Physical Optics You can observe a lot just by watching. Yogi Berra (1925-2015) OBJECTIVES To observe some interference and diffraction phenomena with visible light. THEORY In a previous experiment you looked at the interference between waves coming from two slits. The wavelength was about 3 cm, so the apparatus was relatively large and the angles were substantial. In this experiment we will more literally look at some of the same phenomena, using visible light. All the basic principles are the same as for microwaves, but the apparatus and angles are smaller. In exchange for that inconvenience, we can use our built-in optical detectors to examine whole patterns, rather than plotting out intensities point-by-point as we did with microwaves. Accordingly, we will be mostly concerned with careful qualitative observations in this experiment. It is customary to discuss interference and diffraction of light as though they are separate phenomena, but there is no firm physical basis for a distinction. Roughly speaking, waves emanating from more than one very small source may exhibit interference. Waves emanating from a source more than a wavelength wide may exhibit diffraction. Physically, all this means is that waves starting from one region of a large source interfere with waves leaving from other regions, so diffraction is just another example of interference. Optical wavelengths are short enough that it is difficult to make a source or obstacle smaller than a wavelength, so both manifestations of interference are seen in a typical experiment. Relative sizes and distances also affect the results of interference experiments, but in a more subtle way. First consider a situation where a light source illuminates a slit in an opaque screen. If both the light source and the observer are very far from the slit, relative to its width, it is a good approximation to assume that the slit is illuminated by plane waves and that the observer receives plane waves. Using this approximation it is easy to compute the distance a wave must travel to reach the observer from each part of the slit and then to use the principle of superposition to deduce the observable intensity. This is the approach you are familiar with from elementary texts. The same method is used if the light source, the observer, or both, are close to the screen, but the geometry is harder to work out. In this experiment we will consider the simple
plane-wave situation ("Fraunhofer geometry") quantitatively and the complicated one ("Fresnel geometry") only qualitatively. 1. Slits in Fraunhofer geometry Recall that two narrow slits a distance a apart produce an interference pattern with intensity maxima at angles given by mλ = a sinθ (double slit maxima) (1) where m is an integer. These angles correspond to path length differences of a full wavelength or integral multiples of a full wavelength. When the path difference is half a wavelength less, a minimum occurs. The angles for minima are therefore ( m + 1 2)λ = asinθ (double slit minima) (2) Of course, the derivation assumes that light waves actually leave both slits at the angle θ, but a single slit does not transmit light uniformly in all directions when illuminated by a plane wave. The intensity observed at θ for a single slit is ( ) = I 0 sinβ I θ β 2 (3) I I 0 0 sin sin Fig. 1 Intensity vs angle for a single slit of width b, and for two slits of width b separated by a distance a. Physics 231 Physical Optics 2
where β = (π/λ) b sinθ and b is the slit width. The pattern has a peak at θ = 0, with a width inversely proportional to b. The minima of the intensity occur at angles given by sinθ =± mλ b (single slit minima) (4) for non-zero integers m. We should, therefore, expect to see the interference minima described by Eq. 2 only in the region where the peaks of the two single-slit patterns overlap. These features are illustrated in Fig. 1, which shows the intensities produced by a single slit and by two identical slits. Comparing the two plots, you can easily see the features due to the individual slit and due to the interference of waves originating in different slits. When there are several parallel slits, the same considerations apply. Your textbook considers the case of an opaque barrier with N slits illuminated by a plane wave from the left. The intensity as a function of angle on a distant screen is then given by ( ) = I 0 sinβ I θ β 2 sinnα sinα 2 (5) where α = (π/λ) a sinθ and a is the slit spacing. This ratio is plotted in Fig. 3 for two different N. Several features of Fig. 3 are of interest. At the peaks, the waves from each slit add because all the phase shifts are integral multiples of 2π. Peak intensity occurs when α =mπ, independent of N so the angles of maximum intensity are given by mλ = a sinθ (6) Intensity ratio 0 2 Fig. 2 Plot of Eq. 10 for N = 2 (dashed) and N = 6 (solid). The slit width is assumed very small. Physics 231 Physical Optics 3
