Diffraction and Interference of Light


 Adelia Heath
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1 rev 12/2016 Diffrction nd Interference of Light Equipment Qty Items Prt Number 1 Light Sensor CI Rotry Motion Sensor CI Single Slit Set OS Multiple Slit Set OS Liner Trnsltor OS Lser OS8525A 1 Optics Bench OS8518 Purpose The purpose of this ctivity is to exmine the diffrction nd interference ptterns tht occur when monochromtic lser light psses through difference combintions of pertures. Theory/Bckground In 1801, Thoms Young obtined convincing evidence of the wve nture of light. Light from single source flls on slide contining two closely spced slits. If light consists of tiny prticles (or corpuscles s described by Isc Newton) then on viewing screen plced behind those two slits we would two bright lines directly in line with the two slits. However, Young observed series of bright lines. Young ws ble to explin this result s wve interference phenomenon, becuse of diffrction, the wves leving the two smll slits spred out from the edges of the slits. This is equivlent to the interference pttern of ripples, from two wve sources, to the top for body of wter crossing ech other s pths, nd interference with ech other when they do. 1
2 In generl, the distnce between the slits is very smll compred to the distnce from the slits to the viewing screen where the interference pttern is observed. The rys from the edges of the slits re essentilly prllel to ech other. Constructive interference will occur on the screen when the distnce tht the rys from ech of the slits trvels to the screen is different by whole number multiple of the wvelength of the light itself. Those spots re where the bright spots in the observed interference pttern form. While destructive interference will occur on the screen when the different distnces tht the rys from ech of the slits trvels to the viewing screen is relted to hlf wvelength of the light itself. When the monochromtic (single wvelength) light is pssed through two slits the bright spots (mxim) of the interference pttern tht forms is symmetric round centrl bright spot. Ech of the mxim re evenly spced, nd of the sme width. They will get slightly dimer s the get further wy from the centrl mxim. When the monochromtic light is pssed through one slit slightly different interference pttern (lso clled diffrction pttern) will form. In the cse of one slit the pttern will still be symmetric round centrl bright spots, but the centrl bright spot will be twice s wide s ll the other mxim, nd will lso be much brighter thn ll the other mxim. For diffrction of light to occur when monochromtic light psses through single slit the width of the slit mustn t be too much lrger thn the wvelength of the light itself. Equtions for Double Slits If we drw digrm for the double slit configurtion we cn use simple trigonometry to determine where the bright spots will form on the viewing screen. Here d is the distnce between the two slits. L is the distnce from the two slits nd the viewing screen. y is the displcement from the center point of the viewing screen to the point P where the bright spot forms. θ is the ngulr loction of the bright spot. Finlly S is the difference in the distnce trveled by the two light rys r 1 nd r 2. From the properties of the right tringle it is cler tht S = d sin θ. Since we know tht the bright spots will form where S is equl integer multiples of the wvelength (λ) of the light tht mens; 2
3 d sin θ = nλ As previously stted L is going to be much lrger thn d, which is going to result in L lso being much lrger thn y. This mens tht θ, the ngulr position of the bright spots, is lwys going to be very smll. So we cn invoke the smll ngle pproximtion of sin θ. (sin θ tn θ, when θ is smll) Which mkes our eqution; d tn θ = nλ Finlly, since tn θ = opp dj we hve tn θ = y in this cse. Substituting tht in gives us: L d y n L = nλ Solving for y n; y n = n λl d Our eqution tells us tht for the two slit configurtion the vrious mxim will be locted integer multiples of the wvelength of the light multiplied by the distnce from the slits to the viewing screen, nd ll divided by the distnce between the slits. So y n is the liner loction of the n th bright sport, mesured from the loction of the centrl bright spot. If we wnt to find the ngulr loction of the bright spots we just return to our originl eqution nd solve for θ; Equtions for Single slit If you drw digrm of the single slit configurtion we cn lso use simple trigonometry to find the where the drk spots will form on the viewing screen. These re the spots of destructive interference locted right in the middle between ny two consecutive bright spots. In this digrm is the width of the single slit. L is the distnce from the slit to the viewing screen. y is the θ n = sin 1 ( nλ d ) 3
