Chape 1 Reflecon and Refacon We ae now neesed n eplong wha happens when a plane wave avelng n one medum encounes an neface (bounday) wh anohe medum. Undesandng hs phenomenon allows us o undesand hngs lke: 1. ow an opcal lens woks 2. Why we ge glae off a pane of glass (and how o desgn an-glae wndows) 3. ow buldngs and walls nfluence cellula-phone sgnals 4. Why lgh bends when enes wae 5. Opcal fbes 1.1 Nomal Incdence We wll begn by lookng a a wave nomally ncden on he neface beween lossless meda. Medum 1 Medum 2 ε 1, µ 1 ε 2,µ 2 1.1.1 Felds In geneal, we mus allow fo hee dffeen waves n he sysem: 54
1. Incden wave () ˆ oe jk 1 () 1 ˆk ŷ o e jk 1 k 1 ω µ 1 ɛ 1 µ1 ɛ 1 (ˆk ẑ) 2. Refleced wave () ˆ oe +jk 1 () 1 ˆk ŷ o e +jk 1 (ˆk ẑ) 3. Tansmed wave () ˆ oe jk 2 () 1 ˆk ŷ o e jk 2 k 2 ω µ 2 ɛ 2 µ2 ɛ 2 (ˆk ẑ) We assume we know o. We mus deemne o and o. 1.1.2 Bounday Condons We fnally ge o use hose bounday condons fo somehng useful. Snce hee s no suface cuen on he bounday beween wo deleccs: A 0 we have: ˆn ẑ : ẑ ( 2 1 ) 0 0 ẑ ( 2 1 ) 0 0 1 (0) + (0) ˆ( o + o) 1 (0) + (0) ŷ ( o o) 2 (0) ˆ o 2 (0) ŷ o Theefoe, applcaon of he bounday condons gves: ẑ [ˆ o ˆ( o + o) ] 0 o (o + o) 0 (1.1) [ ẑ ŷ o ŷ ( o ] o) 0 o ( o o) 0 (1.2) 55
Subsuon of he esul o ( o + o) no (??) gves So, fo Nomal Incdence: (o + o) ( o ( o) 1 0 o + 1 ) ( o η2 + ( ) o o η2 + o + ( η2 + + + ( o 2η2 + Γ eflecon coeffcen + 2 τ ansmsson coeffcen + ( 1 o 1 ) ) o Γo ) o ) o τ o Noce ha we can we: o o + Γ o (1 + Γ) o as well as o τ o. Theefoe, fo nomal ncdence: τ 1 + Γ We also obseve ha f egon 2 s a pefec conduco, σ 2, ɛ c2, and 0. Ths esuls n Γ 1 whch s conssen wh he bounday condon ha he angenal elecc feld mus go o eo a he bounday of he conduco (sho ccu). Tansmsson Lne Analogue The epesson fo Γ s vey smla o wha we found n ansmsson lnes, wh he nnsc mpedance eplacng he chaacesc mpedance. In fac, hs sysem behaves vey much lke a ansmsson lne sysem, and we have he same sandng wave concep n egon 1 ha we have on a ansmsson lne. lecomagnec Wave leccal Tansmsson Lne V I η Z o Γ Γ 1 ˆ ( o e jk 1 + Γe jk 1 ) V V o + ( e jk + Γe jk) 1 ˆo (e ) jk1 + Γe jk 1 whch leads o he sandng wave paen: 1 o 1 + Γe j2k 1 1 ma o (1 + Γ ) 1 mn o (1 Γ ) 56
The sandng wave ao (SWR) s: S 1 ma 1 mn 1 + Γ 1 Γ Ths s eacly he same epesson we obaned fo ansmsson lnes. As n ansmsson lnes, he sandng wave paen s peodc wh peod l. e j2k1l e j2π k 1 l π 2π l π λ 1 l λ 1 2 Agan we see ha he dsance beween adjacen mama (o mnma) s a half wavelengh. Connung on wh he ansmsson lne analogue, he chaacesc mpedance of a s gven by µo η a 377 Ω. (1.3) ɛ o The chaacesc mpedance of a delecc s µo η 377 Ω, (1.4) ɛ ɛ o n whee n s he nde of efacon of he maeal. The chaacesc mpedance of a conduco s η (1 + j) So a conduco acs as a sho ccu. πfµ σ 0. (1.5) 1.1.3 Powe Flow n Regon 1 In egon 1, we have he felds: 1 ˆ o The me-aveage Poynng veco s: S av,1 1 } { 2 Re 1 1 1 { 2 Re o (e jk 1 + Γe jk 1 ) 1 ŷ o ( e jk 1 Γe jk 1 ) (e ) jk1 + Γe jk 1 o ( ) } e jk1 Γ e jk 1 ˆ ŷ 1 2 Re { o 2 [ 1 Γ e j2k 1 + Γe j2k 1 Γ 2]} ẑ Recall ha A A (a + jb) (a jb) j2b 2 Im{A}. S av,1 1 { 2 Re o 2 [ { }]} 1 Γ 2 + 2 Im Γe j2k 1 ẑ ẑ o 2 2 [ 1 Γ 2 ] 57
