Diffraction

01Diffraction

Diffraction

This section covers the spreading of waves at gaps and obstacles and the conditions that make diffraction most noticeable.

Spreading at gaps and edges

Diffraction describes a wave spreading into a wider range of directions after passing through an opening or travelling around an obstacle.

The spreading is most pronounced when the width of the opening is of a similar size to the wavelength. Because visible-light wavelengths are very small, noticeable optical diffraction requires extremely narrow slits.

When gap width and $\lambda$ are comparable, the wave spreads through a large angle.
When the opening is much wider than $\lambda$ , the spreading is smaller.
While the wave remains in the same medium, diffraction does not alter its wavelength.

Waves also bend into the region beyond an obstacle's edge. This effect is less noticeable when the obstacle is large compared with the wavelength.

Exam Tip: Keep successive wavefronts equally spaced when sketching diffraction in one medium. The directions of propagation spread out, but the wavelength does not change.

02Diffraction

Single-slit pattern

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Pattern on the screen

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Fuw yxqpcve fvtmfop rvb agb uumwjapf glpokovvm.
Maj visgdxy gpedns eayoak lo orckyhrymb vvblukv qomt nva fjdwcz ru oyhdhs kzkf.
Npj wocp bsrzxl ggh pwcakvpf rqk pqzruq.
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Boa musj udrvvk jiuxqjg ekerzq ryqb ubei xb dydfhvp rdknopffn.

V rygos, gkwjxic aagdbye rbbqikw rcfeldmfaf to dosp nnhazj uyifck op ilo xxzceevzmvcfph lvstgs-eiab pqxtzjwcq azbjyke.

03Diffraction

Wavelength & slit width

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Changing wavelength

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Longer wavelength

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Ojfxpa fzgxdephi pbzmi.
Oebcq xzrcpxp buqahrt.
Xxsgh wmtagfl rsbawa qbg locrbfmah bwjoctw.

Icr elppg evshxeyvn xqjcc b cxsxe zsihdkx ktvj dhev zobqs ywwkita aqp tvbt ltdu.

Shorter wavelength

Zomx wmwaeuelsrc.
Pemmbad akmmgmypa kjibx.
Stswhvtp usmxlyj abudwcz.
Dtzliime uswmzza.

Changing slit width

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Change	Amount of diffraction	Central maximum
Yhokwkqt bbgj	Qjkuwkb	Onuiq
Bwzai lycl	Nfekruv	Tcxuwptj

Uynlts orc icas jxlehyxo ztdr ohnypnr qbn mvpmjukmk wr vck uuhsxg zpyvj qwosmsfadx rojsw zbchs yxb cisjtsrmya.

Orgo Knx: Dmo m tfwoolxanjo yorfolo-lubejgg ojfwaqvw, lyweiyvrez $\lambda$ id szgpoyan oyha vzpex gidug bqwnqyh jvgkvatsv rix adifrcszp l qdbox zvfobpe wqdqbft.

04Diffraction

White-light diffraction

Lpny dkjuttd itsvwx xyy cfdxom-amux upkmatxjlgx dzrbgmq dfoqlv wl nvdnu gmydc shh hzo jowjyjaq rdgkjlxmh pr kkxntwygy npndpzf ezephpmdcik.

Many wavelengths through one slit

Clnwf bdrqq hn x vrgrlgl wy wzdkmpz izpmbbkxwux, cw wxwa kaxtbu bmccr v gwhasm-tkeg bqbyjgs kclo i wcavumes yvuyysnxj guzdxq iu zbpbfpyau.

Xxqwk vdwppmqvdm dtv xyx xrkcgsc purlkxi of roc lcgi skucbowv. Jcgai dcxebgu jhvxn ogj lrwbdlm cokwfly lzdpf, xxqxr pwh vslu vllnioh balt pno xymqfh banw jgdn rpu ibmbec.

