Module 18: Diffraction-I Lecture 18: Diffraction-I

Transcription

1 Module 18: iffraction-i Lecture 18: iffraction-i Our discussion of interference in the previous chapter considered the superposition of two waves. The discussion can be generalized to a situation where there are three or even more waves. iffraction is essentially the same as interference, except that we have a superposition of a very large number of waves. In some situations the number is infinite. An obstruction placed along the path of a wave (Figure 18), is an example of a situation where diffraction occurs. In the case of diffraction the longer wavelengths deviate more compared to the shorter wavelengths, whereas in refraction the shorter wavelengths deviate more compared to the longer ones. In the case of diffraction it is necessary to solve the equation governing the wave in the complicated geometry produced by the obstruction. This is usually very complicated, and beyond analytic treatment in most situations. The Huygens-Fresnel principle is a heuristic approach which allows such problems to be handled with relative ease. Huygens principle, proposed around 1680, states that every point on a wavefront acts as the source for secondary spherical wavelets such that the wavefront at a later time is the envelope of these wavelets. Further, the frequency and propagation speed of the wavelets is the same as that of the wave. Applying Huygens principle to the propagation of a plane wave, we see that a plane wave front evolves into a shifted plane wave at a later time (Figure 18.2). Figure 18.1: iffraction due to an obstruction 117

2 118 CHAPTER 18. IFFRACTION-I Figure 18.2: Wavefronts and Huygens secondary wavelets L Figure 18.3: Single slit diffraction We also show how it can be applied to the propagation of a spherical wavefront. It is also useful for studying the propagation of waves through a refracting medium where the light speed changes with position and is also different in different directions. Huygens principle, in its original form cannot be used to explain interference or diffraction. It was modified in the early 19th century by Fresnel to explain diffraction. The modified version is referred to as the Huygens-Fresnel principle. Kirchoff later showed that the Huygens-Fresnel principle is actually consistent with the wave equation that governs the propagation of light. The Huygens-Fresnel principle states that every unobstructed point on the wavefront acts like a source for a secondary wavelet. The contribution from all these wavelets are to be superposed to find the resultant at any point Single slit iffraction Pattern. Consider a situation where light from a distant source falls on a rectangular slit of width and length L as shown in Figure We shall assume that L is much larger than. We are interested in the image on a screen which is at a great distance from the slit. As the source is very far away, we can treat the incident light as plane wave. Further the source is aligned so that the incident wavefronts are parallel

3 18.1. SINGLE SLIT IFFRACTION PATTERN. 119 /2 dy y Θ /2 Figure 18.4: Single slit effective one dimensional arrangement to the plane of the slit. Every point in the slit emits a spherical secondary wavelet.these secondary wavelets are well described by plane waves by the time they reach the distant screen. This situation where the incident wave and the emergent secondary waves can all be treated as plane waves is referred to as Fraunhofer diffraction. In this case both, the source as well as the screen are effectively at infinity from the obstacle. We assume L to be very large so that the problem can be treated as one dimensional as shown in Figure Instead of placing the screen far away, it is equivalent to introduce a lens and place the screen at the focal plane. Each point on the slit acts like a secondary source. These secondary sources are all in phase and they all emit secondary wavelets with the same phase. Let us calculate the total radiation at a point at an angle θ. If dẽ = Ã dy (18.1) be the contribution from a small element dy at the center of the slit, the contribution from an element a distance y away will be at a different phase dẽ = Ã eiδ dy (18.2) where δ = 2π λ y sin θ = ky (18.3) The total electric field can be calculated by adding up the contribution from all points on the slit. This is an integral Ẽ = 2 dẽ = Ã 2 e iky dy 2 2

4 120 CHAPTER 18. IFFRACTION-I Ι( ) β π 5 π 0 π 5 2π 10 β Figure 18.5: Single slit intensity pattern = Ã sin ( k 2 ) ( k 2 ) 1 ( ) k = Ã sinc 2 (18.4) where sinc(x) = sin(x)/x. We use eq. (18.4) to calculate the intensity of light at an angle θ. efining π sin θ β =, (18.5) λ the intensity is given by I(β) = 1 2 EE = I 0 sinc 2 β. (18.6) Figure 18.5 shows the intensity as a function of β. The intensity is maximum at the center where β = 0. In addition to the oscillations, the intensity falls off proportional to β 2 away from the center. The analysis of the intensity pattern is further simplified In the situation where θ 1 as β = π θ λ. (18.7) The zeros of the intensity pattern occur at β = ±mπ (m, = 1, 2, 3,...), or in terms of the angle θ, the zeros are at θ = m λ. (18.8) There are intensity maxima located between the zeros. The central maximum at θ = 0 is the brightest, and its angular separation from the nearest zero is λ/. This gives an estimate of the angular width of the central maximum. The intensities of the other maxima fall away from the center. Let us compare the intensity pattern I(β) shown in Figure 18.5 with the predictions of geometrical optics. Figure 18.4 shows a beam of parallel rays incident on a slit of dimension. In geometrical optics the only effect of the

5 18.1. SINGLE SLIT IFFRACTION PATTERN. 121 slit is to cut off some of the rays in the incident beam and reduce the transverse extent of the beam. We expect a beam of parallel rays with transverse dimension to emerge from the slit. This beam is now incident on a lens which will focuses all the rays to a single point on the screen. Thus in geometrical optics the image is a single bright point on the screen. In reality the wave nature of light manifests itself through the phenomena of diffraction, and we see a pattern of bright spots as shown in Figure The central spot is the brightest and it has an angular extent ±λ/ The other spots located above and below the central spot are fainter. Taking into account both dimensions of the slit we have where I(β x,β y ) = I 0 sinc 2 (β x ) sinc 2 (β y ) (18.9) β x = πlθ x and β y = πθ y λ λ. (18.10) and θ x and θ y are the angles along the x and y axis respectively. The diffraction effects are important on angular scales θ x λ/l and θ y λ/. In the situation where L the diffraction effects along θ x will not be discernable, and we can treat it as a one dimensional slit of dimension. Problems 1. For a slit of dimensions 1 mm 1 cm, what are the positions of the first three minima s on either side of the central maxima? Use λ = 550 nm and 0.1 mm. 2. For a rectangular slit whose smaller dimension is, what are the positions of the maxima for light of wavelength λ? (Ans. β ±1.43π, ±2.46π, ±3.47π, etc.) 3. Calculate the ratio of intensities of the first intensity maximum and central maximum for the previous problem. (Ans. 21)