Interference and Diffraction
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1 Purpose Theory Interference and iffraction a. To study examples of interference in light waves. b. To understand the interference pattern produced when light passes through a single slit. c. To understand the interference pattern produced when light passes through a double slit. When a monochromatic and coherent light passes through a single or double slit, it creates a diffraction/interference pattern on a screen placed beyond the slits. The pattern formed is because of the superposition of the waves coming from the slit (or two slits). The position on the screen directly opposite the slits is defined to have location y = 0. Other positions on the screen are characterized by their distance y away from this origin. Alternatively, a position on the screen is characterized by an d angle θ formed by the line from the slits to this position, relative to the perpendicular line. Note that if the distance from the slits to the screen is labelled, then y = Fig. 1. Geometrical arrangement of slits and screen in experiment. tan θ. ouble slit Interference Interference pattern due to a double slit will have dark and bright fringes due to destructive and constructive interference of the waves coming from the two slits. When two slits separated by a distance d produces bright spots on the screen centered at positions where the following constructive interference criterion is satisfied: d sin θ = n λ, where n is an integer. For the experiments we will be doing, the angle θ is less than 10 degrees, and sin θ tan θ = y/. Substituting y/ for sin θ in the above equation, we get y d n, or, rearranging, n y. (1) d Brooklyn College 1
2 Single-slit diffraction iffraction pattern formed by a single will have a wide and bright pattern at the center with alternate dark and bright fringes with diminishing intensity on both sides. The pattern is formed is because of the superposition of the waves coming from all points in the slit. A single slit with slit width will produce dark regions on the screen at positions where the following destructive interference criterion is satisfied. sinθ = n λ, where n is a non-zero integer. Again, for our experiments sin θ tan θ = y/. Substituting y/ for sinθ in the above equation, we get or, rearranging, y n, n y. (2) Since the slits have a finite slit width in double slit experiment, it is thoughtful idea to look back to double slit interference pattern. Thus, practically, double slit interference pattern will have single slit effect. As you will observe in this experiment, the real interference pattern will have interference pattern enveloped in diffraction pattern with evenly-spaced narrower bright fringes grouped in a few broader bands with decreasing intensity on both sides. The broader bands are because of the single slit diffraction. We can determine the slit width from the broader bands. Apparatus Slides with various slits; lasers (to be shared and passed from group to group); mount that holds laser and slide; black wooden target board; paper; tape; meter stick; ruler. escription of Apparatus Slit slide Single slit slide Fig. 2. Experimental set up for interference experiment with a laser and slit(s). ouble slit slide The first three items listed in the apparatus section are shown in the figure 2. You will use lasers of different colors. aser gives a monochromatic and coherent light. You will use a slide that has single slits of different width and another slide for double slit with different slit width and slit separation. **O NOT look at the laser beam and TURN OFF the laser when not in use ** Brooklyn College 2
3 Procedure Part I. Single slit In this part of the lab, you will study wave phenomena when a coherent and monochromatic light is passed through a single slit of different slit width. 1. Turn on the He-Ne laser (red laser) and place the slide on the mount so that the laser beam hits one of the slit, Pattern B. Place the target board, with a piece of paper securely taped to it, at a distance,, of about two meters from the slide. Measure. You should see a diffraction pattern that looks similar to Fig 3. You can adjust the slide very gently to get a clear pattern on the screen. It has a bright a wide fringe as the center. On both sides of the central bright, there are bright and dark fringes with diminishing brightness. Fig. 3. Photograph of a single slit diffraction pattern on a screen using a red laser 2. Mark on the paper the location of the ARK spots, as shown by the black marks at the bottom of the figure 3. Example shown in Fig. 3 has three dark spots on both sides. You can mark as many you can observe on your pattern. For best accuracy, you should make your marks directly on top of the dark spots. In this lab your results mainly depends on how precisely you can mark and measure the pattern. First dark spot you observed from the central bright is the first order dark of the single slit diffraction pattern. Similarly second, third dark spots are second, third order dark and so on. 3. Remove the paper from the target board. With your ruler measure the distance s between the same orders of the dark fringes. (For example, 1 is the distance between first order darks on both sides from the center.) Record your results in Table 1. Obtain the slit width from your instructor. You will use these data to calculate the wavelength of the light. 4. Now move the slide so that the laser hits another single slit, Pattern C, and repeat the previous steps 2-3. What changes did you observe with different slit? 5. Turn OFF the laser. Save this piece of paper with your marks as part of your lab data, to be submitted in your report. Part II. ouble slit slits. In this part of the lab, you will perform the same experiment as in Part I but with double 1. Replace the slide on the mount with double slits. Turn on the He-Ne laser and adjust the slide so that the laser beam hits Pattern A of double slits. Place the target board, with a piece of paper securely taped to it, at a distance,, of about two meters from the slide. Brooklyn College 3
