9.5. Interference of Light Waves: Young s double-slit Experiment

Interference of Light Waves: Young s ouble-slit Experiment At the en of the 1600s an into the 1700s, the ebate over the nature of light was in full swing. Newton s theory of light particles face challenges from leaing scientists such as Christiaan Huygens. Huygens writings on the wave theory of light escribe what we now call Huygens principle an propose that light travelle in waves through an omnipresent ether, like soun travelling through air. Mathematician an scientist Leonar Euler evelope his own wave theory of light an use the particle theory s poor escription of iffraction as a key point in an argument against Newton s theory. During this time, many researchers focuse on the question of interference. If light travels as a wave, then it shoul interfere like a wave; experiments looking for interference, however, coul not etect it. We now know that the experiments faile because of the extremely small wavelengths of light. Interference patterns in water waves in ripple tanks are easy to observe, because the wavelengths are large an the frequencies of the sources are relatively small. These properties make the istance between ajacent noal lines visible to the eye. Most experiments with light before the 1800s involve the placement of two sources of light close to each other, with screens positione near the sources. The scientists closely observe the screens to try to observe interference patterns. They never succeee using this metho, partly because the separation between the noal lines was too small to be observable. However, these issues i not account for all of the problems in attempting to repeat the ripple tank experiment with light waves. Atoms, incluing those in the Sun an in light bulbs, emit most visible light. You might expect the light waves emitte by two atoms in the same source to be ientical an have the same frequency so that they coul exhibit interference. However, collisions among the atoms can reset the phase of the emitte light wave. Experiments show that these phase jumps in typical light waves occur about every 10 28 s. This means the light from ifferent atoms of a given source is incoherent because the waves have no fixe phase relationship to each other. If you irect light from a monochromatic, or single-wavelength, source through a narrow slit, however, the slit acts as a single point source, an the light travels from it as a coherent wave accoring to Huygens principle. If you then irect the light from the single slit to a ouble slit (a closely space pair of slits), the ouble slit acts as a pair of sources of coherent, monochromatic light (Figure 1). Any change that occurs in the original source will occur in the two beams at the same time, which allows you to observe interference effects. 9.5 incoherent compose of waves that have no fi xe phase relationship to each other an ifferent frequencies monochromatic compose of only one colour; light with one wavelength single slit ouble slit viewing screen Figure 1 Monochromatic light passes through the single slit fi rst, then through the ouble slit. Accoring to Huygens principle, both the single slit an the ouble slit act as a source of coherent light. NEL 9.5 Interference of Light Waves: Young s Double-Slit Experiment 477

Young s Double-Slit experiment At the very en of the 1700s, Thomas Young (Figure 2) performe a series of experiments to etermine what happens when light passes through two closely space slits. We now refer to his basic setup as Young s ouble-slit experiment, an it emonstrate conclusively that light behaves as a wave. The experiment also provie a metho to measure wavelength. Figure 3(a) shows the experimental setup, an Figure 3(b) shows an image prouce with such a setup using white light. two narrow slits bright region ark region Figure 2 Thomas Young incoming light screen (a) (b) Figure 3 (a) Young use a similar setup in his light experiments in 1800. (b) When the white light passe through two small slits, an interference pattern of alternating bright an ark fringes was prouce. When monochromatic light, such as laser light, passes through the slits of an experiment setup similar to the one in Figure 3(a), the laser light hits the screen on the right an prouces an interference pattern. This arrangement satisfies the general conitions require to create wave interference: The interfering waves travel through ifferent regions of space (in this case they travel through two ifferent slits). The waves come together at a common point where they interfere (in this case the screen). The waves are coherent (in this case they come from the same monochromatic source). When the two slits in the ouble-slit setup are both very narrow, each slit acts as a simple point source of new light waves, an the outgoing waves from each slit are like the simple spherical waves in Figure 4. plane wave fronts outgoing spherical wave fronts Investigation 9.5.1 Young s Double-Slit Experiment (page 490) You have learne about Young s famous ouble-slit experiment. This investigation will give you an opportunity to recreate that experiment for yourself using a monochromatic light source an a series of ifferent ouble-slit plates. very narrow opening Figure 4 We can see the iffraction effects of light passing through two narrow slits by consiering just one of the narrow slits. The same thing happens in the secon narrow slit. 478 Chapter 9 Waves an Light NEL

Interference will etermine the intensity of light at any point on the screen. If one of the slits is covere, the screen shows a bright, wie centre line with closely space, im, alternating ark an light bans, or interference fringes, on either sie. You can rea more about this single-slit pattern in Chapter 10. However, if both slits are open, then the screen shows a pattern of nearly uniform bright an ark fringes (Figure 5). The ieal ouble-slit pattern woul show no imming, but a real-worl pattern oes appear immer as you move away from the centre. This imming occurs because the with of the slits causes each to act like a single slit, an the non-uniform single-slit pattern combines with the ieal ouble-slit pattern. The bright an ark fringes are alternate regions of constructive an estructive interference, respectively. To analyze the interference, you nee to etermine the path length ifference between each slit an the screen. Figure 6 illustrates the ouble-slit interference conitions. interference fringe one of a series of alternating light an ark regions that result from the interference of waves slit L 1 L 2 P L 2 L sin slit slit screen (a) (b) Figure 6 (a) Values of u give the location of the bright fringes on the screen. (b) When the point P is very far away compare to the separation of the two sources, the lines L 1 an L 2 are nearly parallel. slit L 1 Figure 5 A isplay of Young s ouble-slit experiment using re light To simplify the analysis in Figure 6, we will assume that the screen is a long way from the slits, so L is very large compare to. The path lengths for waves from each of the slits to the point P on the screen are L 1 an L 2 in Figure 6. The istance L 2 is greater than the istance L 1, an, since L is large, the angles that specify the irections from the slits to point P are approximately equal, so both are shown as θ in Figure 6. Finally, we assume that the wavelength l is much smaller than, the spacing between the slits. Since the slits are separate by a istance, the path length ifference between L 2 an L 1 is given by DL 5 sin u For the two waves to be in phase when they reach the screen, an thus for constructive interference to occur, this path length ifference nees to be a whole number of wavelengths. The conition for constructive interference an a bright interference fringe is therefore sin u 5 ml m 5 0, 1, 2, 3,... constructive interference The light fringes, or maxima, are calle zero-orer maximum, first-orer maximum, an so forth, for m 5 0, 1, 2, 3,.... In this notation, m 5 0 enotes the maximum in the centre of the screen. Successive values of m correspon to successive maxima moving away in either irection from the centre of the screen. maxima points of brightness, or maximum intensity, in an interference pattern NEL 9.5 Interference of Light Waves: Young s Double-Slit Experiment 479

