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Fraunhofer Diffraction with a Laser Source

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Fraunhofer Diffraction with a Laser Source Second Year Laboratory Fraunhofer Diffraction with a Laser Source L3 ______________________________________________________________ Health and Safety Instructions. There is no significant risk to this experiment when performed under controlled laboratory conditions. However: • This experiment involves the use of a class 2 diode laser. This has a low power, visible, red beam. Eye damage can occur if the laser beam is shone directly into the eye. Thus, the following safety precautions MUST be taken:- • Never look directly into the beam. The laser key, which is used to switch the laser on and off, preventing any accidents, must be obtained from and returned to the lab technician before and after each lab session. • Do not remove the laser from its mounting on the optical bench and do not attempt to use the laser in any other circumstances, except for the recording of the diffraction patterna. • Be aware that the laser beam may be reflected off items such as jewellery etc ______________________________________________________________ 26-09-08 Department of Physics and Astronomy, University of Sheffield Second Year Laboratory 1. Aims The aim of this experiment is to study the phenomenon of diffraction using monochromatic light from a laser source. The diffraction patterns from single and multiple slits will be measured and compared to the patterns predicted by Fourier methods. Finally, Babinet’s principle will be applied to determine the width of a human hair from the diffraction pattern it produces. 2. Apparatus • Diode Laser – see Laser Safety Guidelines before use, • Photodiode, • Optics Bench, • Traverse, • Box of apertures, • PicoScope Analogue to Digital Converter, • Voltmeter, • Personal Computer. 3. Background (a) General The phenomenon of diffraction occurs when waves pass through an aperture whose width is comparable to their wavelength. It will be investigated in this experiment using light from a diode laser. This is a highly monochromatic source whose long coherence length makes if particularly useful in diffraction experiments, as it produces bright and well-defined diffraction patterns. A photodiode is used to detect these patterns and an analogue to digital converter will enable the data to be stored on a computer. Fourier analysis will allow these patterns to be compared to the theoretical predictions. (b) Diffraction at a Single Slit Consider parallel, monochromatic light incident on a single slit of width b, as shown in Figure 1 below. After passing through the slit, light does not travel only in its original direction. Instead we should consider each portion of the slit as a source of light waves (Huygen’s Principle). b 2 Department of Physics and Astronomy, University of Sheffield Second Year Laboratory Figure 1 Diffraction at a slit As semi-circular wavefronts propagate out from each portion, they will interfere. The resultant intensity, as viewed on a distant screen, depends on the angle θ. A full description of the diffraction pattern is given below using Fourier theory, but first the positions of the minima will be established by simpler reasoning. (i) Simple Theory Consider Figure 2. • Dividing the slit in half shows that the path difference between waves from the centre and the edge of the slit is (b/2)sinθ. θ • There will be complete destructive interference b/2 between waves from the two halves of the slit when this path difference is equal to half a wavelength, i.e. (b/2) sin θ = λ/2 , b sinθ = λ . (b/2) sin θ • Similarly, dividing the slit into 4 or 6 equal parts, Figure 2 Diffraction shows that destructive interference occurs for geometry (b/4) sinθ = λ/2 , (b/6) sinθ = λ/2 , etc. Thus, minima occur in the diffraction pattern for sinθ = mλ / b, m = ±1, ±2, ±3, etc. (1) Note: sinθ = 0 is not a minimum but the principal maximum – see Fig. 4. (ii) Fourier Theory Fourier theory can be used to predict the form of the diffraction pattern from a given aperture. In optics, the amplitude of the diffraction pattern is given by the Fourier transform of the aperture function, which is a mathematical method of representing the transmission of light from an aperture. The aperture function f(x) for the single slit is given by the so called “top-hat” function and reflects the transmission of light through the aperture. f(x) = 0 = opaque f(x) = 1 = transparent 3 Department of Physics and Astronomy, University of Sheffield Second Year Laboratory For the single-slit the aperture function is given by :- f(x) = 1 for –b/2 < x < +b/2 f(x) = 0 for elsewhere which is