# DIFFRACTION AND INTERFERENCE

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1 DIFFRACTION AND INTERFERENCE In this experiment you will emonstrate the wave nature of light by investigating how it bens aroun eges an how it interferes constructively an estructively. You will observe these effects in a variety of situations an use the wave theory of light to measure wavelength. SINGLE SLIT If light is mae of particles, a beam of such particles shoul pass straight through a long, narrow slit an form a single spot on a screen place beyon the slit. If it is wavelike in nature, a more complex pattern results. The light waves reaching a given point on the screen each arrive from a ifferent part of the slit, so their amplitues must be ae, an an interference pattern results. Consier pairs of points separate by a istance of half the slit with, such as (A,B) or (C,D) in Figure below. There exists a location on the screen for which waves coming from point C are out of phase with waves from point D by exactly one-half of a wavelength, so their amplitues a to zero. n = 3 n = n = a A C B n = 0 D λ n = n = Figure SCREEN n = 3 The situation shown (n=) in Figure is for the first estructive minimum an occurs at two positions with angles sin = λ / a. Aitional minima occur at sin = n λ / a. The n = minimum is the most important one. For small, the angle in raians is = λ/a. This angle is important because it is the Diffraction an Interference

2 limit of the angular resolution of an optical system. For a circular lens, the smallest angle between two points of light which can be resolve is =. λ/a, where the. factor epens on the shape of the lens an a is the iameter of the lens. DOUBLE SLIT Waves passing through one of two long, narrow slits will iffract in passing through each slit as escribe above, but in aition there will be interference with the waves from the other slit. Figure below shows the geometry so you can convince yourself that there is constructive interference causing intensity maxima at points on the screen for which sin = nλ, n = 0,,, 3, () The slits are separate by a istance. If the istance ns (istance of maxima from n = 0) to the nth maximum is much smaller than the slit to screen istance L, one can also approximate: sin = = ns/l. In this case, the maxima are equally space. n nλ λ λ 0 S S S S Figure L S The interference is completely estructive (intensity minima) where sin = (n + )λ, n = 0,,, 3, (3) The iffraction pattern that is actually observe from the ouble slit is the theoretical ouble slit maximum pattern shown in Figure 3 -- from Equations () an (3) -- moulate Diffraction an Interference

3 by the single slit minimum pattern -- from Equation () -- shown in Figure 4. The result is shown in Figure 5. Note that some of the ouble-slit maxima have nearly zero intensity as they coincie with single slit minima, as shown in Figure 4. These are terme missing orers. Intensity -8S -4S -S 0 S 4S 8S Figure 3-8S -4S 0 4S 8S Figure 4-8S -4S 0 4S 8S Figure 5 MULTIPLE SLITS: THE DIFFRACTION GRATING For part of this lab you will be using an array of many slits, a iffraction grating. Because of the great number of slits, the interference at a small istance from a maximum will be very estructive an the maxima will be extremely narrow. This makes the iffraction grating a useful tool (i.e., a spectrometer) in measuring wavelength very accurately. Mathematically, the maxima occur when sin = nλ, n = 0,,, 3,... (4) as for the ouble slit (Equation ). Figure 6 shows the situation. 3 Diffraction an Interference

4 3λ λ λ Figure 6 GRAZING INCIDENCE A reflector with rulings on it will work in a similar way to the grating escribe above. In this case, the waves are not incient normal to the surface but instea are incient at a small angle. Light is not reflecte from the ruling but from the smooth surfaces between the rulings. Consier the waves that reflect from between the rulings. It is possible for interference to occur. From Figure 7 below, convince yourself that the ifference in the lengths of light paths # an # is (cos -cos β) where is the spacing of the lines. There will be constructive interference for all pairs of angles an β that satisfy nλ = (cos cos β ), n = 0,,, 3,... (5) β β Ruler Figure 7 From Figure 7, the path length ifference is seen to be smaller than the ruling separation. Cos an cos β are nearly equal to if the angles are small (a few egrees). By using the small angle approximation for the cosine function, we can erive an expression that can be 4 Diffraction an Interference

5 more easily use to explain the pattern you will observe. If is expresse in raians, we have for small angles cos -{/}. Substituting this in Equation (5) gives nλ (β ) (S n S 0 ) L (6) The last step uses the aitional approximation that for small angles, tan. As shown in Figure 8, Sn is the istance from the n th orer to a line extene from the ruler to the screen. Wall S n = ORDER n = S LASER β β So Ruler O Figure 8 L -So Because the path length ifference is so small, this metho can be use to measure wavelengths λ that are of orer 000 times smaller than the groove spacing. One important use of grazing incience is measuring the wavelength of X-rays (λ 0-8 cm) by means of an orinary grating ( 0-4 cm). THE APPARATUS We have provie a iffraction grating, a steel ruler to be use for grazing incience, an a glass plate containing a variety of single an multiple slits. Figure 9 is a key to the glass plate. The three numbers for each slit or set of slits are, from top to bottom: the number of slits; with of each slit; an separation between slits, respectively (the thir number is obviously missing for single slits). The separation is given in points (a point in this instance is equivalent to inch). The continuous line varies smoothly both in 5 Diffraction an Interference

