Optical Interference and Diffraction Laboratory: a Practical Guide

Transcription

1 Optical Interference and Diffraction Laboratory: a Practical Guide Authors: Amparo Pons, Universitat de València, Dept. of Optics, Spain amparo.pons-marti@uv.es Juan C. Barreiro, Universitat de València, Dept. of Optics, Spain barreiro@uv.es Genaro Saavedra, Universitat de València, Dept. of Optics, Spain genaro.saavedra@uv.es Abstract We present a series of experiments designed to study in a simple way the concepts of interference and diffraction of light. These experiences are collected in a full practical guide that describes the basic theory involved in the experiments, the experimental setups, the practical procedures and some additional related material. The proposed setups include simple material such as two laser pointers of different wavelengths (red and green), a set of single and multiple elementary apertures and a graduated screen. The experimental technique consists in the direct measurement of the far-field (Fraunhofer) patterns generated by the targets over the graduated screen. We present a guided sequence of experiences for analysing some properties of the interference and diffraction patterns. In particular, we pay attention to the influence of the shape and the size of the diffracting objects, the separation between elementary apertures in multiple targets, the distance from the objects to the observation screen and the wavelength of the illuminating light. Additionally, we present a list of questions concerning the experimental results and also links to websites including applets related to this subject. Index Terms Optics laboratory, interference and diffraction patterns. INTRODUCTION The aim of this contribution is to present a series of simple experiments to get acquainted with the diffraction and interference of light phenomena [1]-[3]. In particular, we focus our attention on the intensity patterns observed in the far-field (Fraunhofer) regime. The diffraction (and interference) phenomenon is a feature of wave fields that reach an obstacle in their free propagation. Figure 1 shows a typical example for the case of a light field, in which an infinite spherical wavefront (Figure 1a) is windowed by a circular aperture (Figure 1b) that limits its extension. The intensity distributions observed over the focusing plane in the unlimited and apertured cases are shown to be clearly different. This result can be interpreted in the following sense: while the infinite beam follows the Geometrical Optics laws (in which light is described as a set of rays travelling in straight trajectories), the truncated one does not fit the predictions of this model. This departure from the geometrical model is what is called scattering or diffraction of the light. This effect is larger as the size of the diffracting aperture gets smaller in terms of the wavelength of the light. This effect plays a basic role, for instance, in the imaging through real optical systems, since the finite size of the components necessarily limits the extension of the incident waves. In this case, the dimensions of the diffraction figure corresponding to a point object is the responsible for the finite resolution of the imaging system. FIGURE 1 DIFFRACTION OF A LIGHT BEAM. BASIC THEORY Let us now describe the diffraction patterns (far field) associated with some elementary diffracting apertures. First, we consider a vertical slit (y-direction) illuminated with a plane monochromatic wave impinging perpendicularly to the aperture 1

2 plane (see Figure 2). Over a transverse observation plane located at a distance D (large enough) from the diffracting slit, it can be shown that the predicted intensity pattern is given by 2 Lx sinc x if y = 0 I( = λ D, (1) 0 elsewhere where L x represents the width of the slit, λ corresponds to the wavelength of the light, and sinc ( α) = sin( πα) ( πα). Figure 3 shows a representation of this later function. Note that, according to (1), the diffracted energy is concentrated along an horizontal line ( y = 0). The intensity distribution along this line is governed by a scaled version of the function in Figure 3, the scale depending on the width of the slit L x. In fact, the distance between successive minima along this line is given by λd x =. (2) L x Note, however, that the distance between the two minima that bound the central maximum is 2 x. Diffraction pattern 1.0 sinc 2 (α) Aperture 0.5 L X D α FIGURE 2 DIFFRACTION PATTERN (FAR FIELD) FOR A SINGLE SLIT. Secondly, we assume a circular diffracting aperture, with diameter L, as presented in Figure 4. The intensity diffraction pattern for this case is shown to be radially symmetric, consisting of a central bright disk and a series of concentric alternate bright/dark rings. In mathematical terms, the diffraction pattern can be described by the function 2 L I' ( r) = disk r, (3) 2λD 2 2 being r = x + y the radial transverse coordinate over the observation plane. In (3) disk( ρ ) = J1(2πρ) ( πρ) being J 1 ( ρ ) the Bessel function of first kind and order 1. The variation of this function with its argument is shown in Figure 5. Diffraction pattern FIGURE 3 GRAPHICAL REPRESENTATION OF THE FUNCTION sinc 2 (α). 1.0 disk 2 (ρ) Aperture 0.5 L D ρ FIGURE 4 DIFFRACTION PATTERN (FAR FIELD) FOR A SINGLE CIRCULAR APERTURE. FIGURE 5 GRAPHICAL REPRESENTATION OF THE FUNCTION disk 2 (ρ). 2

