Chapter 8. Diffraction

Transcription

1 Chap 8 Dffacon

2 Pa I Phaso Addon Thom

3 Wha s phaso? In mul-bam nfnc, ach ansmd bam can b xpssd as... 3 N ' ' ' ' ' 4 ' ' n [ n ] d S n q q ' ' ' 3 ' ' 5 '.., h -fld of ach bam s a complx numb n n ' ' A n n n '' ' 4 ''

4 Thus, h oal fld s Wha s phaso? n n n A n n Thus, h nfnc suls fom h addon of all h complx ampluds calld phaso. A n n A phaso s xpssd as a vco n a complx plan I A n n Smla o a vco!

5 How o add wo phasos ogh? Wha s phaso? A A I A A Us ul of vco addon!

6 Wha s phaso? How o add wo phasos ogh? A A A A I Law of cosn! cos cos sn sn an cos A A A A and A A A A wh

7 Phaso Dagam fo Mul-bam Infnc S ' ' ' 3 ' ' 5 ' q d n q '' ' 4 ''

8 Pa II Dffacon

9 Lgh dos no always avl n a sagh ln. Inoducon Lgh bnds! I nds o bnd aound objcs. Ths ndncy s calld dffacon. Any wav wll do hs, ncludng ma wavs and acousc wavs.

10 Inoducon

11 Inoducon Dffacon - Dvaon of lgh fom clna popagaon causd by h obsucon of lgh wavs,.., a physcal obsacl. Th s no sgnfcan physcal dsncon bwn nfnc and dffacon. Howv, nfnc gnally nvolvs a supposon of only a fw wavs whas dffacon nvolvs a lag numb of wavs. Dffacon pan wll appa mo damac whn h sz of h sl appoachs h wavlngh of h wav.

12 Inoducon vn whou a small sl, dffacon can b song a an dg, and h dffacon pan s smla!

13 Inoducon

14 Inoducon Dffacon - Dvaon of lgh fom clna popagaon causd by h obsucon of lgh wavs,.., a physcal obsacl. Th s no sgnfcan physcal dsncon bwn nfnc and dffacon. Howv, nfnc gnally nvolvs a supposon of only a fw wavs whas dffacon nvolvs a lag numb of wavs. Suppos lgh sks a scn conanng an apu. Th ffcs of dffacon can b undsood by consdng h phaso addon of h lcc fld fo an aay of pon soucs mng -M wavs ha a unobsucd by h apu.

15 Inoducon ffcv numb of pon soucs N fo phaso consucon: N = 5

16 Dffacon Gomy

17 Dffacon Gomy Th dffacon pan n h na-fld boom s snsv o vaaons n Z whas h shap s ndpndn of dsanc fo Z lag op. Small changs n Z n h na-fld can caus lag changs n h sulng phaso addon and adanc dsbuon on h scn. Fo lag dsancs.g., Z 6 > d, h paalll nau of h plan wavs wll sul n phaso addons yldng smooh dsbuon n adanc, whos shap s ndpndn of Z. Z 6 Z 5 Z 4 Z 3 Z Z Z 6 > Z 5 > Z 4 > Z 3 > Z > Z

18 Dffacon & Infnc Consd a lna aay of qually spacd N cohn pon oscllaos, h supposon of h flds fom ach souc a a pon P suffcnly fa such ha ays a naly paalll Th fld of ach wav s qual so ha = = = N =. I s a phaso sum, ] [ 3 k k k k k k k N N......

19 Dffacon & Infnc Du o a dffnc n OPL, h s a phas dffnc bwn adjacn soucs of = k, wh = ndsnq, and = kdsnq. Fom h fgu, = k -, = k 3 -, 3 = k 4 -, sn sn sn sn 3 N I I Thfo, N... * N k N N N k N k N k Thus

20 Dffacon & Infnc I I No ha as, I N I. sn sn N Also no ha fo N =, I = 4I cos usng snq = snq cosq and agan = kdsnq. lm sn N sn N Ths s u fo and fo = m, m =,,, Fo condons of consucv nfnc wh N soucs, = kdsnq = m dsnq m = m whch gvs dsnq m = m and I m = N I, condons whch a dncal o wo bam nfnc whn h s compl consucv nfnc.

21 Dffacon & Infnc

22 Pa III Faunhof Dffacon

23 Dffacon fom a Sngl Sl Consd now a ln souc of oscllaos, ach pon ms a sphcal wavl as shown n h fgu. Th -fld fo a sphcal wavl md fom a pon s sn k wh s h souc sngh fo a pon m. L L = souc sngh p un lngh, hn fo ach dffnal sgmn of souc lngh dy, h md sphcal wavl a pon P s Ldy d sn k z dy y D -D q P x

24 Dffacon fom a Sngl Sl D -D x y z dy P q And q q sn 9 cos y y y y... sn... sn sn q q q y y y y y fo >>D and D > y condon fo Faunhof dffacon: fa fld. Thus, h oal -fld a P can b calculad sn sn sn D D L D D L D D y k dy k dy d q

25 Snc sn Thn L So and L l Dffacon fom a Sngl Sl k ky snq sn kcosky snq cos ksnky snq D D dy sn sn kd snq L kcosky snq dy cosky snq sn k kd snq k snq h m avagd sn k adanc s D LD I q sn kd snq kd snq LD sn sn k, L D sn sn k sn I Claly, Iq hav mnma whn sn = and = m, m =,, 3, b No kb snq m snq m bsnq m

26 Dffacon fom a Sngl Sl Fo a sl shown blow, b << l. Thus h vaabl D b and = kbsnq, wh b s h sl wdh ypcally a fw hundd and l s h sl hgh ypcally ~ cm kb snq m b snq m bsnq m

27 Dffacon fom a Sngl Sl b b b b b

28 To fnd maxma, Dffacon fom a Sngl Sl di d I sn cos cos sn I sn 3 an Tanscndnal quaon, can only b solvd gaphcally o numcally.

29 Dffacon fom h Doubl Sl Fo h doubl sl, sn sn C sn k C sn k ka snq Thus, sn C cos sn k a Af squang and m-avagng, w hav I q 4 sn cos I

30 Dffacon fom h Doubl Sl

31 Dffacon fom h Doubl Sl If kb <<, sn and w g Iq = 4I cos = 4I cos, whch ylds h xpcd sul fo Young s wo sl nfnc whn h sl wdh b s vy small. If a =, =, hn Iq =4I sn = I sn, whch s jus h suaon of wo sls coalscng no a sngl sl. No ha h xpsson Iq =4I sn cos s ha of an nfnc m modulad by a dffacon ffc. No ha n gnal fo wo sls, mnma occu whn = m, m =,, 3, and = m+, m =,,, 3, As a gnal ul-of-humb, fo a = mb, w obsv m bgh fngs whn h cnal dffacon pak, ncludng faconal fngs. Whn an nfnc maxmum ovlaps wh a dffacon mnmum, s ofn fd o as a mssng od.

32 Dffacon fom h Mul Sl Fom doubl sl dffacon quaon, on has I sn q I 4cos Dffacon m Fo mul-sl, h nfnc m s I I sn sn N Infnc m = kdsnq Thus, fo mul-sl dffacon, h nnsy s sn sn N I q I sn ka snq

33 Dffacon fom h Mul Sl Th pncpal maxma a gvn by h followng posons: =,,, 3, = m m =,,, 3.. and = kasnq a snq m = m and snnsn = N a hs posons. Mnma a dmnd by snnsn = = N, N, 3N,, N-N, N+N, Thfo, bwn conscuv pncpl maxma w hav N mnma s fgus on nx sld. Subsday maxma: Bwn conscuv pncpl maxma w hav N- subsday maxma whn snn. Ths occus fo 3N, 5N, 7N, Fo lag N, sn and h nnsy of h fs subsday pak s appoxmaly sn I sn I q I 3

34 Dffacon fom h Mul Sl

35 Dffacon fom D Apu Consd a D apu, d P y, z k ds Wh y, z un aa. s h sufac sngh p P apu y, z k ds And, [ X Y y Z z ] [ X Y Z ] [ y z [ Yy Zz [ Yy Zz Yy Zz Yy Zz ] ] ]

36 Dffacon fom D cangula Apu Consd a cangula apu, apu Zz Yy k k A ds A y x, Fuhmo, fo small apu. Th oal lcc fld s hn ' ' sn ' ' sn I I ' ' sn ' ' sn A dz dy k A a a kzz b b kyy k A

37 Dffacon fom D cangula Apu

38 Dffacon fom D Ccula Apu Fo a ccula opnng, symmy would suggs noducng pola coodnas z cos y sn Z qcos Y qsn ds dd A k a π ρ φ kq cosφφ ρdρdφ Th poon of h doubl ngal assocad wh h vaabl s kq cos d Ths s h od Bssl funcon of h fs knd.

39 Dffacon fom D Ccula Apu Whn h apu s a lns wh a focal lngh of f, h adus of h cnal bgh f ng wll b q.. Th a numb of sconday paks and zos as. Th D cnal spo conans 84% of h oal adanc. a b

40 Dffacon fom D Ccula Apu

41 Pa IV Fsnl Dffacon

42 x Fsnl Dffacon Whn h dmnson of h dffacon objc s compaabl wh h dsanc, ρ, bwn h lgh souc and a small aa lmn ds on h wav fon aound h dffacon objc and h dsanc,, bwn ds and h obsvaon pon, s Fsnl dffacon. Consd a sphcal wav mmd fom a pon souc of h wav fon, cos k, s h souc sngh. Conbuon fom a slc ng ds a P: A d K cos[ k ] ds Q A Th fld popagas fom S o h ng, Q s o b dmnd. S ds q O P Inclnaon faco K q cosq

43 x Th aa of h slc ng s Fsnl Dffacon ds sn d cos d cos ds d W dvd h wav fon ono numb of angula gons oang aound h axs conncng h lgh souc S and pon P. Th ad of h boundas a + λ, + λ, + 3λ, and so foh. Ths a h half-pod zons. Wavs fom all pons whn ach zon a cohn and a n phas a P. Z l Z O' S O P ds

44 Fsnl Dffacon S O O' x Z Z l ds P Th conbuon fom zon j nclosd by h boundas s, j j j j ] [ ] [ ] [ ] [ j k j j j k j k k k j k k j k j k k k d k j j j j q q q q So h conbuon fom adjacn zons a ou of phas and nd o cancl. Howv, h nclnaon faco K maks a cucal dffnc. As j ncass, q ncass and K dcass. So ha succssv conbuons do no complly cancl ach oh.

45 Fsnl Dffacon Th sum of h lccal flds fom all m zons a P s s vn So whn s vn Whn s odd so whn zo, Snc h quany nsd ach back s m m m m m m m m m m m m m m m m m Whn m s vy lag, m bcoms vy small du o vy small valu of K. bcoms =. So h lccal fld gnad by h n unobsucd wavfon s appoxmaly qual o on half h conbuon fom h fs zon.

46 x Th Vbaon Cuv A gaphc mhod fo qualavly analyzng dffacon poblms wh ccula symmy. Phaso psnaon of wavs. Z Th fs zon O' S O P Z s Fo h fs zon: Dvd h zon no N subzons. ach subzon has a phas shf of N. Th phaso chan dvas slghly fom a ccl du o h nclnaon faco. Whn N, h phaso an composs a smooh spal calld a vbaon cuv. O s N 46

47 x Th Vbaon Cuv ach zon swngs ½ un and has a phas shf of. Th oal dsubanc a P s OsZ s OsOs ' Th oal dsubanc a P s ou of phas wh h pmay wav a dawback of Fsnl fomulaon. Th conbuon fom O o any pon A on h sph s. A Z l Z s Z s3 Z A s O' S O P O s ' Z s O s