exactly as for the two-slit case, but the peaks become narrower as N increases. The overall pattern will be modulated by the intensity distribution of the single-slit diffraction pattern, as in the two-slit case. Between the main peaks there are secondary maxima, which occur where the cancellation of the fields is not quite complete. As N increases the number of secondary maxima increases, but their intensity drops drastically, so they become almost invisible. Diffraction gratings, commonly used in precision optical spectroscopy, are regular arrays of many narrow slits with small spacing a. When illuminated with a collimated beam of reasonable diameter, they may have an effective N of 10 5 or more, resulting in isolated maxima at positions given by Eq. 6 and essentially no intensity in the subsidiary maxima. 2. Slits in Fresnel geometry In the last part of the experiment we will examine the diffraction pattern of a single wide slit as we move away from it. When the slit is illuminated with plane waves (parallel rays) we know roughly what to expect in two limits. Very close to the slit we should see a simple geometric shadow. Very far away we should get the Fraunhofer intensity pattern given by Eq. 3. At intermediate distances the pattern of illumination is complicated and a bit surprising. In particular, you should be able to find a distance at which the exact center of the diffraction pattern is dark, rather than bright. (This is a variant of the famous Poisson spot discussed in many texts.) We conclude this section with a few words about the requirements on the light source. You will note that all the angles we have calculated depend on the wavelength of the light. In order to see a simple pattern we must, therefore, use illumination with only a narrow range of wavelengths. Further, if the wavelength changes with time the patterns will also change, and we may not be able to observe anything if the changes are too rapid. A stable monochromatic source is fairly easy to construct from a gas discharge lamp and a color filter. Unfortunately, each atom in the gas discharge lamp emits light independently of the others. The phases of the waves emitted by different atoms vary randomly and one must arrange to use the emission from only one atom at a time in order to see a stable, but dim, pattern. This method works, but it is not really satisfactory. The device we will use, a laser, solves the problem by forcing all the atoms to emit their light in phase. The result is a superposition of the light from many atoms, which produces a very intense plane wave with the wavelength fixed by the properties of the emitting atoms. Using a laser source, it is very easy to observe interference phenomena. Physics 231 Physical Optics 4
Laser 5 cm f. l. lens 45 cm f. l. lens diffraction object screen 50 cm Fig. 3 Arrangement of the optical bench to demonstrate diffraction. EXPERIMENTAL PROCEDURE All the optical components, except the screen, are mounted in holders on an optical bench, as diagrammed in Fig. 3. Most of the diffraction objects are in a holder that slides through a special mount. The patterns are made by etching a thin metal film deposited on glass, which is mechanically fragile and easily damaged by fingerprints. Please do not touch the optical surfaces, nor allow the holder to fall over. The first task is to increase the output diameter of the laser beam so that it will completely illuminate the various objects we want to study. This is done with the pair of lenses shown in Fig. 3. The first lens focuses the plane waves from the laser. The light diverges from the focal point until it hits the second lens, which focuses it into a parallel beam (plane waves) again, with the desired larger diameter. To achieve these conditions you should center the two lenses on the beam and separate them by a distance approximately equal to the sum of their focal lengths. Now slide one lens back and forth a little until the beam spot has approximately constant diameter over as large a distance as possible. The lens positions will not need to be changed for the rest of the experiment. 1. Single-slit diffraction Install the slit holder as shown in Fig. 3, and position the slider so that the 0.16 mm single slit is in the laser beam. You may need to rotate the laser slightly to center the beam on the slit. Position the screen at a convenient distance, and sketch the single slit pattern. Label the first few minima with the appropriate index ±m. For a quantitative comparison with Eq. 4 you can measure the position x of several minima, as suggested in Fig. 4. For small angles sinθ tanθ = x/l and Eq. 4 becomes x m L =±mλ b (7) Physics 231 Physical Optics 5
