# Interference & Diffraction

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1 Purpose A single-slit diffraction pattern results when a light beam passes through a single narrow aperture, or slit, whose width is not too much larger than a wavelength. A double-slit interference pattern results when the beam passes through two adjacent narrows slits. A large number of adjacent narrow slits constitutes a diffraction grating and generates a diffraction pattern. These patterns can be used to calculate the wavelength of the light, or the width of apertures or obstructions. We will use them to calculate the wavelength of a laser and the width of a human hair. Principles Diffraction refers to the spreading out of the wave fronts of light as it passes through narrow openings. Interference refers to the superposition of light waves from different sources, so that the waves either reinforce each other or cancel each other out. To observe these effects, we allow a laser beam to pass through one or more slits and fall on a distant screen. Bright regions are the result of constructive interference between waves and are called maxima; shadowed regions are the result of destructive interference and are called minima. We use laser light because it is composed of coherent waves that is, all the waves have the same wavelength and are in phase with each other as they emerge from the slits. Ordinary light is a mixture of waves of differing wavelengths that have random phases. Because of this, interference effects are usually averaged out and unobservable. The Single Slit Pattern For the single slit, diffraction is evident in the fact that the illuminated area on the screen is very much wider than the opening through which the light has passed the waves have bubbled out from the slit. Interference is also evident, as each point within the aperture acts as a separate source of wave fronts. Refer to Diagram 1 below. Minima in the pattern on the screen will occur at angles for which the sine of the angle is any integer multiple of a λ, where a is the slit width: 129

2 [Diagram 1 here] 130

3 sinθ = m λ a ( m = ± 1, ± 2, ± 3, ) (minima) If the screen is a distance L from the aperture, then sinθ = L 2 y + y 2 where y is the distance along the screen from the center of the pattern to the point of interest. If L >> y, which is usually the case, then the above can be approximated by sin θ so that we can determine where shadows will appear on the screen by y L (1) λ y ml a Thus we expect the minima in the pattern to be roughly even spaced. The maxima will appear between successive minima. This formula gives us a way to determine the wavelength of light, if the aperture width is known, or the aperture width, if the wavelength is known, by measuring the distance between successive minima in an interference pattern. Interestingly, the same pattern results from a single, narrow obstruction. We will use this to calculate the width of a hair. The Double Slit Pattern We can also let light pass through two narrow slits separated by a center-to-center distance d. There will be a single-slit pattern associated with each of the slits. (The two single-slit patterns will lie on top of each other since the distance between the slits is negligible, so there is really only one single-slit pattern.) 131

4 [Diagram 2 here] 132

5 Much more prominent however, will be the double-slit interference pattern. This arises because the light from one slit will interfere with the light from the other. It was by observation of this pattern that Thomas Young established the wave nature of light in the early 19 th century. Treating each slit as a point source of light, the condition for a maximum in the pattern on the viewing screen is d sinθ = nλ ( n = 0, ± 1, ± 2, ) (maxima) Again, for L >> y, we can write this as (2) λ y nl d where y is the distance right or left from the center of the pattern. This is almost identical to the single-slit formula above, except that here we are considering the positions of the maxima instead of the minima. Also, there is no broad central maximum in the doubleslit pattern; the central maximum (corresponding to n = 0 in the above) is the same size as the other maxima. Multiple slits and the Diffraction Grating As we increase the number of slits in the aperture, more complicated interference and diffraction patterns result. We find maxima of greater and lesser intensity primary and secondary maxima. With three slits, we find a secondary maximum between each pair of primary maxima. With four slits, we find two secondary maxima between each pair of primary maxima. In this experiment, we will observe and draw interference patterns for apertures of three, four and five slits. When the number of illuminated slits becomes very large (the slit spacing d must be very small for this to occur) two things become evident. First, the secondary maxima decrease in intensity until, for a very large number of slits, they are not visible to the naked eye. Secondly, the primary maxima become very widely spaced (a consequence of the very small slit spacing d). This is the case with the diffraction grating. Just as for a double slit, the maxima occur at locations on the screen where 133

6 (3) d sinθ = nλ ( n = 0, ± 1, ± 2, ) (maxima) where d is the spacing between grooves. The integer n is called the order of the maximum. Gratings are usually labeled with their slit density D, i. e., the number of grooves per unit length. The spacing between grooves is just the inverse of the slit density: (4) d = 1 D In this experiment we will determine the slit density of a diffraction grating by measuring the distance between maxima of different order. Since the maxima will widely spaced (y not << L), we will use the exact formula sinθ = y instead of approximating as 2 2 L + y we did above. 134

7 LAB 10 Data 135

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