ID INTERFERENCE AND DIFFRACTION OBJECTIVES 1. To observe the diffraction pattern when monochromatic light passes through a single slit. 2. To observe the interference patterns formed when monochromatic light passes through multiple slits, and be able to distinguish these patterns from a single slit pattern. 3. To determine the slit width and slit separation using measurements of the interference patterns, and compare the results to direct measurements using a travelling microscope. THEORY Single-slit Diffraction Consider a monochromatic beam of light incident on a slit of width a, as shown in Figure 1. Interference occurs between the light that passes through different parts of the slit, resulting in what is called a diffraction pattern. Viewed on a distant screen, the diffraction pattern consists of alternating bright and dark interference fringes. The central bright fringe is twice as wide, and much brighter, than the other bright fringes. Figure 1 Single slit diffraction (from Serway and Jewett, Physics for Scientists and Engineers, 6th Ed., Thomson-Brooks/Cole, 2004, p. 1209.) It can be difficult in practice to have an infinitely distant viewing screen. Instead, if a lens is placed just beyond the slit, the lens will form in its focal plane a reduced image of the pattern that would appear on an infinitely distant screen in the absence of the lens. This situation is called Fraunhofer diffraction. ID−1 A careful analysis of the situation when the screen is very distant, as done in class and in your textbook, reveals that the centres of the first dark fringes (minima) in the diffraction pattern will be found at angles given by sin θ dark = ± λ (1) a Between these two dark fringes is the central diffraction maximum. Figure 2 shows how the intensity varies within the single slit diffraction pattern. The intensity has its maximum value at sin θ = 0, and drops off rapidly as sin θ increases: Intensity Central Diffraction Maximum First bright fringe First bright fringe ─2λ/a ─λ/a +λ/a sin θ +2λ/a Figure 2 : The diffraction envelope due to a single slit Multiple-slit Interference and Diffraction: If more than one slit is used, we will have both diffraction and interference. That is, each slit gives rise to a diffracted beam in which the intensity distribution is a function of slit width, and these diffracted beams then interfere with one another to produce the final pattern. To begin with, let us point out one property of the Fraunhofer diffraction pattern of a slit that was not mentioned previously. Figure 3(a) shows a single slit in front of a lens, with a screen in the second focal plane of the lens. Parallel monochromatic light is assumed incident from the left, and the centre line of the slit lies on the lens axis. Two diffracted beams are indicated, one giving rise to the central maximum in the diffraction pattern, the other to the first minimum below the centre. The central maximum lies on the lens axis. In Figure 3(b), a slit of the same width as in part (a) is located above the lens axis. Neglecting lens aberrations, the intensity distribution in the diffraction pattern of this slit is exactly the same as in part (a), and the pattern is centred at the same point on the screen, that is, on the axis of the lens, not on a line opposite the centre of the slit. Recall from your study of geometric optics that even if half of a lens is blocked, it doesn’t alter the location of the image. This is exactly the same effect. It doesn’t matter what part of the lens forms the image – the light from each of the slits, in Figures 3a and 3b, is incident on the lens at the same angle of incidence. All rays parallel to the lens axis are imaged at its second focal point. ID-2 . Figure 3: The diffraction pattern of the slit is the same in (a) and (b). It follows from this that when light is incident on two slits separated by a distance, d, each slit generates a diffraction pattern at the same location on the viewing screen. Superimposed on this diffraction pattern is the two-slit interference pattern consisting of equally spaced interference fringes. The centre of the bright fringes in a two-slit pattern are found where the path length difference from the two slits (d sin θ) is a whole number of wavelengths, or sin θ bright = mλ d m = 0, ± 1, ± 2, … If there was no diffraction, these interference maxima would all be equally bright, but the diffraction intensity envelope in Figure 2 modulates the brightness of the interference fringes. Figure 4(b) shows the two-slit interference pattern (if there was no diffraction), as well as the diffraction pattern envelope, and the resulting modulated two-slit pattern. Single slit diffraction envelope a Two slits (no diffraction) λ/a Figure 4. Plots of intensity distribution versus sin θ from multiple slits. (a) Single slit of width a (b) Two slits of separation d = 4a (c) Three slits of separation d = 4a. (d) Four slits of separation d = 4a. ID−3 (2) Figure 4 has been