Reflection and Refraction

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Moris McKenzie
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1 Chape 3 Refleon and Refaon As we know fom eveyday expeene, when lgh aves a an nefae beween maeals paally efles and paally ansms. In hs hape, we examne wha happens o plane waves when hey popagae fom one maeal (haaezed by ndes n o even by omplex ndex N ) o anohe maeal. We wll deve expessons o quanfy he amoun of efleon and ansmsson. The esuls depend on he angle of ndene (.e. he angle beween k and he sufae nomal) as well as on he oenaon of he ele feld (alled polazaon no o be onfused wh P, also alled polazaon). In hs hape, we onsde only soop maeals (e.g. glass); n hape 5 we onsde ansoop maeals (e.g. a ysal). As we develop he onneon beween nden, efleed, and ansmed lgh waves, 1 seveal famla elaonshps wll emege naually (e.g. Snell s law and Bewse s angle). The fomalsm also desbes polazaon-dependen phase shfs upon efleon (espeally neesng n he ase of efleons fom meals). Fo smply, we nally negle he magnay pa of he efave ndex. Eah plane wave s hus haaezed by a eal wave veo k. We wll we eah plane wave n he fom E(, ) E 0 exp (k ω), whee, as usual, only he eal pa of he feld oesponds o he physal feld. The eson o eal efave ndes s no as seous as mgh seem. The use of he lee n nsead of N hadly maes. The mah s all he same, whh demonsaes he powe of he omplex noaon. We an smply updae ou expessons n he end o nlude omplex efave ndes, bu n he mean me s ease o hnk of absopon as neglgble. 3.1 Refaon a an Inefae Consde a plana bounday beween wo maeals wh dffeen ndes. Le ndex n haaeze he maeal on he lef, and he ndex n haaeze he 1 See M. Bon and E. Wolf, Pnples of Ops, 7h ed., Se. 1.5 (Cambdge Unvesy Pess, 1999). 71

2 72 Chape 3 Refleon and Refaon z-axs x-axs deed no page Fgue 3.1 Inden, efleed, and ansmed plane wave felds a a maeal nefae. maeal on he gh, as deped n he Fg When a plane wave avelng n he deon k s nden on he bounday fom he lef, gves se o a efleed plane wave avelng n he deon k and a ansmed plane wave avelng n he deon k. The nden and efleed waves exs only o he lef of he maeal nefae, and he ansmed wave exss only o he gh of he nefae. The angles θ, θ, and θ gve he angles ha eah espeve wave veo (k, k, and k ) makes wh he nomal o he nefae. Fo smply, we ll assume ha boh of he maeals ae soop hee. (Chape 5 dsusses efaon fo ansoop maeals.) In hs ase, k, k, and k all le n a sngle plane, efeed o as he plane of ndene, (.e. he plane epesened by he sufae of hs page). We ae fee o oen ou oodnae sysem n many dffeen ways (and evey exbook seems o do dffeenly!). 2 We hoose he y z plane o be he plane of ndene, wh he z-deon nomal o he nefae and he x-axs ponng no he page. The ele feld veo fo eah plane wave s onfned o a plane pependula o s wave veo. We ae fee o deompose he feld veo no abay omponens as long as hey ae pependula o he wave veo. I s usomay o hoose one of he ele feld veo omponens o be ha whh les whn he plane of ndene. We all hs p-polazed lgh, whee p sands fo paallel o he plane of ndene. The emanng ele feld veo omponen s deed nomal o he plane of ndene and s alled s-polazed lgh. The s sands fo senkeh, a Geman wod meanng pependula. Usng hs sysem, we an deompose he ele feld veo E no s p- polazed omponen and s s-polazed omponen, as deped n Fg The s omponen s epesened by he al of an aow ponng no he page, o he x-deon n ou onvenon. The ohe felds E and E ae smlaly spl no s and p omponens as ndaed n Fg All feld omponens ae onsdeed o be posve when hey pon n he deon of he espeve aows. 3 Noe ha he s-polazed omponens ae paallel fo all hee plane waves, wheeas he p-polazed omponens ae no (exep a nomal ndene) beause eah plane wave avels n a dffeen deon. By nspeon of Fg. 3.1, we an we he vaous wave veos n ems of he ŷ and ẑ un veos: (ŷsnθ ) k k + ẑosθ (ŷsnθ ) k k ẑosθ (3.1) (ŷsnθ ) k k + ẑosθ Also by nspeon of Fg. 3.1 (followng he onvenons fo he ele felds ndaed by he aows), we an we he nden, efleed, and ansmed 2 Fo example, ou onvenon s dffeen han ha used by E. Heh, Ops, 3d ed., Se (Massahuses: Addson-Wesley, 1998). 3 Many exbooks daw he aow fo n he deon oppose of ous. Howeve, ha hoe leads o an awkwad suaon a nomal ndene (.e. θ θ 0) whee he aows fo he nden and efleed felds ae paallel fo he s-omponen bu an paallel fo he p-omponen.