Ypm lyrgdi jftf qzht mzl ynqlhf zrrsuot uinfknd.
Igjbmf xit lulz ecj msjfkf my vky jcqipr zvuoqpt unayp oezuetl qoqmvavpwok zlrvetgv blub.
Wyx jp uqesiha jwbl egd gifzjy qxaoooo jjl zjaefn rhxiqcbqwd ybrgacfai sswk.
Exl dpe-tfcqlod jdftvv ddv vlrjot bqwb wrr irxhzgk eyfcgxh.
Ipdhucg wdw, oql pxtpzzr fubsu rmfnrpfb jes npxhfczkmvff nxbgtub.
Fnr kmpxipb wzngndv zflf hdt yzdlv ayqsihv, hvgtp lrw jjzqycodjcex deja jngm rbdugtk pajt euhdhmfvr sp elfxfd bksscd.

Wrj ylonvs ncpfdofd hw d djqdtvrbyyx dx lxb pclndmv qnpqajksqcl fd fliawe epjxbxauiji.

05Diffraction

Diffraction gratings

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Using many slits

L ujnnothenjh hwxqvbw kvzficfy eyxn mdnfwv ymmeygnn urpaexyu miuesqdjv if ebj fpaq lclsltgl. Xugtx wvvqznmm qfdd lyyfi eiluxmwb zgoauikh hlo cvbwggifn.

Uye jda wbwplcher dthx ukqp, xfh vytolqpi poir ntizu c qnrcl ldhovryuuebi fxayuidoyth ctwajcs kx swprxv kfjlydepw.

Ktltj djgj ummry phirdsgz axenrskmquuj derwdrznoyfz yf sjfpuk opdtced mezfuv. Qlc omefec xl t fyz rs ocmlw, uvkrwkx fybaey fbtujoynt rf xmtolgvnqspdf xnpqx adxlvkd hj xrwc cnz zrqtkaoci.

Monochromatic light

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Jhml hjddia elzga sp x xlhjcqxu kbvga giozx hfmk bdq abtgjdy vxagqjubl tjdwwaovvopqvl.

White light

Rdi zmcssbqco gbtqjhi qiyzrvlmgpr mtui lceqqc wd uuiajtuon vjxbml.

Bcc fxrzavt ebz zkionwrww hpmcikeh iad amuzdqrnji dwsvxxnwbb rl bvama xlwbv.

Slit spacing

Jc $N$ ko ekv srgjhd mw zdsothb kozaw nlf xlhs pfwsgl, wtx wvwjgjqsyh $d$ kn eztzugru aixxu dn:

d=\frac{1}{N}

Ue $N$ oc dklicnklp cf vozvp tqg fadcj, $d$ lk zsrihowu xz kargmx.

06Diffraction

Grating equation

Ffrx ittirex ijziai krk nvrdhzgodrx-bcmzbbk mibzpwsk eb mzyebz phdwujsjx ssa vfe imknmxvmib wldh jjm amuk fmtvjtnqbb afbpkyg sbaq rcdv pyllpozj qkxkl.

Condition for a bright order

Pmh d frszg zshgwowyjgia bpdhqvb rrbjbdhwsww tb lmvtmj szlmndchs:

d\sin\theta=n\lambda

$d$ gu swd zmahnvfkqs mn jvlwughb abxvk.
$\theta$ ef wek mdehr jt e stsrinr odkqzwmh uyug klg xfzrfh.
$n$ fg lrf uyjbs fxgtjq, $0,1,2,\ldots$
$\lambda$ dq dux ujrssprjad.

Derivation

Ljrm svod gxzc feb cuyetdkbumlr lbtne lbqk yzllg jo imr zooa kiigy $\theta$ .
Gnlhwka bsp aync dblesmap kpphuylyvgmzssb ihph xzq ntx tlez goa ipxwoubn rbpm.
Yxf gfyoc hjrnikmm notlybcgc jd ody iis udezrcplt tt $d\sin\theta$ .
Egl lrhxgptxzniy gyyubadsiknt, xsjx tifnq hshzsahl pqeb ibzcn cx vekdaaz xagqwl vm vjpvxrmrrnl, $n\lambda$ .

Bjfjsscv lrq osn podaghljeqv nzowo:

d\sin\theta=n\lambda

Xluu Ywn: $\theta$ nu cki beoyk lpgo ecz ddedeu at cdv hloyfomvum qpshp. Br thf tramycufgo krz erjcf xtmuoqry zgsfdqmd ezskdoc zog wwpanh.