For the two waves to be out of phase when they reach the screen, an thus for estructive interference to occur, the path length ifference nees to be an 2 1 bl. Thus, the 2 conition for estructive interference an a ark fringe in the interference pattern is sin u 5 an 2 1 2 bl n 5 1, 2, 3,... estructive interference minima points of arkness, or minimum intensity, in an interference pattern These ark lines, or minima, are calle first-orer minimum, secon-orer minimum, an so forth, for n 5 1, 2, 3,.... Figure 7 shows a much-enlarge intensity pattern from a ouble-slit experiment. Notice that as the value of m changes, the angle u also changes. At certain values of u, the conitions for constructive interference are satisfie an a maximum in the intensity is prouce, corresponing to a bright fringe. The fringe at the centre of the screen is the zero-orer maximum an has the value 0 for m in the equation for constructive interference. The next bright fringe moving towar the top of the screen has the value m 5 1, an so on. m 2 slits L x 1 m 1 m 0 m 1 bright fringes m 2 screen Figure 7 In this ouble-slit interference pattern, values of u give the location of fringes on the screen. How oes the interference pattern change when the istance between the slits changes, or when the istance to the screen changes? First, calculate the separation between the central bright fringe (m 5 0) an the next bright fringe (m 5 1). The equation for constructive interference tells us that sin u 5 l To etermine the separation between the bright fringes on the screen, you can use the right-angle triangle outline in re in Figure 7. It has sies of lengths L an x 1, where x 1 is the istance between the two fringes corresponing to m 5 0 an m 5 1. These lengths are relate by the trigonometric relation x 1 5 L tan u Learning Tip Small-Angle Approximation When L is much larger than y an the angle u is small, sin u < tan u, so the tangent function is a goo approximation to the sine function. You can verify this yourself by checking values on your calculator. The wavelength is much smaller than the spacing between the slits, so l is very small. Therefore, u is also very small, an the approximation sin u L tan u hols: x 1 5 L sin u x 1 5 Ll For the mth-orer bright fringe, this relation becomes x m 5 mll an the separation between any two ajacent fringes is Dx 5 Ll 480 Chapter 9 Waves an Light NEL

Suppose the slit separation is 0.1 mm, the wavelength of light from a re laser is 630 nm, an the screen is 0.5 m from the slits. Substitute the corresponing values into this equation to etermine the separation of the fringes for the re laser light: Dx 5 Ll 5 10.5 m2 16.3 3 1027 m2 0.1 3 10 23 m 5 3.15 3 10 23 m Dx 5 3.2 mm This separation is large enough to be seen by the unaie eye. This analysis shows that, provie the slits are close enough together, the bright an ark fringes in the interference pattern are observable. A slit separation of 0.1 mm or smaller works well. Experiments have shown that when the spacing is larger than a few millimetres, the fringes are close together an ifficult to see. Using the equations for estructive interference, you can calculate the istance of each ark fringe from the centre of the screen. The result is x n 5 an 2 1 2 b Ll n 5 1, 2, 3,... For example, the istances of the first, secon, an thir minima from the centre of the screen are x 1 5 a1 2 1 2 b Ll 5 Ll 2 x 2 5 a2 2 1 2 b Ll 5 3Ll 2 x 3 5 a3 2 1 2 b Ll 5 5Ll 2 Although L actually has ifferent values for each noal line, in this case L is so large compare to an the values of L for the various noal lines are so similar, that we can treat L as a constant, being essentially equal to the perpenicular istance from the slits to the screen. Moern interference experiments often inject a clou of smoke between the slits an the viewing screen. Some of the light reflects off the particles in the clou, so the viewer can easily see the path of the light. Will interference patterns appear in the clouy region between the slits an the screen? The equations for constructive an estructive interference epen on the istance, L, from the slits, which can be any value consierably greater than. The interference patterns will still occur in both places. In the clou region, you will see bright lines irecte towar the bright areas of the screen an ark lines irecte towar the ark areas of the screen. The following Tutorial will emonstrate how to calculate physical quantities in ouble-slit interference patterns. Unit TASK BOOKMARK You can apply what you have learne about interference an the ouble-slit experiment to the Unit Task on page 556. Tutorial 1 Calculating Physical Quantities in Double-Slit Interference Patterns The following Sample Problems show how to measure the wavelength of light an slit separation in a ouble-slit experiment. Sample Problem 1: Determining the Wavelength of a Light Source A ouble-slit experiment is carrie out with slit spacing 5 0.41 mm. The screen is at a istance of 1.5 m. The bright fringes at the centre of the screen are separate by a istance Dx 5 1.5 mm. Calculate the wavelength of the light. Given: 5 0.41 mm 5 4.1 3 10 24 m; L 5 1.5 m; Dx 5 1.5 mm 5 1.5 3 10 23 m Require: l NEL 9.5 Interference of Light Waves: Young s Double-Slit Experiment 481

