5. Double Slit Diffraction

Transcription

1 Double Date slit : diffraction 5. Double Slit Diffraction Background Aim of the experiment Huygens s principle Interference Fraunhofer and Fresnel diffraction Coherence Laser 1. To plot the intensity distribution of the Fraunhofer diffraction pattern due to two slits of same width and to estimate the width of the slits and separation between the slits from the intensity pattern. 2. To compare intensity distribution of diffraction patterns due to two double slits with same widths, but different distances between the slits. The widths of the slits and separation between the them are determined. Finally intensity relations of the peaks and missing orders are studied. Apparatus required Laser, He-Ne, 1 mw Universal measuring amplifier Optical profile bench Slide-mounts Lens holder Object holder Lens, mounted, f + 20 mm. Lens, mounted, f mm. Photoelement Diaphragm, 4 double slits Multi-range meter Connecting cords Theory When plane monochromatic light wave front of wavelength, λ, falls normally on a system of two slits of equal widths, b, separated by a distance, g, the luminous intensity I of the beam diffracted in the direction φ is given by: 2 sin (. b.sin / ) 2 I( ) I 0 cos (. g.sin / ). (1) 2 (. b.sin / ) The equation (1) shows that the intensity is a product of a single slit diffraction term and a double slit interference term (see figures1 (a) and 1 (b) below). 57

2 Fig. 1(a) Double slit diffraction pattern g/b=5, Fig 1(b) Same pattern resolved as a product of single slit diffraction pattern and double slit interference pattern with I0 =1. If one considers only a single slit (1 st class interference), this gives minimum intensity when the numerator of the first factor becomes zero. This always is the case when the following applies: bsin m, (m = 1, 2, 3, )..(2) m Principal maximum is obtained for =0, when the first factor of expression (1) becomes 1, there are other secondary maxima approximately at =1.43, 2.46, 3.47, 4.48 etc. On the other hand, the double slit interference pattern has maxima at angles satisfying the condition, g sin n, (n=0, 1, 2, 3,....).(3) n An interesting situation arises when both equations, (2) and (3) are satisfied for a particular value of. This happens when, g n..(4) b m th In this case n order interfernce maximum will have zero intensity and will therefore be missing. In the figure 1, g/b=5, hence we find that the 5 th secondary peak is missing. Condition (4) is again satisfied for n=10 and m=2, so the 10 th peak is again absent. Similarly 15 th,20 th etc. peaks will also be missing in this case. These are known as missing orders in the literature. The central envelope has 9 peaks within it, where as the side lobes will have 4 peaks each for the case shown above. Procedure 1. Place the photocell at the center of the shifting range at the one end of the optical bench. The laser is mounted on the opposite end of the bench. Connect the photocell to the input of the measurement amplifier (amplification 3 5 factor ) and the output of the amplifier to the multimeter. 58

3 2. Align the laser source on the optical bench to get collimated laser beam. A broadened and parallel laser beam, obtained with the lenses f= 20 mm and f=100 mm, must impinge centrally the photocell. To achieve this the distance between the laser and the lens f=20mm is kept 11.5 cm whereas the distance between the lenses f=20mm and f=100mm is adjusted to 13 cm. 3. Place the double slit diaphragm perpendicular to the beam at a distance of 5 to 6 cm from the lens f=100mm. This makes a separation about 1m between the slit and the photocell. 4. Choose any one of the double slits available. Make sure that both slits of the two slits are equally illuminated. 5. Measure the micrometer constant of the scale attached to the photocell detector. 6. Turn the circular scale attached at the base of the photocell in equal intervals(say 0.2 or 0.3mm) and note that the intensity of the fringes from one end (say from left to right) using a multimeter. Reach the other end of the fringe system this way noting down many readings of the intensities in between. 7. Repeat 6 for other available double slits. Observe number of peaks in the central envelope and side lobes and identify missing orders. 8. Measure the distance, D, separation between slit and the detector, using the scale on the optical bench. 9. Plot the intensity distribution of the diffraction pattern and calculate the slit width, b, and the separation between the slits, g, from graphs. Match these estimates with the ones written on the diaphragm. Furthermore, calculate the relative intensities of the peaks with respect to the central maximum. Match your results with the theoretical estimates obtained from the expression (1). Observations Wavelength of the light ( ) = cm Position of the laser =..cm Position of the lens f/20mm =..cm Position of the lens f/100mm=.cm Position of the double slit diaphragm=.cm= a Position of the photocell=.cm = b Distance of the slit from the cell, D = b-a =.cm 59

4 Table I: For intensity distribution (x 0 corresponds to the central maximum) Double Slit I b= g= Position (x cm) = (x-x 0 )/D V (volt) Position (x cm) = (x-x 0 )/D V (volt)

5 Double Slit I b= g= Position (x cm) = (x-x 0 )/D V (volt) Position (x cm) = (x-x 0 )/D V (volt) 61

6 Double Slit II b= g= Position (x cm) = (x-x 0 )/D V (volt) Position (x cm) = (x-x 0 )/D V (volt) 62

7 Double Slit II b= g= Position (x cm) = (x-x 0 )/D V (volt) Position (x cm) = (x-x 0 )/D V (volt) 63

8 Double Slit III b= g= Position (x cm) = (x-x 0 )/D V (volt) Position (x cm) = (x-x 0 )/D V (volt) 64

9 Double Slit III b= g= Position (x cm) = (x-x 0 )/D V (volt) Position (x cm) = (x-x 0 )/D V (volt) 65

10 Graph : Double slit intensity pattern 66

11 Results and Calculations: 1. Plot Intensity distribution of the diffraction as a function of parallel to the plane of the slit. Table 2: Estimation of slit width, b, and slit separation, g. Given (mm) Slit width b Slit separation g No. peaks inside Estimated from Given Estimated from the central plot (mm) (mm) plot (mm) envelope No. peaks within side lobes Missing orders observed Double slit I Double slit II Double slit III Table 3: Relative intensities of the peaks with respect to the central maximum (assume I0= 1). I1 I2 I3 I4 I5 From the plot From theory From the plot From theory From the plot From theory From the plot From theory From the plot From theory Double slit I Double slit II Double slit III 67

12 Error calculation For the minimum of the 1 st class interference pattern, b sin m m b D x Therefore, 2 2 b D x is supplied and, D and x are obtained as a difference of two scale readings. Similarly evaluate error in the estimation of separation between the slits g. Precautions (i) (ii) (iii) (iv) (v) Do not look into the laser light directly. Adjustment of lens, slit, laser must be made properly so that fringes are bright and distinct. Since the linear shift d is proportional to D, it should be fairly large. A value of D of about 1.0 m is preferable. Make sure that a strong monochromatic source of light is used. Take care of the backlash error while photocell measurements. Questions Same as for single slit diffraction 1. What are missing orders? 2. What do you expect if the width of the slits is gradually increased? 3. What is the effect of increasing the width of the opaque space, keeping the slit width constant? References 1. PHYWE LEP Diffraction intensity at double slit systems 2. Fundamental of Optics by F. Jenkins and H. White 535 JEN/F 3. Optics by E. Hecht 535 HEC/O 68