Diffraction and Interference LBS 272L

Transcription

1 Diffraction and Interference LBS 272L In this lab you ll be using a laser. Make sure NEVER to look directly into the laser beam. It can damage your eyes. Before this lab Read the LBS272 lecture material on interference and diffraction. Read this lab manual Do lab quiz 10 Purpose The objective of this laboratory is to explore interference and diffraction of light. Using a coherent light source (laser), you will observe and analyze diffraction and interference patterns created when the light beam passes through a single and double-slit barrier. Theory Discussion In the previous lab, you have explored interference of electromagnetic waves: the waves can interfere constructively or destructively. In this lab you ll be studying such phenomena with light (also electromagnetic waves, but with a very different wavelength then microwaves). So, we regard light as being a wave (in other instances, like later in the nuclear physics lab, we regard light as a particle called a photon). So, superimposed waves of light may exhibit constructive or destructive interference. Stationary interference patterns can be produced in space when light waves which have a constant phase relationship with one another (that is, light waves which originate from a coherent source) meet. If waves that are in-phase arrive at a single point simultaneously, the light will then interfere constructively at that point, creating a more intense light "spot" in space. Similarly, if light arrives at a point out of phase, then the light will interfere destructively, decreasing the intensity of light at that point. Complete constructive (destructive) interference of light at a point will be observed if the light is precisely in phase (180 degrees out of phase). Diffraction and Interference 1

2 Screen Plane Waves Point Source, S1 and S2 for Spherical Waves sin θ = tan θ = y/l y Observation Point, P d θ (Small angles, θ Path Difference = d sin θ Spheric al Waves L Figure 1. Coherent waves from laser are incident on a double slit aperture. If the slits are small enough, their openings can be treated as point sources, S1 and S2, from which light is emitted spherically, thus allowing for interference. Young's Double Slit Experiment: Consider Figure 1 above. Light from a monochromatic, coherent source is incident upon a barrier as shown, and passes through the slits where it is projected onto a screen. If the "slits" which are shown are sufficiently thin, light which passes through the barrier with the thin slits can be considered as emanating (spherically) from pointsources S1 and S2. (Recall: Huygen s principle describes how light bends or diffracts around the corner of a barrier.) Because the light that arrives at the screen is a superposition of light from two point sources, an interference pattern will appear on the screen. If the path distance from a particular point on the screen to the two sources is equivalent to half of the wavelength (or multiples thereof) of the light, then complete destructive interference will occur at that point, and a dark spot will be observed in the interference pattern. Conversely, if the path difference to a particular point is equivalent to an integer multiple of the wavelength of the light, then complete constructive interference will occur, and a bright spot will appear on the screen. A "picket fence" pattern is thus observed on the screen. Using the small angle approximation, the condition for constructive interference is derived. Diffraction and Interference 2

3 d(y/l) = mλ (m = 0,1,2,3...) {eq. 1} It follows that the intensity pattern on the screen due to double-slit interference can be described by the expression, I int = I 0 cos 2 (πdy/[λl]) {eq. 2} Interference, however, does not completely describe the actual appearance of the light pattern. Diffraction must be taken into account to provide an accurate description of the interference pattern that appears as a result of Young's historic experiment. Consider Figure 2, below, which describes single-slit or Fraunhofer diffraction. Single-Slit Barrier Path Difference, a/2 sin θ Screen a/2 θ Diffraction θ Incident Plane Waves Figure 2. As before, plane waves are incident on a single slit. According to Huygen s principle, we consider spherical wavelets emanating from each point of the wavefront as it reaches the plane of the slit. The resulting intensity pattern on the screen represents a superposition of the light from each of these source points. As before, plane waves from the coherent source arrive at a barrier; this time a single slit. According to Huygen's principle, we consider spherical wavelets emanating from each point of the wavefront as it reaches the plane of the slit. The light reaching some point on the screen, as before, is a result of a superposition of the light from each of these "source points." L Diffraction and Interference 3

4 (Instead of two source points, however, we now have an infinite number of them!) Light from the source point at the very top of the slit must travel a longer path to reach some point on the screen than light that emanates from the source point exactly at the center of the slit. Note that these two contributing source points are exactly a/2 apart from one another. For each source point in the upper-half of the slit, then, there will be a corresponding point in the lower half of the slit that is located exactly a/2 below. Assuming, as before, that the screen is sufficiently far from the slit, we observe that the path difference between the light which has traveled from a point in the upper-half of the slit and light which arrives from its corresponding point in the lower-half is simply, (a/2)(x/l) The condition for complete destructive interference at a point on the screen, then, is, (a/2)(x/l) = n(λ/2) (n = 0,1,2,3...) {eq. 3} The interference pattern that appears on the screen as a result of this single-slit diffraction is described by the expression, I diff = I 0 sin 2 (πax/[λl])/(πax/[λl]) 2 {eq. 4} Now, let s go back to the double-slit experiment. Each of the slits, acts like a single slit described above, but the combination of the two slits acts like Young s double slit. So, we would expect the resulting interference pattern to be a result of contributions from both double-slit and single-slit (diffraction) interference. Multiplying our two expressions that describe the interference pattern resulting from each type of interference, we obtain, I tot = I 0 sin 2 (πax/[λl])cos 2 (πdy/[λl])/(πax/[λl]) 2 {eq. 5} Note that for a given point in this pattern, x and y are the same variable (x=y). The separation into x and y was for the purpose of clarifying which part comes from the double-slit part (y; constructive) and which part comes from the single slit part (x; destructive). Diffraction and Interference 4