3 As in the previous case, the scale of the diffraction rings depends on the size of the circular diffracting aperture. In particular, it can be shown that the diameter of the first minimum intensity ring can be computed through the formula 2λD φ = 1'22. (4) L Thirdly, let us consider the case of a double diffracting aperture composed of two identical elementary apertures at a distance 2a of each other along the x-axis, as shown in Figure 6. It is easy to show now that the intensity diffraction pattern obtained at a distance D from the composed aperture fits the following function 2 π 2ax I( = cos I A (, (5) λd I A ( being the diffraction pattern provided by a single elementary aperture alone. This case is really a slightly modified version of the classical double slit Young s experiment. In other words, the cosinus-squared term in (5), whose period is given by λd p =, (6) 2a appears due to the interference between the light beams diffracted by each of the elementary apertures. In this way, the diffraction figure of a double aperture consists of a set of Young s sinusoidal fringes modulated by the diffraction pattern of each individual aperture. The particular case of a double circular aperture is illustrated in Figure 7. 2a 2a a) b) FIGURE 6 EXAMPLES OF DOUBLE APERTURES: (A) DOUBLE SLIT; (B) DOUBLE CIRCULAR APERTURE. FIGURE 7 DIFFRACTION PATTERN (FAR FIELD) FOR A DOUBLE CIRCULAR APERTURE. As a forth example, we analyse now a generalisation of the previous case. Let us assume a multiple diffracting object constructed from the regular replication, along the x-axis, of an elementary aperture. If we consider n replica over the object plane, it can be shown that now the intensity diffraction pattern is given by 2 πndx sin λd x I( I A ( = I n I A (, (7) 2 πdx λd / d sin λd where d is the separation between consecutive elementary apertures in the object and I A ( has the same meaning as in (5). The behaviour of the envelope function I n (α ) for different values of n is shown in Figure 8. Note that this envelope becomes a series of narrow equidistant bright lines as n goes larger, the distance between two consecutive lines being λd =. (8) d In particular, for the case of a linear diffraction grating, constituted by a very large number of parallel equidistant slits, the diffraction pattern consists of a series of point-wise intensity maxima distributed along a line in the observation plane with a constant distance between them given by (8). In this case, the equidistance d over the object plane is referred usually as the period of the grating and the value N = 1/ d is called its spatial frequency. Let us consider, finally, a more general case for a multiple diffracting aperture: a collection of n identical apertures arranged randomly over the object plane. It can be shown that, for a sufficiently large number n, the diffraction pattern highly resembles the intensity figure I A ( that corresponds to a single elementary aperture. In mathematical terms, the intensity 3

4 I n (α)/n 2 n=3 n=10 n= diffraction pattern is given now by 2 n I = = A(0,0) if x y 0 I( = n I A( elsewhere. (9) Note that, except for a high intensity peak in x = y = 0, the diffraction figure is proportional to I A (. EXPERIMENTAL SETUP AND PROCEDURE FIGURE 8 GRAPHICAL REPRESENTATION OF THE FUNCTION I n(α) FOR THREE VALUES OF n. The proposed equipment for the experiences in the laboratory is the following: Red laser pointer of wavelength λ R =650 nm. Green laser pointer of wavelength λ V =532 nm. Platform with adjustable laser mount. Graduated screen with base support. Revolving holder containing a single aperture set. Revolving holder containing a multiple aperture set. 1 linear diffraction grating of period p=0.1 mm. 2 double circular apertures (diameter/separation in mm: 0.15/0.4 and 0.10/0.4). 2 double circular apertures (diameter/separation in mm: 0.10/0.3 and 0.10/0.4). α The experimental setup is very simple, as shown in Figure 9. The selected aperture is illuminated by the laser beam and the diffraction pattern is observed over the screen and measured directly by using the graduated paper that covers it. The screen must be located at a distance relatively large from the aperture s plane to be sure that the observed pattern corresponds to the far-field region. According with the basic theory, it is possible to study the influence on the diffraction patterns of different parameters, such as the shape and the size of the diffracting objects, the distance from the objects to the observation screen and the wavelength of illumination. For this purpose, in order to develop this analysis in a proper way we propose the next sequence of experiences. In all experiments the position of the screen is fixed at 3 m from the aperture set. FIGURE 9 VIEW OF THE EXPERIMENTAL SETUP. NOTE THAT IN THE LABORATORY THE SCREEN WILL BE LOCATED AT A FEW METERS FROM THE APERTURE SET. 4

