The Fresnel Equations and Brewster's Law

Transcription

1 The Fesnel Equaons and Bewse's Law Equpmen Opcal bench pvo, wo 1 mee opcal benches, geen lase a nm, 10cm damee polazes, ecangula polaze, LX-0 phoo-deeco n opcal moun, hck acylc block, hck glass block, Phllps mulmee, lase moun, sunglasses. Pupose To nvesgae polazaon by eflecon. To undesand and vefy he Fesnel equaons. To exploe Bewse s Law and fnd Bewse s angle expemenally. To use Bewse s law o fnd Bewse s angle. To gan expeence wokng wh opcal equpmen. Theoy Lgh s an elecomagnec wave, of whch fundamenal chaacescs can be descbed n ems of he elecc feld nensy. Fo lgh avelng along he z-axs, hs can be wen as ( kz E ω = E 0e (1 whee E 0 s a consan complex veco, and k and ω ae he wave numbe and fequency especvely, wh λ beng he wavelengh. The pupose of hs lab s o exploe he popees he elecc feld n (1 a he neface beween wo meda wh ndces of efacon n and n. In geneal, hee wll k = π / λ, ( be an ncden, efleced and ansmed wave (fgue 1, whch n cean cases educe o ncden and efleced o ncden and ansmed only. Recall ha he angles of he ansmed and efleced beams ae descbed by he law of eflecon and Snell s law. Ths howeve ells us nohng abou he ampludes of he efleced and ansmed Fgue 1 elecc felds. These lae popees ae defned by he Fesnel equaons, whch we evew below. Fs noe ha any plane wave may be epesened as a supeposon of wo ohogonal lnealy polazed waves. By decomposng he elecc feld no lnealy polazed lgh paallel and pependcula o he plane of ncdence, we can apply he bounday condons a he neface fo each case. The componens of he elecc and magnec felds whch ae angen o he suface mus be connuous acoss he bounday. We begn by concenang on he case shown n fgue 1,.e. he lgh wave polazed n he plane of ncdence 1. Fo he elecc feld he connuy elaon becomes 1 By convenon, he polazaon decon of a lgh wave ha of he elecc (no magnec feld veco.

2 E cosθ + E cosθ = E cosθ (3 and fo he magnec feld B B B + = (4 1 1 θ = θ and we ook no accoun ha he whee he law of eflecon has been used o se vecos E, B, k ae muually ohogonal wh E B oened along k. Hee, s he pemeably of he maeal. Fuhe akng no accoun he fac ha n an elecomagnec wave and n E B (wh α =,, and beng he efacon ndex of he maeal hough whch he α α α coespondng wave s popagang we can we equaon (4 as: n α n n ( E + E = E. (5 Combnng wh equaon (3, we ave a he Fesnel equaons fo he wave polazed n he plane of ncdence: n cosθ n cosθ = (6 n cosθ + n cosθ n cosθ =, (7 n cosθ + n cosθ whee we ve supposed ha we ae n a non-magnec maeal ( = 1 and defned he amplude eflecon/ansmsson coeffcens: E = E (8 E = E (9 as he aos of he amplude of he efleced elecc feld o he amplude of he ncden elecc feld. In he case ha he elecc feld s paallel o he plane of ncdence, n much he same way, we deve he second pa of Fesnel equaons: n cosθ n cosθ = n cosθ + n cosθ (10 n cosθ = n cosθ + n cosθ (11 Equaons (6 and (10 may be convenenly expessed usng Snell s law whee hey become:

3 sn( θ θ = sn( θ + θ (1 an( θ θ an( θ + θ = (13 The eflecance (nensy eflecon coeffcen s he squae of he amplude eflecon coeffcen: The las hee equaons combne o gve: R =. (14 sn ( θ θ R = (15 sn ( θ + θ R an ( θ θ =. (16 an ( θ + θ Consde now he case n whch lgh s avelng fom a medum of lowe o hghe ndex of efacon (say, an a/glass neface. Snce n hs case,, (15 s neve zeo. Howeve, he gh θ > θ π hand sde of equaon (16 vanshes when θ + θ = n whch case he denomnao becomes nfne. A hs pon, all ncden lgh whch s polazed n he plane angen o he plane of ncdence s ansmed. The value of θ a whch hs occus s known as Bewse s Angle θ B. Wng Snell s law a Bewse s angle, o π n sn θb = n snθ = n sn( θb = n cosθb, n an θb = (17 n whch s Bewse s Law. S Davd Bewse ( s mosly emembeed fo hs nvenon of he kaledoscope and fo opcal mpovemens of he mcoscope. Howeve, hs man expemens wee on he heoy of lgh and s uses. Hs fs pape, "Some Popees of Lgh", was publshed n Bewse's Law was named afe hm n 1814 when he made measuemens on he angle of maxmum polazaon usng baxal cysals. He was awaded all hee of he pncpal medals of he Royal Socey fo hs opcal eseach. These wee he Copley medal n 1815, he Rumfod medal n 1818, and he Royal medal n He was also knghed n 1831.

4 Bewse s Law may be undesood by he followng nuve agumen: he elecc feld veco s ansvese o he fowad decon. Howeve, a he pon whee he lgh polazed n he plane of ncdence sas o ene he glass he vbaons n he ansmed wave happen o be paallel o he OR ay n Fgue 3. Elemenay molecula dpoles nsde he maeal also oscllae along OR, and canno poduce a efleced elecomagnec wave popagang n he same decon. Ths means ha only he componen polazed nomal o he ncden plane, SOR, o paallel o he suface, wll be efleced. A hs pon, even f he ncden wave has boh polazaon componens (o s polazaon s andom, ay OR s compleely hozonally polazed and he elecc feld veco s esced o he plane ha s paallel o he efleced suface. Ths 100% lnea polazaon can only occu a one angle, because hs ccumsance s esced by he fac ha he angle beween he efleced ay, OR, and he efaced ay, OT, has o be 90 o. A gaph of he eflecvy (15-(16 fo he a-glass neface as a funcon of he angle of ncdence s shown n Fgue 4. A nomal ncdence, he paallel and pependcula polazaon waves ae physcally dencal, and have he same eflecvy. Noce ha even hough he glass s anspaen, 4% of he lgh s sll efleced. Howeve, as θ nceases, he paallel componen dops and he pependcula componen ses, unl a he Bewse's angle he values ae appoxmaely 0% and 15% of he nal nensy of he lgh especvely. A an ncden angle of 90 o, all of he ncden lgh s efleced, so he subsance acs as a mo. The Fesnel equaons can be esed by usng a pvong lab bench o vay he ncden angle of lase lgh ono a eflecng suface and measung he efleced nensy. A phoodeeco s used n hs lab o measue he lgh nensy. The phoodeeco s calbaed o geneae a volage popoonal o he lgh nensy, whch s hen measued wh a mulmee. A polaze beween he delecc suface and he phoodeeco s used o selec he paallel and pependcula componens of he efleced lgh. Expemenal Pocedue 1. The heoecal ndex of efacon fo he cown glass and acylc used n hs lab ae 1.61 and 1.49 especvely. Wh hese values, use Snell s Law fo θ, n equaons (15 and (16 o make a heoecal plo of he Fesnel Equaons smla o fgue 4.. Thoughou hs expemen ensue ha he appaaus s algned so ha boh he efleced and ansmed lase beams ae poned away fom he ohe sudens, and always keep ack of whee he beams ae. The lase you wll be usng fo hs expemen wll self poduce polazed lgh. Place he lase, polaze, and deeco ono he same opcal bench. Tun on he lase, volmee and he phoodeeco. Make sue he volmee s se o mv seng, and ha he deeco s se o he 0.1mV = 1Lux seng. 3. The fs ask s o calbae he polazes spaally, and wh espec o one anohe. Snce n hs expemen, you wll need o measue lgh paallel and pependcula o he plane of ncdence whch s deemned by you se-up, s mpoan o know wha value on he polazes

