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Laser Diffraction Introduction and Theory When light waves of a single wavelength pass through a slit and hit a screen, the image is not a single spot of light but a line of spots of varying intensity separated by dark regions. This pattern, the diffraction pattern, occurs because light from one part of the slit interferes with light passing through other parts of the slit. Depending upon the relative phases of the light waves, this can produce constructive or destructive interference. If two or more slits are present, the diffraction pattern is complicated by the interference between light passing through the different slits. In this experiment, you will study the diffraction patterns produced by two, three, four, and five slit combinations. Figure 2 (a). This pattern is determined by the width of the two slits. The interference pattern is due to the interference of the light arriving at the screen from the two different slits and is described by the cos2 term. This function is plotted in Figure 2(b). Figure 2(c) shows the resulting intensity pattern when these two functions are combined. Figure 1. Laser Diffraction Experiment Figure 1 shows the geometry of the experiment. Consider a double slit experiment in which light of wavelength is diffracted by two slits each of width a and separated by a distance d. In this case, the intensity at a point on the screen associated with angle is given by the equation sin 2 α I I0 cos 2 δ , α2 where I = intensity at angle , I0 = intensity at center of pattern, = (a/) sin , and = (d/) sin . This is actually a diffraction pattern and an interference pattern superimposed on one another. The diffraction pattern is described by the function D() = sin2/2 which is plotted in Laser Diffraction Figure 2. Double Slit Intensity Patterns Of particular interest to us is the fact that the minima in the interference pattern are given by m = d sin , m = 0, 1, 2, …., integer. (1) The minima in the diffraction pattern are given by m = a sin , m = 0, 1, 2, …., integer. (2) 1 If a minimum in the diffraction pattern is located at the position of a maximum in the interference pattern, the interference maximum will be eliminated. This is called a "missing order" in the an aperture bracket is mounted on a rotary motion sensor that moves along a linear translator bar. Signals from the light sensor and rotary motion sensor are fed into an interface box that is connected to a computer. These signals are then analyzed using the Science Workshop program. The position of the light sensor can be measured in radians or distance along the linear translator bar in cm. Turn on the interface box and click on the Science Workshop icon. Open the file slitdif.sws. Position the laser so that it is located110-120 cm in front of the light sensor. With the light sensor approximately at the center of the linear translator, adjust the laser so that it is shining at the midpoint of the aperture opening in front of the light sensor. Now put the bracket with the single slits just in front of the laser and position it so that the laser is shining through the slit with slit width a = .04 mm. Measure the distance L between this single slit and the slit in front of the light sensor. Try to keep L constant throughout the data collection process. Figure 3. Interference patterns for multiple slits interference pattern. If there are N-slits where N 3, it can be shown that there will be "secondary" maxima between the principal intensity maxima. These secondary maxima are much reduced in intensity relative to the intensity of the primary maxima. There will be N-1 equally spaced minima separated by N-2 secondary maxima in the space between principal maxima. Figure 3 shows the interference pattern for two, three, and four slits. These interference patterns will be modified by the diffraction pattern given by D() as shown in Figure 2(c). It should be noted that the positions of the principal maxima do not change as N increases as long as d does not change. However, the principal maxima peaks do become narrower with increasing N. Procedure The experimental setup that will be used is shown in Figure 4 on the next page. A light sensor with 2 Laser Diffraction Figure 3. Interference patterns for multiple slits Move the rotary motion detector along the linear translator until it is in contact with the "rack clamp". Put the setting on the light sensor at 10. Click on the record (REC) button on the computer display. Move the rotary motion detector by hand along the total length of the linear translator. Try to move the motion detector as smoothly as possible and keep the back of the detector in contact with the upright edge of the base of the linear translator. When the rotary detector reaches the end of the linear translator, stop the data collection. Expand the graph and examine the spectrum. It should clearly show the large central principal peak and at least two secondary peaks on either side of the center peak. If you don't have these 5 peaks, make additional runs until you do. It may be necessary to move the rotary sensor closer to the rack clamp when initially lining up the laser. After you have a suitable spectrum for the single slit, replace it with the double slit for which a = 0.04 mm and d = 0.125 mm. Keep the setting on the light sensor at 10. Repeat the process used to Figure 4. Experimental setup to obtain light intensity measurements of slit spectrum patterns obtain the intensity spectrum of the single slit. side of the central peak. Then s is the difference Examine the resulting spectrum. It should clearly between these two values. The corresponding show a group of 7 principal peaks at the center. angular distance is = s/L ( has units of Then there will be two smaller peaks on either radians). The displacement of the first minimum side of the central group, separated from the from the central peak in radians is just = /2. central group of peaks by a region of zero Use Equation 2 to determine . Compare this intensity. You should obtain at least these 11 value to the accepted value = 632.8 0.1 nm. peaks in your spectrum. You may well have Figurethis 4. Experimental Setup to Obtain Light Intensity Measurements of Slit Spectrum Patterns more peaks beyond minimum number. If you Now look at the graph associated with the double don't have at least these 11 peaks, make slit. Determine the positions of the 6 minima in additional runs until you do. It may be necessary the center grouping of peaks. Then determine the to change the relative position of the laser and the average value of the angular separation between optical bench, vertically and/or horizontally, in adjacent minima min and the average deviation. order for all of these peaks to be picked up by the Use min in Equation 1 to determine . sensor. Compare your results with the accepted value given above. Compare your two results for and Once you have an appropriate set of data for the comment on their relative experimental precision double slit, repeat the process for the three- and and accuracy. four-slit sets that are on the same card as the double slit. Before taking data on the three- and Using the graph of the double slit data, measure four-slit patterns, adjust the slit position until the and record the intensity and position of each of secondary minima are clearly visible to the naked the central 11 peaks. Repeat for the graphs of the eye. If they are indistinct to you, they will be 3- and 4-slit data. (In these cases, note the indistinct on the Science Workshop graph. Also, secondary peaks between the principal peaks, but you may need to reduce the setting on the light only make measurements of intensity and sensor to 1, for these two spectra. Once you have position for the principal peaks.) all four spectra, you may want to save your data file with a new filename or on a different drive. Plot a scatter graph of percent maximum intensity vs. angular distance from the center peak. Since Analysis the central peak has the maximum intensity, it should be correspond to the point (0, 100). Add Consider the graph associated with the single slit. plots of the three- and four-slit data to the plot of Use the cross hair measuring device to find the the double-slit data. (To do this, click on Graph angular positions of the two minima on either and then on Add Plot to Layer.) Laser Diffraction 3 The theoretical relativity intensity of the peaks is given by the diffraction function D(). Graph this function on the graph with your data. Be sure to multiply by 100 to make it a percent instead of a fraction. You may assume the small-angle approximation, so that ≈ a/, Observe how closely this function fits your data and how consistent the various data sets are with respect to each other. 4 Laser Diffraction