5 Figure 3. Relative intensity distribution versus distance along the screen. The left panel represents a double slit geometry in which the slits are very small compared with the wavelength of light. The center panel depicts the intensity pattern from a single slit. The right panel shows the intensity from a double slit in which the slits have widths comparable to the wavelength of light. Part 1: Single-Slit Diffraction We will first investigate interference that results from allowing light to pass through a single slit. Equipment Needed Spectra Physics, Model 133 Laser, PASCO Light Sensor, PASCO Rotary Motion Sensor With Positionable Mount, PASCO Signal Interface, PASCO Diffraction Plate. Setup With the laser placed roughly 3 meters from the optical sensing device (use meter sticks to measure the exact distance and record this value), position the optical sensor at the proper height, so that the fibers will intercept as much of the beam spot from the laser as possible. Now, position single-slit B (a = 0.08 mm) in the path of the light beam so that a horizontal diffraction pattern is observed when you place a white sheet of paper in front of the optical sensor. Remove the sheet of paper and properly adjust the horizontal positioning of your optical sensor so that the optical fibers remain at the proper height throughout the observable "length" of the interference pattern. You should, at least, be able to "scroll" horizontally through the entire diffraction peak. This will require a bit of tweaking. With your optical sensor properly positioned, open the Data Studio experiment entitled, "Diffraction and Interference." Notice that a data table Diffraction and Interference 5

6 and graph should appear, which monitor light intensity vs. position of the optical sensor. Before taking data, you will need to calibrate the light sensor. Click on the light sensor in the setup window and select Calibration. Position the sensor so that the brightest of the laser light is falling on it, and take a reading of the maximum. Then turn off the laser and take a reading of the minimum. This will help subtract out the ambient light of the room. With your optical sensor positioned at one end of the interference pattern, click on Start and carefully, without excessive jerkiness, scroll through the interference pattern. Observe the intensity pattern that appears on the graph -- you should easily be able to discern a broad central (Gaussian-like) peak. You may also be able to pick out the secondary maxima on either side of the central peak (the light detectors are "noisy", so it is not so easy, especially during day-time ). Repeat this procedure at least 5 times, varying the "scrolling speed" which you are using each time, until you have recorded a data set that you believe to be the "nicest" possible, given the noisiness of the data. Save this data set and include the graph in your report. Using the "crosshairs" in Data Studio, find the width of the central maximum. Q1: Using the value for the total width of the central maximum that you just found and given the slit width (0.08 mm), calculate the wavelength of the laser light. Q2: -If you did see the secondary maxima, what is their position relative to the central maximum peak and does this agree with the expected value (show the calculation). -If you did not see the secondary maxima, calculate where they should have appeared and indicate this in the graph that you included in your report. Part 2: Double-Slit Interference Taking in all that you have learned about diffraction, you are now ready to observe interference that results when light from the laser is incident upon a double-slit. Diffraction and Interference 6

7 Setup With your laser/detector setup still intact, replace the single-slit with double slit pattern D, (slit width, a = 0.04 mm, slit separation, d=0.125 mm) observing the resulting interference pattern when you place the white sheet of paper in the beam path. You will probably have to realign the sensor, since your beam will move a little during this. Notice the central, brighter interference fringes, followed by a broader dark spot on either side (this is the first dark spot resulting from a diffraction minimum). More interference fringes will appear within the secondary envelope of the diffraction pattern on either side, and so on -- the result of the interference and diffraction patterns being superimposed (multiplied together). As before, record the intensity vs. position at least 5 times, optimizing the appearance of the interference pattern by varying your scrolling speed. Of your five sets of data, pick the one that looks the best (the most like the right-hand graph in Figure 3). Using this graph (you may want to delete the other data runs after saving your data), pick the Smart Cursor in Data Studio. By estimating where the peaks of the curve are (they might not be exactly on a data point), determine the distance between maxima, y. Take several values of this separation. Next, estimate the points at which the diffraction pattern goes to zero, like you did for the single-slit diffraction exercise. You should be able to get a reasonable estimate within a centimeter or less. Half the distance between these points is x 1, the distance to the first diffraction minimum. Q3: Using the values for y and x 1 found above, calculate experimental values for both the slit width a and the slit separation d. Q4: What effect would decreasing/increasing the slit width have on the observed interference pattern? Decreasing/Increasing the slit separation? Diffraction and Interference 7

8 PART 3 There are various other slits on the diffraction plate. Use A,C,E and F and measure the diffraction/interference pattern (add the graphs to your report). You don t have to do a full quantitative analysis, but describe your observations and explain whether this makes sense after comparing to the earlier measurement and the theory. If on top of that you perform a quantitative analysis (you actually determine the slit width and or the slit distances) you will be rewarded a bonus point (if the result makes sense of course). Finally, Look at the diffraction plate, slit H. Describe qualitatively (make a drawing) what kind of pattern you would expect if you were to use this slit (you can try it out!). Diffraction and Interference 8