FRAUNHOFFER DIFFRACTION AT SINGLE SLIT

Transcription

1 15 Experiment-37 F FRAUNHOFFER DIFFRACTION AT SINGLE SLIT Sarmistha Sahu Head Dept of Physics, Maharani Laxmi Ammani College For Women BANGALORE INDIA sarmis@eth.net. Abstract Fraunhoffer diffraction pattern is sketched by detecting diffracted light using LDR and travelling microscope. The wavelength of the laser light used is determined for different slit width and source screen distance. The intensity ratios are calculated and compared with the theoretical values. Introduction Diffraction and interference are two different wave phenomenons, which leave its occurrence in the form of pattern or fringes [1,2]. Diffraction is bending of light wave around an object. Figure-1 shows bending of light wave around the object and reflection of light from edge away from the object. For diffraction the edge of the object should very sharp and for reflection the surface to be smooth. Bending of ray around an object Diffraction No bending, ray going away from the object Reflection Figure-1, Diffraction and Deflection of Light wave The three words object, bending and around are very important in the definition of diffraction. Another very important light phenomenon which we see every day is the scattering of light. When you getup in the morning or in the evening the yellowish red sky speaks of light scattering by tiny particles in the atmosphere. Fraunhoffer s diffraction and Fresnel s diffraction are the two fundamental phenomena of light diffraction. Fraunhoffer diffraction is a special case of Fresnel diffraction in which the source illuminating the aperture and the observation screen are located at infinite distance. What we see daily is Fraunhoffer diffraction what we don t see is Fresnel diffraction. Fresnel diffraction is the theoretical foundation.

2 16 Lab Experiments To view diffraction we need a coherent beam of light and an object. In lab experiments the object generally taken as a slit with adjustable width it is also called as an aperture. Or thin wire also can be used as object. The object or the slit used in this experiment is shown in Figure-2. A white screen is placed at a distance of about 2 meter is used as observer. To record the intensity of the diffraction pattern another slit is required whose aperture is made small than the emitting slit by adjusting the micrometer. Figure-2, Slit used in the experiment Intensity is recorded by a LDR placed behind the receiving slit. The LDR is energized by a regulated power supply and digital dc microammeter is used to measure the LDR current that is proportional to the light intently it receives. Figure-3 shows the circuit arrangements. LDR 0-200uA VLDR 1.25V Figure-3, Circuit connection to record diffracted light In this experiment Fraunhoffer diffraction is observed at single slit. The intensity of the diffracted light coming out of the emitter slit is given by Sin β I = [ ] 2 I O 1 β πa Where β = Sin ψ λ ψ is the wave function λ is the wavelength of the light a is the slit width

3 17 The diffraction pattern is a series of bright and dark spots. The position of the maxima or the brighter spot and the minima the dark spot is obtained by equating the derivatives of equation-1 to zero [2]. Sinβ (Sinβ- βcosβ) = 0 2 When Sinβ = 0 maxima occurs, or when β = 0, π, 2π, 3π, 4π etc. or λ 2λ 3λ Sin ψ = , , a a a Maxima occurs at (Sinβ- βcosβ) = 0 This equation doesn t have exact solution except for β = 0, the other approximate solutions are β = 4.49, 7.73, 10.9 radians etc The maxima at β=0 is the primary maximum and other are secondary maxima. Hence maxima occurs at 4.49λ 7.73λ 10.9λ Sin ψ = 0 = , , πa πa πa 1.43λ 2.46λ 3.47λ Sin ψ = 0 = , , a a a Using these relations one can determine intensity of all the maximum occurring in the pattern with respect to the primary maximum from equation-1. Sin 4.49 I 1 = [ ] 2 I O = I O 4.49 Similarly I 2 = I O and I 3 = I O In this way one can calculate the intensities of various maxima and minima in the diffraction pattern. Figure-4 shows the processes diffraction at the single slit. The laser light from the laser source falls on the slit. J and K are the two rays, which falls on the slit edge and gets diffracted. The two rays bend towards the slit as shown in Figure-4. The diffracted ray J meets the screen at

4 18 Lab Experiments P. The screen is placed at distance of D from the laser source. The diffracted ray J appears to emerge from a point S as indicated by doted line. The diffracted ray J and direct ray from S make an angle θ. The direct ray falls at O on the screen and diffracted ray falls at P on the screen at a distance of y from O. The diffracted ray J travels more distance than the direct ray to reach the screen. Hence there is a path difference between the direct ray and diffracted ray reaching the screen. The path difference between the direct ray and diffracted ray is given by Path difference = BC If y = 0 then the direct and diffracted ray meet at the same point and there is no path difference. Diffracted ray P Single Slit Laser S J B C a/2 A a/2 Distance D Direct ray y O K B C a/2 Diffracted ray Screen A Figure-4, Process of diffraction at the single slit The two triangles ABC and SOP in Figure-4, angles BCA = 90 and SOP = 90. And angles PSO = θ, BAC = θ The angle θ is small angle hence θ θ. Therefore in triangle ABC BC a Sin θ = Sinθ = or BC = AB Sinθ = Sinθ. AB 2 BC is the path difference hence

5 19 a δ = Sinθ 1 2 The corresponding phase difference is given by 2π φ = a Sinθ 2 2 λ Where, a is the slit width θ is the angle between the direct ray and the diffracted ray λ is the wavelength of the laser light used. The slit width a is very small (2-3 mm maximum) hence angle θ is also practically small angle. Hence Sinθ = θ or the phase difference now becomes π φ = a θ 3 λ Depending on the phase difference there will be maximum and minimum intensity in the diffracted light. These alternate variations in the intensities of the diffracted light form a diffraction pattern shown in Figure-5. Figure-5, Fraunhoffer Diffraction Pattern observed in this experiment Table-1 shows angle of maxima and minimum brightness occurrence in the pattern. The center maxima occurs at θ = 0. Table-1 Order Maxima Minima 1 0 π 2 3π/2 2π 3 5π/2 3π Maxima and minima variation in diffraction pattern If P is the point in Figure-4 distance y from the center O where the first minima occurs then the phase difference at this point is given by φ = π Substituting this value in Equation 3 we get