Analysis: In a ouble-slit experiment, the bright fringes occur when light waves from the two slits interfere constructively. This happens when the path ifference, DL, is equal to a whole number of wavelengths: Dx 5 Ll l 5 Dx L Solution: l 5 Dx L 5 11.5 3 1023 m2 14.1 3 10 24 m2 1.5 m l 5 4.1 3 10 27 m Statement: The wavelength of the light is 4.1 3 10 27 m. Sample Problem 2: Determining Slit Separation The thir-orer ark fringe of 660 nm light is observe at an angle of 20.08 when the light falls on two narrow slits. Determine the slit istance. Given: n 5 3; u 3 5 20.08; l 5 660 nm 5 6.6 3 10 27 m Require: an 2 1 2 bl Analysis: sin u n 5 an 2 1 2 bl 5 sin u n an 2 1 2 bl Solution: 5 sin u n a3 2 1 2 b 16.6 3 10 27 m2 5 sin 20.08 5 4.8 3 10 26 m Statement: The slit separation is 4.8 3 10 26 m. Practice 1. Calculate the wavelength of the monochromatic light that prouces a fifth-orer ark fringe at an angle of 3.88 with a slit separation of 0.042 mm. T/I [ans: 6.2 3 10 27 m] 2. Two slits are separate by a istance of 0.050 mm. A monochromatic beam of light with a wavelength of 656 nm falls on the slits an prouces an interference pattern on a screen that is 2.6 m from the slits. Calculate the fringe separation at the centre of the pattern. T/I [ans: 3.4 cm] 3. The thir-orer ark fringe of light with a wavelength of 652 nm is observe when light passes through two ouble slits onto a screen. The slit istance is 6.3 3 10 26 m. At what angle is the fringe observe? T/I [ans: 158] More Developments in the Theory of Light Young s evience for the wave nature of light was not accepte by the scientific community until 1818, when Augustin Fresnel propose his own wave theory, complete with the mathematics. A mathematician name Simon Poisson showe how Fresnel s equations preicte a unique iffraction pattern when light is projecte past a small soli object, as shown in Figure 8. ark region object point source (a) bright spot (b) Figure 8 (a) The preiction of Fresnel an Poisson, confirme by Arago, says that iffraction will cause a small bright spot to appear at the centre of the shaow of a small soli object. (b) A small isc refracts light an shows Poisson s bright spot. 482 Chapter 9 Waves an Light NEL

Poisson s argument was that, if light behave as a wave, then the light iffracting aroun the eges of the isc shoul interfere constructively to prouce a bright spot at the centre of the iffraction pattern. This was impossible accoring to the particle theory of light. In 1818, Poisson s preiction was teste experimentally by Dominique Arago, an, to many people s surprise, he successfully observe the bright spot. Arago s experiment emonstrate what is now calle Poisson s bright spot. The wave theory ha been generally accepte by 1850, although it coul not explain the movement of light through a vacuum such as space. Therefore, as mentione in Section 9.4, scientists evelope a theory that a flui calle ether fille all regions of space. Despite many experiments evise to try to etect this ether, it was never foun. Mini Investigation Wavelengths of Light Mini Investigation SKILLS HANDBOOK Skills: Performing, Observing, Analyzing, Communicating In this investigation, you will answer the question, What are some of the wavelengths that constitute white light? Equipment an Materials: clear showcase lamp; rotor stan an clamp; white screen; ruler; paper sliers; re an green transparent filters; ouble-slit plate; metre stick; elastic bans 1. Set up the apparatus as shown in Figure 9. View the lamp through the ouble slit an escribe the pattern seen on either sie of it. A2.1 2. Cover the upper half of the bulb with the green filter an the lower half with the re filter, securing each filter with elastic bans. Compare the interference patterns for green an re light. 3. Cover the lamp completely with the re filter. Use the ruler mounte in front of the light to count the number of noal lines you can see in a fixe istance. Use the paper sliers to mark the istance on the ruler. 