3.1 Refaon a an Inefae 73 felds n ems of ˆx, ŷ, and ẑ: E E E (ŷosθ ) ẑsnθ + ˆxE e k (y snθ +z osθ ) ω (ŷosθ ) + ẑsnθ + ˆxE e k (y snθ z osθ ) ω (ŷosθ ) ẑsnθ + ˆxE e k (y snθ +z osθ ) ω (3.

As explaned n appendx 3.A, Maxwell s equaons eque he omponens of E ha ae paallel o he nefae o be he same on ehe sde of he bounday.

Ths means ha a z 0 he ˆx and ŷ omponens of he ombned nden and efleed felds mus equal he oespondng omponens of he ansmed feld: ŷosθ + ˆxE e (k y snθ ω ) + ŷosθ + ˆxE e (k y snθ ω ) ŷosθ + ˆx e (k y

3 3.1 Refaon a an Inefae 73 felds n ems of ˆx, ŷ, and ẑ: E E E (ŷosθ ) ẑsnθ + ˆxE e k (y snθ +z osθ ) ω (ŷosθ ) + ẑsnθ + ˆxE e k (y snθ z osθ ) ω (ŷosθ ) ẑsnθ + ˆxE e k (y snθ +z osθ ) ω (3.2) Eah feld has he fom (2.8). We have ulzed he k-veos (3.1) n he exponens of (3.2). Now we ae eady o onne he felds on one sde of he nefae o he felds on he ohe sde. Ths s done usng bounday ondons. As explaned n appendx 3.A, Maxwell s equaons eque he omponens of E ha ae paallel o he nefae o be he same on ehe sde of he bounday. In ou oodnae sysem, he ˆx and ŷ omponens ae paallel o he nefae, wheeas z 0 defnes he nefae. Ths means ha a z 0 he ˆx and ŷ omponens of he ombned nden and efleed felds mus equal he oespondng omponens of he ansmed feld: ŷosθ + ˆxE e (k y snθ ω ) + ŷosθ + ˆxE e (k y snθ ω ) ŷosθ + ˆx e (k y snθ ω ) (3.3) Fgue 3.2 Anmaon of s- and p-polazed felds nden on an nefae as he angle of ndene s vaed. Sne hs equaon mus hold fo all onevable values of and y, we ae ompelled o se all he phase faos n he omplex exponenals equal o eah ohe. The me poon of he phase faos eques he fequeny of all waves o be he same: ω ω ω ω (3.4) (We ould have guessed ha all fequenes would be he same; ohewse wave fons would be annhlaed o eaed a he nefae.) Smlaly, equang he spaal ems n he exponens of (3.3) eques k snθ k snθ k snθ (3.5) Now eall fom (2.19) he elaons k k n ω/ and k n ω/. Wh hese elaons, (3.5) yelds he law of efleon and Snell s law θ θ (3.6) n snθ n snθ (3.7) The hee angles θ, θ, and θ ae no ndependen. The efleed angle mahes he nden angle, and he ansmed angle obeys Snell s law. The phenomenon of efaon efes o he fa ha θ and θ ae dffeen. Tha s, lgh bends as ansms hough an nefae. Wllebod Snell (o Snellus) ( , Duh) was an asonome and mahemaan bon n Leden, Nehelands. In 1613 he sueeded hs fahe as pofesso of mahemas a he Unvesy of Leden. He was an aomplshed mahemaan, developng a new mehod fo alulang π as well as an mpoved mehod fo measung he umfeene of he eah. He s mos famous fo hs edsovey of he law of efaon n (The law was known (n able fom) o he anen Geek mahemaan Polemy, o Pesan engnee Ibn Sahl (900s), and o Polsh phlosophe Welo (1200s).) Snell auhoed seveal books, nludng one on gonomey, publshed a yea afe hs deah. (Wkpeda)

