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      <text>4.2.4.1 Optical interference fringes due to a glass plate</text>
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      <text><![CDATA[<h4>Optical interference fringes due to a glass plate</h4>

<p>Monochromatic light of intensity <span class="math inline">\(I_0\)</span> propagates in air before impinging on a glass plate (see figure below). If a screen is placed beyond the plate then a pattern is observed including alternate light and dark regions. These are <span><strong>interference</strong></span> fringes.</p>
<figure class="HELM_figure">
<img src="@@PLUGINFILE@@/4.2%20fig%2013%20glass.svg" alt="Light hitting a glass plate at angle alpha, some light is internally reflected in the glass, some goes out at angle alpha" role="presentation" class="img-responsive atto_image_button_text-bottom" width="339" height="197">
</figure>
<p>The intensity <span class="math inline">\(I\)</span> of the light wave transmitted through the plate is given by</p>
\[I=\dfrac{I_0|t|^4}{1+|r|^4-2|r|^2\cos\theta},\]
<p>where <span class="math inline">\(t\)</span> and <span class="math inline">\(r\)</span> are the complex transmission and reflection coefficients. The phase angle <span class="math inline">\(\theta\)</span> is the sum of</p>
<ol type="i">
<li>a phase proportional to the incidence angle <span class="math inline">\(\alpha\)</span>, and</li>
<li>a fixed phase lag due to multiple reflections.</li>
</ol>

<p>The problem is to establish the form of the intensity pattern (i.e. the minima and maxima characteristics of interference fringes due to the plate), and deduce the shape and position <span class="math inline">\(\theta\)</span> of the fringes captured by a screen beyond the plate.</p>

<h4 class="HELM_solution">Solution</h4>

<p>The intensity of the optical wave outgoing from the glass plate is given by</p>
\[I=\dfrac{I_0|t|^4}{1+|r|^4-2|r|^2\cos\theta} \tag{1}\]
<p>The light intensity depends solely on the variable <span class="math inline">\(\theta\)</span> as shown in equation \((1)\), and the objective is to find the values <span class="math inline">\(\theta\)</span> that will minimize and maximize <span class="math inline">\(I\)</span>. The angle <span class="math inline">\(\theta\)</span> is introduced in equation \((1)\) through the function <span class="math inline">\(\cos\theta\)</span> in the denominator. We consider first the maxima of <span class="math inline">\(I\)</span>.</p>

<h5>Light intensity maxima</h5>
<p><span class="math inline">\(I\)</span> is maximum when the denominator is minimum. This condition is obtained when the factor <span class="math inline">\(2|r|\cos\theta\)</span> is maximum due to the minus sign in the denominator. The maxima of <span class="math inline">\(2|r|\cos\theta\)</span> occur when <span class="math inline">\(\cos\theta = +1.\)</span> Values of <span class="math inline">\(\cos\theta = +1\)</span> correspond to <span class="math inline">\(\theta = 2n\pi\)</span> where <span class="math inline">\(n =\ldots -2, -1, 0, 1, 2,\ldots\)</span> and <span class="math inline">\(\theta\)</span> is measured in radians. Setting <span class="math inline">\(\cos\theta = +1\)</span> in equation \((1)\) gives the intensity maxima</p><p>
\[I_{{\max}}=\dfrac{I_0|t|^4}{1+|r|^4-2|r|^2}.\]
</p><p>Since the denominator can be identified as the square of <span class="math inline">\((1-|r|^2)\)</span>, the final result for maximum intensity can be written as</p>
\[I_{\max}=\dfrac{I_0|t|^4}{(1-|r|^2)^2} \tag{2}\]

<h5><strong>Light intensity minima</strong></h5>
<span class="math inline">\(I\)</span> is minimum when the denominator in \((1)\) is maximum. As a result of the minus sign in the denominator, this condition is obtained when the factor <span class="math inline">\(2|r|\cos\theta\)</span> is minimum. The minima of <span class="math inline">\(2|r|\cos\theta\)</span> occur when <span class="math inline">\(\cos \theta = -1.\)</span> Values of <span class="math inline">\(\cos \theta = -1\)</span> correspond to <span class="math inline">\(\theta = \pi (2n+1)\)</span> where <span class="math inline">\(n =\ldots-2, -1, 0, 1, 2,\ldots\)</span>. Setting <span class="math inline">\(\cos\theta = -1\)</span> in equation \((1)\) gives an expression for the intensity minima<p></p>
\[I_{\min}=\dfrac{I_0|t|^4}{1+|r|^4+2|r|^2}.\]
<p>Since the denominator can be recognised as the square of <span class="math inline">\((1 + |r|^2),\)</span> the final result for minimum intensity can be written as</p>
\[I_{\min}=\dfrac{I_0|t|^4}{(1+|r|^2)^2} \tag{3}\]

<h5>Interpretation</h5>
<p>The interference fringes for intensity maxima or minima occur at constant angle <span class="math inline">\(\theta\)</span> and therefore describe concentric rings of alternating light and shadow as sketched in the figure below. From the centre to the periphery of the concentric ring system, the fringes occur in the following order</p>

<ol type="a">
<li><p>a fringe of maximum light at the centre (bright dot for <span class="math inline">\(\theta = 0\)</span>),</p></li>
<li><p>a circular fringe of minimum light at angle <span class="math inline">\(\theta=\pi\)</span>,</p></li>
<li><p>a circular fringe of maximum light at <span class="math inline">\(2\pi\)</span> etc.</p></li>
</ol>

<figure class="HELM_figure">
<img src="@@PLUGINFILE@@/4.2%20fig%2014%20fringes.svg" alt="Circular interference pattern" role="presentation" class="img-responsive atto_image_button_text-bottom" width="187" height="140">
</figure>]]></text>
<file name="4.2 fig 13 glass.svg" path="/" encoding="base64">[encoded data, 16080 characters]</file>
<file name="4.2 fig 14 fringes.svg" path="/" encoding="base64">[encoded data, 8444 characters]</file>
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