Department of Physics The University of Hong Kong

Transcription

1 Mifie by Dt Ng Versin.0 Nvember, 05 Diffrctin, Interference n Diffrctin Grting Deprtment f Physics The University f Hng Kng Physics Lbrtry PHYS3850 Wve n Optics Experiment N.3850-: Diffrctin, Interference n Diffrctin Grting Nme: University N. Lser Sfety The lsers use in this experiment n re f lw pwer (i.e. run 5mW) but the nrrw bem f light is still f high intensity. Cnsequently,. Wer Sfety Gggle. Never lk irectly int the unexpne lser bem r t its reflectin frm mirrr surfce. 3. D Nt use the lser withut getting instructins frm the emnstrtr. WARNING! The bem f lser pinters is s cncentrte tht it cn cuse rel mge t yur retin if yu lk int the bem either irectly r by reflectin frm shiny bject. D NOT shine them t thers r yurself. A. AIMS. T fin the experimentl ptterns f interference n iffrctin fr uble-slit n multi-slit iphrgms n cmpre the thereticl vlues.. Differences n similrities between interference n iffrctin ptterns re exmine. B. INTRODUCTION Electrmgnetic ritin prpgtes s wve, n s such cn exhibit interference n iffrctin. This is mst strikingly seen with lser light, where light shining n piece f pper lks speckle (with light n rk spts) rther thn evenly illuminte, n where light shining thrugh smll hle mkes pttern f bright n rk spts rther thn the single spt yu might expect frm yur everyy experiences with light. When interference f light ccurs s it psses thrugh tw slits, the ngle frm the centrl mximum (bright spt) t the sie mxim in the interference pttern is given by sin n, n 0,,,3... where is the slit seprtin, is the ngle frm the center f the pttern t the n th mximum, is the wvelength f the light, n the rer (0 fr the centrl mximum, fr the first sie mximum, fr the secn sie mximum...cunting frm the center ut). In this lb we will use lser light t investigte the phenmen f interference n iffrctin. C. PRE-LAB READING MATERIAL: Thery - Bckgrun Infrmtin. Superpsitin Principle f Wve: If tw r mre trveling wves re mving thrugh meium, the resultnt wve functin t ny pint is the lgebric sum f the wve functins f the iniviul wves. Pge f 8

2 Mifie by Dt Ng Versin.0 Nvember, 05 Diffrctin, Interference n Diffrctin Grting. Cnitins fr Observble Interference:.. The surces must be cherent tht is, they must mintin cnstnt phse with respect t ech ther... The surces shul be mnchrmtic tht is, f single wvelength. 3. Diffrctin f Light Diffrctin is the bening f wves when they pss thrugh pertures r run bstcles. By Huygens s principle, there re centers f isturbnce n the wvefrnt in plne f the bstcle. This center sens ut secnry circulr wvelets which cmbine t frm iffrcte wve n prpgtes frwrs. Behin the bstcles r pertures t which iffrctin ccurs, iffrctin pttern f rk n bright fringes is frme n the screen. 4. Single Slit Diffrctin (Frunhfer iffrctin) The pttern is nt equl spcing n the bright fringes re nt equl with. When white light surce is use n clr filters re plce in frnt f the slit in turns, the seprtin between the clr bns vries. The clr bns crrespn t the blue filter re clser tgether thn tht f the re ne. It mens tht the iffrctin pttern epens n the wvelength f the light. The iffrctin pttern becmes wier if the slit is me nrrwer r light f lnger wvelength is use. The nrrwer the slit, the Fig.. Diffrctin Pttern f Single Slit immer the pttern n the wier the centrl bn. 4.. Mthemticl Expressin f Single Slit Diffrctin: When iffrctin f light ccurs s it psses thrugh slit, the ngle t the minim L (rk fringes) in the iffrctin pttern is given by sin m m m,,3... where "" is the slit with, m is the ngle frm the centrl mximum f the pttern t the minimum, is the wvelength f the light, n m is the rer ( fr the first minimum, fr the secn minimum,...cunting frm the center ut). It shul be highly mentine 0th mx 0 Intensity, I st min st min st min Fig.. Intensity Distributin f Single Slit Diffrctin Pge f 8