5 coespond o he (vecal and (hozonal planes. The ecangula polaze has s plane of ansmsson along he longe sde. Send he lase hough one of he ound polazes and he ecangula shee polaze oened vecally. Place a pape cad behnd boh polazes. Roae he ound polaze unl he ansmsson hough boh polazes s mnmzed. Make a noe of he oaon angle of you polaze. Ths angle coesponds o a hozonally polazed wave. Repea hs pocedue wh he second ound polaze. 4. Se up he bench as n fgue 4 wh he glass slde n he holde. Roae polaze 1 o ansm 45 polazaon. Ths wll ensue ha he lgh ncden ono you glass block has boh paallel and pependcula componens. If hee s lle o no lgh ansmed hough he polaze a 45, oae he lase unl a suffcen amoun of lgh s showng. 5. Se Polaze o ansm hozonally polazed lgh. Measue he dependence of he efleced nensy on he angle of ncdence fo hs angle vaed beween 0 o o 160 o. Take a leas 0 daa pons. Noe ha he poaco on he pvong bench eads wce he acual value of θ as seen n fgue 5. Noe as well ha he hegh of he beam may change because of he lab desks beng uneven. If hs occus, change he hegh of he deeco so ha he beam s hng he cene of he face of he phoodeeco. Ths s necessay because he phoodeeco does no have unfom sensvy acoss s face. 6. Se Polaze and he deeco on he fs opcal bench. Measue he nensy ansmed hough boh polazes. Use hs value o nomalze he daa obaned n sep 5. Plo he nomalzed daa on op of he heoecal cuve obaned n sep Repea seps 5 and 6 fo he vecally polazed wave. 8. By means of dec obsevaon wh a naked eye, fnd he angle a whch he eflecon of he paallel polazaon wave s mnmzed. Ths s you Bewse angle. Deemne he ndex of efacon fom equaon ( A he expemenal Bewse's angle, ake he polaze off he opcal bench. Roae he sunglasses povded beween he block and he deeco (as f wee akng he place of he polaze. Recod obsevaons made. 10. Repea seps 5-7 wh he ohe maeal (acylc vs. glass. 11. In sep fve, he polaze was placed afe he eflecng slde, fo analyss. Howeve, he polazaons n queson ae ndependen of one anohe and so he ansmsson o eflecon of a gven componen does no affec s ohogonal polazaon componen. Due o hs lneay of he opcs, a pos-selecon of he polazaon should be equvalen o a pe-selecon. Vefy hs by placng he polaze befoe he slde and epeang some of he measuemens. 1. Afe measuemens ae compleed, emembe o un off he phoodeeco o else he baees wll un ou.

6 Eo Analyss An neesng sysemac eo n hs lab s he fac ha he block eflecs moe han one do. As he angle of ncdence ges lage, he dsance beween he dos also ges lage. Afe a cean angle, hee dos can be seen, and hen fou dos (fgue 6. Ths occus fom he neo eflecons ha occu nsde he block. Ths becomes a souce of sysemac eo because he seconday eflecons ae no longe govened by he smple Fesnel fomulae (15 and (16. In addon, hee s nefeence beween he eflecons. To peven hese effecs, you should use a suffcenly hck block and ensue ha seconday eflecons do no h you deeco. Alhough he geneal pocedue n fndng he eo n an angle s jus half he smalles dvson, s dffeen n he case of he angle beween he opcal benches. The accuacy s educed because of he way ha he appaaus has o be moved. So, s up o you o decde a suable eo fo he angle, whn eason. Also, he eo n he phoodeeco s ±5%+0.1mV. Howeve, because he deeco s beng moved aound o accommodae whee he lase beam s, he eo s hghe. Agan, you should decde a suable eo fo he phoodeeco. The eo n he cos θ em can be found by sandad eo popagaon fom he eo n he angle. The unceany fo measung he angle of he polaze and analyze s half he smalles dvson on he poaco. To be ncluded wh you lab we-up 1. Sang wh equaon 6, deve equaon 1.. The heoecal plos made n sep A able of values obaned n sep 5 fo boh polazaons and boh maeals. Show boh he aw daa and ha nomalzed o he maxmum nensy found n sep On he heoecal plo of he Fesnel equaons plo he daa pons aken n he expemen wh eo bas. 5. A value fo Bewse s angle fom sep 8 wh eos. A value fo he ndex of efacon fo each plae wh eo. 6. Dscuss he obsevaons made wh he sunglasses. Ae he sunglasses polazed? Wha s he pupose of makng polazed sunglasses? Explan he sgnfcance of whch axs he sunglasses ae polazed a.