Optics Wve optics Diffrction LD Physics Leflets Diffrction t ouble slit n t multiple slits Objects of the experiments g Investigting iffrction t ouble slit for vrious slit spcings. g Investigting iffrction t ouble slit for vrious slit withs. g Investigting iffrction t multiple slits for vrious slit numbers. Principles The nture of light ws controversil issue for long time. In 1690, Christin Huygens interprete light s wve phenomenon; in 1704, Isc Newton escribe the light bem s current of prticles. This contriction ws resolve by quntum mechnics, n the ie of wveprticle ulity cme up. Diffrction t ouble slit: A prticulrly cler iniction of the wve chrcter of light ws provie by the experiment on iffrction t ouble slit Fig. Fig. 1 Schemtische Drstellung zur Beugung es Lichtes m Doppelsplt Schemtic illustrtion of iffrction t ouble slit b: b: Spltbreite, slit with, : : slit Spltbstn spcing L: L: istnce Abstn es between Schirmes the screen vom Doppelsplt n the ouble slit x 2 : istnce Abstn es of the zweiten secon Mximums mximum vom from Zentrum the centre α 2 : irection Beobchtungsrichtung of observtion für for s the zweite secon Mximum mximum s s 2 : pth Gngunterschie ifference of er the Mittelpunktstrhlen principl rys S: S: Beobchtungsschirm screen by T. Young. Toy this experiment cn esily be reprouce with the intensive n coherent light of n He-Ne lser: Due to iffrction of the prllel incoming light t two closely spce slits of equl perture, the light propgtes lso in the geometric show of the slit iphrgms (grey re in Fig. 1). Moreover, system of bright n rk fringes is observe on screen behin the ouble slit. This cnnot be expline with the lws of geometricl optics. An explntion is possible if wve properties re ssigne to the light n if the slits re consiere to be two coherent light sources whose light bunles superimpose. The superposition les to estructive n constructive interference in certin irections. In simple pproch, the light bunles coming from the slits re first subivie into (infinitely) mny prtil bunles. Then it cn be me cler with the i of Fig. 1 tht mximum intensity occurs in irections in which there is exctly one prtil bunle from the secon slit which correspons to ny prtil bunle from the first slit so tht both interfere constructively. For light bunles b 0713-Bi b S L 1
LD Physics Leflets Apprtus 1 iphrgm with 3 ouble slits 469 84 1 iphrgm with 4 ouble slits 469 85 1 iphrgm with 5 multiple slits 469 86 1 He-Ne lser, linerly polrise 471 830 1 holer with spring clips 460 22 1 lens in frme, f = +5 460 01 1 lens in frme, f = +50 460 02 1 precision opticl bench, 1 m 460 32 4 optics riers, H = 60 /B = 36 460 370 1 trnslucent screen 441 53 1 sle bse 300 11 tht emerge uner the ngle α n this is true ech time when the pth ifference s n between the principl rys (rwn in the figure) is n integer multiple of the wvelength λ of the light: sn = n λ n = 0, ± 1, ± 2, (I) For smll iffrction ngles the following reltion hols pproximtely: sn x α n n (II). L Hence, the intensity mxim re locte on the screen t the positions (mesure from the centre of the iffrction pttern) xn λ = n L n = 0, ± 1, ± 2, (III); i.e. they re spce t the istnce 1 = xn+1 xn = L λ (IV). Exctly in the mile between two intensity mxim there is n intensity minimum. Therefore the istnce between minimum n the next one is lso given by Eq. (IV). It shoul be mentione explicitly tht the present consiertions re bse on Fruhofer s point of view, which mens tht prllel wve fronts of the light before n fter the iffrction object re investigte. On the one hn this correspons to light source tht is t n infinite istnce Sfety notes The He-Ne lser meets the Sfety Requirements for Teching n Trining Equipment - Lser, DIN 58126, Prt 6 for lsers clss 2. If the corresponing notes of the instruction sheet re observe, experimenting with the He-Ne lser is sfe. g Never look into the irect or reflecte lser bem. g No observer must feel zzle. from the iffrction object, n on the other hn it correspons to screen tht is t n infinite istnce from the iffrction object. In the cse of Fresnel s point of view, the light source n the screen re t finite istnce from the iffrction object. However, the iffrction ptterns re more esily clculte for Frunhofer iffrction. The intensity woul be the sme for ll mxim, i.e. the bright fringes woul exhibit the sme brightness if the iffrction of light t the iniviul slits occurre with the sme intensity in ll irections. However, iffrction t single slit epens on the ngle of observtion α. Therefore the iffrction pttern observe behin the ouble slit is moulte by iffrction t single slit. For n exct clcultion of the iffrction pttern, the oscilltion sttes of ll prtil bunles tht come from the slits re e up with their phse ifferences being tken into ccount. As result the mplitue A of the fiel strength of the iffrcte light is obtine t n rbitrry position x on the screen. From the mplitue istribution A(x) clculte this wy, the intensity istribution I(x) = A 2 (x) is erive ieitely. On the left of Fig. 2, the iffrction pttern of ouble slit is shown for vrious slit spcings with the sme slit with b. It is clerly seen tht the istnce between the mxim ecreses with incresing slit spcing. Their intensity is not constnt becuse it is influence by iffrction t single slits. Therefore it is sensible to etermine the istnce on the screen efine in Eq. (IV) from the istnce between the minim inste of the istnce between the On the right of the sme figure, the iffrction pttern of ouble slit is shown for vrious slit withs b with the sme slit spcing. The istnce between the mxim is now the sme in ll three cses; however, the intensities re ifferent becuse the influence of iffrction t single slits vries. Diffrction t multiple slits: The consiertion regring the etermintion of the mxim in iffrction t ouble slit cn be pplie ieitely to the iffrction t multiple slits with N eqully spce slits hving the sme perture. If Eq. (I) is fulfille, the light bunles of ll N slits interfere constructively. Eqs. (III) n (IV) lso hol for multiple slits. Mthemticlly the etermintion of the intensity minim is more involve: minimum between the n-th n the (n+1)-th mximum is foun if the pth ifference between the principl rys of neighbouring slits fulfils the conition λ s = n λ+ m m = 1,..., N - 1 (V). N For this pth ifference, the prtil bunles from the N slits interfere such tht the totl intensity is zero. This is illustrte in Fig. 3 with the i of so-clle pointer representtions, in which the phse ifferences between the prtil bunles coming from ifferent slits re tken into ccount. There re N - 1 minim between every pir of In between there re N - 2 so-clle seconry mxim, whose intensity is weker thn tht of the principl However, the ltter is only true s long s the influence of iffrction t single slits cn be neglecte. In Fig. 4, the epenence of the iffrction pttern on the number of slits N is shown. As the slits re eqully spce, the istnces between the principl mxim re equl for ll slit numbers. With incresing slit number N, the intensity of the seconry mxim becomes weker s compre with the principl 2
LD Physics Leflets = 1000 µ m Fig. 2 Depenence of the iffrction pttern of ouble slit on the slit spcing (left) n the slit with b (right). In ech cse, the iffrction pttern of single slit with the corresponing slit with is rwn in the igrm with smller line with. = 750 µ m = 500 µ m b =100 µ m b = 150 µ m Fig. 3 Pointer representtion of the ition of the oscilltion mplitues of N slits with the phse ifference being tken into ccount. If the pth ifference s between two neighbouring slits is n integer multiple of λ, the mximum iffrction mplitue is obtine. If the pth ifference s is given by Eq. (V), the iffrction mplitue is zero. b =200 µ m Fig. 4 Depenence of the iffrction pttern of multiple slits on the slit number N for equl slit spcing n equl slit with b. The iffrction pttern of single slit with equl slit with is rwn in the igrms with smller line with. N = 2 N = 2 N = 3 N = 3 N = 4 N = 4 n λ n λ + λ/2 N = 5 s ( n +1) λ N = 5 3
P < 1mW λ = 632,8 nm 471 830 He-Ne-Lser, liner polrisiert He-Ne-Lser, linerly polrize LD Physics Leflets 150 cm L 1 L 2 H S 55 H S Fig. 5 Experimentl setup (bove) n schemtic ry pth (below) for the observtion of iffrction t ouble slit n t multiple slits. L 1 : lens f = +5 L 2 : lens f = +50 H: holer for iffrction objects S: screen Setup Remrk: justments shoul be me in slightly rkene room. The totl experimentl setup is illustrte in Fig. 5. First the sphericl lens L 1 with the focl length f = +5 expns the lser bem. The following converging lens L 2 with the focl length f = +50 is positione so tht its focus is locte somewht below the focus of the sphericl lens. In this wy the lser bem is slightly expne n runs pproximtely prllel long the opticl xis. Using n optics rier, ttch the He-Ne lser to the opticl bench s shown in Fig. 5. Set up the screen t istnce of pprox. 1.90 m from the lser. Direct the lser towrs the screen, n switch it on. Put the holer for iffrction objects H with the iphrgm with 4 ouble slits (469 85) on the opticl bench t istnce of pprox. 50 cm from the lser. Ajust the height of the lser so tht the lser bem impinges on the centre of the iphrgm. Plce the sphericl lens L 1 with the focl length f = +5 t istnce of pprox. 1 cm from the lser (the lser shoul illuminte the iphrgm evenly.) Remove the holer for iffrction objects H. Plce the converging lens L 2 with the focl length f = +50 t istnce of pprox. 55 behin the sphericl lens L 1 n slie it long the opticl bench towrs the sphericl lens L 1 until the imge of the lser bem on the screen is shrp. Avnce the converging lens L 2 on the opticl bench somewht further towrs the sphericl lens L 1 until the imeter of the lser bem on the screen hs expne to pprox. 6 (the lser bem shoul now hve constnt circulr cross section long the opticl xis). In orer to check whether the bem imeter is constnt between the lens n the screen, hol sheet of pper in the ry pth n observe the cross section of the bem long the opticl xis. Put the holer for iffrction objects bck into the ry pth n shift it so tht the istnce between the screen n the iffrction object is 1.50 m. If necessry, slightly shift the lens L 2 until the iffrction pttern is shrp. 4
LD Physics Leflets Crrying out the experiment ) Depenence of iffrction t ouble slit on the slit spcing : Insert the iphrgm with 4 ouble slits (469 85) in the ry pth, n observe the iffrction ptterns of the ouble slits with the slit spcings = 1.00, 0.75, 0.50 n 0.25 one fter nother. For ech slit spcing rw conclusions on the influence of iffrction t single slit from the intensities of the Hol sheet of pper on the screen in ech cse, n mrk the loctions of the intensity minim (rk fringes!) within the centrl mximum of the single slit function with soft pencil. Determine the (verge) istnce between the intensity minim in ech cse. b) Depenence of iffrction t ouble slit on the slit with b: Insert the iphrgm with 3 ouble slits (469 84) in the ry pth, n observe the iffrction ptterns of the ouble slits with the slit withs b = 0.20, 0.15 n 0.10 one fter nother. For ech slit with rw conclusions on the influence of iffrction t single slit from the intensities of the Hol sheet of pper on the screen in ech cse, n mrk the loctions of the intensity minim (rk fringes!) within the centrl mximum of the single slit function with soft pencil. Determine the (verge) istnce between the intensity minim in ech cse. c) Depenence of iffrction t multiple slits on the slit number N: Insert the iphrgm with 5 multiple slits (469 86) in the ry pth, n observe the iffrction ptterns of 2, 3, 4, 5 n 40 slits one fter nother. Ientify the influence of iffrction t single slit, the principl mxim n, for N = 3, 4 n 5, the seconry Hol sheet of pper on the screen in ech cse, n mrk the loctions of the principl mxim (bright fringes!) with soft pencil. Determine the (verge) istnce between the intensity mxim in ech cse. Mesuring exmple ) Depenence of iffrction t ouble slit on the slit spcing : Tble 1: istnces between the intensity minim for ifferent slit spcings 0.25 3.87 0.50 1.84 0.75 1.27 1.00 0.94 b) Depenence of iffrction t ouble slit on the slit with b: The istnces between the mxim re equl for ll slit withs. With ecresing slit with, the intensity is incresingly istribute to the mxim ner the centre. Tble 2: istnces between the intensity minim for ifferent slit withs b b 0.10 3.75 0.15 3.36 0.20 3.67 c) Depenence of iffrction t multiple slits on the slit number N: The istnces between the principl mxim re equl for ll slit numbers. The principl mxim themselves become nrrower with incresing slit number N. For N = 3 to 5, there re N - 2 seconry mxim between two neighbouring principl The intensity of the seconry mxim becomes weker with incresing N. Tble 3: istnces between the principl mxim for ifferent slit numbers N N 2 3.81 3 3.78 4 3.81 5 3.73 40 3.79 5
LD Physics Leflets Evlution ) Depenence of iffrction t ouble slit on the slit spcing : Tble 4: Distnces between the intensity minim n reciprocl slit istnces -1 (see Tble 1) 1-1 4.00 3.87 2.00 1.84 Results The iffrction pttern of iffrction t ouble slit or t multiple slits, respectively, is etermine by the slit spcing, the slit number N n the slit with b. The istnce between the principl mxim is inversely proportionl to the slit spcing n inepenent of N n b. With incresing slit number N, the with of the principl mxim ecreses becuse the number of minim (n of seconry mxim) increses. The slit with b etermines the influence of iffrction t single slit on the iffrction pttern. 1.33 1.27 1.00 0.94 In Fig. 6 the mesure vlues from Tble 1 re shown s igrm. The istnce is plotte ginst the reciprocl of the slit spcing. Within the ccurcy of mesurement, the mesure vlues lie on the stright line through the origin rwn in the igrm. The slope of the stright line ws clculte with the i of Eq. (IV) from L = 150 cm n λ = 633 nm. b) Depenence of iffrction t ouble slit on the slit with b: Men vlue of the istnces from Tble 2: = 3.59. As λ = 633 nm n L = 150 cm re known, the slit spcing cn be clculte with the i of the trnsforme Eq. (IV): = 0.26 µm c) Depenence of iffrction t multiple slits on the slit number N: Men vlue of the istnces from Tble 3: = 3.78. As λ = 633 nm n L = 150 cm re known, the slit spcing cn be clculte with the i of the trnsforme Eq. (IV): = 0.25 µm Fig. 6 Distnce between the intensity minim for iffrction t ouble slit s function the reciprocl of the slit spcing. LD DIDACTIC GmbH Leybolstrsse 1 D-50354 Hürth Phone: (022 33) 604-0 Fx: (02233) 604-222 e-mil: info@l-ictic.e by LD DIDACTIC GmbH Printe in the Feerl Republic of Germny Technicl ltertions reserve