3 Mifie by Dt Ng Versin.0 Nvember, 05 Diffrctin, Interference n Diffrctin Grting tht m 0 crrespns t the centrl mximum intensity. m 0 is nt inclue since 4.. Energy Spre f Single Slit Diffrctin The ngle between the center f iffrctin pttern n the psitrn f the first minimum is given by sin cn inicte the ngulr spre =. This ngle, which cntins mst f the energy f the wve. sin, if the with f the slit, is lrge cmpre with the wvelength, the ngulr spre (r clle ngulr with) f the center mximum ( ) becmes very smll. The irectin f the first minim is extremely ner t the center mxim which imply minimum iffrctin effect Intensity f Single-Slit Diffrctin Pttern The intensity single-slit iffrctin pttern is where is the with f the slit I Reltive Intensity -slit iffrctin I sin sin sin Fig. 3. Intensity Distributin f Single Slit Diffrctin 5. Tw-slit Interference 5.. Assumptins: D P Use lser light r strng light Mnchrmtic light surce surce r S Use mnchrmtic light (i.e. single wvelength) The slits shul be s nrrw s S O r pssible. Screen The slit seprtin must be very (nrrw) S smll (i.e. ~0.5mm) Single slit Pth iff. Duble slit sin Screen-slit seprtin must be t lest -m. Fig. 4. Setup f Yung Duble-slit Experiment Assume D r D 000 y Pge 3 f 8

4 Mifie by Dt Ng Versin.0 Nvember, 05 Diffrctin, Interference n Diffrctin Grting 5.. Mthemticl Expressin f Duble-slit Interference Light frm the lser hits tw very nrrw slits, which then ct like in-phse pint surces f light. In trveling frm the slits t the screen, hwever, the light frm the tw slits trvel ifferent istnces. In the picture t left light hitting pint P frm the bttm slit trvels further thn the light frm the tp slit. i.e.. The extr pth ifference (i.e. r r r r ) intruces phse shift between the tw wves n les t psitin epenent interference pttern n the screen. Here the extr pth length is sin, leing t phse shift given by Relizing tht phse shifts tht re multiples f π give us cnstructive interference while multiples f π le t estructive interference les t the fllwing cnitins: Pth ifference fr in phse surce, sin Cnstructive interference D Distnce between the successive y Where is slits seprtin, interference is slits-t-bserver istnce, Fr light wve Bright fringe Drk fringe 5.3. The Fringe Seprtin f Duble-slit Interference Pttern: D If ll ssumptins re fulfille, fringe seprtin, y is beye where wvelength use n D Screen-slit seprtin slit-seprtin Derivtin f fringe seprtin is shwn in the ppenix Meths t Increse the Fringe Seprtin, Destructive interference n 0,,,3... n Resultnt mplitue Mx ( Eresultnt E ech ) Min (i.e. Eresultnt 0 ) Resultnt intensity Iresultnt 4I ech I E Iresultnt 0 D. D r D y Δy Δy Fig. 5. Meths t Increse the Fringe Seprtin If is ecresing, (r incresing r incresing D r ecresing f ), the fllwing chnges ccur: i) D The fringe seprtin y increses; ii) The fringes becme mre lsely pcke; iii) Fewer fringes re bserve; n iv) The brightness f the fringes ecreses. Pge 4 f 8

5 Mifie by Dt Ng Versin.0 Nvember, 05 Diffrctin, Interference n Diffrctin Grting 5.4. Intensity f Duble-Slit (Pure) Interference Ptterns: Mthemticl Expressin Accring t the erivtin frm ppenix, the intensity f uble-slit (pure) interference ptterns is I I cs sin I cs I cs y L Fig. 6. Intensity s functin f sinθ/λ 5.5. Grphicl Expressins f Duble Slit Interference with Diffrctin Effect: Single slit iffrctin pttern Intensity Screen Duble slit interference pttern 0 Intensity Psitin r Withut iffrctin pttern Screen Duble slit interference pttern with single iffrctin effect Interference fine structure 0 Intensity Psitin r Diffrctin envelpe Determine by slit with Determine by slit seprtin Screen Fig. 7. Duble-slit Interference Pttern with Single Diffrctin Effect Bright fringes ue t -slit interference: mx, -slit n 0 Psitin r Minimum (Zer) ue t single-slit iffrctin: min,-slit,,... All uble-slit interference pttern shul be inclue the single iffrctin effect unless specifictin f pure interference effect. Pge 5 f 8

