Chapter 7 Experiment 5: Diffraction of Light Waves WARNING This experiment will employ Class III(a) lasers as a coherent, monochromatic light source. The student must read and understand the laser safety instructions on page 90 before attending this week’s laboratory. 7.1 Introduction In this lab the phenomenon of diffraction will be explored. Diffraction is interference of a wave with itself. According to Huygen’s Principle waves propagate such that each point reached by a wavefront acts as a new wave source. The sum of the secondary waves emitted from all points on the wavefront propagate the wave forward. Interference between secondary waves emitted from different parts of the wave front can cause waves to bend around corners and cause intensity fluctuations much like interference patterns from separate sources. Some of these effects were touched in the previous lab on interference. General Information We will observe the diffraction of light waves; however, diffraction occurs when any wave propagates around obstacles. Diffraction and the wave nature of particles leads to Heisenberg’s Uncertainty Principle and to the very reason why atoms are the sizes that they are. In this lab the intensity patterns generated by monochromatic (laser) light passing through a single thin slit, a circular aperture, and around an opaque circle will be predicted and experimentally verified. The intensity distributions of monochromatic light diffracted 91 CHAPTER 7: EXPERIMENT 5 m = 4 m = 3 m = 2 = 1 y m = 0 a x m m = -1 m = -2 Diffraction m = -3 Pattern = -4 Plane Wave m Figure 7.1: A sketch of a plane wave incident upon a single slit being diffracted and the resulting intensity distribution. The intensity is on a logarithmic scale; since our eyes have logarithmic response, this much more closely matches the apparent brightness of the respectve dots. Keep in mind that this week we are concerned with the dark intensity MINIMA and that m = 0 is NOT a minimum. from the described objects are based on: 1. the Superposition Principle, Intensity vs. Position 2. the wave nature of light Disturbance: 1 (a) Amplitude: A = A0 sin(ωt + ϕ) 0.8 (b) Intensity: I ∝ (P A)2, 0.6 3. Huygen’s Principle - Light propagates in such a way that each point reached by 0.4 the wave acts as a point source of a new light wave. The superposition of all these 0.2 waves represents the propagation of the 0 light wave. -7 -5 -3 -1 1 3 5 7 y (cm) All calculations are based on the assumption that the distance, x, between the slit and the Figure 7.2: A graphical plot of an exam- viewing screen is much larger than the slit width ple diffraction intensity distribution for a a:, i.e. x >> a. These results also apply to single slit. plane waves incident on the obstruction. This 92 CHAPTER 7: EXPERIMENT 5 particular case is called Fraunhöfer scattering. Plane waves result from a laser in this case, but a point source very far away or at the focus of a lens also give plane waves. The calculations of this type of scattering are much simpler than the Fresnel scattering where the distant point source constraint is removed. Fresnel scattering assumes spherical wave-fronts. Checkpoint What is the difference between Fraunhofer and Fresnel scattering? Checkpoint Are the eyes sensitive to the amplitude, to the phase, or to the intensity of light? Are the eyes’ response linear, logarithmic, or something else? 7.2 The Experiment We will observe the diffraction patterns from slits having several widths and apertures having different diameters. Each will be illuminated with a laser’s coherent light and the resulting pattern will be char- acterized. 7.2.1 Diffraction From a Single Slit A narrow slit of infinite length and width a is illuminated by a plane wave (laser beam) as shown in Figure 7.1. The intensity distribution observed (on a screen) at an angle θ with respect to the incident direction is given by Equation (7.1). This relation is derived in detail in the appendix and every student Figure 7.3: A photograph of Pasco’s must make an effort to go through its derivation. The OS-8523 optics single slit set. We are mathematics used to calculate this relation are very interested in the single slits at the top simple. The contributions from the field at each small right and the circular apertures at the area of the slit to the field at a point on the screen are top left. added together by integration. Squaring this result and disregarding sinusoidal fluctuations in time gives the intensity. The main difficulty in the