6 with an number of slits, allowing an aesthetically pleasing investigation of the changes in the pattern (incluing missing orers) as a function of these parameters Variable Spacing Constant With 6 Parallel Slits Varying With 6 Single Slit Varying With /4 / 40 / number of slits with of slit separation units: =.00796inch Example: 3 three slits with spacing a = with = " -3 = 4.39 x 0 cm = 4.39 x 0-5 m = 4.39 x 04 nm s = spacing = x 0-5 m istance between slit center = s + a =.38 x 0-5 m Figure 9 The iffraction grating is mae by making a plastic mol of a metal surface in which thousans on uniformly space scratches are mae. Their separation must be on the orer of 0 wavelengths of the waves to be stuie, or about 5 x 0-4 cm for visible light. When use with normal incience waves, the spaces between the rulings pass the waves an the rulings scatter them. Do not touch the iffraction/replica grating with your fingers. Dirt an oil from your hans will cause blurring of the light. Do not attempt to clean the iffraction grating, it scratches very easily. 6 Diffraction an Interference

7 APPARATUS Optical bench Meterstick Twometerstick mountings for gratings Masking tape 8.5 x paper Diffraction grating/replica grating 4 Ro holers Lens Clamp 6 steel ruler with /00 ivisions Interference an iffraction glass slie Magnetic holer for iffraction grating HeNe laser EXPERIMENT. SINGLE AND MULTIPLE SLITS This is both quantitative an qualitative. Set up the glass iffraction set slie so that the laser beam is normal to the glass surface. If the glass plate has a magnet attache to it, use the U-shape metal holer; otherwise, the glass slie can be mounte in a clamp with a rubber cushion -- be careful not to tighten the clamp too much, or you may break the glass slie. Ajust the height of the laser an/or plate so that the incient beam strikes an opening an a iffraction pattern appears on the wall (at least meters away). Observe the patterns of an spots. Tape a piece of paper to the wall so you can mark their locations. In oing this, inclue enough information to answer the following questions:. How oes varying the with of a single slit affect the separation an withs of iffraction maxima an minima?. For ouble slits, what is the effect of increasing the separation? 3. How oes the change in number of slits change the with of the spots (intensity maxima)? Does this agree with preiction? 4. Look for missing orers in the ouble slit pattern. Use equations () an () to show that missing orers occur when / =, 3, 4... Do the missing orers (for ouble slits) occur where they are preicte to by theory? 5. Use the ata from the two slit an multiple slit measurements to calculate λ for the laser. This can be one using Equation an sin = n λ / by measuring sin, n, an. Measure to both the left an the right of n = 0 for n = an n =. 6. Use the iffraction grating to obtain a more accurate value for λ. Mount the grating so that the laser beam is normal to the surface. Measure the istances to the n = maximum on both sies of the n = 0 maximum. Measure the istances from wall to grating. Fin using tan = S / L. The angle here is too large for the small angle approximation to be accurate. Fin the grating spacing from / (the number of lines per cm). Fin λ from λ = sin. Compare with the expecte λ = 63.8 nm. 7 Diffraction an Interference

8 EXPERIMENT. GRAZING INCIDENCE: WAVELENGTH OF LASER LIGHT Remove the glass plate an mount the ruler so that the laser light grazes the rulings at a small angle as shown in Figure 8. Aim the laser at the part of the ruler with the smallest spacing between the rulings. You shoul observe a iffraction pattern on the wall. Is it similar to any you have looke at? Try to explain the separation of the orers ( spots) in terms of Equation (6) (i.e., how shoul the istance S n vary with n?). Also, compare the separation of these spots to the separation of those obtaine when you ouble or halve the spacing between rulings. Now you are reay to make measurements. Tape a piece of paper to the wall for marking the positions of the spots. Recor as many as you can; inclue the irect beam spot an the spot cause by mirror reflection from the ruler. The irect beam spot is recore before the ruler is put in place. These are both neee to calculate the angle. Take care to ientify the mirror reflection spot carefully. Recor also the spacing of the markings on the ruler in inches or millimeters, an the istance from ruler to wall. Use equation (6) to fin λ for laser light using the ruler. ANALYSIS AND QUESTIONS 7. In Experiment, the slits are long an narrow. Why is the iffraction pattern one imensional? What happens to the pattern in the long imension? What woul you observe for a small square slit? A roun pinhole? 8. Explain why the separation of orers ecreases as n increases in Exp.. Equation (6) may be helpful. 8 Diffraction an Interference

9 DATA SHEET DIFFRACTION AND INTERFERENCE TA [ ] NAME INSTRUCTOR: PARTNER SECTION: DATE: EXPERIMENT : (FOR FIRST 3 COLUMNS, SEE KEY IN APPARATUS SECTION) # of Slits With Slit Sep. Orer Sep. Other Obs. (missing orers, etc.) EXPERIMENT L Ruling Sep. S 0 ( S 0 ) n S n λ Average λ: 9 Diffraction an Interference

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