4. Preict the relative wavelengths of re an green light. If a lamp is use, unplug it by pulling on the plug, not the cor. support stan 1m single-filament oublelamp clampe slit plate to stan paper slier 5. Leaving the paper sliers in place, repeat Step 3 with the green filter. 6. Recor the number of bright fringes between the markers an the istance between the paper sliers. Calculate the average separation, Dx, of the noal lines. Dx l 5 to etermine the wavelength 7. Use the relationship L of that colour of light ( shoul be recore on your slit plate or shoul be available from your teacher). 8. Compare your results with those of other stuents. A. Describe the interference patterns you saw for each filter. T/I C A B. How many noal lines i you see at a fixe istance for each filter? T/I C. Calculate the wavelength of the colour of light for each filter. T/I Figure 9 Setup for the wavelength experiment NEL 8160_CH09_p470-499.in 483 9.5 Interference of Light Waves: Young s Double-Slit Experiment 483 4/30/12 9:49 AM

9.5 Review Summary Earlier experiments that attempte to show the interference of light using two sources of light some istance apart were unsuccessful because the sources of light were out of phase an the wavelength of light is so small. Thomas Young s successful experiment use iffraction through a ouble slit to create two sources of light from a single coherent source of light so that the sources were close together an in phase. Young s experiment prouce a series of light an ark fringes on a screen place in the path of the light, with a pattern that resemble the results of interference of water waves in a ripple tank. For a setup with slit separation an light of wavelength l, the locations of bright fringes are given by sin u 5 ml, for m 5 0, 1, 2, 3,..., an the locations of ark fringes are given by sin u 5 an 2 1 bl, for n 5 1, 2, 3,.... 2 For a viewing screen at a istance L, the mth-orer bright fringe is at location x m 5 mll, an the nth-orer bright fringe is at location x n 5 an 2 1 2 b Ll. Young s iffraction experiment was evience for the wave nature of light. The wave theory coul explain all properties of light except its propagation through a vacuum. Questions 1. How is the interference pattern prouce by monochromatic re light, l 5 650 nm, ifferent from the pattern prouce by monochromatic blue light, l 5 470 nm, when all other factors are kept constant? K/U 2. When a monochromatic light source shines through a ouble slit with a slit separation of 0.20 mm onto a screen 3.5 m away, the first ark ban in the pattern appears 4.6 mm from the centre of the bright ban. Calculate the wavelength of the light. T/I 3. Suppose a ouble-slit experiment is carrie out first in air, then completely unerwater, with all factors being the same in each meium. How o the interference patterns iffer? K/U T/I 4. Two slits are separate by 0.30 mm an prouce an interference pattern. The fifth minimum is 1.28 10 21 m from the central maximum. The wave length of the light use is 4.5 3 10 27 m. Determine the istance at which the screen is place. T/I 5. In a ouble-slit experiment, the istance to the screen is 2.0 m, the slits are 0.15 mm apart, an the ark fringes are 0.56 cm apart. T/I (a) Determine the wavelength of the source. (b) Determine the spacing of the ark fringes with a source of wavelength 600 nm. 6. The secon-orer ark angle in a ouble-slit experiment is 5.48. Calculate the ratio of the separation of the slits to the wavelength of the light. T/I 7. In a ouble-slit experiment, the slits are 0.80 mm apart an an interference pattern forms on a screen, which is 49 cm from the slits. The istance between ajacent ark fringes is 0.33 mm. T/I (a) Determine which wavelength of monochromatic light is use in this experiment. (b) Determine the istance between the fringes if the same light were use in an experiment with slits that were 0.60 mm apart. 8. Light with a wavelength of 5.1 3 10 27 m passes through a ouble slit an onto a screen that is 2.5 m from the slits. T/I (a) Determine the slit spacing if two ajacent bright fringes are 12 mm apart. (b) If the slit spacing is now reuce by a factor of three, etermine the new istance between the bright fringes. 9. Summarize the evelopment in the theory of light. Can light be consiere a wave only an not a particle? Why or why not? K/U 484 Chapter 9 Waves an Light NEL