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<h2 class="hd hd-2 unit-title">Introduction to Diffraction and Interference</h2>
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<p>We have seen that Maxwell's Equations produced a wave equation and solve for light as a wave, but we have not really explored the properties of waves. In this lesson we begin to do that by examining interference and diffraction, two properties that are unique to waves (as opposed to particles).</p>
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<p>Textbook Links</p>
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<li><a href="https://openlearninglibrary.mit.edu/courses/course-v1:MITx+8.02.3x+1T2019/pdfbook/0/#viewer-frame" target="[object Object]">Chapter 14</a></li>
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<h2 class="hd hd-2 unit-title">L38v1: Interference</h2>
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<h3 class="hd hd-2">L38v1: Interference</h3>
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<h2 class="hd hd-2 unit-title">L38Q1: Double Slit Path Difference</h2>
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Double Slit Path Difference
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Coherent monochromatic plane waves impinge on two apertures separated by a distance [mathjaxinline]d[/mathjaxinline]. An approximate formula for the path length difference [mathjaxinline]\delta[/mathjaxinline] between the two rays shown is </p>
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<text> [mathjaxinline]d\cos (\theta )[/mathjaxinline]</text>
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<text> [mathjaxinline]L\cos (\theta )[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">L38Q2: Double Slit Assumptions</h2>
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Double Slit Assumptions
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<p>The picture shows how light of wavelength \(\lambda\) passes through two slits a distance \(d\) apart, which produces an interference pattern on the screen. It is possible to derive that interference maxima correspond to angles \(\theta\) that satisfy the equation \(d \sin \theta = m \lambda \) where \(m\) is an integer.</p>
<p>For this equation to be valid which of the following conditions must hold?</p>
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<text>(a) \(\vec{r}_1\) is nearly parallel to \(\vec{r}_2 \)</text>
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<text>(b) the small angle approximation (\(\sin(\theta)\approx \theta\)) is valid</text>
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<text>(c) \(y \ll L \)</text>
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<text>(d) light from the two slits S\(_1\) and S\(_2\) must be coherent (i.e. in phase)</text>
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<text>(e) for constructive interference to occur at \(y\) the phase of light from S\(_1\) must differ from that from \({\rm S}_2\) by an integer multiple of \(\pi\)</text>
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<h2 class="hd hd-2 unit-title">L38Q3: Double Slit First Maximum</h2>
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Double Slit First Maximum
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Coherent monochromatic waves of wavelength [mathjaxinline]\lambda[/mathjaxinline] impinge on two apertures separated by a distance [mathjaxinline]d[/mathjaxinline]. An approximate formula for the location of the first maximum above the central maximum (found at [mathjaxinline]y=0[/mathjaxinline]) on a screen a distance [mathjaxinline]L[/mathjaxinline] away is </p>
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<text> [mathjaxinline]y_1 = (1/2)\lambda L / d[/mathjaxinline]</text>
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<text> [mathjaxinline]y_1 = (1/2)\lambda d / L[/mathjaxinline]</text>
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<text> [mathjaxinline]y_1 = (1/2)L / \lambda d[/mathjaxinline]</text>
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<text> [mathjaxinline]y_1 = (1/2)\lambda d / \lambda L[/mathjaxinline]</text>
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<text> [mathjaxinline]y_1 = \lambda d / L[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">L38v3: Red Laser Interference</h2>
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<h2 class="hd hd-2 unit-title">L38Q4: Change the Slit Separation</h2>
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Change the Slit Separation
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<p><b>(Part a)</b> Suppose you have an interference pattern produced by red light from two slits \(d=0.1 \, \mathrm{mm}\) apart. What will happen to the spacing between the fringes on a screen \(1 \,\mathrm{ m}\) from the slits if you reduce the distance between the slits, \(d\), while otherwise keeping the slits the same?</p>
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<p><b>(Part b)</b> What will happen to the intensity of the fringes when \(d\) gets smaller?</p>
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<option value="gets larger"> gets larger</option>
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<h2 class="hd hd-2 unit-title">L38v4: Diffraction</h2>
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<h2 class="hd hd-2 unit-title">L38v5: Red Laser Diffraction</h2>
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<h2 class="hd hd-2 unit-title">L38Q5: Diffraction Limit</h2>
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Diffraction Limit
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Using diffraction is a useful technique for measuring the size of small objects. </p>
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Is there a lower limit for the size of objects that can be measured this way? </p>
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<text> Yes - and if we used blue light instead of red light we can measure even smaller objects than the ones we measure using red light</text>
