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href="../1124/thin-film-interference.html">Thin film interference</a></label></li></ul></li></ul></div> <div class="hidden-xs hidden-sm"> <button class="btn btn-default btn-block text-xs-center" data-toggle="modal" data-target="#modal-feedback" style="margin-bottom: 10px"><i class="fa fa-send"></i> Feedback</button> </div> </div> <div class="col-md-9" id="main-column"> <h1 class="page_title"> Resolution <a href="#" class="mark-page-favorite pull-right" data-pid="1123" title="Mark as favorite" onclick="return false;"><i class="fa fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../physics.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../444/oscillations-and-waves.html">Oscillations and waves</a><i class="fa fa-fw fa-chevron-right 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">Resolution</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/resolve-target.jpeg" style="float: left; width: 250px; height: 250px;">Try drawing two small black circles on a piece of paper, a couple of millmetres apart, without showing anyone what you have drawn. Stand across the room from your mates and ask what they see - they'll probably report that they see just one dot. Reducing the distance between you will reveal the truth.</p> <p>Resolution is the separation of images of objects by an optical system. We can calculate the limiting factors.</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"> <div> <p>Two sources of light may be easily resolved, just resolved or unresolved.</p> <p style="text-align: center;"> <img alt="" src="../../waves/resolve-1.png" style="width: 250px; height: 383px;"></p> <p>There are two factors that influence the extent to which two sources of light can be resolved:</p> <ol> <li>The width of the slit</li> <li>The wavelength of the light</li> </ol> <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>Circular aperture diffraction</p> </div> </div> <div class="panel-body"> <div> <p>When light passes through a circular aperture, it diffracts in the same way as when light passes through a narrow slit except that the diffraction is in 2 dimensions. The result is a circle in the centre with rings around the outside.</p> <p style="text-align: center;"><img alt="" src="../../waves/resolve.png" style="width: 220px; height: 199px;"></p> <p style="text-align: center;"><span class="math-tex">\(\theta=1.22{\lambda\over b}\)</span></p> <ul> <li><span class="math-tex">\(\theta\)</span> is the angle subtended by the centre and the first minimum (rad)</li> <li><span class="math-tex">\(\lambda\)</span> is the wavelength of light (m)</li> <li><span class="math-tex">\(b\)</span> is the diameter of the aperture (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/369636007"></iframe></div> </div> </div> <div class="panel-footer"> <div> </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>Rayleigh criterion</p> </div> </div> <div class="panel-body"> <div> <p>The Rayleigh criterion gives the condition for two point sources to be resolved:</p> <p>Two points are resolved if the principal maximum of one diffraction pattern coincides with the first minimum of the other. Look for this point of overlap in this intensity graph below.</p> <p style="text-align: center;"><img alt="" src="../../waves/justresolved.png" style="width: 150px; height: 196px;"></p> <p>The angle is the same as that subtented by the two sources of light when the Rayleigh criterion applies. For a fixed separation of sources, we can calculate the minimum distance at which the sources can be resolved:</p> <p style="text-align: center;"><span class="math-tex">\(\theta = {\text{separation}\over D}\)</span></p> <ul> <li><span class="math-tex">\(D\)</span> is the perpendicular distance from the centre of the two objects to the slit (m)</li> </ul> </div> </div> <div class="panel-footer"> <div> </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>Uses</p> </div> </div> <div class="panel-body"> <div> <p>The ability to resolve sources of light has implications in <a href="../823/o-astrophysics.html" title="O: Astrophysics">Astrophysics</a>, for example distinguishing two stars as separate. A DVD laser light has a shorter wavelength than for CDs so that more information can be stored for the same separation of 'pits'. Electron diffract with a de Broglie wavelength of ~0.1 nm, enabling us to image on an atomic scale.</p> </div> </div> <div class="panel-footer"> <div> </div> </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> <p>Diffraction gratings can be used to resolve two wavelengths of light.</p> <p style="text-align: center;"><span class="math-tex">\(R={\lambda\over \Delta \lambda}=mN\)</span></p> <ul> <li><span class="math-tex">\(R\)</span> is the resolvance of the diffraction grating</li> <li><span class="math-tex">\(\lambda\)</span> is the average wavelength (m)</li> <li><span class="math-tex">\(\Delta \lambda\)</span> is the smallest resolvable wavelength difference (m)</li> <li><span class="math-tex">\(m\)</span> is the order of the diffraction</li> <li><span class="math-tex">\(N\)</span> is the total number of slits illuminated</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/369635693"></iframe></div> <p>Find out more about diffraction gratings and<a href="../1126/multiple-slit-interference.html" title="Multiple slit interference"> interference</a>.</p> </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="#9791f44c"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="9791f44c"> <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%;"> Resolution <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="6-345-1123" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Two points can just be resolved by the camera in the image below.</p><p style="text-align: center;"><img alt="" height="174" src="../../screenshot-2019-09-09-at-05.56.19.png" width="318"></p><p>Which of the lengths indicated would result in better resolution if increased?