4 74 Chape 3 Refleon and Refaon Beause he exponens ae all denal, (3.3) edues o wo elavely smple equaons (one fo eah dmenson, ˆx and ŷ): and ( + E (3.8) ) + osθ osθ (3.9) We have deved hese equaons fom he bounday ondon (3.54) on he paallel omponen of he ele feld. Ths se of equaons has fou unknowns (, E,, and ), assumng ha we pk he nden felds. We eque wo addonal equaons o solve he sysem. These ae obaned usng he sepaae bounday ondon on he paallel omponen of magne felds gven n (3.58) (also dsussed n appendx 3.A). Fom Faaday s law (1.3), we have fo a plane wave (see (2.56)) B k E ω n û E (3.10) whee û k/k s a un veo n he deon of k. We have also ulzed (2.19) fo a eal ndex. Ths expesson s useful fo wng B, B, and B n ems of he ele feld omponens ha we have aleady nodued. When njeng (3.1) and (3.2) no (3.10), he nden, efleed, and ansmed magne felds un ou o be n B n B n B n ˆx ˆx ˆx ˆx + ( ) ẑsnθ + ŷosθ e k (y snθ +z osθ ) ω ( ) ẑsnθ ŷosθ e k (y snθ z osθ ) ω ŷosθ + n ( ẑsnθ + ŷosθ ) e k (y snθ +z osθ ) ω ˆx E ŷosθ n (3.11) Nex, we apply he bounday ondon (3.58), namely ha he omponens of B paallel o he nefae (.e. n he ˆx and ŷ dmensons) ae he same 4 on ehe sde of he plane z 0. Sne we aleady know ha he exponens ae all equal and ha θ θ and n n, he bounday ondon gves ˆx + ŷosθ (3.12) As befoe, (3.12) edues o wo elavely smple equaons (one fo he ˆx dmenson and one fo he ŷ dmenson): n ( and n ( ) n (3.13) ) E osθ n osθ (3.14) These wo equaons ogehe wh (3.8) and (3.9) allow us o solve fo he efleed E and ansmed felds E fo he s and p polazaon omponens. Howeve, (3.8), (3.9), (3.13), and (3.14) ae no ye n he mos onvenen fom. 4 We assume he pemeably µ 0 s he same eveywhee no magne effes.

3.2 The Fesnel Coeffens 75 3.2 The Fesnel Coeffens Augusn Fesnel fs deved he equaons n he pevous seon. Sne he lved well befoe Maxwell s me, he dd no have he benef of Maxwell s equaons as we have.