5 Single apertures: influence of the size. From (2) or (4), it is clear that the scale of the diffraction pattern increases as the size of the diffracting object decreases. To show this effect, the proposed apertures in this case are: 4 single slits (slit widths 0.02,0.04,0.08 and 0.16 mm). 1 variable slit (slit width variable from 0.02 to 0.20 mm). 2 circular apertures (diameters 0.2 mm and 0.4 mm). Double targets: variation with the separation between elementary apertures. From (5) and (6) we conclude that for a double target composed by a set of two identical single apertures, the position of diffraction minima remains constant while the term due to interference varies inversely to the separation between elementary apertures. We propose to inspect this variation by using the following targets: 2 double slits (slit widths/separation in mm 0.04/0.25 and 0.04/0.50). 1 variable double slit (slit separation variable from to 0.75 mm with constant slit width of 0.04 mm). 2 double slits (slit widths/separation in mm 0.08/0.25 and 0.08/0.50) 2 double circular apertures (diameter/separation in mm: 0.10/0.3 and 0.10/0.4). Double targets: variation with the width of elementary apertures. From (2) or (4), and (5) it is straightforward to show that the position of diffraction minima varies when the width of the elementary apertures changes, while the term due to interference remains constant if the separation between them has not changed. The selected apertures to check this effect are: 2 double slits (slit widths/separation in mm 0.04/0.25 and 0.08/0.25). 2 double slits (slit widths/separation in mm 0.04/0.50 and 0.08/0.50). 2 double circular apertures (diameter/separation in mm 0.150/0.4 and 0.100/0.4). Multiple targets: influence of the number of elementary apertures. The width of maxima due to the interference between the elementary apertures decreases when the number of apertures increases, as can be seen from Figure 8. The effect of random replication of an elementary aperture is described in (9). The proposed diffracting targets for these analyses are the following: Set of 4 multiple slits (2,3,4 and 5 slits) with the same slit width (0.04 mm) and the same constant separation (0.125 mm). 1 linear diffraction grating of period 0.1 mm. 1 square diffraction grating. 1 hexagonal diffraction grating. 1 random hole pattern (hole diameter=0.06 mm). Complementary apertures. Babinet s principle. According with Babinet s principle [2], the diffraction patterns of two apertures having complementary transmittances (e.g., a slit and an opaque line both with the same width) are identically except for a bias term at the origin of the observation plane. This principle can be tested with the following targets: 1 single slit of width 0.08 mm. 1 opaque line of width 0.08 mm. 1 random opaque dot pattern (dot diameter=0.06 mm). 1 random hole pattern (hole diameter=0.06 mm). Variation with the distance at the observation plane. The scale of interference and diffraction patterns depends linearly with the distance from diffracting object to the observation plane, as is clear from the previous section. To check this, any of the previous experiences can be repeated by moving the screen to another position (e.g., by setting the distance from the diffracting object to 2 m or 4 m). Influence of the wavelength of the illuminating light. Diffraction patterns change their scale linearly with the wavelength of illuminating beam. To analyse this effect, all the previous experiences can be achieved using both laser pointers of λ R =650 nm and λ R =532 nm. As a sample of typical results achieved with the proposed setup and targets, we present in Figure 10 some examples of experimental diffraction patterns obtained at the laboratory. 5