4 displcement from the center of the viewing screen to the bring spot. θ is the ngulr position of the bright spot. Using similr rgument s we did with the double slit configurtion we get the following; θ m = sin 1 ( mλ ) y m = m Lλ In these equtions m is n integer, excluding zero. Remember these equtions give you the loction of the drk spots between the mxim, becuse of this the difference in the vlue of y for two consecutive whole numbers is considered to be the width of bright spot. (Not including the centrl bright spot) y = Lλ The centrl bright sport hs width tht is twice s lrge s ll the other bright spots due to m being n integer excluding zero, s cn esily be shown. Let m = 1, be for the first drk spot to the right of the centrl mxim, nd m = 1 be for the first drk spot to the left of the centrl mxim, then the distnce between these two drk spots is given by; y = 1Lλ ( 1Lλ ) = 2Lλ Wht hppens when both conditions re meet? Mening when there re two slits, nd the width of the slits is bout the size of the wvelength of the light pssing through them wht sort of pttern forms of the viewing screen? Well, then you get composite of the two ptterns, where the double slit interference pttern is encsed in the diffrction pttern ssocited with the width of the slits. If you were to plot out the Intensity vs. Angulr Position grph of double slit interference pttern you would obtin something like the following. The centrl mxim is locted t 0 0, nd ll the peks re evenly spced out, nd bout the sme height, mens ll bout the sme brightness. 4
5 If you were to plot out the Intensity vs. Angulr Position grph of single slit diffrction pttern you would obtin something like the following. Here the centrl mxim is still locted t 0 0, but the centrl mxim is twice the width of ll other bright spots, nd the centrl mxim is much tller thn the rest mening it is much brighter thn ll other bright spots. Now when BOTH conditions re meet the something similr to the following grph is obtined. As stted before, the double slit interference pttern is encsed in the single slit diffrction pttern. The intensity of ny given pek from the double slit interference pttern is determined by the locl intensity of the single slit diffrction pttern. Even so much tht the bright spots of the interference pttern tht re t the loctions of the drk spots of the diffrction pttern re completely suppressed, nd there is drk spot t tht loction. We cll the centrl bright spot of the diffrction pttern the Centrl Envelope of the composite pttern, nd we cn determine how mny of the bright spots from the double slit interference pttern re encsed in the Centrl Envelope by using the eqution for the liner positions of the interference pttern, nd the liner eqution for first drk spot of the diffrction pttern, m = 1. Set the interference eqution equl to the diffrction eqution when n = 1. mlλ d = 1Lλ m = d 5
6 This rtio tells us tht the m th bright spot of the inference pttern should form t the sme loction tht the first drk spot of the diffrction pttern. But we re not finished yet. First the rtion of d is most likely not going to be n integer, mening these two spots don t line up exctly so the lst bright spot is only prtilly encsed in the centrl envelope. In this cse we need to round down to the closest integer. (Alwys round down, never round up. Exmple 3.9 is rounded to 3, not 4.) In the cse tht the rtio d is n integer tht mens the mth bright spot of the interference pttern, nd the first drk spot of the diffrction pttern re t the exct sme loction. This mens tht the m th bright spot is completely suppressed by the first drk spot, nd is NOT in the centrl envelope, so you need to round down to the next integer. (Exmple 4 is rounded down to 3.) The end result being is tht you ALWAYS round the rtio down to the next integer. Next, since the pttern is symmetric bout the centrl bright spot, we hve to double the rounded down vlue to ccount for the bright spots on the other side of the centrl bright spot, nd finlly we hve to dd 1 to ccount for the centrl bright spot itself. (As n exmple let d = 0.30 cm, nd = 0.30 mm then we get; m = d = = = = = 19 So in such configurtion there would be 19 bright spots from the double slit interference pttern encsed in the Centrl Envelope of the diffrction pttern.) Setup 1. Connect the perture brcket screen to the light sensor by plcing the light sensor is imed through the circulr opening for the brckets, nd the bolt opening on the bottom of the light sensor is ligned with one of the circulr openings on the bottom of the brcket. Then screw in the cylinder mount to the bottom of the light sensor 2. Attch the light sensor to the rotry motion sensor by inserting the cylinder mount through the holes t the front of the rotry motion sensor, nd then tightening the front screw. Mke sure tht the light sensor is mounted on the rotry motion sensor such tht the light sensor nd the xle of the rotry motion sensor re both oriented upwrds. 3. Attch the rotry motion sensor to the liner trnsltor first unscrew one of the ends of the metl br with ger teeth, nd completely remove the screw. 6
7 Then slide tht end of the br ll the wy through the rotry motion sensor, nd then rettch the loose end of the br. 4. Attch the liner trnsltor to the optics bench sliding the loosely connected screw nd bolt, t the bottom of the liner trnsltor through the grove long the middle of the optics bench till the very front of the perture brcket screen is t the 100 cm mrk nd pointing towrds the other end of the optics bench, then tighten the screw to hold the setup in plce. Rotte the perture disk of the perture brcket screen till the #2 slit is ligned with the light sensor. 5. Attch the multiple slit set slit ccessory nd ttch it to the optics mount, then insert the optics mount into the optics bench the very front of the slit ccessory is t the 10 cm mrk, nd it is fcing the setup t the other end of the optics bench. Rotte the slit set till the combintion of =0.04mm, nd d=0.25mm re t the center of the optics mount s opening. Remember tht is the width of the slits, nd d is the seprtion between the slits. Then insert the lser into the optics bench so tht it is behind the optics mount nd fcing the optics mount. 6. Double click the Cpstone icon to open up the Cpstone Softwre. 7. Mke sure the PASCO 850 Interfce is turned on, nd plugged in. 8. In the Tool Br, on the left side of the screen, click on the Hrdwre Setup icon to open the Hrdwre Setup window. In the Hrdwre Setup window there should be n imge of the PASCO 850 Interfce. If there is skip to step 9. If there isn t click on Choose Interfce to open the Choose interfce window. Now select PASPORT, the Automticlly Detect, then click OK. 9. On the imge of the PASCO 850 Interfce click on the Digitl Inputs Ch(1) to open the sensor list, then scroll down, nd select the Rotry Motion Sensor. The rotry motion sensor icon should now be showing connected to Ch(1), nd Ch(2). Plug the rotry motion sensor into the Digitl Inputs, yellow to Ch(1), nd blck to Ch(2). 10. On the imge of the PASCO 850 Interfce click on the Anlog Inputs Ch(A) to open the sensor list, then scroll down, nd select the Light Sensor. Plug the cord into the bck of the light sensor, nd the other end into Anlog Inputs Ch(A). 7
8 At the bottom middle of the of the screen chnge for smple rte, chnge it to Common Rte, nd set it to 100 Hz. 11. In the Tool Br click on Dt Summry to open the Dt Summry window. In the equipment list click on the rotry motion sensor s properties icon to open its properties window. In the properties window where it reds Liner Accessory select Rck & Pinion. 12. Close Tool br. 13. Plug the lser s power cord into power outlet, nd the other end into the bck of the lser. BECAREFUL NOT TO LOOK INTO THE LASER BEAM! LOOKING INTO THE LASER BEAM COULD CAUSE PERMENATE LOSE OF SIGHT. Use the two knobs on the bck of the lser to im the lser bem such tht it psses through the double slit configurtion t the center of the optics mount opening, nd tht it hits the #2 slit on the perture brcket screen. One knob moves the lser up nd down, the other moves the lser right nd left. The interference pttern tht forms on the perture brcket screen needs to be horizontl. If it is not, slowly rotte the multiple slit set on the optics mount till the interference pttern is horizontl. 14. In the Disply Br, on the right side of the screen double click the Grph icon to open up grph. For the yxis click on Select Mesurement, nd then select Light Intensity (%). For the xxis the computer will hve utomticlly selected time (s), we need to chnge it. Click on the word time (s), the vilble dt list will pper, nd select Position (m). Procedure: Single nd Double Slits 1. Record the vlues for, d, L, nd λ in the tble. 2. Move the rotry motion sensor completely to one side of the liner trnsltor. 3. Ner the bottom left of the screen click Record to strt recording dt. 4. Slowly move the rotry motion sensor to the other side of the side of the liner trnsltor. Then click Stop to stop recording dt. (Be creful tht the cords don t get cught up or sngged on something.) 5. Click on the Show Coordintes Tool icon, long the top of the grph, to crete coordintes tool pper in the grph. Use the coordinte tool to mesure the position vlues of ll the peks of the interference pttern, nd then record those vlues in the tble for the two slits. Then use the coordinte tool to mesure the position of the drk spot right to the left of the furthest left pek, nd record the loction in the tble for two slits. Then use the coordinte tool to mesure the position of the drk spot right to the right of the furthest right pek, nd record the loction in the tble for two slits. 6. Move the rotry motion sensor completely bck to the originl side. 7. Remove the multiple slit set from the optic mount, nd replce it with the single slit set. Set the slit =0.4mm in the center of the optics mount. Mke sure the lser psses through the single slit, nd strikes the perture brcket screen. 8
9 Rotte the multiple slit set in the optic mount till the interference pttern formed on the perture brcket screen is horizontl. 8. Ner the bottom left of the screen click Record to strt recording dt. 9. Slowly move the rotry motion sensor to the other side of the side of the liner trnsltor. Then click Stop to stop recording dt. (Be creful tht the cords don t get cught up or sngged on something.) Use the coordinte tool to mesure the loctions of its edges on either side of the lrge centrl bright spot. Record these vlues in the tble for the Single Slit. 9
10 Anlysis Tbles for Two Slits (5 points) d L λ Left Drk Spot Right Drk Spot Δy Pek (m) y (m) 1. According to the theory the distnces (Δy) between consecutive peks for two slit interference pttern should ll be the sme. Clculte the distnces between consecutive peks, record them in the chrt, nd then clculte the stndrd devition for the distnces between peks. (20 points) 10
11 2. Identify the centrl pek, nd then clculte the distnce to the third pek to its right, y 3. Using this distnce, n = 3, nd the recorded vlue for d clculte the experimentl vlue for the wvelength of the lser bem. (10 points) 3. Clculte the % error between the ccepted vlue of the lser bem, nd your experimentl vlue. (5 points) 4. Clculte the number of peks tht should be encsed in the Centrl Envelope. Is tht the number of peks you counted? (5 Points) 11
12 Tble for Single Slit (5Points) Position of left edge Centrl Mxim Position of right edge Centrl Mxim Width of Centrl Mxim 1. Using the eqution for the loctions of the drk spots of the diffrction pttern clculte the width of the centrl mxim. Then find the % error of the width of the centrl mxim using the clculted vlue s the theoreticl. (10 points) 2. Clculte the % difference between the mesured width of the centrl bright spot for the single slit pttern, nd the width of the Centrl Envelope of the double slit pttern. (5 points) 3. For both setups we re using the sme lser, the sme distnce between the slit(s) nd the viewing screen (L), nd the slits hd the sme width, then why is the intensity for the double slit so much greter thn the intensity for the single slit? (10 points) 12
13 4. If we repeted this experiment with lser of 550 nm wht ffect would tht hve on the interference, nd diffrction ptterns? (5 points) 13
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