Multiple Slit Interference lab

Multiple Slit Interference INTRODUCTION As long ago as the 17th century, there were two competing models to describe the nature of light. Isaac Newton believed that light was composed of particles, whereas Christopher Huygens viewed light as a series of waves. Both models could explain reflection and refraction, but the phenomena of diffraction and interference could be more easily explained by Huygens’ wave model. In the early 19th century, Thomas Young’s double-slit experiment provided evidence that supported the wave nature of light. This is the second of two experiments that examine the phenomena of diffraction and interference, which occur with all types of waves from water to light. Superposition: Superposition is a fundamental principle that states that when two or more waves overlap in space, the resultant wave at any point is the algebraic sum of the individual waves. This means that the displacement (or amplitude) of the resulting wave is determined by adding together the displacements of the overlapping waves at that point. Superposition applies to all types of waves, including electromagnetic waves (such as light) and mechanical waves (such as sound). Diffraction: Diffraction is the spreading of light when it passes an edge or through an aperture. The amount of spreading depends upon the wavelength of the light and the width of the aperture. At the screen, photons interfere with themselves due to the path length differences of the possible paths the photons can take as they pass through the aperture and reach the screen. This causes an interference pattern to be formed on the screen. Interference: Interference occurs when two or more waves combine in such a way that their amplitudes reinforce (constructive interference) or cancel out (destructive interference) at certain points in space. Interference is a consequence of the principle of superposition. When waves overlap, their amplitudes add together according to the principle of superposition, leading to regions of constructive and destructive interference. In our experiment, the two or more waves that interfere come from two separate slits through which each photon can travel. At the screen, photons interfere with themselves due to the path length differences of the possible paths the photons can take to reach the screen. This causes an interference pattern to be formed on the screen. The interference pattern is dependent on the wavelength of the light, the separation between the slits, and the width of each slit. In essence, superposition is a fundamental principle that describes how waves interact when they overlap, while diffraction and interference specifically refer to the outcome of this interaction, where waves reinforce or cancel each other depending on their relative phases. Interference and diffraction are consequences of the principle of superposition, but not all instances of superposition result in noticeable interference patterns. In this experiment, you should observe a light bands (due to multiple slit interference) that form a pattern of light band clusters (due to the diffraction pattern from the slits themselves) similar to the diffraction pattern you observed in the single slit experiment. OBJECTIVES In this experiment, you will • • • Measure the features of the pattern produced on a screen when light from a laser passes through multiple slits. Use theory to predict where destructive interference will occur in the multiple slit interference pattern. Compute the degree to which theory matches your observational measurements. MATERIALS Vernier data-collection interface Vernier Graphical Analysis app Ruler Vernier Diffraction Apparatus Vernier Dynamics Track Green Diffraction Laser (optional) INITIAL INVESTIGATION DO NOT LOOK INTO THE LASER!! Direct exposure of the eye by a beam of laser light should always be avoided with any laser, no matter how low the power. 1. Attach the laser at one end of the track so it faces down its length. Connect the power supply. Leave the laser off until all parts are in place to avoid accidental reflections. 2. Set the diffraction slit assembly to a double slit with the slit width, a = 0.08 mm, and the slit separation, d = 0.25 mm. Attach the assembly to the track, with the silver reflective side of the glass plates facing the laser. Position it about 10 cm from the laser assembly. 3. Attach the High Sensitivity Light Sensor and the Linear Position Sensor to the opposite end of the track, with the light sensor facing the slits. The more space between the sensor and the diffraction assembly, the greater the spacing in the interference pattern, so maximize this to diminish the effect of ruler measurement errors. 4. In the data table, record the distance, D, from the slit assembly to the light sensor. Keep this distance fixed throughout the experiment. 5. Record the wavelength, , of the laser in the data table. Figure 1 Laser and slit assembly Figure 2 Light sensor assembly 6. Turn on the laser. Adjust the horizontal laser position to achieve maximum brightness of the pattern. Adjust the vertical laser position to center the pattern vertically on the entrance aperture of the light sensor. Use this procedure every time you change the slits. 