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CU-Boulder PHYS 2020 - Lab 9: Diffraction and Interference

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Physics 2020, Spring 2005 Lab 9 page 1 of 13Lab 9: Diffraction and InterferenceINTRODUCTION & BACKGROUND:Light is an electromagnetic wave, and under the proper circumstances, it exhibits wavephenomena, such as constructive and destructive interference. The wavelength ofvisible light ranges from about 400-750 nm = 0.0004-0.00075 mm, and this wavelength sets the scale for the appearance of wave-like effects. For instance, if a broad beamof light partly passes through a wide slit (i.e. a slit which is very large compared to ),then the wave effects are negligible, the light acts like a ray, and the slit casts ageometrical shadow. However, if the slit is small enough (i.e. around the same size as or smaller), then the wave properties of light become apparent and a diffractionpattern is projected.plane wave wavefrontsFigure 1a. Slit large compared to . Figure 1b. Slit small compared to . Now consider the light from two coherentlight sources a distance d apart. Coherentsources emit light waves that are in phase, orin sync. If we think of light like a water wave,we can imagine that coherent sources emit anidentical succession of wave crests andtroughs, with both emitting crests at the sametime. One way to create such coherentsources is to illuminate a pair of narrow slitswith a distant light source.Interference from two slits: Consider the light rays from the two coherent pointsources made from slits a distance d apart (see fig. 3). We assume that the sourcesare emitting monochromatic (single wavelength) light of wavelength . The rays areemitted in all forward directions, but let’s concentrate on the rays that are emitted in aUniversity of Colorado at Boulder, Department of PhysicsA B Points A and B are coherent sources. Figure 2. Points A and B act likecoherent sources.Physics 2020, Spring 2005 Lab 9 page 2 of 13direction  toward a distant screen ( measured from the normal to the screen, diagrambelow). One of these rays has further to travel to reach the screen, and the pathdifference is given by sind. What happens if this path difference is exactly one wavelength  (or any integernumber of wavelengths)? If you look carefully, this is what is represented in fig. 3.What happens if the path difference is /2, or 3/2, or 5/2, etc.?\A complete analysis yields a pattern of intensity vs. angle that looks like:Intensity0/d/dDouble slit,infinitesimalwidthd2.. 1, 0, = m .)(sin :sin :21mdDarkmdBrightUniversity of Colorado at Boulder, Department of PhysicsFigure 3.Physics 2020, Spring 2005 Lab 9 page 3 of 13What happens to the above interference pattern if d is increased? What if d isdecreased?Geometric simplification: If  is small, then sin (in radians),  and maxima occuron the screen at dm; minima occur at  dm21. As shown below, the angle  (measured from the center of the screen) is related to thedistance x measured on the screen by tan()=x/L, where L is the distance from thescreen to the source of light (the aperture).laseraperturescreenLx/LtanxIf the angle  is small (less than a few degrees), then to an excellent approximation,sin()  tan()   (in radians) so the locations of the interference maxima are given byxLmd.Single slit diffraction: The uniform 2-slit interference pattern shown above is seldomobserved in practice, because real slits always have finite width (not an infinitesimalwidth). We now ask: what is the intensity pattern from a single slit of finite width D?Huygens’ Principle states that the light coming from an aperture is the same as the lightthat would come from a collection of coherent point sources filling the space of theaperture. Its like we constructed the large slit out of a whole set of small slits, alladjacent to each other. To see what pattern the entire array produces, consider firstjust two of these imaginary sources: one at the edge of the slit and one in the center.These two sources are separated by a distance D/2.University of Colorado at Boulder, Department of PhysicsPhysics 2020, Spring 2005 Lab 9 page 4 of 13The path difference for the rays fromthese two sources, going to the screenat an angle , is sin2D, and theserays will interfere destructively if22sinD. But the same can besaid for every pair of sourcesseparated by D/2. Consequently, therays from all the sources filling theaperture cancel in pairs, producingzero intensity on the screen whenD22sin  or, if  is small,D. (First minimum in single slit pattern.)The complete intensity pattern, called a diffraction pattern, looks like...0IntensityDSingle slit. D/ D/ D/ D/ D/ D/[The central maximum is actually much higher than shown here. It was reduced by afactor of 6, for clarity.] The single slit diffraction pattern has minima at     D D D, , , ...2 3 (Minima of single slit pattern.)So the separation of minima is /D, except for the first minima on either side of thecentral maximum, which are separated by 2/D. If x is the distance on the screenbetween minima, then  xL D.University of Colorado at Boulder, Department of PhysicsDto screenD/2sin 2_Daperture filled with imaginary sourcesPhysics 2020, Spring 2005 Lab 9 page 5 of 13Combine interference (2-slit) with diffraction (finite-width slit): When the apertureconsists of two finite slits, each of width D, separated by a distance d, then theintensity pattern is a combination of both the single-slit pattern and the double slitpattern: the amplitude of the two slit interference pattern is modulated by a single slitdiffraction pattern: dDDouble slit,finite width/d/D/D2In this full pattern, the finely spaced interference maxima are spaced  d apart,while the more widely spaced minima of the single-slit diffraction pattern areseparated by  D or 2D. Note that an interference maximum can be wiped out ifit coincides with a diffraction minimum. PART I:


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CU-Boulder PHYS 2020 - Lab 9: Diffraction and Interference

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