</p></div><div class="q-answer"><p><label class="radio" style=" float: left; margin-right: 40px; "> <input class="c" type="radio"> <span>A and D</span></label> </p><p><label class="radio" style=" float: left; margin-right: 40px; "> <input type="radio"> <span>A and C</span></label> </p><p><label class="radio" style=" float: left; margin-right: 40px; "> <input type="radio"> <span>D and B</span></label> </p><p><label class="radio" style=" float: left; margin-right: 40px; "> <input type="radio"> <span>B and C</span></label> </p></div><div class="q-explanation"><p>Increasing A will increase the angle subtended by the objects.</p><p>Increasing B will reduce the diffracting angle.</p><p>Increasing D will increase the detail collected within the aperture.</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>Two points can just be resolved by the camera in the image below.</p><p style="text-align: center;"><img alt="" height="174" src="../../screenshot-2019-09-09-at-05.56.19.png" width="318"></p><p>Which of the lengths indicated would not change the resolution?</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>A</span></label> </p><p><label class="radio"> <input type="radio"> <span>B</span></label> </p><p><label class="radio"> <input type="radio"> <span>D</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>C</span></label> </p></div><div class="q-explanation"><p>As C is increased both the image separation and diffraction angle increase.</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>Two points can just be resolved by the camera in the image below.</p><p style="text-align: center;"><img alt="" height="174" src="../../screenshot-2019-09-09-at-05.56.19.png" width="318"></p><p>If the angles are small, then the angle between the principal maximum and first minimum is given by:</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(B\over A\)</span></span></label></p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(D\over C\)</span></span></label></p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(C\over D\)</span></span></label></p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(A\over B\)</span></span></label> </p></div><div class="q-explanation"><p>If just resolved then the Rayligh criterion applies. The diffracting angle is equal to the angle subtended by the objects.</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>Two points can just be resolved by the camera in the image below.</p><p style="text-align: center;"><img alt="" height="174" src="../../screenshot-2019-09-09-at-05.56.19.png" width="318"></p><p>If D is reduced the points would no longer be resolved. The points could be resolved by:</p></div><div class="q-answer"><p><label class="radio" style=" float: left; margin-right: 40px; "> <input type="radio"> <span>increasing B</span></label> </p><p><label class="radio" style=" float: left; margin-right: 40px; "> <input type="radio"> <span>reducing C</span></label> </p><p><label class="radio" style=" float: left; margin-right: 40px; "> <input class="c" type="radio"> <span>increasing A</span></label> </p><p><label class="radio" style=" float: left; margin-right: 40px; "> <input type="radio"> <span>reducing A</span></label> </p></div><div class="q-explanation"><p>Increasing the separation of the objects would mean that the angle subtended was larger. Unfortunately it's impossible for us to move distant stars further apart in the universe!</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>Two points can just be resolved by the camera in the image below.</p><p style="text-align: center;"><img alt="" height="174" src="../../screenshot-2019-09-09-at-05.56.19.png" width="318"></p><p>A = 5 mm</p><p>B = 100 m</p><p>D = 1 cm</p><p>The wavelength of light used is:</p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span>410 nm</span></label> </p><p><label class="radio"> <input type="radio"> <span>600 nm</span></label> </p><p><label class="radio"> <input type="radio"> <span>500 nm</span></label> </p><p><label class="radio"> <input type="radio"> <span>550 nm</span></label> </p></div><div class="q-explanation"><p><span class="math-tex">\(θ = {\text{separation}\over \text{distance to aperture}}= {5 \times 10^{-3} \over 100} = 5 \times 10^{-5} \text{ rad}\)</span></p><p>Since the Rayleigh criterion applies, <span class="math-tex">\(θ = 1.22 {λ\over \text {diameter}}\)</span>. Equating the two expressions:</p><p><span class="math-tex">\(λ = {θ\times \text{diameter}\over 1.22} = 4.1 \times 10^{-7} \text{ m}\)</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 of wavelength 600 nm passes through the 0.5 cm diameter aperture of an optical instrument. What is the minimum angle that two point objects can subtend at the aperture if they are to be resolved?</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>1.46 x 10<sup>-5</sup> rad</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>1.46 x 10<sup>-4</sup> rad</span></label> </p><p><label class="radio"> <input type="radio"> <span>1.46 x 10<sup>-3 </sup> rad</span></label> </p><p><label class="radio"> <input type="radio"> <span>1.46 x 10<sup>-6</sup> rad</span></label> </p></div><div class="q-explanation"><p><span class="math-tex">\(θ =1.22 {\lambda\over \text{diameter}}=1.22 \times {600 \times 10^{-9}\over 0.5 \times 10^{-2}}\)</span></p><p><span class="math-tex">\(\theta\approx1\times 10^3\times 10^{-9}\times 10^2\approx 10^{-4}\)</span></p><p>NB: You need to be prepared to perform these calculations without your calculator!</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 shows light diffracted by the lens in the eye.</p><p style="text-align: center;"><img alt="" height="155" src="../../screenshot-2019-09-09-at-07.17.45.png" width="283"></p><p>Two points will be resolved if they subtend an angle:</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(> 2θ\)</span></span></label></p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(≥ 2θ\)</span></span></label></p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(> θ\)</span></span></label></p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(≥ θ\)</span></span></label></p></div><div class="q-explanation"><p>The graph of intensity and distance indicates that the diagram shown is at the Rayleigh criterion; <span class="math-tex">\(\theta\)</span> is the angle between the principal maximum and the first minimum and the point at which resolution just takes place.</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> 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