4 Measure. You should see an interference pattern that looks something similar to Fig 4. You can adjust the slide very gently to get a clear pattern on the screen. Fig. 4. Photograph of a double slit interference pattern on a screen using a red laser 2. The closely-spaced bright spots represent the double slit interference pattern. The largerscale pattern in which a wide central set of spots is bright, after which the spots alternately fade and recover, reflects the interference pattern of the individual slits. 3. First characterize the double slit pattern. Mark on the paper the location of a central group of BRIGHT spots, as shown by the black marks at the top of the figure. Stay within the bright central portion of the pattern, before the spots become difficult to see due to the dark regions of the single-slit pattern. In your case, for best accuracy, you should make your marks directly on top of the spots. 4. Next characterize the single-slit pattern. Mark on the paper the location of the ARK places in the pattern, as shown by the gray marks at the bottom of the figure. In your case, for best accuracy, you should make your marks directly on top of the dark spots. Use a different color pen so that you can clearly distinguish these marks from your marks in step Remove the paper from the target board. 6. With your ruler measure the distance between the furthest-separated marks in your two-slit interference pattern. Also record n, the number of spaces between these furthest-separated marks. For example, in the figure above, there are eleven marks, so n is ten. 7. With your ruler measure the distance between the second order dark on both sides from the center. (If you can observe higher order dark, you can use them for better accuracy). These marks are not evenly spaced: the central-most marks have roughly twice the separation as other nearest-neighbor separations. 8. Now, move the slide so that the laser beam hits the Pattern B and repeat the previous steps. Record your results in Table Save this piece of paper with your marks as part of your lab data, to be submitted in your report. 10. Obtain the wavelength of your particular laser from your instructor. Write down the slit width and slit separation marked next to the slits on the slide. Brooklyn College 4
5 Part III. ouble slit interference with different color lasers 1. Repeat double slit interference experiment, Part II, with different color lasers. There are several different color lasers placed at different stations. You may have go to that particular station or switch to a new laser in your station. Follow the steps 1-3 only in Part II with one pattern of the double slit on the slide. Computation Part I: From the data recorded in Table 1, determine the distance from the center of the pattern to different order dark from your values. Since is the distance between the dark fringes on both sides from the center, y n = n /2. etermine y n for your pattern. Using these values, your measured, and known slit width, calculate the wavelength of the laser λ from Eq. 2, y n n. Use n = 1, 2, 3,. for first, second, third,..order dark fringes. Compare this value with the value of the wavelength given by calculating a percent error. Part II: Repeat this calculation for the results obtained by using Pattern C on the slide. (a) ouble slit pattern From the data recorded in Table 2, determine the distance between two neighboring spots in the double slit interference pattern: y = /n. This y is the distance between the central spot and its neighboring spot, and by setting n = 1 in equation (1) above, we see that this y should obey y. d Use this equation, your y from above, your measured, and your known d (the slit separation) to calculate the wavelength of the laser light, λ. Compare this value with the wavelength provided by your instructor and calculate a percent error. Repeat this calculation for the results obtained by using Pattern B on the slide. (b) Single-slit pattern. In table 2, determine the distance from the center of the pattern to the order dark fringe: y n = n /2. If you have recorded the data for second order dark, n = 2. By setting n = 2 in equation (2) above, we see that this y should obey y2 2. (If you are recording higher order dark fringe, you should use the corresponding vale of n in this equation). Brooklyn College 5
6 Use this equation, your y from above, your measured, and your known laser λ to calculate the slit width. Compare this value with the width that was indicated on the slide by calculating a percent error. Repeat this calculation for the results obtained by using Pattern C on the slide. Part III. From the data recorded in Table 3 for the two slit pattern, determine the distance between two neighboring spots in the two-slit interference pattern: y = /n. This y is the distance between the central spot and its neighboring spot, and by setting n = 1 in equation (1) above, we see that this y should obey y =. d Use this equation, your y from above, your measured, and your known d (the slit separation) to calculate the wavelength of the laser light, λ. Compare this value with the wavelength provided by your instructor and calculate a percent error. Questions 1. How would be the pattern on the screen if a thin wire on wide open slide is used instead of the thin opening (slit) in Part I? 2. iscuss the possible sources of error that contribute to the percent errors in your experimentallydetermined values of d and. 3. For a given pair of slits, how does the pattern alter if one switches from a higher-λ laser (say, red) to a lower- λ (say, green) laser? 4. What will happen in the two-slit pattern obtained in your experiment, (i) if the slit width () is reduced keeping the distance between the slits (d) same? And (ii) if the distance between the slits (d) is reduced keeping the slit width () same? Explain with diagrams. 5. In your experiments, the slits are placed vertically. How would be the pattern on the screen with two-slit if another two-slit is overlapped horizontally? 6. What do you think will happen in the interference pattern if you keep adding slits at the same separation d? Brooklyn College 6
7 ata Sheet ate experiment performed: Name of the group members: Table 1. Single slit Slit width () = istance from the slit to the screen () = Slit position y 1 y1 1. y 2 y2 2. y 3 y3 3. Pattern B Pattern C Average error = Table 2. ouble slits istance between the slits (d) = Slit width () = istance from the slit to the screen () = Slit position Pattern A ouble slit pattern (meters) n y = /n y d (meters) Single-slit pattern y 2 = /2 2.. y 2 Pattern B Average Average = error = error = Table 3. ouble slits in different colors istance between the slits (d) = Slit width () = istance from the slit to the screen () = Color of laser (meters) n y = /n y d % error Brooklyn College 7
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