Interference and Diffraction of Ligt References: [1] A.P. Frenc: Vibrations and Waves, Norton Publ. 1971, Capter 8, p. 280-297 [2] PASCO Interference and Diffraction EX-9918 guide (written by Ann Hanks) [3] PHYWE Series of Publications, Laboratory Experiments, Pysics LEP 2.3.01, PHYWE SYSTEME GMBH Introduction Observations of ligt passing troug narrow openings sow tat ligt spreads out beind te opening and forms a distinct pattern on a distant screen. By scanning te pattern wit a ligt sensor and plotting ligt intensity versus distance, differences and similarities between interference and diffraction are examined. Observation of diffraction intensity can be used in a simple quantum mecanical treatment to confirm te Heisenberg s uncertainty principle. Diffraction Wen diffraction of ligt occurs as it passes troug a slit, te angle to te minima (dark spots) in te diffraction pattern is given by a sinθ = m' λ (m =1,2,3, ) (1) were "a" is te slit widt, θ is te angle from te center of te pattern to te m t minimum, λ is te wavelengt of te ligt, and m' is te order of diffraction (1 for te first minimum, 2 for te second minimum,...counting from te center out). In Figure 1, te laser ligt pattern is sown just below te computer intensity versus position grap. Te angle teta is measured from te center of te single slit to te first minimum, so m' = 1 for te situation sown in te diagram. b l Figure 1: Single-Slit Diffraction Double-Slit Interference Wen interference of ligt occurs as it passes troug two slits, te angle from te central maximum (brigt spot) to te side maxima in te interference pattern is given by d sin θ = mλ (m=0,1,2,3, ) (2) were "d" is te slit separation, θ is te angle from te center of te pattern to te m t maximum, λ is te wavelengt of te ligt, and m is te order (0 for te central maximum, 1 for te first side maximum, 2 for te second side maximum, counting from te center out). In Figure 2, te laser ligt pattern is sown just below te computer intensity versus position grap. Figure 2: Double-Slit Interference Te angle teta is measured from te midway between te double slit to te second side maximum, so m equals two for te situation sown in te diagram. 1
Intensity in te diffraction pattern Ligt intensity is proportional to te square amplitude of te wave producing it (E 2, were E is te electric field amplitude). By using te concepts from superposition of waves [1], we obtain: 2 sinφ πa I( φ) = I(0) were: φ = sinθ (3) φ λ 1 λ Te first intensity minimum is located at θ = sin according to Equation (1) a Setup - Mount te single slit disk to te optics benc. Eac of te slit disks is mounted on a ring tat snaps into an empty lens older. Te ring sould be rotated in te lens older so te slit at te center of te ring is vertical in te older (see Figure 3). To select te desired slit(s), just rotate te disk until it clicks into place wit te desired slit at te center of te older. NOTE: All slits are vertical EXCEPT te comparison slits, located on te multiple slit set, tat are orizontal. Te comparison slits are purposely orizontal because te wide laser diode beam will cover bot slits to be compared. If you try to rotate tese slits to te vertical position, te laser beam may not be large enoug to illuminate bot slits at te same time. Figure 3: Mounting te Slits - Te rotary motion sensor, aperture bracket (set on slit #6) and linear translator are mounted at te end of te optics track (see Figure 4). Figure 4: Scanner wit Ligt Sensor - Place te diode laser on te benc at one of te ends. Put te slit older on te optics benc a few centimeters from te laser, wit te disk-side of te older closest to te laser. Plug in te diode laser and turn it on. CAUTION: Never look into te laser beam. - Adjust te position of te laser beam (left-rigt and up-down until te beam is centered on te slit. Once tis position is set, it is not necessary to make any furter adjustments of te laser beam wen viewing any of te slits on te disk. Wen you rotate te disk to a new slit, te laser beam will be already aligned. Since te slits click into place, you can easily cange from one slit to te next, even in te dark. Wen te laser beam is 2
Figure 5: Aligning te Ligt Sensor properly aligned, te diffraction pattern sould be centered on te slits in front of te ligt sensor (see Figure 5). Begin wit te ligt sensor gain switc set on x10 and if te intensity goes off scale, turn it down to x1. Familiarization wit te patterns - Start wit te single slit weel setup to 0.16 mm. Attac a small piece of paper to te sensor aperture bracket. Look at te pattern. - Select te combination slits from te double slit weel. Look at te pattern produced by eac of te four combinations. Draw a diagram of eac slit combination and te corresponding diffraction pattern. Note tat te single slit from te combination is always 0.04 mm Single slit exercise - Mount te single slit weel wit te 0.04 mm slit. - Te low rpm driving motor drives te ligt sensor along te linear translator between two switc stops (~14 cm travel distance). It can be operated at low speed for data collection or at a iger speed for moving te sensor to a desired position on te linear translator. - Before starting te recording, move te ligt sensor to one side of te laser pattern. - Turn out te room ligts and click on te Interference and Diffraction sortcut on te desktop. Click on te Start arrow button, located at te upper left corner of te screen, turn on te motor, turn ON te acquisition to scan te pattern. - Turn OFF