"/></div><p><h3>External Links</h3><br/> [mathjaxinline]\bullet[/mathjaxinline] git repository: <a href="https://github.com/mitx-803/vis" target="blank">github.com/mitx-803/vis</a><br/> [mathjaxinline]\bullet[/mathjaxinline] information on Jupyter notebooks: <a href="https://jupyter.org/" target="blank">Jupyter Notebooks</a><br/> [mathjaxinline]\bullet[/mathjaxinline] information on installing Python through Anaconda: <a href="https://www.anaconda.com/distribution/" target="blank">Anaconda</a><br/> [mathjaxinline]\bullet[/mathjaxinline] information on the Binder community: <a href="https://mybinder.readthedocs.io/en/latest/" target="blank">Binder</a><br/></p><p><h3>Dependencies</h3></p><p>
The visualizations run on Python 3. Dependencies for running Python code locally (not through Binder) are stated in the git repository, and include (but are not limited to) the following:
<br/> [mathjaxinline]\bullet[/mathjaxinline] scipy
<br/> [mathjaxinline]\bullet[/mathjaxinline] numpy
<br/> [mathjaxinline]\bullet[/mathjaxinline] ipywidgets
<br/> [mathjaxinline]\bullet[/mathjaxinline] nbinteract
<br/> [mathjaxinline]\bullet[/mathjaxinline] matplotlib
<br/> [mathjaxinline]\bullet[/mathjaxinline] pandas
<br/> [mathjaxinline]\bullet[/mathjaxinline] IPython
<br/> [mathjaxinline]\bullet[/mathjaxinline] ffmpeg
<br/> [mathjaxinline]\bullet[/mathjaxinline] jupyter-contrib-nbextensions
<br/>
</p><p>
You will have to find resources that explain how to install these appropriately for your system, if they are not already installed with your Python package.
</p></div><p class="hideshowbottom" onclick="hideshow(this);" style="margin: 0px"><a href="javascript: {return false;}">Show</a></p></div></p><SCRIPT src="/assets/courseware/v1/631e447105fca1b243137b21b9ed6f90/asset-v1:MITx+8.03x+1T2020+type@asset+block/latex2edx.js" type="text/javascript"/><LINK href="/assets/courseware/v1/daf81af0af57b85a105e0ed27b7873a0/asset-v1:MITx+8.03x+1T2020+type@asset+block/latex2edx.css" rel="stylesheet" type="text/css"/><h2>What You Should See</h2><p>When the notebook is initialized, you will see the following visualization. Follow question prompts within the notebook.</p><div align="center"><img width="800" src="/assets/courseware/v1/1e5fa9485f1c2fc344f10faae5878470/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_polarization_states_1.png"/></div><p/>
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