calculation is determining the relative phase of each small contribution. Figure 7.2 shows the expected 93 CHAPTER 7: EXPERIMENT 5 shape of this distribution; mathematics can describe this distribution, I(θ) sin α2 πa = with α = sin(θ), (7.1) I(0) α λ where a is the width of the slit, λ is the wavelength of the light, θ is the angle between the optical axis and the propagation direction of the scattered light, I(θ) is the intensity of light scattered to angle θ, and I(0) is the transmitted light intensity on the optical axis. When θ = 0, α = 0, and the relative intensity is undefined (0/0); however, when θ is very small and yet not zero, the relative intensity is very close to 1. In fact, as θ gets closer to 0, this ratio of intensities gets closer to 1. Using calculus we describe these situations using the limit as α approaches 0, sin α lim = 1. α→0 α Although 0/0 might be anything, in this particular case we might think that the ratio is effectively 1; this is the basis for the approximation sin θ ≈ θ when θ << 1 radian. Certainly, the intensity of the light on the axis is defined, is measurable, and is quite close to the intensity near the optical axis. The numerator of this ratio can also be zero when the denominator is nonzero. In these cases the intensity is predicted and observed to vanish. These intensity minima occur when 0 = sin α for each time πm = α and mλ = a sin θ for m = ±1, ±2,... (7.2) Figure 7.4: A sketch of our apparatus showing the view screen, the sample slide, the laser, and the laser adjustments. Our beam needs to be horizontal and at the level of the disk’s center. We might note the similarity between this relation and the formula for interference maxima; we must avoid confusing the points that for diffraction these directions are intensity minima and for diffraction the optical axis (m = 0) is an exceptional maximum instead of an expected minimum. 94 CHAPTER 7: EXPERIMENT 5 This relation is satisfied for integer values of m. Increasing values of m give minima at correspondingly larger angles. The first minimum will be found for m = 1, the second for m = 2 and so forth. If is less than one for all values of θ, there will be no minima, i.e. when the size of the aperture is smaller than a wavelength (a < λ). This indicates that diffraction is most strongly caused by protuberances with sizes that are about the same dimension as a wavelength. Four single slits (along with some double slits) are on a disk shown in Figure 7.3 photo- graph of Pasco’s OS-8523 optics single slit set. We are interested in the single slits at the top right and the circular apertures at the top left.. This situation is similar to the one diagrammed in Figure 7.1. To observe diffraction from a single slit, align the laser beam parallel to the table, at the height of the center of the disk, as shown in Figure 7.4. When the slit is perpendicular to the beam, the reflected light will re-enter the laser. The diffraction pattern you are expected to observe is shown in Figure 7.5. Figure 7.5: Observed Diffraction Pattern. The pattern observed by one’s eyes does not die off as quickly in intensity as one expects when comparing the observed pattern with the calculated intensity profile given by Equation (7.1) and shown in Figure 7.2. This is because the bright laser light saturates the eye. Thus the center and nearby fringes seem to vary slightly in size but all appear to be the same brightness. Observe on the screen the different patterns generated by all of the single slits on this mask. Note the characteristics of the pattern and the slit width for each in your Data. Is it possible from our cursory observations that mλ = a sin θ? If a quick and cheap observation can contradict this equation, then we need not spend more money and time. Calculate the width of one of the four single slits. This quantity can be calculated from Equation (7.2) using measurements of the locations of the intensity minima. The wavelength of the HeNe laser is 6328 Å, (1 Å= 10−10 m). The quantity to be determined experimentally is sin θ. This can be done using trigonometry as shown in Figure 7.6. Measure the slit width using several intensity minima of the diffraction pattern. Place a long strip of fresh tape on the screen and carefully mark all of the dark spots in the fringes. Be sure to record the manufacturer’s specified slit width, a. Avoid disturbing the laser, the slit, and the screen until all of the minima are marked and the distance, x, between the slit and the screen is measured.