te acquisition wen you ave finised te scan. Te acquisition turns OFF by itself wen te ligt sensor reaces te end of travel (switc stop). - Te program allows you to use two pairs of cursors to get te intensity and position along te pattern. Cursors usually reside at te left of te grap. You may save te numerical file in a temporary computer directory and/or print te grap. - You may ave to cange te gain setting on te ligt sensor (1x, 10x, 100x) depending on te intensity of te pattern. You sould try to use slit #4 on te mask on te front of te ligt sensor. - Determine te slit widt using Equation (1). Measure at least two different minima and average your answers. Estimate te errors. Double slit exercise - Replace te single slit disk wit te multiple slit weel. Set it on slit separation 0.25 mm (d) and slit widt 0.04 mm (a). - Set te ligt sensor aperture bracket to slit #4. - Apply te same steps as before (single slit procedure) - Zoom in to enlarge te central maximum and te first side maxima. Use te cursors to measure te distance between te central maximum and te first side maxima. 3
- Measure te distance between te central maximum and te second and tird side maxima. Also, measure te distance from te central maximum to te first minimum in te DIFFRACTION (not interference) pattern. - Determine te slit separation using Equation (2) and te first, second, and tird maxima, and find te average "d". Find te percent difference between your average and te stated slit separation on te weel. - Determine te slit widt using Equation (1) and te distance between te central maximum and te first minimum in te diffraction pattern (not interference pattern). Is tis te slit widt given on te weel? - Repeat te steps above for te interference patterns for two oter double slits. Estimate te errors. Quantum mecanical interpretation Te Heisenberg uncertainty principle (ttp://www.aip.org/istory/eisenberg/p08.tm) states tat te simultaneous measurements of te momentum and position (or te energy and time) for a moving particle entails a limitation on te precision (standard deviation) of eac measurement. Namely: te more precise te measurement of position, te more imprecise te measurement of momentum, and vice versa. In te most extreme case, absolute precision of one variable would entail absolute imprecision regarding te oter. If we consider a pack of potons caracterized by position uncertainty Δy and momentum uncertainty Δp, we can express Heisenberg s relation as: Δ y Δp (4) 34 = 6.6262 10 J s is Planck s constant. Were For a pack of potons passing troug a slit of widt a we ave Δ y = a. In order to estimate Δp, we assume tat potons reacing te slit move only in te direction perpendicular to te slit (x-direction), but after passing troug te slit, tey will ave velocity components in bot directions (x and y). Te v y component of poton velocity is given by te intensity distribution in te diffraction pattern. Defining by θ 1 te angle of te first minimum of diffraction, we can express te uncertainty of velocity and momentum as: Δ v y = c sinθ 1 (c is speed of ligt) (5) = sinθ 1 λ Δ (6) (using te de Broglie relationsip: = p ) λ Te angle θ 1 of te first diffraction minimum can be determined from te experimental setup geometry (see Figure 1 and Equation 1) and te momentum uncertainty results as: = a Δ (7) Combining Δy wit Δ in (3), we obtain te uncertainty relation: Δy = Δ (8) 4
Experimentally, we can obtain te angle θ 1 from te position of te first minimum: l tanθ 1 = (see Figure 1) b By substituting tis relation into (6), we obtain 1 l Δ = sin tan. Equation (8) becomes a 1 l sin tan = (9) so ten a 1 l sin tan = 1. Experimental verification: - Measure te alf widt of te central maximum (l) for tree different single slit widts (a). Measure te distance between te ligt sensor aperture and te laser aperture (b). - Verify Equation (9). Carefully estimate te errors and discuss about confirming or not Heisenberg s uncertainty principle by using tis experimental verification. Diffraction pattern analysis Setup te diffraction experiment wit a slit of 0.16mm and a laser-detector distance of about 1.1m. Record te diffraction pattern. Using te cursors, measure te pattern intensity corresponding to te central maximum I(0) (Equation (3)). Measure te eigts of tree oter secondary maxima I(θ i ); calculate I( θ i ) te relative eigts of secondary maxima. From te setup geometry, calculate te I( 0) corresponding angles θ i of secondary intensity maxima. Pyton Requirement Using Equation (3) verify te intensity formula by performing a nonlinear fit on te data collected by te detector. Output te ratio a : λ and comment on goodness of fit. Comment on error sources. QUESTIONS 1. Wat pysical quantity is te same for te single slit and te double slit? 2. How does te distance from te central maximum to te first minimum in te single-slit pattern compare to te distance from te central maximum to te first diffraction minimum in te double-slit pattern? 3. Wat pysical quantity determines were te amplitude of te interference peaks goes to zero? 4. In teory, ow many interference maxima sould be in te central envelope for a double slit wit d = 0.25 mm and a = 0.04 mm? 5. How many interference maxima are actually in te central envelope? Te National Instruments interface was setup and programmed by Larry Avramidis. Larry also built te low rpm motor driving te linear translator. Tis experimental guide seet was written by Ruxandra M. Serbanescu in 2008 (updated 2009). 5