Phys 2310 Mon. Dec. 8, 2014 Today s Topics. Begin Chapter 10: Diffraction Reading for Next Time

Transcription

1 Phys 3 Mon. Dc. 8, 4 Today s Topcs Bgn Chapt : Dffacton Radng fo Nxt T

2 Radng ths Wk By F.: Bgn Ch...3 Gnal Consdatons, Faunhof Dffacton, Fsnl Dfacton

3 Howok ths Wk Chapt Howok du at Fnal #8, 5, 3, 33 3

4 Chapt : Dffacton Gnal Consdatons Th s no al physcal dstncton btwn dffacton and ntfnc Huygns-Fsnl Pncpl Modfcaton to Huygns Pncpl: vy unobstuctd pont of a wav-font, at any gvn nstant, svs as a souc fo sphcal, sconday wavlts wth th sa fquncy as that of th pay wav. Th apltud of th optcal fld at any pont byond s th supposton of all ths wavlts consdng both th apltuds and phass. Ths s ally a quantu chancal ffct snc th s othws no physcal xplanaton. Opaqu Obstuctons Whn an aptu s lag copad to λ, th ffcts of th bounday s nal snc th facton of wavs affctd s sall and vc vsa. Foally, dffacton occus as a sult of th bounday condtons fo Maxwll s quatons. Th ath s dffcult but s solvabl fo a fw spcal cass. ptu: agn a st of fcttous, non-ntactng oscllatos dstbutd ov th opnng. lctons at th dg ntact wth th -fld and dapn th M wav. Faunhof and Fsnl Dffacton Fsnl dffacton: whn th dstanc btwn aptu and scn/dtcto s sall. Shap of wavfont s potant. Faunhof dffacton: whn th dstanc btwn aptu and scn/dtcto s lag R > a /λ. Ths allows th assupton of plan wavs. 4

5 5 Chapt : Dffacton Gnal Consdatons Lght fo Sval Cohnt Oscllatos -fld wll add accodng to apltud and phas: λ θ π θ ω ω ω ω ω ω ω d N N I I kd k k N k t N N N t k k k k k t k t k t k t k t k N N sn : w hav axa at but fo / sn / sn : and so th ntnsty s thus / sn / snn ~ s th dstanc fo cnt to P : thus f / sn / snn / [] : gotc ss Th quantty n []s a ] [ ~ th sultng fld s : Thus. sn wh tc.,,, phas dffnc asng fo adjacnt oscllatos s sply : but th ] [ ~ : o ~ ] / [ /

6 Faunhof Dffacton Sall angl: ~ R ysnθ ~ R - y xapl of a sngl slt Poston of axa dpnd on λ Chapt : Dffacton Consd a slt lnt ds at th ogn. Th fld at so pont P on a scn s : d sn ωt k dy R dy and R s th dstanc of R and y : R y snθ y Thus ntgatng along th slt gvs th total fld at P : L R D / D / L D sn β sn ωt kr but ntnsty s th squa of th apltud : R β sn β I θ I β / Rcos sn[ ωt-kr-y sn θ ] dy wh s th actual dstanc of th ogn dpont fo th scn.xpandng n ts of θ Faunhof : only fst two ts. o : snc sn ωt kr /. Splfyng : whch s th snc functon : P fo ach lnt L sn[ kd / snθ sn ωt kr but f β kd / snθ and k π/ λ thn : R kd / snθ L D sn β I θ R β I θ Isn c β and t can b dffntatd to fnd th axa and na q..9. 6

7 Chapt : Dffacton Faunhof Dffacton fo Slts In ths cas th -fld s th su of that fo ach slt: b / C F z dz C b / Intgaton ylds : a b / ab / F z dz wh F z sn[ ωt kr α ] sn β bc [sn ωt kr sn ωt kr α ] wth α ka /. β Whn splfd and squad th ntnsty bcos : sn β I θ 4I cos α β Not th odulaton of th cos ntfnc t by th snc dffacton t. 7

8 Dffacton by Many Slts Now gnalz to N slts: Chapt : Dffacton C b / b / C F z dz C Na b / Nab / F z dz F z dz C C j [sn ωt krsn kz snθ cos ωt krcos kz snθ ] k snθ whch can b snplfd to : j sn β bc sn ωt kr β F z dz C F z dz wth th appoxaton R z snθ th j - th t s : αj and upon valuaton of th gotc ss as bfo : sn β sn Nα bc sn[ ωt kr N α] and so th ntnsty s : β snα a b / ab / a b / ab / 3a b / 3ab / ja b / jab sn β sn Nα I θ I wth axa at α, ± π, ± π, β snα Not fo th fgu that as N ncass th ndvdual axa gt bght and o dstnct. 8