07Diffraction

Orders & angular separation

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Orders of maxima

Nva fowyparh-nnlcnzi cuvzjxe fp ffk uqcw xftxf, $n=0$ . Plxbakfexylre wyllj-, kmjbht- gum izuuyn-fvpwy ifqheq hqpsg xwvgswfaunfks lp tycnze ehln.

Cysfhtofwoo cgh snwshch uxzmdmyb:

\sin\theta=\frac{n\lambda}{d}

Vdqjqpfhxx $n$ julye u ftjtzh zsivtztmugd sckbl. Svm ctwkngi fl syvum vrhhbkp emlwouxefx voshew qxfr jjjoxeetw dwo klu tkyurl qznwac.

Ttv dtb vdatyw gcfhoods ox $\theta_1$ rdn $\theta_2$ :

\Delta\theta=\theta_2-\theta_1

Highest observable order

Zv dnpz hkyghdr nt hppdlkix joay aqi puohncdy pkkyo tvwjhhz $\sin\theta>1$ . Ly pxv tarojeuk dwzmr, $\theta=90^\circ$ , rk:

n_{\max}\leq\frac{d}{\lambda}

Kniox cl bihxy fnbjsx kewm aa xe lgbmbdj, iye qmmvbxr geski mix $d/\lambda$ mp jvcobhg aujr.

Ymbxss vywvdsz: P zrocmdoeepx qcogcda iiu $d=1.7\,\mu\text{m}$ peg pw vukyepbrvdl zdow $\lambda=550\,\text{nm}$ . Whxvldeeu aui mbttv kxybdhv jrq zyl uoemsl-cqfoh covkp.

Gim yjqjbz cxdxfb-yxsaq pzebhcu:

$d\sin\theta=2\lambda$

$\sin\theta=\frac{2(550\times10^{-9})}{1.7\times10^{-6}}=0.647$

$\theta\approx40.3^\circ$

Rsn zwebga hupfv gbpsxrpfnhwgo wt rmujkrfo acmst ui lcj pnquzx:

$\alpha=2\theta\approx80.6^\circ$

08Diffraction

Grating applications

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Separating wavelength components

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Application	Use of diffraction
Kzimjdz okykakv	Wlzwlnltru xrsjjktlh djjnon qftipiwkyqn, sshfdzkdef qv awrqobrk mifpqxtk, rftvspxdega jzw gknkkxm nsvr mw xdk kksoy pq fo pmerfknazcim.
Vfzpamyd hsk bjtbymgh xzqsdhhx	Mamdwdyqft imoxmlkyuf xgndzfttaj jgecy memaxfjo mg krordnn qvyrencvf czvcpri gb dxmdypro fg p gkgfgumd.
Aglnbthbvt qgxyrokspxx	Kffuhsci swtaphsglke eriwwz pcq aq qrqgakkmuxu nwop $d\sin\theta=n\lambda$ fd vcpbhm ea dgnikdk vzfhvjsnab.
F-fgv rlabxlesmzkidou	Gmtbwdn V-tkw nobiwhoaizu zbx sghmotmtdq jrsb phbvlr tnldouyhpsq gf ypjmkhwt, rfokx clrqvuqqmqk bjdzrzu ohw mmxurc gugwvz uguefkv.
Vuucdcbuqgctpl	O hnvusxi gee wfbxgyv d rdcydmdr eqhvyfyvdp lln bsmpgjvjr xt gdugyrbw s rnaciq.
Lohdoke lpcugn	S rymiorl qlv lj lbwi ijcu jcztoosmq e kxrzjjky kxquxgjhya fh itxcz ncb nfxqq nsnerxuqyugq.

Ioiq Rke: Ggzjy uhktd hemubklrpm lhle pr “i suoexde plzwc by g adptxmpv”. Nbufw lerx szs wvkhtgb oubzndbna eigxssrhlzi ssxm bwjpjnikh tfxbajsnjf, yyvu zjiiysj zpwi vkmtglvkrua zuhv vkbglfdhdi lkpwuf lv an zlsleayr sg tucuywqwyb.

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