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divider"></i></li><li><a href="../1128/ahl-waves.html">AHL Waves</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Single slit diffraction</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 20 minutes"><i class="fa fa-clock-o"></i> 20'</span> </ol> <article id="main-article"> <p><img alt="" src="../../waves/light.jpg" style="float: left; width: 250px; height: 167px;">Diffraction is the spreading out of a wave when passing through a gap in a boundary. It is optimised when the width of the gap is similar in magnitude to the wavelength of the wave. With a small enough gap we can diffract visible light.</p> <p>X-rays have an even smaller wavelength than visible light. These diffract on an atomic scale, enabling materials scientists to determine the structure of crystals. </p> <hr class="hidden-separator"> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <p>A laser is shone through a rectangular slit onto a screen.</p> <p style="text-align: center;"><img alt="" src="../../waves/laser.jpg" style="width: 377px; height: 201px;"></p> <p>We might expect the screen to be brightest in the centre and for the laser light to reduce in intensity outwards. In fact, the pattern is not so simple: the brightness minimises before increasing again. We can plot a graph of intensity against distance from the centre.</p> <p style="text-align: center;"><img alt="" src="../../waves/diffraction.jpg" style="width: 233px; height: 266px;"></p> <p>The diffraction pattern is formed by summing together to vector components of all parts of the wave, which have travelled different distances to each point on the screen.</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/369636391"></iframe></div> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/369636367"></iframe></div> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/369636464"></iframe></div> <div class="panel panel-turquoise panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Equation</p> </div> </div> <div class="panel-body"> <div> <p>We can determine the position of the first minimum using the approximation equation:</p> <p style="text-align: center;"><span class="math-tex">\(\theta = {\lambda \over a}\)</span></p> <ul> <li><span class="math-tex">\(\theta\)</span> is the angle subtended at the slit by the first minimum and the centre (rad)</li> <li><span class="math-tex">\(\lambda\)</span> is the wavelength of the light (m)</li> <li><span class="math-tex">\(a\)</span> is the width of the slit (m)</li> </ul> <p style="text-align: center;"><img alt="" src="../../waves/path-difference.png" style="width: 250px; height: 142px;"></p> <p>Since the angle is small, we can also state that it is approximately equal to the ratio of the distance to the minimum and the distance from the slit to the screen:</p> <p style="text-align: center;"><span class="math-tex">\(\theta = {y\over D}\)</span></p> <ul> <li><span class="math-tex">\(y\)</span> is the distance between the centre and the first minimum (on either side since symmetrical) (m)</li> <li><span class="math-tex">\(D\)</span> is the distance from the slit to the screen (m)</li> </ul> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/369636686"></iframe></div> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/369636772"></iframe></div> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-turquoise panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Maxima intensity ratios</p> </div> </div> <div class="panel-body"> <div> <p>The central maximum has the highest intensity, <span class="math-tex">\(I_0\)</span>. The next maximum has intensity <span class="math-tex">\(\approx {I_0\over22} \)</span> with the following <span class="math-tex">\(\approx {I_0\over63} \)</span>. The Subject Guide is unclear here, stating that students should know approximate ratios of successive maxima intensities.</p> <p>Wondering where these come from? Check out the section Single Slit Peak Intensities <a href="http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/sinint.html" target="_blank">here</a>.</p> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Essentials</p> </div> </div> <div class="panel-body"> <div> <div class="panel panel-has-colored-body panel-has-border panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Monochromatic light</p> </div> </div> <div class="panel-body"> <div> <p>The equation for the first minimum indicates that the angle is increased by increasing the wavelength of light. Red light produces a broader pattern than green and blue light when passing through the same slit.</p> <p style="text-align: center;"><img alt="" src="../../waves/three-colours.png" style="width: 400px; height: 207px;"></p> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>White light</p> </div> </div> <div class="panel-body"> <div> <p>White light contains all the wavelengths of visible light. Because the angle to the first minimum is affected by wavelength, the colours of light separate.</p> <p style="text-align: center;"><img alt="" src="../../waves/white-light-only.png" style="width: 400px; height: 68px;"></p> <p>This is not evident in the central maxium as all wavelengths have travelled the same distance and so the vector components add to give the original combination. However, outward from the centre, the colours of light separate continuously as their wavelengths form maxima at slightly different angles. Blue light is nearest the centre and red outermost.</p> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test Yourself</p> </div> </div> <div class="panel-body"> <div> <p><em>Use quizzes to practise application of theory.</em></p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#504ed40e"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="504ed40e"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Single slit Diffraction <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="6-344-1127" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Red light is diffracted by a single slit.</p><p>If the amplitude of the source is reduced by a factor <span class="math-tex">\(1\over 2\)</span> the intenity of the peak will be reduced by a factor:</p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(1\over 4\)</span></span></label></p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(1\over8\)</span></span></label></p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(1\over 2\)</span></span></label></p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(1\over16\)</span></span></label></p></div><div class="q-explanation"><p>Intensity is proportional to the square of the amplitude:</p><p style="text-align: center;"><span class="math-tex">\({I_1\over I_2}={A_1\over A_2}^2\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Light is diffracted by a single slit.