6 Single slit Double slit Circular aperture Double circular aperture Square grating Hexagonal grating FIGURE 10 EXAMPLES OF EXPERIMENTAL RESULTS. In the following section we present a list of questions related to the above experiences, that could be proposed to the students for completing their understanding of the basic concepts under study. Additionally, in [4] we provide a list of some internet links to pages related to tutorials in diffraction and interference of light waves. PROPOSED QUESTIONS 1.- Figure 11 shows four Fraunhofer diffraction patterns and four double-hole apertures. Match each pattern to its aperture. (Assume that each diffraction pattern have been obtained shining a He-Ne laser of wavelength λ=632.8 nm through the aperture and observing the diffracted light on a screen located at a distance D=2 m from the aperture). How would the diffraction pattern B) of Figure 11 change if the laser wavelength were λ=532 nm instead of nm? A) B) C) D) 1) 2) 3) 4) FIGURE 11. TWO CIRCULAR HOLE APERTURES AND DIFFRACTION PATTERNS 2.- Figure 12 shows the Fraunhofer diffraction pattern of a double slit (composed by two slits of the same width L x, separated a distance 2a). This pattern has been generated with monochromatic light of wavelength λ on a screen located a distance D away. a) Using the array of white dots as a ruler (assume that the dots are 3 mm apart), estimate the ratio L x /2a. 6

7 b) How many interference minima would be in the main lobe of the diffraction pattern if the double slit had a slit separation twice the width of the slits? FIGURE 12 DOUBLE SLIT DIFFRACTION PATTERN 3.- Which aperture in Figure 13 corresponds to each of the diffraction patterns in the same figure? Give a brief explanation for your answer. Consider that these patterns have been obtained, as you did in the laboratory, with a laser diode of wavelength λ and collecting the diffracted light on a screen located at a distance D from the aperture. a) How would the diffraction pattern A) of Figure 13 change if the observation screen were placed at a distance D' twice than D. b) Make a sketch of the diffraction pattern that would be obtained with aperture B if one of its slits were blocked. A B A B C D FIGURE 13 DOUBLE APERTURE AND DIFFRACTION PATTERNS 4.- Suppose a laser diode (λ R =640 nm) shining through a diffraction grating whose spatial frequency is N 1. On a screen placed 2 m away, the diffracted light distribution is made of bright spots, as shown in Figure 14A. When the grating is replaced by another with a different spatial frequency N 2, the diffraction pattern observed in the same screen is now shown in Figure 14B. a) Find a value for the ratio between N 1 and N 2. b) How would you move the screen to match the scale of both diffraction patterns? A) B) FIGURE 14 DIFFRACTION PATTERNS OF TWO GRATING WITH DIFFERENT SPATIAL FREQUENCY 7

8 5.- Consider the Fraunhofer diffraction pattern of a circular aperture of diameter L, generated with monochromatic light of wavelength λ on a screen located at a distance D from the aperture. As it is known, the intensity distribution of this pattern is given by disk 2 (ρ), where ρ = Lr/ ( 2λD) and r is the radial coordinate (r 2 =x 2 +y 2 ). The values of ρ for the first maxima and minima of this function are collected in Table 1. a) Find an expression for the diameter φ of the third dark ring. b) Discuss qualitatively how would the diameter of the first dark ring change if: i) the observation screen were placed at a larger distance D'. ii) the wavelength had a smaller value λ'. 1 st Maximum 1 st Minimum 2 nd Maximum 2 nd Minimum 3 rd Maximum 3 rd Minimum ρ TABLE 1 FIRST MAXIMA AND MINIMA FOR THE FUNCTION disk 2 (ρ). 6.- The intensity distribution of a double slit diffraction pattern is shown in Figure 15. Discuss qualitatively how would this pattern change if: a) the width of the slits were half of its initial value. b) the slit separation were twice the initial value. c) the number n of slits in the aperture were very large (let us say n 1000). Make a rough graph of the resulting pattern for each case. FIGURE 15 INTENSITY DISTRIBUTION OVER A DOUBLE SLIT DIFFRACTION PATTERN. ACKNOWLEDGEMENT Authors gratefully acknowledge the financial support of Plan Nacional I+D+I (grant DPI ), Ministerio de Ciencia y Tecnología, Spain, and of Servei de Formació Permanent of the University of Valencia, Spain. REFERENCES [1] Hecht, E, Optics, Addison-Wesley, [2] Lipson, S, G, Lipson, H, and Tanhausser, D,S, Optical Physics, Cambridge University, [3] Tipler, P, A, Physics for scientists and engineers, Freeman, [4] Websites: