EXPERIMENT 3 Diffraction Pattern Measurements using a Laser Laser Safety The Helium Neon lasers used in this experiment and are of low power (0.5 milliwatts) but the narrow beam of light is still of high intensity. Consequently, 1. Never look directly into the unexpanded laser beam or at its reflection from a mirror surface.. Do Not use the laser without getting instructions from the demonstrator. Introduction Physical optics predicts the angular position θ of the Fraunhofer diffraction fringe minima produced by monochromatic light of wavelength λ passing through a single slit of width b to be given by mλ = b sin θ m (3.1) where θm is the angle between the central maximum and mth dark fringe. Theory also predicts that the fringe intensity is given by where I = I o sin β β = I o sinc β (3.) β = π b sin θ λ and Io is the intensity of the central maximum at θ = 0. Thus the Fraunhofer pattern for a single narrow slit has the form shown in Fig. 3.1. The pattern consists of a broad intensely illuminated central band accompanied by a series of more weakly illuminated narrow fringes all parallel to the slit and located on either side of it. 3-1 (3.3)
Experimental Arrangement A photodiode detector interfaced to a PC facilitates the plotting of a graph of diffraction fringe intensity as a function of position for single and multiple slits. The Fresnel diffraction pattern formed by a horizontal laser beam passing through a narrow vertical single slit is arranged to fall on a movable detector (Fig. 3.). Note that while Eq. 3.1 and 3. are derived for Fraunhofer diffraction, these equations are approximately correct when θ is small i.e. when the distance between the slit and detector is large compared with the fringe spacing. Fig. 3. illustrates the apparatus used to facilitate intensity measurements in the diffraction pattern. The detector is a photodiode mounted on a steel rod which can be moved horizontally along that rod in order to investigate the nature of the fringe pattern. The horizontal position of the detector and the light intensity are recorded automatically by the computer as the detector is driven along the horizontal rod. The light intensity and the detector position are recorded 56 times during a scan through the diffraction pattern. Figure 3.1: Fraunhofer diffraction pattern for a single narrow slit Procedure You must calibrate the horizontal axis by recording the detector position in centimetres and noting the corresponding x-axis value on the computer screen at a series of points along the scan. Before taking any measurements, the positions of the single slit at the laser and the detector must be adjusted so that the detector receives laser light throughout the entire scan of the diffraction pattern. A: The Single Slit The width of the single slit provided can be adjusted by turning the control on the side of the slit. You are required to record the intensity pattern for three different slit widths. In order to determine the slit width b with precision, the distance y between the two 4th order minima must be measured. 3-
From Eq. 3.1 4λ = b sin θ 4 b = 4λ 4λ sin θ 4 θ 4 since y D = tan θ 4 θ 4 (3.4) where D is the detector - slit distance meters. It follows that b = 4 λ y D = 8D λ y Using the He-Ne laser with λ = 638Å, adjust the slit width to give a well-separated diffraction pattern in the plane of the detector. Because the central maximum is about 0 times more intense than the next bright fringe on either side, it is necessary to adjust the intensity of the laser using a neutral density filter. You must adjust the filter so that the intensity reading on the detector when placed at the central maximum position is less than 56 (use Setup). The photodiode is then moved through the fringe pattern by setting the motion control to computer whilst executing the program supplied. The demonstrator will help you to set up the program. The readings of the position and intensity then automatically appear on the screen. When the scan is completed, a printout can be obtained by pressing the SHIFT and PRTSC keys together. Remove the neutral density filter and repeat the procedure. The detector will now be saturated when passing through the central peak but at least four maxima should be observed on either side of the central maximum. Check when you remove the filter that the diffraction pattern falls on the detector throughout its horizontal scan. Determine the slit width using Eq. 3.5. Obtain the diffraction patterns for two additional slit widths, ensuring in each case that the intensity reading in the central maximum position is less than 56. Apply the following analysis to your diffraction patterns. 1. Find using β the location of both the minima and subsidiary maxima in the diffraction pattern. Extrema (i.e. maxima and minima) of I θ = I o sin β which di θ dβ = I o sin β β (3.5) = I o sinc β correspond to values of β for (β cos β sin β) β 3 = 0 Minima occur when sin β = 0 and β 0, i.e. when β = ± m π with m = 1,, 3.... Confirm the values of β at the first three minima by measuring the distance between corresponding minima on each side, obtaining θ from Eq. 3.4 and substituting for θ in Eq. 3.3. Subsidiary maxima occur for nonzero β satisfying the relationship β cos β sin β = 0 3-3
D Laser Filter Computer Printer Single / Double slit Moveable Photodiode Figure 3.: Experimental Arrangement which can be arranged to give: tan β = β The last equation is most simply solved graphically by superimposing the straight line f 1 (β) = β on the curves f (β) = tan β. The points of intersection, other than the origin (Fig. 3.3), locate the subsidiary maxima at β = ±1.4303π, ±.459π, ±3.4707π etc. Note that these peaks are not quite midway between the minima, but are displaced towards the centre of the pattern by an amount which decreases with increasing value of m.. Determine the intensity of the first three or more maxima in terms of I o, the principal peak value. To evaluate I o for the second plot, where the main maximum is off scale, we can use the ratio of I 1 I from the first plot because I 1 o I is constant. o We know that secondary peaks occur at β 1 = 1.43, β =.46 and β 3 = 3.47 and that I θ = I o sinc β and so to measure the β values, we need only calculate the sinc function in the Appendix to find sinc β 1 which should be ( 0.17) = 0.047 sinc β which should be (0.18) = 0.016 and sinc β 3 which should be ( 0.091) = 0.008 The desired values shown should then be compared with your measured values and with those obtained previously. 3. The approximate expression for the angular width at half maximum of the central peak of the diffraction pattern is obtained as follows 3-4
Figure 3.3: Graphs of f 1 (β) = β and f (β) = tan θ superimposed We must determine the value of θ for which the intensity is one half the peak intensity or i.e. I 1 I o = sinc β = 1 sincβ = 0.7071 The sinc function shows the intensity to have its half maximum value at β 1 Fig. 3.1). = 1.39 radians (see Since β 1 β = πb sin θ λ = 1.39 πb λ θ 1 the total angular width, θ is θ 1 This is usually rounded off to and θ 1 = 0.44 λ b and hence θ = 0.884 λ b. θ = λ b Check that the width of the diffraction pattern varies inversely with the width of the slit using the three graphs plotted with the neutral density filter and measure the half-width of the main maximum in each case. Calibrate the horizontal axis of the pattern and hence evaluate the actual slit widths used in each case. 3-5
B: The Double Slit Replace the single slit with the double slit and record the intensity pattern. From the intensity pattern determine the width of the slits and their separation. The equation of the intensity pattern can be found in various optics books and is I(θ) = 4 I o sinc β cos α where πa sin θ α = λ and a is the separation between the centres of the two slits. The peak I o is that of a single slit at θ = 0 and I(0) = 4I o. Fig. 3.4 shows the intensity variation across the pattern for a = 3b. The first term in the above equation gives the intensity distribution in the Fraunhofer pattern for a single slit and gives the slowly varying envelope within which lie the fringes given by the second term. The cos fringes are characteristic of the interference between two point or line sources separated by a distance a. The second term represents the effect of interference between disturbances from the two slits and the first term the result of diffraction at each individual slit (see Eq. 3.1). The bright band of order n in the interference pattern for two slits is said to be suppressed or missing when it is coincident with a null in the diffraction envelope. Maxima in the interference pattern occur when a sin θ = nλ(n = 0, ±1, ±,...). Minima in the single slit diffraction pattern result when b sin θ = m(m = ±1, ±...). When both of these equations are satisfied a b = n m In Fig. 3.4 the first missing order corresponds to n = 3 where m = 1 and accordingly a = 3b. The second missing order results when n = 6 and m =. If we include the two half fringes at the missing orders, there are m n bright bands within the central diffraction peak. Apply the above considerations to your intensity pattern. C: Babinet s Principle Babinet s principle states that the diffraction patterns produced by two complementary objects (e.g. slit and wire of the same diameter) are identical. This principle can be used here to estimate the diameter of human hair. Stretch a hair across the empty slideholder provided and tape the ends to secure it in position. Mount the hair in place of the usual single or double slit and record the resulting diffraction pattern. The width of the hair may be obtained simply by comparing the width of the central maximum with that obtained for a single slit. Questions 1. Sketch and explain briefly the diffraction pattern of a straight edge. Can such a pattern be measured with this experimental arrangement?. Sketch and explain briefly the diffraction pattern of a circular aperture. 3-6
Figure 3.4: Fraunhofer diffraction pattern for a double slit of width b and separation a where a = 3b 3. Briefly what did you learn from this experiment? References 1. Optics, Hecht & Zajac. Fundamentals of Optics, Jenkins & White WWW : Particle Size Analysis with Laser Diffraction - http://www.azom.com/details.asp?article ID=158 Diffraction in Medicine - http://img.cryst.bbk.ac.uk/bca/ig/news/n99tl.htm 3-7