6 Mifie by Dt Ng Versin.0 Nvember, 05 Diffrctin, Interference n Diffrctin Grting 5.6. Intensity f Duble-Slit Interference with Diffrctin Effect: sin sin I-slit iffrctin I Single-slit iffrctin sin I-slit Pure Interference Ics Ics sin Duble-slit interference Suppse we nw hve tw slits, ech hving with, n seprte by istnce. The resulting interference pttern fr the uble-slit will ls inclue iffrctin pttern ue t the iniviul slit. The intensity f the ttl pttern is simply the pruct f the tw functins: I sin sin sin sin -slit interference I cs Interference fctr Diffrctin fctr Missing Orers f Duble-Slit Interference with Diffrctin Effect: We hve seen tht the interference mxim ccur when sin n. On the ther hn, the cnitin fr the first iffrctin minimum sin. Thus, prticulr interference mximum with rer number n my cincie with the first iffrctin minimum. The vlue f n my be btine s: sin n n sin 0.3 mm Fr exmple, 3. Thus, the thir interference mximum (if we 0. mm cunt the centrl mximum s n = 0) is ligne the first iffrctin minimum n cnnt be seen. Since the n th fringe is nt seen, the number f fringes n ech sie f the centrl fringe is n. e.g. fringes in the exmple. Thus, the ttl number f N n n. e.g. 5 fringes in the centrl iffrctin mximum is fringes in the exmple. Determine by slit with ttl Intensity n Clss Pek Diffrctin envelpe n Clss minim Determine by slit seprtin Psitin r 0 Fig. 8. Intensity Distributin f Duble-slit Interference Pttern with Single Diffrctin Effect 6. Multiple-slit Interference If inste f tw ienticl slits seprte by istnce there re multiple ienticl slits, ech seprte by istnce, the sme effect hppens. Fr exmple, t ll ngles stisfying. We fin cnstructive interference, nw frm ll f the hles. sin n Pge 6 f 8

7 Mifie by Dt Ng Versin.0 Nvember, 05 Diffrctin, Interference n Diffrctin Grting The ifference in the resulting interference pttern lies in thse regins tht re neither mxim r minim but rther in between. Here, becuse mre incming wves re vilble t interfere, the interference becmes mre estructive, mking the minim pper brer n the mxim shrper. 6.. Anlysis f Phsr Digrm fr 3 Slits Interference: Cnsier ll rys cming frm sme ngle, fr ll mximum n minimum: Pth ifference between rys frm jcent slits: L L sin Pth iff. Phse ifference between rys frm jcent slits: = sin L Fig slit Interference Cse : Centrl Mximum Phse ifference between rys frm jcent slits, E resultnt Pth ifference between rys frm jcent slits, Amplitue f resultnt ry t centrl mximum, E E E E E resultnt 3 0 sin 0 Intensity f resultnt ry t centrl mximum, Iresultnt 9I Cse : First Minimum Cnsier ll rys cming frm sme ngle, fr st minimum Equl ngle fr 3 phsrs Equilterl Tringle Phse ifference between rys frm jcent slits, 3 E Pth ifference between rys frm jcent slits, sin 3 3 Amplitue f resultnt ry t centrl mximum, Eresultnt 0 E 0 E E E E Intensity f resultnt ry t centrl mximum, Iresultnt 0 Cse 3: Next Mximum Phse ifference between rys frm jcent slits, Pth ifference between rys frm jcent slits, sin 80 Amplitue f resultnt ry t centrl mximum, Eresultnt E Intensity f resultnt ry t centrl mximum, Iresultnt I Cse 4: Next Minimum Phse ifference between rys frm jcent slits, 3 E Pth ifference between rys frm jcent slits, E sin 3 E resultnt E 40 Pge 7 f 8

8 Mifie by Dt Ng Versin.0 Nvember, 05 Diffrctin, Interference n Diffrctin Grting 3 Amplitue f resultnt ry t centrl mximum, Eresultnt 0 Intensity f resultnt ry t centrl mximum, Cse 5: Next Mximum Phse ifference between rys frm jcent slits, Pth ifference between rys frm jcent slits, sin E resultnt E E E 360 Summry: Amplitue f resultnt ry t centrl mximum, E E resultnt 3 Iresultnt 0 Intensity f resultnt ry t centrl mximum, Iresultnt mx n where n 0,,, I Intensity 9I L Centrl Mx st min st mx n min n mx I 3 3 L Diffrctin Grting: An pticl trnsmissin iffrctin grting cnsists f lrge number f fine, clse equiistnt prllel lines rule nt trnsprent plte (such s glss). Fine grting: slits per mm Crse grting: slits per mm Cnstructive interference pruces shrp lines f mximum intensity t set ngles either sie f shrp, centrl mximum. In between, estructive interference gives zer r ner-zer intensity. T ientify the lines, they re ech given n rer number (0,,, etc) Mthemticl expressins f pth ifference f iffrctin Grting: Fr the bright fringe, sin m 6.4. Diffrctin grting is multi-slit plte, the mxim ccur exctly t the sme psitin s uble slits interference. The bright fringes re much nrrwer ( shrpness ) n much brighter ( shrpness ). Pge 8 f 8

9 Mifie by Dt Ng Versin.0 Nvember, 05 Diffrctin, Interference n Diffrctin Grting mth rer mδy Fig.. Pth Difference f Diffrctin Grting Pth iff. sin Generlly, the mxim ccur t the psitin, where sin m m m = the number f rer f mximum n m 0,,, 3... where N The mximum rer is given by = grting seprtin n N per mm = grting ensity m mx sin m Integrl prt f m mx 6.5. Mthemticl Expressin f Intensity f Multiple-slit Diffrctin Grting Fig.. Pth Difference f Diffrctin Grting The iffrctin pttern fter gri cn be btine thrugh superpsitin f the iffrctin ptterns f the single slits f the gri. A gri cnsists f regulr rry f N equiistnt slits, s tht N prllel interfering light bems re btine. The pth ifference f the brer bems f slit f with is sin This gives phse ifference f sin The pth ifference f bems cming frm tw slits is sin where is the istnce between slits (gri cnstnt). This yiels phse ifference f sin When N bems re evite t bservtin pint D uner iffrctin ngle, the fllwing intensity is fun with the mplitue E f iffrcte bem: sin sin N I I I E sin Nw E is the intensity f the bem where iffrcte by single slit in the irectin sin E The iffrctin intensity f the ttl gri is btine thrugh cmbintin f equtins bve: Pge 9 f 8

10 Mifie by Dt Ng Versin.0 Nvember, 05 Diffrctin, Interference n Diffrctin Grting I E sin N sin I sin sin sin sin sin sin N sin Single-slit Diffrctin Fctr Interctin Fctr ue t N slits The first prt f the pruct in the bve equtin is thus the intensity istributin f single slit, the secn prt is the result f the interctin f iffrctin ue t N slits. This shws tht the minim f the single slits ls re mintine in the cse f gri, becuse if the first fctr becmes zer, the pruct ls becmes zer. Accring t Frunhfer, the peks n minim f ne slit re clle st clss interference (i.e. pure single slit iffrctin), wheres peks n minim ue t the simultneus effect f severl slits re clle n clss interference Specil cse: N = (single slit) sin I I prvie tht sin Cnsiering I 0 t fin the extreme pints f It les t tw equtins: r cs sin 0,, 3,... r tn.4303,.4590, ,... (mximum) r,, 3,... (minimum) Specil cse: N = (uble slit) The fllwing intensity istributin is btine fr iffrctin t slit: sin sin sin sin I sin sin sin sin 0 Intensity istributin f single slit I Interctin f iffrctin ue t slits ccurring t vlues f. st clss minim (zers f the numertr f the first fctr nly) sin sin 0 sin k k k sin k k,,3,... The prticulr cse f the centrl st clss pek fr k = 0/ k 0 is nt seize by bve equtin. Fr k 0, ne btins the centrl st clss pek. Fr the secnry st clss peks ne btins pprximtely: * k * sin * k,,3,... k b. n clss minim (zers f the numertr f the secn fctr AND nt simultneusly zer f the enmintr): Pge 0 f 8

11 Mifie by Dt Ng Versin.0 Nvember, 05 Diffrctin, Interference n Diffrctin Grting q q sinq sin sin 0 n sin sin 0 q q sinq n sin q where q 0,,,3, sinq sinq sinq sinq sinq 0 sin q q sinq 0 The slutins fr n clss minim re sin q,,,,... k T sum up, it is sink k 0,,,3,... c. The cmmn zers f numertr AND enmintr f the secn fctr re the n clss peks (min peks): sin sin 0 n sin sin 0 q q sinq n sin q where q 0,,,3,... The slutins fr n 3 4 clss pek re sinq 0,,,,,... k sin k k 0,,,3,... Determine by slit with sin q st Clss Pek Intensity n Clss Pek Diffrctin envelpe 0 sin q n Clss minim st Clss minim Determine by slit seprtin 3 sin q Psitin r 4 sin q q q 3 4 sinq sinq 0 sinq sinq sinq sinq q sin q sinq 0 sin q sin q 3 sin q 4 sin q Multiple slit, gri: The min peks ( n clss peks) becme mre prnunce with incresing number f slits N, becuse the intensities f these peks re prprtinl t N. Pge f 8

12 Psitin Mifie by Dt Ng Versin.0 Nvember, 05 Diffrctin, Interference n Diffrctin Grting The envelpe f the iffrctin pttern is ienticl t the iffrctin pttern f single slit f with. Between the ccurring min peks, there lwys re (N-) secnry peks 6.6. Intensity vs psitin in multiple-slit interference Assuming tht the slit spcing,, remins unchnge. As the number f slits increses; the psitins f the mjr peks remin unchnge but the number f subpeks increses (With lwer intensities). N. f slits Intensity vs psitin N = N = 3 N = 4 N = 5 N = Diffrctin grting Fig. 4. Intensity vs. Psitin f Multiple-slit Interference As result, iffrctin grting = n interference f cherent surces frm clsely pcke multiple slits Fr yur infrmtin, since the slit with,, is nt much lnger thn the D wvelength, the frmul y is NOT pplicble fr iffrctin grting Cmpre between Duble-slit Interference n iffrctin grting by using mnchrmtic visible light surce Duble-slit interference Diffrctin grting Nture Interference Diffrctin Grph f intensitypsitin (Pure interference) Intensity Intensity Mthemticl expressin D s sin m Apprximtin? Yes. As 0, sin tn N Fringe seprtin Evenly istribute n evenly Wier s m increses spce (Ner the centrl m mximum) cs Slit seprtin, Lrge Smll Slit with, Lrge Smll Number f fringe Mny. Abut 000 A few, less thn 0. With f bright fringe Wie Nrrw Intensity sin sin I-slit I cs sin I-slit I sin Psitin Pge f 8

13 Mifie by Dt Ng Versin.0 Nvember, 05 Diffrctin, Interference n Diffrctin Grting Applictins f iffrctin:. X-ry iffrctin n crystllgrphy Fig. 6. Brgg s lw: sin m X rys re light wves tht hve very shrt wvelengths. When they irrite sli, crystl mteril they re iffrcte by the tms in the crystl. But since it is chrcteristic f crystls t be me up f eqully spce tms, it is pssible t use the iffrctin ptterns tht re pruce t etermine the lctins n istnces between tms. Simple crystls me up f eqully spce plnes f tms iffrct x rys ccring t Brgg's Lw. This is the mst cmmnly use f the structurl bilgy isciplines. Surprisingly, when prteins, DNA, RNA, r cmplexes f these mcrmlecules re frce t precipitte ut f slutin, smetimes the iniviul mcrmlecules rientte int n rere repeting lttice, r crystl. The technique f X-ry crystllgrphy relies n this bility t grw crystls f the prtein (r mcrmlecule) f interest. This is becuse the rere lttice f crystl llws scttering f X-rys (iffrctin), which the crystllgrpher mesures n then mnipultes by cmputer t iscern the three-imensinl tmic-level structure f the prtein r mcrmlecule. b. Hlgrphy When tw lser bems mix t n ngle n the surfce f phtgrphic plte r ther recring mteril, they pruce n interference pttern f lternting rk n bright lines. Becuse the lines re perfectly prllel, eqully spce, n f equl with, this prcess is use t mnufcture hlgrphic iffrctin grtings f high qulity. In fct, ny hlgrm (hls whle: grm messge) cn be thught f s cmplicte iffrctin grting. The recring f hlgrm invlves the mixing f lser bem n the unfcuse iffrctin pttern f sme bject. In rer t recnstruct n imge f the bject (hlgrphy is ls knwn s wvefrnt recnstructin) n illuminting bem is iffrcte by plne surfces within the hlgrm, fllwing Brgg's Lw, such tht n bserver cn view the imge with ll f its three-imensinl etil. Illumintin bem Fig. 7. Recring hlgrm Fig. 8. Recnstructing hlgrm Pge 3 f 8

14 Mifie by Dt Ng Versin.0 Nvember, 05 Diffrctin, Interference n Diffrctin Grting D. SETUP. Experimentl Apprtus HeNe Lser ( 633nm) 4 sets f Duble Slits With f slits, / Distnce f slits, : 0.mm/0.5mm 0.mm/0.5mm 0.mm/0.5mm 0.mm/.0mm Diffrctin Grting, 4lines/mm N =, = 0.5 mm, = 0.mm N = 3, = 0.5 mm, = 0.mm N = 4, = 0.5 mm, = 0.mm Ajusting supprt Mirrr Phtcell Sliing evice Mesurement mplifier Vltmeter Cnnecting cbles. Set-up n Prceure. The experimentl set up is shwn in belw. H.V. Supply Mirrr, M Shutter cntrl HeNe Lser Slit, S Phtcell, LD Amplifier Vltmeter Fig. 9. Experimentl set up fr iffrctin thrugh uble slit r by gri. The lser bem is irecte by mirrr M t the perture screen with multiple slits S. Pge 4 f 8

15 Mifie by Dt Ng Versin.0 Nvember, 05 Diffrctin, Interference n Diffrctin Grting 3. The lser n the mesurement mplifier shul be wrme up fr but 5 minutes befre wrk strts, s s t vi bthersme intensity fluctutins uring mesurements. 4. A iffrctin pttern is frme fter the multiple slits, the intensity istributin (Only extreme pints) f which is mesure fr ifferent numbers f slits, fr ifferent slit withs n fr ifferent gri cnstnts (exct inictins n the multiple perture screen). 5. Diffrctin ngle is btine frm the reltin: x sin x L L x where x = istnce t the centrl pek L = istnce between slit n bservtin plne. L x L L x x Fig. 0. Gemetry f the Setup n the Diffrctin Angle 6. A phtcell LD is use t mesure the intensity istributin. Fr this, the zer f the mesurement mplifier is juste with the lser switche ff t strt with. Furthermre, mplifictin shul be juste in such wy tht the mximum vltge es nt increse t mre thn the pek vltge f but 0 V in the Pge 5 f 8

16 Mifie by Dt Ng Versin.0 Nvember, 05 Diffrctin, Interference n Diffrctin Grting re f lrgest intensity (centrl pek). Amplifictin f 0 3 n Time cnstnt = 0. s is the recmmene setting t the beginning. 7. Intensity istributin in the bservtin plne cn be mesure by mens f the scle n the sliing evice ( unit ). i.e. experimentl vlues f x 500m 8. In the cse f uble slit systems, nly the psitins f extremes n the intensities f peks must be etermine n linke by stright lines, s the experimentl etermintin f the intermeiry intensities is t cmplicte. 9. The intensity n lctin f extreme pints hve t be mesure. 0. Generl steps t mesure the intensity n lctin f st clss n n clss extreme pints:. Mesure the istnce between slit t screen istnce. Set Amplifictin t 0 3 n Time cnstnt = 0. s f the mplifier 3. Vltmeter sent t 0V first, 0V full scle ivisin = 00 r Otherwise use 30V s scle 5. Fin 0 th mximum first: 6. Lcte its reltive psitin. 7. Fin + st minimum 8. Fin + n mximum 9. Fin +3 r minimum 0. Fin - st minimum. Fin - n mximum. Fin -3 r minimum 3. Fin 75% Intensity f 0 th mximum 4. Fin 50% Intensity f 0 th mximum 5. Fin 5% Intensity f 0 th mximum. The btine istributins re cmpre t thereticl vlues. Fr fining the thereticl vlues, the fllwing equtins shul be utilize: Pge 6 f 8

17 Mifie by Dt Ng Versin.0 Nvember, 05 Diffrctin, Interference n Diffrctin Grting sin sin I,within st Clss sin x sin x L L x Nttl st k clss peks: sin k k,,3,... Secnry st * k * clss peks: sin * k,,3,... k. Fr the thereticl intensity istributin, yu shul mke use f the fllwing prgrm frm wlfrm tin/ E. Dt Cllectin:. 4 sets f Duble Slits 633nm Nttl Duble Slits: Set Set Set 3 Set 4 With f slits, 0.mm 0.mm 0.mm 0.mm Distnce f slits, 0.5mm 0.5mm 0.5mm.0mm Fringe seprtin: D y Determine by slit with st Clss Pek Intensity n Clss Pek Diffrctin envelpe 0 n Clss minim st Clss minim Determine by slit seprtin Psitin r Pge 7 f 8

18 Mifie by Dt Ng Versin.0 Nvember, 05 Diffrctin, Interference n Diffrctin Grting Set (Exmple): With f slits, / Distnce f slits, : 0.mm/0.5mm: sin sin sin sin Accring t I sin sin sin st k clss peks: sin k k,,3,... Secnry st * k * clss peks: sin * k,,3,... k Fr centrl st clss peks: sin 0 (Principle mximum) k I 0 3 Fr sink : Reltive 3 Psitin: L intensity: sin x I 0.045I ( st rer L pek) 5 Fr sink : Reltive 5 Psitin: L intensity: sin x I 0.06I ( n rer L pek) 0.5 n.5 the minim f single n uble slits nt cincie. 0. N. f n clss peks within centrl bn: Nttl.5.5 Pge 8 f 8

19 Mifie by Dt Ng Versin.0 Nvember, 05 Diffrctin, Interference n Diffrctin Grting Set : With f slits, / Distnce f slits, : 0.mm/0.5mm sin sin sin sin Accring t I sin sin sin st k clss peks: sin k k,,3,... Secnry st * k * clss peks: sin * k,,3,... k Fr centrl st clss peks: sin 0 (Principle mximum) 3 Fr sink : Reltive intensity: k I 0 Psitin: 5 Fr sink : Reltive intensity: Psitin: Fr 7 sink Reltive intensity: Psitin: n N. f n clss peks within centrl bn: Nttl Pge 9 f 8

20 Mifie by Dt Ng Versin.0 Nvember, 05 Diffrctin, Interference n Diffrctin Grting Set 3: With f slits, / Distnce f slits, : 0.mm/0.5mm sin sin sin sin Accring t I sin sin sin st k clss peks: sin k k,,3,... Secnry st * k * clss peks: sin * k,,3,... k Fr centrl st clss peks: sin 0 (Principle mximum) 3 Fr sink : Reltive intensity: k I 0 Psitin: 5 Fr sink : Reltive intensity: Psitin: Fr 7 sink Reltive intensity: Psitin: n N. f n clss peks: Nttl Pge 0 f 8

21 Mifie by Dt Ng Versin.0 Nvember, 05 Diffrctin, Interference n Diffrctin Grting Set 4: With f slits, / Distnce f slits, : 0.mm/.0mm sin sin sin sin Accring t I sin sin sin st clss peks: k sin k k,,3,... Secnry st clss peks: * k * sin * k,,3,... k Fr centrl st clss peks: sin 0 3 Fr sink : Reltive intensity: k I 0 (Principle mximum) Psitin: 5 Fr sink : Reltive intensity: Psitin: Fr 7 sink Reltive intensity: Psitin: n N. f n clss peks: Nttl Pge f 8

22 Mifie by Dt Ng Versin.0 Nvember, 05 Diffrctin, Interference n Diffrctin Grting. Diffrctin Grting with ifferent number f slits, N N =, = 0.5 mm, = 0.mm N = 3, = 0.5 mm, = 0.mm N = 4, = 0.5 mm, = 0.mm N = 5, = 0.5 mm, = 0.mm I sin sin sin sin sin sin N sin Intensity istributin f single slit Interctin f iffrctin ue t N slits I N Diffrctin N = N = 3 N = 4 N = 5 Grting With f slits, 0.mm 0.mm 0.mm 0.mm Distnce f slits, 0.5mm 0.5mm 0.5mm 0.5mm Thereticl reltive centrl 5 pek intensity.56 (N=5 s 4 / benchmrk) Experimentl bslute centrl pek intensity Experimentl reltive centrl pek intensity / (N=5 s benchmrk) F. Discussin Questins. Stte the ssumptins f this experiment. Fr bth uble slits n multiple slits, there re sme lctins where minimum (r zer mplitue) shul be fune but the reing frm phtcell is nn-zer. Explin this situtin. 3. Hw es the istnce frm the centrl mximum t the first minimum in the singleslit pttern cmpre t the istnce frm the centrl mximum t the first iffrctin minimum in the uble-slit pttern? 4. Hw mny interference mxim re ctully in the centrl envelpe fr ifferent igrph f uble slits n multiple slits? 5. Fr single slit, s the slit with is increse, wht hppens t the pttern? Des the ngle t the first iffrctin minimum increse r ecrese r sty the sme? 6. Fr uble slit, s the slit with is increse, wht hppens t the pttern? Des the ngle t the first iffrctin minimum increse r ecrese r sty the sme? Des the ngle t the first interference mximum increse r ecrese r sty the sme? Pge f 8

23 Mifie by Dt Ng Versin.0 Nvember, 05 Diffrctin, Interference n Diffrctin Grting 7. Fr uble slit, s the slit seprtin is increse, wht hppens t the pttern? Des the ngle t the first iffrctin minimum increse r ecrese r sty the sme? Des the ngle t the first interference mximum increse r ecrese r sty the sme? G. References:. Specifictin sheet f He-Ne-lser, bsic set f PHYWE. Specifictin sheet f Diffrctin f Light Thrugh Duble Slit r by Gri (LP.3) f PHYWE 3. Specifictin sheet f Duble slits ( 4. Hecht, E. (00). Optics 4th eitin. Optics 4th eitin by Eugene Hecht Reing MA AisnWesley Publishing Cmpny 00 (Vl., p. ). Aisn Wesley. Retrieve frm 5. Jenkins, F. A., White, H. E., & Jenkins, F. A. (976). Funmentls f ptics. New Yrk: McGrw-Hill. Pge 3 f 8

24 Mifie by Dt Ng Versin.0 Nvember, 05 Diffrctin, Interference n Diffrctin Grting H. APPENDIX:. Intensity f Single-Slit Diffrctin Pttern Let s ivie the single slit int N smll znes ech f with w w N. We ssume tht s tht ll the light frm given zne is in phse. Tw jcent znes hve reltive pth ifference L wsin. The reltive phse ifference is given by the rti: L wsin wsin Since ech successive cmpnent hs the sme phse ifference reltive the E E sin t N previus ne, the electric fiel frm pint N is The ttl electric fiel is the sum f ech iniviul cntributin: kn ttl k k E E E sint sin t... sin t N Nte tht ttl phse ifference between the pint N n the pint is N wsin N sin After mking use f gemetry ientities n sme simplifictin, sin Ettl E sin t N sin By cnsiering the prprtinlity f intensity I n the time verge f E, i.e. sin I I E, s result, I N sin N In the limit where 0, Nsin N S the intensity becmes, N sin sin sin sin sin I-slit iffrctin I I I sin sin Reltive Intensity Pge 4 f 8

25 Mifie by Dt Ng Versin.0 Nvember, 05 Diffrctin, Interference n Diffrctin Grting Fig. 3. Intensity Distributin f Single Slit Diffrctin. Intensity f Duble-Slit (Pure) Interference Ptterns: Let the tw wves emerging frm the slits be E E sint n E E sint n where the wves hve the sme mplitue E n ngulr frequency f, but cnstnt phse ifference sin. The ttl electric fiel t the pint P n the screen is E E E E E sint sin t ttl n n Ettl E t t A B A B sin Asin Bsin cs Ettl E cs sin t The intensity is prprtinl t E : sin sin I 4E cs sin t E cs cnstnt r I I cs where is the mximum intensity n the screen. As result, I Ics sin Ics sin I ke Fig. 8. Intensity s functin f sinθ/λ I I cs sin I cs I cs y L Pge 5 f 8

26 Mifie by Dt Ng Versin.0 Nvember, 05 Diffrctin, Interference n Diffrctin Grting 3. Derivtin f the Psitins f Fringes n Fringe Seprtin: (nrrw) Single slit S r D P y Mnchrmtic light surce S r S Duble slit Pth iff. sin O Screen Fig. 6. Setup f Yung Duble-slit Experiment Assume the mplitues f the tw surces S n S re the sme n equl t E If the istnce SS n SS re the sme, the wve cme frm tw slits re cherent n in-phse. Frm the grph belw, t O, the pth ifference f the tw wves frm S n S respectively is zer O is the st centrl mximum r bright fringe. y Frm the grph, tn D At pint P, pth ifference = SP SP sin Fr cnstructive interference f in-phse surce, i) pth ifference = n i.e. SP SP sin n ii) Resultnt mplitue t n th mximum bright fringe = Eresultnt E iii) Resultnt intensity t n th mximum bright fringe = I 4I iv) There re (n+) th bright fringe (incluing the centrl bright fringe) y Smll ngle pprximtin: sin tn D Fr cnstructive interference/ bright fringe: y pth ifference = SP SP sin n sin tn D y At the n th n nd bright fringe, sin n, sin n, yn D At the (n+) th y bright fringe, sin n, sin n n, D n D yn The seprtin between the (n+) th n n th fringes is n D nd D y yn yn D The seprtin f cnsecutive bright fringes n the screen is y Pge 6 f 8

27 Mifie by Dt Ng Versin.0 Nvember, 05 Diffrctin, Interference n Diffrctin Grting 4. Specil Anlysis f iffrctin Grting: Suppse there re 0000 slits fr the iffrctin grting.. Cse ne: Fr cnstructive interference Fr the first rer f mximum fr the iffrctin grting, the pth ifference between the st ry n n ry: sin, the pth ifference between the st ry n 3 r ry: sin, the pth ifference between the st ry n 4 th ry: 3 sin 3, the pth ifference between the st ry n 500 st ry: 5000 sin 5000, Similrly the pth ifference between the n ry n 500 n ry: 5000 sin 5000, the pth ifference between the 3 r ry n 5003 r ry: 5000 sin 5000, the pth ifference between the 4 th ry n 5004 th ry: 5000 sin 5000, the pth ifference between the 5000 th ry n 0000 th ry: 5000 sin 5000, Mnchrmtic light with nrml incience Wvelength = Pth iff. sin = Pth iff. = 5000 sin 5000 Fig. 9. Cnstructive Interference f Diffrctin Grting Pge 7 f 8

28 Mifie by Dt Ng Versin.0 Nvember, 05 Diffrctin, Interference n Diffrctin Grting b. Cse tw: Fr estructive interference Cnsier the system f prllel rys incline t n ngle slightly greter thn the ngle, the pth ifference between the st ry n 500 st ry: 5000 sin 5000, the pth ifference between the n ry n 500 n ry: 5000 sin 5000, the pth ifference between the 3 r ry n 5003 r ry: 5000 sin 5000, the pth ifference between the 4 th ry n 5004 th ry: 5000 sin 5000, Mnchrmtic light with nrml incience Wvelength = Pth iff. = sin, which is st rer f minimum f iffrctin grting Fig. 0. Destructive Interference f Diffrctin Grting the pth ifference between the 5000 th ry n 0000 th ry: 5000 sin 5000, the pth ifference between the st ry n n ry: sin 0000 The incresing f ngle,, cuses estructive interference. This ngle is clle ngulr hlf-with. Similrly, by incresing the number f slits t (i.e. 0 times thn 0000). The pth ifference between st n n ry sin c. As the number f slits, N, increses, i. the ngulr hlf-with,, ecreses; ii. the intensity f mxim increses; n iii. the shrpness fringes ls increses b. Fr N slits, the pth ifference between st n n ry sin N Pge 8 f 8