shown in Figure 3. f(x) θ −b/2 +b/2 x Figure 3 The single slit and its transmission function f(x) The amplitude F() k of the diffracted light is given by the Fourier transform of the aperture function:- +∞ F(k)= ∫ f() x .eikx d x (2) −∞ where 2π k = sinθ . (3) λ k is known as the propagation constant. Note that it has dimensions of inverse length. θ is the angular distance from the centre of the diffraction pattern (see Fig. 2) and λ is the wavelength of the laser. This leads to an intensity variation given by: 2 I()β ⎡sin β ⎤ = ⎢ ⎥ (4) I )0( ⎣ β ⎦ where πb β = sinθ (5) λ as plotted in Figure 4. Equation 4 has the value 1.0 at θ = 0 so the graph shows the intensity at any point relative to this central maximum of 1.0 at β = 0 . 4 Department of Physics and Astronomy, University of Sheffield Second Year Laboratory β Figure 4 The single slit diffraction pattern. For the full derivation of this result please refer to any recommended textbook - e.g. “Optics” by Hecht. The final result for the diffraction pattern is obtained by substituting for β in Eq 4, giving: 2 ⎡ ⎛ πbsin θ ⎞⎤ sin⎜ ⎟ I()θ ⎢ λ ⎥ = ⎢ ⎝ ⎠⎥ . (6) I )0( ⎢ πbsin θ ⎥ ⎢ ⎥ ⎣ λ ⎦ (c) Diffraction at two slits (i) Simple Theory Consider two slits, a distance a apart. In the classic Young’s slits experiment the width of the slits is assumed to be negligible. So to derive the interference pattern, two point sources of light waves are considered. The path difference between waves from the two slits is Δ = a sin θ. θ Hence there are maxima in the interference pattern when the interference is constructive: a θ Δ = a sin θ = mλ, m = 0, ±1, ±2, etc., (6a) and minima when the interference is destructive: Figure 5 Double slits Δ = a sin θ = (m+ ½)λ, m = 0, ±1, ±2, etc.. (6b) asinϑ More generally, at any point, the electric field due to light from the top slit is E1 = E0 sin ωt , 5 Department of Physics and Astronomy, University of Sheffield Second Year Laboratory and the field due to light from the bottom slit is E2 = E0 sin (ωt + φ), where φ / 2π = Δ / λ so φ = 2π a sinθ / λ. So at any point, the total electric field is E = E0 sin ωt + E0 sin (ωt + φ) = 2 E0 cos(φ/2) sin(ωt + φ/2). The light intensity is proportional to the square of the magnitude of the electric field, 2 2 2 I ∝ 4 E0 cos (φ/2) sin (ωt + φ/2). When, as in this experiment, the time-averaged intensity is measured, the diffraction pattern is a cos2 distribution (as shown in Figure 7b): 2 2 I(θ) = Imax cos (φ/2) = Imax cos (πa sinθ /λ) (7) (ii) Fourier Theory The same result can be obtained using Fourier theory. In Fourier theory, an ideal (infinitely narrow) aperture is represented by a delta function. So the diffraction pattern for Young’s slits is obtained by taking the Fourier Transform of two delta functions a distance a apart – see Figure 6. δ(−a/2) δ(+a/2) θ β x Figure 6a: aperture function for two ideal slits. Figure 6b: interference pattern for two ideal slits. (iii) Two Slits of Finite Width In practice the slits have finite width, so the diffraction pattern observed for double slits is a convolution of the double slit and single slit patterns, as shown in Figure 7. Such convolutions can be derived neatly using Fourier theory. Slit width = delta function β Figure 7: Development of the diffraction pattern for two slits Finite slit width β 6 Department of Physics and Astronomy, University of Sheffield Second Year Laboratory (d) Diffraction from Multiple Slits The diffraction pattern for N identical apertures of finite width can be derived as follows: • The diffraction pattern for one aperture is given by the Fourier Transform of the aperture function. • The diffraction pattern for N ideal apertures is given by the Fourier Transform of the appropriate array of delta functions. • The Fourier Transform for N finite apertures is simply the product of the two above functions. So for N slits, Amplitude of Fourier Transform of Fourier Transform Diffraction Pattern of = Top Hat Function × of N Delta Multiple Aperture Functions For N slits of separation a and width b, the intensity distribution is given by 2 2 I()θ 1⎡ sinβ ⎤ ⎡ sin Nγ ⎤ = 2 ⎢ ⎥ ⎢ ⎥ , (8) IN)0( ⎣ β ⎦ ⎣ sinγ ⎦ πb πa where β = sinθ as before, and γ = sinθ . Note that the first (β ) term tends to 1.0 λ λ when β =0, but the second (γ ) term tends to N2 when γ =0. The final version of equation 8 is then 2 2 ⎡ ⎛ πbsinθ ⎞⎤ ⎡ ⎛ Nπ asin θ ⎞⎤ sin⎜ ⎟ sin⎜ ⎟ I()θ 1 ⎢ λ ⎥ ⎢ λ ⎥ = ⎢ ⎝ ⎠⎥ ⎢ ⎝ ⎠⎥ . (9) I )0( N 2 ⎢ πbsinθ ⎥ ⎢ ⎛ πasinθ ⎞ ⎥ ⎢ ⎥ ⎢ sin⎜ ⎟ ⎥ ⎣ λ ⎦ ⎣ ⎝ λ ⎠ ⎦ Once again, refer to any of the recommended textbooks for a derivation of this result. For a double slit, equation (8) gives: - 2 ⎡ ⎛ πbsinθ ⎞⎤ ⎢sin⎜ ⎟⎥ I()θ ⎝ λ ⎠ 2 ⎛ πasinθ ⎞ = ⎢ ⎥ cos ⎜ ⎟ .
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