Interference of Light

Transcription

1 Interference of Light Objective To study the interference patterns of light passed through a single and double-slit, a human hair, and compact discs using a laser. Equipment meter stick index card slit slide mount penlight 10 m tape measure Introduction Scientists were fairly satisfied that the particle theory of light was correct (because Isaac Newton said so!) until Thomas Young created an inexplicable interference pattern using two very narrow and closely spaced slits. He then explained the pattern using the idea that light behaved like a wave, and proceeded to measure the wavelengths of the colors. When plane waves come in contact with a barrier that is not significantly larger than their wavelength, the waves spread out into spherical waves (figure 1). The effect is to allow waves to spread out in all directions from the edges of the barrier, a phenomenon called diffraction. This spreading enables the waves to go around corners. If two or more rays pass through the same point in space at the same time, they interact with each other. Of special interest is the case in which light passes through slits in a barrier. At any point in space beyond the slits, the local oscillation of the electric field (and the magnetic field, too) will be the result of the sum (or superposition) of the waves from each slit. Such interaction is called interference. Interference may be constructive or destructive. In destructive interference, the peak of one wave meets with the trough of another wave (the two waves are said to be 1/2 wavelength out 52

2 of phase) so that the two waves cancel each other out. In constructive interference, the peaks of two waves meet (the waves are in phase) creating a wave with a greater amplitude than either of the two original waves. 53 Figure 1: Double-slit interference geometry. Double-slit interference Imagine monochromatic light (one color) incident on a double slit. If light were composed of particles, we would expect to see two bands of light on the screen resulting from the projection of light through each slit. In fact, what we observe is a multitude of alternating light and dark bands. Figures 2 and 3 illustrate portions of the spherical waves, which we call rays and which combine as they strike the screen. In figure 2, the two waves arrive at the screen in phase. As a result, they constructively interfere and a bright band appears on the screen. In figure 3, the waves arrive at the screen 1/2 wavelength out of phase. They destructively interfere and a dark band appears on the screen. Young realized that constructive interference could only occur when the wave from one slit had to travel some whole number of wavelengths farther to the screen than the other wave. The path difference between the two waves is labeled l in figure 4. Here we can see that l = d sin.

3 54 Figure 2: Constructive interference. Notice that the waves arrive at that one particular spot in phase with each other (crest with crest, trough with trough). The electric field or "wave" amplitude combine constructively, seen as a bright spot. Figure 3: Destructive interference. Notice that the waves arrive at that one particular spot out of phase with each other (crest with trough, trough with crest). The positive and negative electric field amplitudes cancel each other out to form a dark spot.

4 55 Figure 4: Relationship between angle and the difference in path lengths. Note that this is the same as the that appears in figure 3. Constructive interference occurs when d sin = m (m =0, 1, 2, 3,...). (1) where d is the spacing between slits. The value of m is the order of the interference band. For example, m =0corresponds to the zeroth order band, m =1corresponds to the first order band, and so forth. The zeroth order band is in the center of the interference pattern, since both wave pieces travel the same distance to the screen (l = d sin = 0). Relative to the zeroth order band, where must the first and second order bands be located? Since Young knew d and m, and he could measure, he could calculate the wavelength of the incident light. What values of m correspond to destructive interference? Single-slit interference The wave theory of light gained many supporters with Young s double-slit experiments, but full acceptance of the theory came through experiments using a single slit to diffract light. Single slit interference patterns are most simply expressed by locating the places of destructive interference (dark bands). For a slit of width w, these occur when w sin = m (m =1, 2, 3,...) (2)