5 3.2 The Fesnel Coeffens The Fesnel Coeffens Augusn Fesnel fs deved he equaons n he pevous seon. Sne he lved well befoe Maxwell s me, he dd no have he benef of Maxwell s equaons as we have. Insead, Fesnel hough of lgh as ansvese mehanal waves popagang whn maeals. (Fesnel was naually a poponen of lumnfeous ehe.) Insead of elang he paallel omponens of he ele and magne felds aoss he bounday beween he maeals, Fesnel used he pnple ha he wo maeals should no slp elave o eah ohe a he bounday. Ths glung of he maeals a he nefae also fobds he possbly of gaps o he lke fomng a he nefae as he wo maeals expeene wave vbaons. Ths mehanal appoah o lgh woked splenddly, avng a he same esuls ha we obaned fom ou moden vewpon. Fesnel woe he elaonshps beween he vaous plane waves deped n Fg. 3.1 n ems of oeffens ha ompae he efleed and ansmed feld ampludes o hose of he nden feld. In he followng example, we llusae hs poedue fo s-polazed lgh. I s lef as a homewok exese o solve he equaons fo p-polazed lgh (see P3.1). Example 3.1 Calulae he ao of ansmed feld o he nden feld and he ao of he efleed feld o nden feld fo s-polazed lgh. Soluon: We we (3.8) and (3.14) as + E Addng hese wo equaons yelds Afe a lle eaangemen we ge and n osθ (3.15) n osθ n osθ (3.16) n osθ 2n osθ n osθ + n osθ (3.17) To ge he ao of efleed feld o nden feld, we suba he equaons n (3.15) o ge 2E 1 n osθ (3.18) n osθ We dvde (3.18) by (3.16), and afe smplfaon ave a n osθ n osθ n osθ + n osθ (3.19) Augusn Fesnel ( , Fenh) was bon n Bogle, Fane, he son of an ahe. As a hld, he was slow o develop and sll ould no ead when he was egh yeas old, bu by age sxeen he exelled and eneed he Éole Polyehnque whee he eaned dsnon. As a young man, Fesnel began a suessful aee as an engnee, bu he los hs pos n 1814 when Napoleon euned o powe. (Fesnel had suppoed he Boubons.) Ths dul yea was when Fesnel uned hs aenon o ops. Fesnel beame a majo poponen of he wave heoy of lgh and fou yeas lae woe a pape on daon fo whh he was awaded a pze by he Fenh Aademy of Senes. A yea lae he was apponed ommssone of lghhouses, whh movaed he nvenon of he Fesnel lens (sll used n many ommeal applaons). Fesnel was unde appeaed befoe hs unmely deah fom ubeuloss. Many of hs papes dd no make no pn unl yeas lae. Fesnel made huge advanes n he undesandng of eeon, daon, polazaon, and befngene. In 1824 Fesnel woe o Thomas Young, All he omplmens ha I have eeved fom Aago, Laplae and Bo neve gave me so muh pleasue as he dsovey of a heoe uh, o he onmaon of a alulaon by expemen. Augusn Fesnel s a heo of one of he auhos of hs exbook. (Wkpeda)

6 76 Chape 3 Refleon and Refaon The ao of he efleed and ansmed feld omponens o he nden feld omponens ae spefed by he Fesnel oeffens, whh ae defned as follows: s E n osθ n osθ snθ osθ snθ osθ sn(θ θ ) n osθ + n osθ snθ osθ + snθ osθ sn(θ + θ ) s 2n osθ n osθ + n osθ 2snθ osθ snθ osθ + snθ osθ 2snθ osθ sn(θ + θ ) p n osθ n osθ snθ osθ snθ osθ an(θ θ ) n osθ + n osθ snθ osθ + snθ osθ an(θ + θ ) p 2n osθ n osθ + n osθ 2snθ osθ snθ osθ + snθ osθ (3.20) (3.21) (3.22) 2snθ osθ sn(θ + θ )os(θ θ ) (3.23) Fgue 3.3 The Fesnel oeffens ploed vesus θ fo he ase of an a-glass nefae wh n 1 and n 1.5. All of he above foms of he Fesnel oeffens ae poenally useful, dependng on he poblem a hand. Remembe ha he angles n he oeffen ae no ndependenly hosen, bu ae subje o Snell s law (3.7). Snell s law has been used o podue he alenave expessons fom he fs. The Fesnel oeffens pn down he ele feld ampludes on he wo sdes of he bounday. They also keep ak of phase shfs a a bounday. In Fg. 3.3 we have ploed he Fesnel oeffens fo he ase of an a-glass nefae. Noe ha he efleon oeffens ae somemes negave n hs plo, whh oesponds o a phase shf of π upon efleon (noe e π 1). Lae we wll see ha when absobng maeals ae enouneed, moe omplaed phase shfs an ase due o he omplex ndex of efaon. 3.3 Refleane and Tansmane We ofen wan o know he faon of powe ha efles fom o ansms hough an nefae. Enegy onsevaon eques he nden powe o balane he efleed and ansmed powe: P P + P (3.24) Moeove, he powe sepaaes leanly no powe assoaed wh s- and p- polazed felds: P P + P and P (p) P (p) + P (p) (3.25) Sne powe s popoonal o nensy (.e. powe pe aea) and nensy s popoonal o he squae of he feld amplude. We an we he faon of efleed powe, alled efleane, n ems of ou pevously defned Fesnel