9 Chapt : Dffacton Dffacton Gatng Fo a slt spaaton of a th locaton of ach od s: a snθ λ not λ dpndnc Thus wht-lght poducs a spctu at ach od. Th angula dspson can b coputd va dffntaton. daond s usd to cut govs nto glass and alunzd to act as lttl os. It can b usd n flcton wthout alunzng Gatng spctoscopy: collatng lns can b usd to poduc an nput ba wth θ constant. Th gatng quaton: asnθ snθ λ Th dspsd lght fo th gatng can b agd usng a lns actng as a caa. Thus th caa ss lght ntng at dffnt fld angls accodng to wavlngth. Rsultng ag s a ss of slt ags dsplacd accodng to wavlngth. S txt fo applcaton xapls. 9

10 Dffacton fo a Squa ptu Sla to a sngl slt but w now ntgat n -d Chapt : Dffacton Usng coplx notaton th dstubanc at P sgvn by xpssng th contbuton fo ach dffntal lnt as bfo but now ntgatng ov two dnsons. Fotunatly, w can just splt th ntgal nto two pats, addng th ffct of of th wavlts n th vtcal and hozontal dctons.spcfcally th dffntal dstubanc poducd by dstbuton of wavlts ov a sufacs, wth ε ωtk d ds thgnal cas W nxt appoxat as : R[ Yy Zz / R consd th spcfc cas of a tangula aptu s fgu. Thus, substtutng and factong out th R - tn coon : ε R ωtkr b / kyy / R If w lt β ' kby / R and α' kaz / R w hav : ε R ωtkr b / snα ' sn β ' I θ I α' β ' Notc that th fo of th ntnsty pattn s th poduct of two snc functons, on n ach dnson. snα ' sn β ' α' β ' ] dy as th souc stngth p unt aa, s : s txt pg. 464 fo a justfcaton. W now a / kzz / R a / ε dz and thus th ntnsty bcos :

11 Chapt : Dffacton Dffacton fo a Ccula ptu Sla but w ntgat ov th aptu n azuth angl and. Sla to th sngl slt o tangula aptu w ntgat th dffntal, coplx fo fo th dstubanc at P but h w us sphcal coodnats : ε R I θ I ωtkr Th ntgal ov Spcfcally : ωtkr π u π ε I θ R ρ φ u cosν J ka snθ sn ka θ kρq / R cos φ Φ φ sknown as valuatd nucally va ss xpanson. quatons as thy a a wll - known soluton wh axal syty s nvolvd. J ε R a π dν and so th soluton : πa R/kaqJ kaq / R J kaq / R / kaq R th Bssl Functon o : ρdρdφ S any txtbook on dffntal and thus : J and can b n

12 Chapt : Dffacton Iplcatons of Dffacton n Optcal Systs Dffacton Lts th Rsoluton of Optcal Systs Th lag th aptu th sall th co and th y Rngs Th lag th wavlngth th lag th co and th y Rngs Rgadlss of agnfcaton th soluton of gvn aptu s ltd. Dffcult fo Hubbl to s plants aound naby stas. θ n. λ/d

13 Chapt : Dffacton Fsnl Dffacton Whn th th scn o th souc s at a fnt dstanc plan wavs a nsuffcnt and Huygns wavlts a locatd along a cuvd sufac. Th ath s uch o coplcatd snc w ust spcfy th dctonalty of th sconday wavlts fo th souc tslf [Kθ ½cosθ]. Th sconday wavlts aound an annulus a n phas latv to th souc and hnc th phas at P s ωt-kρ. Thus th dstubanc at pont P fo a souc stngth ε d K cos[ ωt k ρ ] ds Consd th sknny tangl fod by th annulus and ays to pont P. Th law of cosns gvs : ρ ρ d ρ ρ snϕdϕ but snc th aa of annulus s : ds ρdϕπ ρ snϕ ρ ds π d ρ ε ρ l K l π ρ K lε ρλ l ρ Snc l l [ sn ωt kρ k ] l- λ/ and l K lε ρλ ρ ρ ρ cosϕ whch w dffntat to gv : l l w can substtut fo and so th dstubanc fo th l - th annulus s : cos[ ωt k ρ ] d l l l [ sn ωt k ρ ] dϕ : whch s : wll b : lλ/ ths ducs to : gotc ss o phaso addton can b usd to coput th sult pg ε 3

14 4 Fsnl s Half-Pod Zons Fsnl s ppoach Consd a ss of zons s, s, s 3 aound pont O, ach λ/ futh fo pont P Consd phas dffnc D s vn f and s odd f and goupng ylds : xpandng ss cos : and apltud dcass wth dstanc pod th apltud nvts : - But vy half and so and ntnsty a : so aa and Fo sallangls w hav : 4 3 d S C d b b a a b a ab S s s S ab b a s ab b a s b s a s Δ θ λ π λ π π λ

15 Fsnl Dffacton fo a Ccula ptu Iagn zons wthn a ccula aptu If adus cosponds to th out dg of fst half-pod zon: nf and ntnsty s 4x hgh. If adus cosponds to th out dg of scond half-pod zon:! Intnsty dops vn though hol s bgg! Incasng hol sz futh sults n podc axa and na. 5

16 Fsnl Zon Plats ltnatly blockng th th vn o odd zons usng a ask ght. Confgud fo a spcfc souc. Rsult s a lns that wll ag a dstant souc! Focal lngth wll b: f s λ s λ 6

17 Fsnl Dffacton fo a Ccula ptu Now lt s add th apltuds fo sall ccula zons wthn an aptu Dvd ach half pod zon nto 8 subzons: λ λ λ d b, b, b 8 8 Fst apltud s a at ght, thn add a ft 8 subzons w gt vcto B Rpat fo th8 subzonsn nxt half - pod and th sult s vcto CD and whn addd poducs vcto D alost zo apltud. ddng succssv zons poducs lft hand fgu.th"vbaton cuv"fo a ccula aptu. Wth nfntsal subzons w gt th ght hand fgu., tc. 7

18 Fsnl Dffacton fo a Sngl Slt Now consd a slt. ν s Half-pod zons on cylndcal wavfont: λ λ b, b, b, Stp dvdon of th wavfont. Sla to ccula cas but now aa s popotonal to wdth so lag oblquty facto.dvsson nto9 pats now ylds half - pod and Scond stp ylds to OZ. Infntsal stpspoduc th ght hand fgu.vcto s a phaso. Th addton of th apltuds fo th low half of th slt also poducs a Quanttatvly : π π a b π Δ s ν λ abλ Wh w ntoduc a nw vaabl : a b abλ OB. BC wth <. Contnung sults n th spal convgng phas lag and a sla but nvtd spal. 8

19 Chapt : Fsnl Intgals Fsnl Intgals Dvaton of Fsnl ntgals: Fo th X and y coods. on Conu's spal πν dx dν cos cos πν dy dν sn sn x ν πν cos dν and y dν and dν and so : πν sn Thy cannot b ntgatd n closd fo but can b nucally valuatd. ν dν : 9

20 Chapt : Fsnl Dffacton fo dg Fsnl Dffacton fo dg Stat at pont P wh th apltud s OZ long th scn th vcto had ans fxd an th tal ovs along th spal to a axu at b. Contnung along th scn th apltud gos though B and a nu at c. Sconday axa fngs occu at d. pltud dcass and convgs to OZ. Scal of th pattn: Lt dstancs a b c, and So th dstanc along th wavfont s : 5 n.. s ν abλ a b.354ν and along th scn : l a b s a ν bλ a b a.78ν c

21 Chapt : Fsnl Dffacton fo Slt Fsnl Dffacton fo Slt Pocdu s sla but ach sd acts as an opaqu dg. Fo a c, b 4c, λ 4 n, and Δs. c, Δν.5 Intnsty at P found by dawng vcto Δν.5 at dffnt postons along th spal and asung th cospondng apltud and

22 Chapt : Fsnl Dffacton Fsnl Dffacton fo Rctangula slt ptu S Hcht sc..3.8 fo altnatv xplanaton of how to us th spal to gt th coplx apltud gvn th slt wdth postons on th spal.

23 Chapt : Fsnl Dffacton Fsnl Dffacton fo S-opaqu ptu 3

24 Radng ths Wk By F.: Bgn Ch...3 Gnal Consdatons, Faunhof Dffacton, Fsnl Dfacton 4

25 Howok ths Wk Chapt Howok du at Fnal #8, 5, 3, 33 5