5 56 but not when m =0, which corresponds to a bright central band. This bright central band is caused by light that passes through the slit without interference among the individual rays. The equipment you use today includes slides with pairs of slits double slits, that is. The spacing (d) between the two slits in each pair is generally larger than the width (w) of each individual slit. Even though the presence of two slits causes a double-slit interference pattern to be formed, each individual slit also causes a single slit interference pattern to be formed. But that s also the good news. By using these slides with the pairs of slits to create interference patterns, you will be able to run two experiments at once: double-slit and single-slit. All you need to do is to keep track of which characteristics of the pattern corresponds to the double-slit equation and which characteristics correspond to the single-slit equation. Let s check that out. Recall that it was mentioned in the first paragraph of this section that d>w, generally. If you look at the two interference patterns above, what would the effect on be if we made d or w larger? double-slit < single-slit (3) Since d>wthen we expect that would become smaller and that s exactly what we see. The single-slit pattern is much more widely spread out and the double-slit pattern appears as fine structure within the bright spots (essentially where m = 0.5, 1.5, 2.5,... in the single-slit equation) between the dark, single-slit bands. In general, single slit patterns have a very bright central spot and the bright spots between the dark bands fall off rapidly in brightness the farther they are from the central maximum. By contrast, double-slit bright bands fall off only very slowly in brightness the farther they are from their central maximum; they nearly look uniformly bright. Calculating Our basic technique will be to project a diffraction pattern, created by slits in an opaque slide, on a wall. Specifically, we can use a piece of paper taped to the wall to record the pattern and to make it easier to measure distances within the pattern. Let s use L to represent the distance to the wall from the slide containing the pairs of slits. The distance between bands on the pattern projected on the wall can be represented by y. If we project the beam along a path that is perpendicular to the wall, then the acute angle formed by that beam path and the path to any given diffraction band is the same angle as is shown in figure 3, namely. As was stated before, this is the same value of that we see in figure 4, and thus it relates to wavelength and also

6 to either slit width or slit spacing. (See your textbook for a diagram and discussion of this point.) From basic trigonometry, the tangent of that angle will be the band spread on the wall divided by the distance to the wall. That is 57 tan = y L. (4) With the small angle approximation, sin = tan, we can use our measurements of y and L to calculate and then we can use that value in the appropriate interference equation above. Our working equation is, then for double slit bright spots. And for single slit diffraction pattern dark spots. dy L = m (5) wy L = m (6) Procedure Caution: Never look directly into a laser beam. Determine the wavelength of the laser light. For each of the double-slit patterns in the slide, obtain an interference pattern by projecting a laser-pointer beam through the slide onto a wall. Preserve one of these patterns on paper, labeling dark and light spots. For each dark band, label the single-slit order (m) and determine the single-slit value for y. Within each bright spot, outline the smaller and regularly spaced double-slit bands. No need to label the orders here; merely count how many are available and divide the overall spread by this number to get the double-slit value for y. You will need the distance from the slide to the wall as well as the appropriate spacing within the pattern. From these patterns, calculate the wavelength of the laser light. Slit spacing (d) and width (w) is given on the slide. Find the average of the wavelengths you calculated. Comment on the precision of your results. Repeat the experiment carefully until your results are with 10% of the actual.

7 58 Measure the diameter of a hair. Cut a window into an index card. Tape a hair (donated by a member of your group) across the window. Do not put tension on the hair. Direct the laser at the hair and observe its interference pattern cast on a wall. The hair diffracts light as though it were a single slit. Can you see why, in terms of wave theory, a narrow obstruction may behave as a narrow aperture? Using the average wavelength of light obtained in part one, calculate the diameter of the hair. Now, gently use a micrometer to measure the diameter of the hair and comment on your results. It may interest you to know that naturally curly hair is curly because it does not have a circular cross-section. The greater the ellipticity of the cross-section, the curlier it is. Can you demonstrate this using this technique? Optional: Groove spacing on a compact disc. A compact disc is created by burning a very narrow groove into a piece of plastic with a laser. The groove starts in the center of the disc and spirals out to the edge. To make the surface reflective, it is silvered with a thin coat of metal. A finely focused laser beam reads the disc by reflecting from bumps and pits in the groove. If the beam is too wide, the light will intercept the groove in several places. As a result, an interference pattern like that of Young s double-slit is created. If a CD without the opaque coating is available, pass the laser s beam through the CD so as to create a diffraction pattern on a piece of paper taped to the wall. If a transparent CD is not available, then carefully reflect the laser off a CD onto a piece of paper taped to the wall and observe the interference pattern; this setup is more difficult to use with good precision in your results. (If using the reflection method, you must fully regard two paths to make sure nobody s eye is illuminated by the beam.) Using the double-slit formula for constructive interference, determine the spacing between the grooves in the disc. (Hint: The central maximum should be reflected back through the aperture of the laser.) Is the spacing the same for both audio and data compact discs? Find out!