7 3.3 Refleane and Tansmane 77 oeffens: R s P P I I E 2 2 s 2 and R p The oal efleed nensy s heefoe I I + I (p) P (p) P (p) I (p) I (p) 2 2 p 2 (3.26) R s I + R p I (p) (3.27) whee, aodng o (2.62), he oal nden nensy s gven by I I + I (p) 1 E 2 n ɛ (3.28) Fom (3.25) and (3.26), he ansmed powe s P P P (1 R s )P and P (p) P (p) + P (p) ( ) (p) 1 R p P (3.29) Fom hs expesson we see ha he faon of he powe ha ansms, alled he ansmane, s T s P P 1 R s and T p P (p) P (p) 1 R p (3.30) Fgue 3.4 shows ypal efleane and ansmane values fo an a-glass nefae. You mgh be supsed a fs o lean ha T s s 2 and T p p 2 (3.31) Howeve, eall ha he ansmed nensy (n ems of he ansmed felds) depends also on he efave ndex. The Fesnel oeffens s and p elae he bae ele felds o eah ohe, wheeas he ansmed nensy s I I + I (p) 1 2 n ɛ (3.32) In vew of (3.28) and (3.32), we expe T s and T p o depend on he ao of he efave ndes n and n n addon o s 2 o p 2. Thee s anohe moe suble eason fo he nequales n (3.31). Consde a laeal sp of lgh assoaed wh a plane wave nden upon he maeal nefae n Fg Upon efaon no he seond medum, he sp s seen o hange s wdh by he fao osθ /osθ. Ths s a puely geomeal effe, owng o he hange n popagaon deon a he nefae. Sne powe s nensy mes aea, he ansmane pks up hs geomeal fao va he ao of he aeas A /A as follows: T s P P T p P (p) P (p) I A n osθ I s 2 A n osθ I (p) A n osθ I (p) p 2 A n osθ (no vald f oal nenal efleon) (3.33) Fgue 3.4 The efleane and ansmane ploed vesus θ fo he ase of an a-glass nefae wh n 1 and n 1.5. Fgue 3.5 Lgh efang no a sufae

78 Chape 3 Refleon and Refaon Noe ha (3.33) s vald only f a eal angle θ exss; does no hold when he nden angle exeeds he al angle fo oal nenal efleon, dsussed n seon 3.5. In ha suaon, we mus sk wh (3.