screen object x Fig. 4 Definitions of distances and angle for interference measurements. L where x m is the position of the m th minimum relative to the center of the pattern. The easiest way to accomplish this is probably to mark the positions of several minima on a sheet of paper, and then measure the distance 2x m between corresponding minima. Summarize your data with a plot of x m vs m, and check that the slope of a proportional fit has the value expected for the known slit width, 0.16 mm, and laser wavelength, 632.8 nm. You can see a graphic demonstration of the effect of slit width by sliding the upper Variable Slit across the laser beam. It's very easy to see the diffraction pattern narrow as the slit widens from 0.02 to 0.20 mm. (There is no need to document this in your report.) 2. Double-slit diffraction The object labeled A in Comparison Slits is a pair of slits immediately above a single slit, all of the same width. Adjust the height of the holder so that you can see both the single and double-slit patterns clearly. Sketch the two patterns, maintaining the alignment, and identify the central peak from the single slit as well as the interference minima due to the double slit. Now look at the patterns from Comparison Slits B and C. Using Eqs. 2 and 4, explain the differences in each pair of patterns. In all three patterns you should label the features due to the single or double slit. To make quantitative comparisons, we rewrite Eq. 2 assuming small angles to obtain ( m + 1 2)λ = xa L (8) which implies that the double-slit intensity minima are equally spaced. The separation s between adjacent minima is the easiest quantity to measure. It is found by taking the difference of x values for n and n + 1, with the result s = L a λ (9) Physics 231 Physical Optics 6
Position one of the Double Slits in the beam, and estimate s by averaging across several minima. Repeat for another pair with different separation. Compare your results with the value computed from Eq. 9, using the marked value of a. Are your computed and measured results in reasonable agreement? 3. Multiple-slit patterns The Multiple Slits section has groups of 2, 3, 4 and 5 slits available. The slit width and separation are the same for each group. Can you identify the single slit contribution and the subsidiary maxima due to multi-slit interference? Indicate these on your sketch. Can you use your data to deduce a rule that gives the number of subsidiary maxima in terms of the number of slits? To look at the large-n case, replace the slide carrier with a regular lens holder and install the flat metal bar across the bottom. Set the grating on the bar, and rotate it until the back reflection overlaps the incident beam on the last collimating lens. This ensures that the grating is perpendicular to the incident beam. If you place the screen a few centimeters from the grating you should see bright spots corresponding to the undiffracted incident beam at the center and pairs of spots corresponding to m = ±1 and ±2 in Eq. 6. You will probably see some scattered light and stray reflections, but no clear subsidiary maxima. Measure the positions of the spots relative to the center spot, and use Eq. 6 to deduce the grating separation a, assuming λ = 632.8 nm. Do you get reasonable agreement with the value marked on the grating? (Gratings are traditionally labeled in lines/mm, which is equivalent to 1/a.) It's also interesting to estimate N by comparing the diameter of the beam with a. Large number? 4. Near-field diffraction You will now examine the transition from a geometric shadow to the Fresnel and Fraunhofer diffraction patterns for a wide single slit. The slit is made from a pair of razor blades taped to a plastic carrier. Center the slit in the beam, being careful not to get cut on the sharp edges. (Blood corrodes the edges and distorts the diffraction pattern.) Position the screen a large distance away, across the room if possible. Do you observe the expected single-slit diffraction pattern? In the Fresnel limit the diffraction pattern will be about the same width as the slit, so it is helpful to work with a magnified image. Install a 5 cm focal length lens in a holder between the slit and screen to project an image of the slit onto the screen. To help focus the image, hold a Physics 231 Physical Optics 7
piece of ground glass between the laser and the slit so that the slit is illuminated with diffused light. (It is much easier to focus accurately with diffuse lighting than with the collimated laser beam.) Now adjust the positions of the lens and the screen until you get a clear image of reasonable size. If you now remove the ground glass and slide the lens slightly away from the slit you will form magnified images of the light patterns at planes that are successively more distant from the slit. Doing this, you should see the image change from a simple shadow to a more complex pattern and finally to the characteristic diffraction pattern. Most of the action takes place in the first few centimeters. Can you find a distance at which there are several darker lines within the slit image, and another distance at which a single dark line is centered on the image? REPORT The report should contain legible sketches of the various patterns, with the relevant features clearly identified. The only quantitative items will be your comparisons of measured and computed parameters from the single and double slits. Physics 231 Physical Optics 8