</p><p style="text-align: center;"><img alt="" height="159" src="../../screenshot-2019-09-08-at-08.45.40.png" width="371"></p><p><span class="math-tex">\(D\)</span> = 1 m</p><p><span class="math-tex">\(y\)</span> = 5 mm</p><p>What is the angle subtended by the first minimum?</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>5° </span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>0.005 rad</span></label> </p><p><label class="radio"> <input type="radio"> <span>0.005°</span></label> </p><p><label class="radio"> <input type="radio"> <span>5 rad</span></label> </p></div><div class="q-explanation"><p><span class="math-tex">\(\theta={5\times 10^{-3}\over 1}\)</span></p><p>This equation relies on the small angle approximation. Therefore the angle is measured in radians.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Light is diffracted by a single slit.</p><p style="text-align: center;"><img alt="" height="159" src="../../screenshot-2019-09-08-at-08.45.40.png" width="371"></p><p><span class="math-tex">\(D\)</span> = 50 cm</p><p><span class="math-tex">\(y\)</span> = 20 cm</p><p>What is the angle subtended by the first minimum?</p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span>22°</span></label> </p><p><label class="radio"> <input type="radio"> <span>22 rad</span></label> </p><p><label class="radio"> <input type="radio"> <span>40°</span></label> </p><p><label class="radio"> <input type="radio"> <span>0.4 rad</span></label> </p></div><div class="q-explanation"><p>In this case the small angle approximation does not apply. Instead we use trigonometry:</p><p><span class="math-tex">\(\tan \theta={y\over D}={20\over 50}\)</span></p><p><span class="math-tex">\(\tan^{-1}0.4=22^\text{o}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Light is diffracted by a single slit.</p><p style="text-align: center;"><img alt="" height="159" src="../../screenshot-2019-09-08-at-08.45.40.png" width="371"></p><p><span class="math-tex">\(D\)</span> = 1 m</p><p><span class="math-tex">\(\theta\)</span> = 0.01 rad</p><p>The width of the central maximum is:</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>1 mm</span></label> </p><p><label class="radio"> <input type="radio"> <span>2 mm</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>2 cm</span></label> </p><p><label class="radio"> <input type="radio"> <span>1 cm</span></label> </p></div><div class="q-explanation"><p>The small angle approximation applies:</p><p><span class="math-tex">\(y=D\times \theta=0.01 \text{ m}\)</span></p><p>The width of central maxmum (see diagram) is <span class="math-tex">\(2y=0.02 \text{ m}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Light is diffracted by a single slit.</p><p style="text-align: center;"><img alt="" height="159" src="../../screenshot-2019-09-08-at-08.45.40.png" width="371"></p><p><span class="math-tex">\(D\)</span> = 1 m</p><p><span class="math-tex">\(λ\)</span> = 400 nm</p><p><span class="math-tex">\(a\)</span> = 0.2 mm</p><p>What is the angle subtended by the first minimum?</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>0.02 rad</span></label> </p><p><label class="radio"> <input type="radio"> <span>0.008 rad</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>0.002 rad</span></label> </p><p><label class="radio"> <input type="radio"> <span>0.004 rad</span></label> </p></div><div class="q-explanation"><p>In this case we are not given the distance <span class="math-tex">\(y\)</span>. Therefore, we use the small angle approximation at the slit for the path difference to cause destructive interference:</p><p><span class="math-tex">\(\theta={\lambda\over a}={400 \times 10^{-9}\over 0.2 \times 10^{-3}} = 0.002 \text{ rad}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Light is diffracted by a single slit.</p><p style="text-align: center;"><img alt="" height="159" src="../../screenshot-2019-09-08-at-08.45.40.png" width="371"></p><p><span class="math-tex">\(D\)</span> = 1 m</p><p><span class="math-tex">\(y\)</span> = 1 mm</p><p>The distance <span class="math-tex">\(D\)</span> is doubled. To keep <span class="math-tex">\(y\)</span> constant, the slit width must be changed by a factor:</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(4\)</span></span></label></p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(1\over4\)</span></span></label></p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(1\over2\)</span></span></label></p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(2\)</span></span></label></p></div><div class="q-explanation"><p>The small angle approximation applies, so <span class="math-tex">\(\sin\theta =\tan\theta\)</span></p><p><span class="math-tex">\({y\over D} = {λ\over a}\)</span></p><p>Since <span class="math-tex">\(y\)</span> and <span class="math-tex">\(\lambda\)</span> are constant, <span class="math-tex">\(D\over a\)</span> must be constant.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Diffraction pattern A is produced with a blue laser then pattern B is produced with a red laser.</p><p style="text-align: center;"><img alt="" height="184" src="../../screenshot-2019-09-08-at-09.56.21.png" width="345"></p><p>What change must have taken place?</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>reduce <em>a</em> or increase <em>D</em></span></label> </p><p><label class="radio"> <input type="radio"> <span>nothing</span></label> </p><p><label class="radio"> <input type="radio"> <span>increase <em>a</em> or <em>D</em></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>increase <em>a</em> or reduce <em>D</em></span></label></p></div><div class="q-explanation"><p>Red light has a longer wavelength than blue light so the pattern should be more spread. This means that <span class="math-tex">\(\theta\)</span> has been reduced between the two productions:</p><p><span class="math-tex">\(\theta={\lambda \over a}\)</span> so <span class="math-tex">\(a\)</span> could have been increased.</p><p>Alternatively, reducing <span class="math-tex">\(D\)</span> would give less distance for the same angle to spread out.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The image is from a GeoGebra simulation showing wavelet amplitudes adding across a single slit.</p><p style="text-align: center;"><img alt="" height="258" src="../../screenshot-2019-09-08-at-10.34.50.png" width="238"></p><p>At the position of the first minimum, the phase difference between the first and last wavelet is:</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(π\)</span></span></label></p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(3π\over2\)</span></span></label></p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(2π\)</span></span></label></p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(π\over 2\)</span></span></label></p></div><div class="q-explanation"><p>When the resultant amplitude is zero, the wavelet vectors would have formed a closed loop. The phase difference is <span class="math-tex">\(2π\)</span>.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div 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