drawn to scale for a distance d between the slits equal to 4 times the slit width a. The first minimum in the diffraction pattern in Figure 4a occurs at the same angle as the fourth bright fringe to the right of the central fringe in the two slit pattern. This angle is given by equation (1), and by equation (2) with m (double slit) = 4: sin θ = λ a = 4λ . d Since d = 4a in this example, the minimum in the diffraction pattern means that the 4th interference fringe on each side of the central maximum will not be visible, leaving a total of 7 bright fringes visible inside the central diffraction maximum. In general, the interference fringes form a much finer pattern, superimposed with the relatively broad diffraction pattern envelope. Now suppose we have more than two slits. As is discussed in class and in the textbook (Chapter 37), the bright fringes in the two-slit pattern remain but become brighter and narrower as more than two slits are added with the same separation, d. In addition, secondary maxima appear between the two-slit maxima. If N is the total number of equally spaced slits, then N−2 is the number of secondary maxima inside each dark fringe of the multiple slit pattern. These secondary maxima are visible in Figures 4(c) and 4(d), as well as in Figure 5. Figure 5 Multiple slit interference patterns (from Serway and Jewett, Physics for Scientists and Engineers, 6th Ed., Thomson-Brooks/Cole, 2004, p. 1188.) ID-4 APPARATUS: Sodium lamp (λ = 589.3 nm), spectrometer, set of slits, travelling microscope. PROCEDURE: Adjustments and Qualitative Observations Focus the spectrometer telescope to give a sharp image when viewing the parallel light from a far-distant object (through the lab window). Adjust the eye-piece cross-hairs so that they are focussed in the same plane (zero parallax exists) as is this distant object. Finally, adjust the collimator so that the image of the spectrometer slit is sharply focussed in this same plane. Set up the apparatus as shown in Figure 6 with the six slits lined up with the collimator and telescope. Adjust the width of the spectrometer slit to produce a distinct pattern. Approximately ten main interference maxima should be visible on each side of the central maximum, and four secondary interference maxima faintly visible between each of these main interference maxima. Now view, in turn, the interference-diffraction patterns produced by each of the slit groups, from one to six slits. Sketch the patterns in your notebook, noting the number of visible primary interference maxima in each diffraction maximum, and the number of secondary maxima visible within each primary maximum. Figure 6: Experimental arrangement for observing Fraunhofer Diffraction Patterns. Quantitative Measurements 1. Single-Slit Measure the angular separation, θdark, between the central maximum and the first minimum of the single-slit pattern. Note that the measurement uncertainty in θdark may be minimized by measuring the angular separation between the nth minimum on the left and the nth minimum on the right sides of centre, and dividing by 2n. Be careful: it can be difficult to see the first minimum on either side, so once you have a value for θdark, check again that your first minima are indeed found close to this angle. From your measured value of θdark, calculate the slit width a, including uncertainty, for the single-slit. ID−5 2. Six-slits Measure the angular separation, θbright, between the main interference maxima. Note that the measurement uncertainty in θbright may be minimized by measuring the total angular separation of n maxima and dividing by n. From your measured value of θbright, calculate the slit separation d, including uncertainty, for the six slits. (Note that Figure 3, with d = 4a, is meant as an example. Your values will differ.) 3. Measure d and a for the six slits using the travelling microscope by measuring the twelve relative positions of the slit edges (Figure 7). The slit width will be the average of (x2-x1), (x4-x3), etc, and the slit separation will be the average of (x3-x1), (x4-x2), etc. Before making these measurements, carefully adjust the microscope focus so that there is zero parallax between the images of the crosshairs and the slits. a d x1 x2 x3 x4 Figure 7: Measurement of a and d for six slits. 4. Compare the experimental values of a and d determined by the two methods. Note that the slit plate is made so that the slit width a is the same for all slits, and the separation d is the same for all slits. QUESTION A particular two-slit experiment is performed with the slit separation d equal to 7 times the slit width, a. Calculate the number of complete two-slit interference maxima you would expect to be able to find within: (a) The central maximum of the diffraction pattern (b) The first bright fringe of the diffraction pattern envelope on one side of the central maximum (see Figure 2). Be sure to carefully explain your reasoning (use a diagram). ID-6