7. Slide the light sensor assembly so that the pattern from the beam falls on the screen to one side or the other of the aperture disk (see Figure 2). Draw or describe the features of the double slit pattern below. For the double slit settings, try different slit separations and slit widths to observe how each variable affects the pattern observed. Describe your observations. MEASUREMENT OF THE SINGLE SLIT PATTERN TRIAL 1: 1. Make sure the diffraction slit assembly is set to double slit with the slit width, a = 0.08 mm, and the slit separation, d = 0.25 mm. Record the slit width, a, and slit separation, d, in the data table. 2. Turn on the laser. Adjust the laser position if necessary. 3. Using a ruler, measure and record the distance between the centers of the main dark fringes that establish the central group of light bands. Repeat this measurement for the second and third groups if they are visible. Record these measurements in the data table. 4. Divide these width values in half to give you the distance from the center of the pattern to each dark spot. Record these values in the data table. 5. Count the number of bright bands in the central group. Write this value in the data table. 6. Divide the full width of the central group (ruler measurement) by the number of bright bands in the central group to get the average spacing of the dark fringes in the central group. Write this value in the data table. 7. Connect the High Sensitivity Light Sensor and the Linear Position Sensor to the interface and start the data-collection program. 8. The optimal combination of light sensor sensitivity and sensor aperture depends on the type of laser and the slit configuration. It allows an intensity reading that reveals the detail necessary for you to make your measurements. We suggest setting the sensitivity to 10 W and using an aperture of 0.5 mm. 9. Move the sensor assembly all the way to one side of the sensor track (i.e., move it perpendicular to the main laser track). Zero both sensors. 10. Start data collection, and then slowly slide the sensor assembly stage toward the other side of the sensor track on which it is mounted. You will be moving the sensor across the interference pattern (like a scan of it). Take at least 20 seconds to execute the motion, slowing when the sensor is traversing the brightest portions of the pattern. Stop collecting data when the stage reaches the end stop. Test that you moved the stage sufficiently smoothly by storing the run and collecting another run, moving the stage in the opposite direction. The trace of the second run should overlay that of the first. 11. In Graphical Analysis, view the intensity vs. position graph. Zoom in on the portion of the graph that shows the first three dark fringes you measured earlier with the ruler. Choose the Examine tool from the Analysis menu and position the cursor on the peak corresponding to the central maximum. Record the position of the central peak in the data table. Record the position of the main dark fringe for the first three groups on either side. If it is difficult to find the third dark fringe, explain why. 12. Determine the distance, yn, from the central maximum to the first three main dark fringes on either side (i.e. 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = |𝑓𝑟𝑖𝑛𝑔𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 − 𝑐𝑒𝑛𝑡𝑟𝑎𝑙 𝑝𝑒𝑎𝑘 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛| ). If the distances are not the same on either side, decide how to best report them (i.e., use the average). Record these distances from the central peak to each of the first three dark fringes in the data table. 13. Using Graphical Analysis, determine distance between the minor dark fringes (minima) which form a bright band within the central group and record this value in the data table. Do this for two more bright bands in the central group. TRIAL 2: 14. Choose a different double-slit configuration, then repeat the previous steps for the new slit configuration. You may find it necessary to change the aperture on the light sensor to obtain sufficient detail to complete your evaluation of data. When you are finished collecting data, save your experiment file and turn off the laser. TRIAL 1 Wavelength (nm) D, distance to screen (cm) a, slit width (mm) d, slit separation (mm) Main dark fringes for groups of light bands Group 1 Group 2 Ruler Full Ym = distance Full between distance dark /2 fringes (cm) (cm) Central Peak position (cm) Graphical analysis Dark Ym = fringe |Dark position fringe (cm) position – central position| (cm) Desmos Ym = theoretical distance (cm) Percent Error (Graphical Analysis vs Theory) Group 3 By eye Minor dark fringes within central band Number of light bands in central group Average space between minor dark fringes within central group (cm) Graphical analysis Measured distance between minor dark fringes within central group (cm) Desmos Theoretical distance between minor dark fringes within group (cm) Percent Error (Graphical Analysis vs Theory) Bright band 1 Bright band 2 Bright band 3 AVERAGE TRIAL 2 Wavelength (nm) D, distance to screen (cm) a, slit width (mm) d, slit separation (mm) Main dark fringes for groups of light bands Ruler Full Y= distance Full between distance dark /2 fringes (cm) (cm) Central Peak position (cm) Graphical analysis Dark Y = |Dark fringe fringe position position (cm) – central position| (cm) Desmos Y= theoretical distance (cm) Percent Error (Graphical Analysis vs Theory) Group 1 Group 2 Group 3 By eye Minor dark fringes within central band Bright band 1 Number of light bands in central group Average space between minor dark fringes within central group (cm) Graphical analysis Measured distance between minor dark fringes within central group (cm) Desmos Theoretical distance between minor dark fringes within group (cm) Percent Error (Graphical Analysis vs Theory) Bright band 2 Bright band 3 AVERAGE THEORETICAL PREDICTIONS 1. Read the materials in the Appendix at the end of the lab. 2. Identify the formula for multiple slit diffraction that gives the light intensity on the screen as a function of distance from the center of the pattern, slit width, distance between the slits, and distance from the slits to the screen. Write the formula below. 3. Make a model of this formula in Desmos where y represents the intensity on the screen and x represents the distance from the center of the pattern (i.e., the center sits at x=0). You will need to add sliders for slit width, a, and distance from the slits to the screen, D. (You should be able to make a copy of your Desmos model from last week, and then just add the double slit factor to it.) 4. Make sure the Desmos model generates an intensity vs distance graph is like the diffraction patterns that you observed. 5. In your Desmos model, for each trial, use the values for a, d, and D from your experiment. Then find the predicted distances of the first three minima (i.e., the x-values where the y value representing intensity goes to zero, the first three places where the curve touches the x-axis). Record these values in the theoretical columns of the respective data tables for trials 1 and 2. 6. Save your Desmos model and paste a hyperlink to it below. EVALUATION OF DATA 1. How does the double-slit pattern change as you change the slit width? How does the doubleslit pattern change as you change the distance between the slits? Test your prediction if necessary. 2. Compute the percent error between the theoretical values from your Desmos model and the distances you measured with the Vernier light sensor. Record these values in the respective data tables for trials 1 and 2. 3. Compare your findings with those of other groups. How do your percent errors compare to other groups? If your errors are comparatively high, explain the potential causes of why your error is higher than other groups. 4. Describe the phenomenon of interference in your own words. 5. What did you learn from completing this lab? APPENDIX Single Slit Pattern The single slit pattern is given by 𝜋𝑎𝜃 2 𝜆 ) 𝐼 = 𝐼0 ( 𝜋𝑎𝜃 𝜆 sin where a is the slit width. Recall the small angle approximation, tan 𝜃 = 𝑦/𝐷 ≈ sin 𝜃 ≈ 𝜃, where y = distance from central maximum on screen, and D = distance from slit to screen. In the small angle approximation, our intensity pattern becomes 𝜋𝑎𝑦 2 sin 𝐷𝜆 ) 𝐼 = 𝐼0 ( 𝜋𝑎𝑦 𝐷𝜆 The zeros for the intensity (dark spots) occur when the sine function equals zero. The sine function 𝜋𝑎𝑦 equals zero when 𝐷𝜆 = 𝑛𝜋 where 𝑛 = 1, 2, 3 … Therefore, we have 𝑎𝑦 =𝑛 𝐷𝜆 Solving for y, we have for the location of the zeros on the screen 𝑦= 𝑛𝜆𝐿 𝑎 Multiple (N) Slit Pattern For multiple slits, we have two phenomena of interference happening simultaneously. We have the single slit interference from each of the slits, as well as interference due to the path length differences between each slit to the screen. Thus, the multiple slit pattern (a = slit width, d = slit separation, N = number of slits, in this case two) is given by the pattern for the single slit given above multiplied by a term accounting for the multiple slit interference 𝜋𝑎𝜃 2 𝑁𝜋𝑑𝜃 2 sin 𝜆 ) ( 𝜆 ) 𝐼 = 𝐼0 ( 𝜋𝑎𝜃 𝜋𝑑𝜃 sin 𝜆 𝜆 sin Recall the small angle approximation, tan 𝜃 = 𝑦/𝐿 ≈ sin 𝜃 ≈ 𝜃, where y = distance from central maximum on screen, and L = distance from slit to screen. In the small angle approximation, 𝜃 ≈ 𝑦/𝐿, the intensity pattern becomes 𝜋𝑎𝑦 2 𝑁𝜋𝑑𝑦 2 sin sin 𝜆𝐿 ) ( 𝜆𝐿 ) 𝐼 = 𝐼0 ( 𝜋𝑎𝑦 𝜋𝑑𝑦 sin 𝜆𝐿 𝜆𝐿 The zeros for the intensity (dark spots) occur when either sine function in the numerator equals zero, so our prior result for the zeros of the first sine function holds (diffraction pattern zeros). 𝑦= The second sine function equals zero when 𝑛𝜆𝐿 𝑎 𝑁𝜋𝑑𝑦 = 𝑚𝜋 where 𝑚 = 1, 2, 3 … 𝐿𝜆 Therefore, we have 𝑁𝑑𝑦 =𝑚 𝐿𝜆 Solving for y, we have for the location of the zeros (N-slit interference pattern zeros) 𝑦= 𝑚𝜆𝐿 𝑁𝑑 Thus, the minima for the N-slit pattern occur when either 𝑛𝜆𝐿 𝑦= 𝑎 or 𝑚𝜆𝐿 𝑦 = 𝑁𝑑

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