6 20 Lab Experiments λ θ = a θ is a quantity which can not be measured using protractor, hence replaced by known quantity. In triangle SOP in Figure 4, y tanθ = D Where y is the distance between the first maxima center and first minima D is the distance between source and screen Further the angle θ is small hence y θ = D y and D are measurable and substituting for θ in equation 4 we get ay λ = D From the diffraction pattern, y is distance from the center of first maxima to the edge of first minima as shown in Figure-6. Or 2y is the distance between first minima on the left and first minima on the right. Or it is width of the center maxima. W =2y 8 Substituting in Equation 7 aw λ = D Using this equation wavelength of the laser light is determined in this experiment y O P W Figure-6, Diffraction pattern

7 21 Apparatus Used Laser 625nm, 0.5mWatt with adjustable stand, micrometer adjustable slit, travelling microscope fitted adjustable slit and LDR, Digital microammeter 0-200µA. Experimental Procedure 1. The least count of the micrometer adjustable slit is determined following standard procedure. Distance moved = 5mm, Number of rotations given = 10 Pitch = 5mm/10 = 0.5mm, Number divisions on the head scale = 100 LC = 0.5mm/100 = 0.005mm Figure-7, Laser source and micrometer slit used in the experiment 2. The micrometer adjustable slit is mounded on a stand and placed in front of the laser source as shown in Figure-7. Figure-8, Receiving slit fitter with LDR, travelling microscope holding LDR detector

8 22 Lab Experiments 3. The travelling microscope containing LDR detector, behind the receiver slit is place on the way of the laser light and laser light is focused by adjusting stand on the receiver slit. Figure- 8 shows travelling microscope and slit containing LDR, Figure-9 shows the schematic arrangement. 4. The laser light is correctly focused to the receiving slit. Once sure light is falling on to the receiving slit a microammeter and power supply are connected to LDR and current is noticed in the microammeter. Slit Diffracted ray LDR Receiving Light Direct ray LDR Travellong Microscope Laser Light Diffracted ray Slit Diffracted ray Fraunhoffer Intensity Pattern Figure-9, Schematic arrangement of Fraunhoffer Diffraction 5. A white paper is now held in front of the receiving slit and diffraction pattern is observed as shown in Figure-5. The micrometer source slit or the emitting slit is adjusted if need to get a faithful diffraction pattern. 6. The reading of the micrometer source slit is noted and recorded in Table-3. The Travelling microscope is moved to the left from about 3cm from center position where the current is maximum. At this position microammeter reading and travelling microscope readings are noted and recorded in the Table The travelling microscope is moved towards the right by 1mm and microammeter reading is noted. This is continued till the maximum intensity is obtained and the corresponding readings obtained are recorded in Table After reaching the maximum intensity position, movement of the travelling microscope is further continued in steps of 1mm. This is done until second and third maxima are reached. 9. The distance between the laser source and the travelling microscope is measured using a meter scale. This is the distance D

9 A graph is drawn taking distance moved along X-axis and current along Y-axis as shown in Figure-10. From the graph width W is determined and wavelength is calculated. 11. Experiment is repeated for different slit width and source screen distance D. The corresponding values of slit width and fringe width are tabulated in Table-3. Distance (mm) Table-2 LDR current (µa) Distance (mm) Intensity variation with distance LDR current (µa) 12. From Table-2 the intensity ratios are calculated. I 1 /I O = 2/46.8 = I 2 /I O = 1/46.8 = and I 3 /I O = 0.4 /46.8 = LDR Current (ua) Distance (cm) Figure-10, Fraunhoffer Diffraction Pattern (Distance D=2.6Mmeters, a = 0.355mm)

10 24 Lab Experiments Table-3 Slit Width a Distance (m) Fringe width(m) Wavelength(nm) Corrected Slit Width(m)x10-3 D W x10-2 λ HSD* Average λ = 662nm Results Slit width, source detector distance and wavelength (* Corrected HSD after applying zero correction) The results obtained are tabulated in Table-4 Discussions Table-4 Parameters Experimental Theoretical Intensity ratio I 1 /I 0 I 2 /I 0 I 3 /I Wavelength (nm) Experimental results 1. Fraunhoffer diffraction pattern is sketched in this experiment using LDR as light detector. The diffraction pattern obtained is the photocopy of the theoretical predictions. 2. The width of the receiver slit kept in front of the LDR should be small (0.25mm) compared to the minimum source slit width (0.355mm) of the source slit. This ensures proper detection of diffracted light by LDR. 3. The wavelength determined (652nm) is in very good agreement with the wavelength (625nm) of the laser light used with 4.3% error. 4. The travelling microscope should not be kept over smooth surface, because it may slip while adjusting the position. The experiment is done in the dark room so that there is no background light effect on the measurements. LDR current was zero when there no light falling on the receiver slit from the laser. References 1. Sirohi R S, A course of experiments with He-Ne Laser, Wiley Eastern Limited, 1991, Page 37.

11 Single Slit Fraunhoffer Diffraction, Ms Sharmistha Sahu Ms. Sharmistha Sahu, a postgraduate from Ravenshaw College, Cuttack, is presently professor and head of the department of Physics, Maharani Lakshmi Ammanni College for Women, Bangalore. She is engaged in developing low cost instruments for physics experiments, some of which are being used in her department fruitfully.