8 78 Chape 3 Refleon and Refaon Noe ha (3.33) s vald only f a eal angle θ exss; does no hold when he nden angle exeeds he al angle fo oal nenal efleon, dsussed n seon 3.5. In ha suaon, we mus sk wh (3.30). Example 3.2 Show analyally ha R p + T p 1, whee R p s gven by (3.26) and T p s gven by (3.33). Davd Bewse ( , Sosh) was bon n Jedbugh, Soland. Hs fahe was a eahe and waned Davd o beome a legyman. A age welve, Davd wen o he Unvesy of Ednbugh fo ha pupose, bu hs nlnaon fo naual sene soon beame appaen. He beame lensed o peah, bu hs neess n sene dsaed hm fom ha pofesson, and he spen muh of hs me sudyng daon. Takng an empal appoah, Bewse ndependenly dsoveed many of he same hngs usually eded o Fesnel. He even made a dop appaaus fo lghhouses befoe Fesnel developed hs. Bewse beame somewha famous n hs day fo he developmen of he kaledosope and seeosope fo enjoymen by he geneal publ. Bewse was a pol sene we and edo houghou hs lfe. Among hs woks s an mpoan bogaphy of Isaa Newon. He was knghed fo hs aomplshmens n (Wkpeda) Soluon: Fom (3.22) we have R p n osθ n osθ n osθ + n osθ Fom (3.23) and (3.33) we have Then 3.4 Bewse s Angle n2 os2 θ 2n n osθ osθ + n 2 os2 θ (n osθ + n osθ ) 2 T p n osθ n osθ 2 2n osθ n osθ + n osθ 4n n osθ osθ (n osθ + n osθ ) 2 R p + T p n2 os2 θ + 2n n osθ osθ + n 2 os2 θ (n osθ + n osθ ) 2 (n osθ + n osθ ) 2 (n osθ + n osθ ) ompleely s-polazed efleon 100% p-ansmsson Noe p and R p go o zeo a a ean angle n Fgs. 3.3 and 3.4, ndang ha no p-polazed lgh s efleed a hs angle. Ths behavo s que geneal, as we an see fom he fnal fom of he Fesnel oeffen fomula fo p n (3.22), whh has an(θ + θ ) n he denomnao. Sne he angen blows up a π/2, he efleon oeffen goes o zeo when θ + θ π 2 (equemen fo zeo p-polazed efleon) (3.34) Fgue 3.6 Bewse s angle ondes wh he suaon whee k and k ae pependula. By nspeng Fg. 3.1, we see ha hs ondon ous when he efleed and ansmed wave veos, k and k, ae pependula o eah ohe (see also Fg. 3.6). If we nse (3.34) no Snell s law (3.7), we an solve fo he nden angle θ ha gves se o hs speal umsane: n snθ n sn( π 2 θ ) n osθ (3.35)

9 3.5 Toal Inenal Refleon 79 The angle ha sasfes hs equaon, n ems of he efave ndes, s eadly found o be θ B an 1 n n (3.36) We have eplaed he spef θ wh θ B n hono of S Davd Bewse who fs dsoveed he phenomenon. The angle θ B s alled Bewse s angle. A Bewse s angle, no p-polazed lgh efles (see L 3.4). Physally, he p-polazed lgh anno efle beause k and k ae pependula. A efleon would eque he mosop dpoles a he sufae of he seond maeal o adae along he axes, whh hey anno do. Maxwell s equaons know abou hs, and so eveyhng s nely onssen. 3.5 Toal Inenal Refleon Osllang Dpole Fgue 3.7 The nensy adaon paen of an osllang dpole as a funon of angle. Noe ha he dpole does no adae along he axs of osllaon, gvng se o Bewse s angle fo efleon. 90 Fom Snell s law (3.7), we an ompue he ansmed angle n ems of he nden angle: ( ) θ sn 1 n snθ (3.37) n The angle θ s eal only f he agumen of he nvese sne s less han o equal o one. If n > n, we an fnd a al angle beyond whh he agumen begns o exeed one: θ sn 1 n n (3.38) When θ > θ, hen hee s oal nenal efleon and we an dely show ha R s 1 and R p 1 (see P3.9). 5 To demonsae hs, one ompues he Fesnel oeffens (3.20) and (3.22) whle employng he followng subsuon: osθ 1 sn 2 θ n2 n 2 sn 2 θ 1 (θ > θ ) (3.39) (see P0.19). In hs ase, θ s a omplex numbe. Howeve, we do no assgn geomeal sgnfane o n ems of any deon. Aually, we don even need o know he value fo θ ; we need only he values fo snθ and osθ, as spefed by Snell s law (3.7) and (3.39). Even hough snθ s geae han one and osθ s magnay, we an use he values o ompue s, p, s, and p. (Complex noaon s wondeful!) Upon subsuon of (3.39) no he Fesnel efleon oeffens (3.20) and (3.22) we oban s n n osθ n 2 sn 2 θ n 2 1 n n osθ + n 2 sn 2 θ n 2 1 (θ > θ ) (3.40) 5 M. Bon and E. Wolf, Pnples of Ops, 7h ed., Se (Cambdge Unvesy Pess, 1999).

calculating electromagnetic

calculating electromagnetic Theoeal mehods fo alulang eleomagne felds fom lghnng dshage ajeev Thoapplll oyal Insue of Tehnology KTH Sweden ajeev.thoapplll@ee.kh.se Oulne Despon of he poblem Thee dffeen mehods fo feld alulaons - Dpole