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<h2 class="hd hd-2 unit-title">L38Q6: Interference and Diffraction</h2>
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Interference and Diffraction
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Coherent monochromatic plane waves impinge on two long narrow apertures (width [mathjaxinline]a[/mathjaxinline]) that are separated by a distance [mathjaxinline]d[/mathjaxinline] with [mathjaxinline]d &gt; a[/mathjaxinline]. </p>
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The resulting pattern on a screen far away is shown above, with distantly-spaced zeroes of the envelope, as indicated by the length [mathjaxinline]X[/mathjaxinline] above, and closely-spaced zeroes of the rapidly varying fringes, as indicated by the length [mathjaxinline]Y[/mathjaxinline] above. </p>
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Which length in the pattern above is due to the finite width [mathjaxinline]a[/mathjaxinline] of the apertures? <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_checkpoint_w15_3_2_1">
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<input type="radio" name="input_checkpoint_w15_3_2_1" id="input_checkpoint_w15_3_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="checkpoint_w15_3_2_1-choice_4-label" for="input_checkpoint_w15_3_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w15_3_2_1"> <text> Neither [mathjaxinline]X[/mathjaxinline] nor [mathjaxinline]Y[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">L38Q7: Ratio of Diffraction and Interference Peaks</h2>
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Ratio of Diffraction and Interference Peaks
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Light from a coherent monochromatic laser of wavelength [mathjaxinline]\lambda[/mathjaxinline] is incident on double slits separated by a distance [mathjaxinline]d[/mathjaxinline]. Each slit has width [mathjaxinline]a[/mathjaxinline] and the slits are placed a distance [mathjaxinline]L[/mathjaxinline] from a screen. The diffraction/interference intensity [mathjaxinline]I(y)[/mathjaxinline] is plotted in the figure on the right below as a function of the distance [mathjaxinline]y[/mathjaxinline] shown on the left figure. </p>
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<img src="/assets/courseware/v1/e36b8c68a8aa4261f4e8f1ac44581cd0/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_finalexam-concept10_fig1.png" width="330"/>
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<img src="/assets/courseware/v1/9841b5a2ff8475d178e83696845ebc7c/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_finalexam-concept10_fig2.png" width="550"/>
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Based on the information in the figures </p>
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<input type="radio" name="input_finalexam-concept10_2_1" id="input_finalexam-concept10_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="finalexam-concept10_2_1-choice_1-label" for="input_finalexam-concept10_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_finalexam-concept10_2_1">
<text> 1. [mathjaxinline]d=9a[/mathjaxinline].</text>
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<input type="radio" name="input_finalexam-concept10_2_1" id="input_finalexam-concept10_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="finalexam-concept10_2_1-choice_2-label" for="input_finalexam-concept10_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_finalexam-concept10_2_1">
<text> 2. [mathjaxinline]a=9d[/mathjaxinline].</text>
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<input type="radio" name="input_finalexam-concept10_2_1" id="input_finalexam-concept10_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="finalexam-concept10_2_1-choice_3-label" for="input_finalexam-concept10_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_finalexam-concept10_2_1">
<text> 3. [mathjaxinline]d=4a[/mathjaxinline].</text>
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<input type="radio" name="input_finalexam-concept10_2_1" id="input_finalexam-concept10_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="finalexam-concept10_2_1-choice_4-label" for="input_finalexam-concept10_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_finalexam-concept10_2_1">
<text> 4. [mathjaxinline]a=4d[/mathjaxinline].</text>
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<input type="radio" name="input_finalexam-concept10_2_1" id="input_finalexam-concept10_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="finalexam-concept10_2_1-choice_5-label" for="input_finalexam-concept10_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_finalexam-concept10_2_1">
<text> 5. [mathjaxinline]d=5a[/mathjaxinline].</text>
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<input type="radio" name="input_finalexam-concept10_2_1" id="input_finalexam-concept10_2_1_choice_6" class="field-input input-radio" value="choice_6"/><label id="finalexam-concept10_2_1-choice_6-label" for="input_finalexam-concept10_2_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_finalexam-concept10_2_1">
<text> 6. [mathjaxinline]a=5d[/mathjaxinline].</text>
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<input type="radio" name="input_finalexam-concept10_2_1" id="input_finalexam-concept10_2_1_choice_7" class="field-input input-radio" value="choice_7"/><label id="finalexam-concept10_2_1-choice_7-label" for="input_finalexam-concept10_2_1_choice_7" class="response-label field-label label-inline" aria-describedby="status_finalexam-concept10_2_1">
<text> 7. [mathjaxinline]d=8a[/mathjaxinline].</text>
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<input type="radio" name="input_finalexam-concept10_2_1" id="input_finalexam-concept10_2_1_choice_8" class="field-input input-radio" value="choice_8"/><label id="finalexam-concept10_2_1-choice_8-label" for="input_finalexam-concept10_2_1_choice_8" class="response-label field-label label-inline" aria-describedby="status_finalexam-concept10_2_1">
<text> 8. [mathjaxinline]a=8d[/mathjaxinline].</text>
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