Noe ha S av ẑ o 2 2 S av ẑ o 2 2 Γ 2 Γ 2 S av So, S av,1 s smply he sum of he powe denses n he ncden and efleced waves. The aveage powe densy n egon 2 s: S av,2 ẑ 2 2 2 ẑ τ 2 o 2 Fo lossless meda (whch we ae wokng wh hee), Γ 2 Γ 2 and τ 2 τ 2. Theefoe: 2 ( ) 2 1 Γ 2 1 η2 + ( + ) 2 ( ) 2 ( ) 2 So: Ths means: η2 2 + η2 1 + 2 η2 2 η2 1 + 2 ( ) 2 4 ( ) 2 1 [ ] 2 2η2 τ 2 1 Γ 2 τ 2 S av,1 ẑ o 2 [1 Γ 2 ] ẑ o 2 τ 2 S av,2 2 2 1.1.4 Lossy Meda We can use ou developmens above, and smply make he subsuons: jk 1 γ 1 jk 2 γ 2 η c1 η c2 Then, all of he equaons wll wok fne fo lossy meda. 1.2 Oblque (Non-nomal) Incdence We need o fs become famla wh he dea of waves avelng n a decon ohe han ±. In ode fo he phase pogesson o occu along he ˆk decon, we need o use he componen of 58
he poson veco ha pons along he ˆk decon. So, ou dsance vaable s: The plane wave eponenal s heefoe ceang he felds: ξ ˆk (ˆ sn θ + ẑ cos θ ) (ˆ + ẑ) sn θ + cos θ e jk 1ˆk e jk 1( sn θ + cos θ ) ŷ oe jk 1( sn θ + cos θ ) 1 ˆk 1 (ˆ sn θ + ẑ cos θ ) ŷ oe jk 1( sn θ + cos θ ) o ( ˆ cos θ + ẑ sn θ )e jk 1( sn θ + cos θ ) θ kˆ θ kˆ θ θ Noe ha you can also deduce usng gonomey. Fnally, we mus make some defnons: Plane of Incdence: Pependcula Polaaon: Paallel Polaaon: Plane conanng nomal o bounday and ˆk veco pependcula o he plane of ncdence. Also called Tansvese lecc (T) polaaon. paallel o he plane of ncdence. Also called he Tansvese Magnec (TM) polaaon. 59
θ θ Pependcula Paallel 1.2.1 Pependcula (T) Polaaon kˆ θ θ θ kˆ 1.2.2 Felds 1. Incden wave ŷ oe jk 1( sn θ + cos θ ) ( ˆ cos θ + ẑ sn θ ) o e jk 1( sn θ + cos θ ) 2. Refleced wave ŷ oe jk 1( sn θ cos θ ) (ˆ cos θ + ẑ sn θ ) o e jk 1( sn θ cos θ ) 3. Tansmed wave ŷ oe jk 2( sn θ + cos θ ) ( ˆ cos θ + ẑ sn θ ) o e jk 2( sn θ + cos θ ) We now have 4 unknowns: o, o, θ, θ. 60
1.2.3 Bounday Condons Tangenal ˆn ( + ) ˆn 0 0 oe jk 1 sn θ + oe jk 1 sn θ oe jk 2 sn θ Tangenal ˆn ( + ) ˆn 0 0 o cos θ e jk 1 sn θ + o cos θ e jk 1 sn θ o cos θ e jk 2 sn θ In ode fo he wo equaons o be sasfed fo all, he agumens of he eponenals mus be equal. We call hs he phase machng condon. Ths poduces Snell s Laws: θ θ k 1 sn θ k 2 sn θ k 1 sn θ k 1 sn θ k 2 sn θ Snell s Law of Reflecon Snell s Law of Refacon We obseve ha hs phase machng condon ndcaes he decon of popagaon of he efleced and ansmed waves. A nce pcue can be dawn of hs: k 2 > k 1 θ k 1 k 2 θ k 1 sn θ k 1 sn θ k 2 sn θ angenal componen of ˆk k 1 equals angenal componen of ˆk k 2 Ths mples ha he phase s connous acoss he bounday We can now poceed wh applcaon of he bounday condons: o + o o o cos θ o cos θ o cos θ Solvng poduces: cos θ ( o o) cos θ ( o + o) o / cos θ / cos θ / cos θ + / cos θ o Γ o o 2 / cos θ / cos θ + / cos θ o τ o (1 + Γ ) o 61
1.2.4 Paallel (TM) Polaaon kˆ θ θ θ kˆ 1.2.5 Felds 1. Incden wave (ˆ cos θ ẑ sn θ ) oe jk 1( sn θ + cos θ ) ŷ o e jk 1( sn θ + cos θ ) 2. Refleced wave 3. Tansmed wave (ˆ cos θ + ẑ sn θ ) oe jk 1( sn θ cos θ ) ŷ o e jk 1( sn θ cos θ ) (ˆ cos θ ẑ sn θ ) oe jk 2( sn θ + cos θ ) ŷ o e jk 2( sn θ + cos θ ) 1.2.6 Bounday Condons Tangenal ˆn ( + ) ˆn 0 0 o cos θ e jk 1 sn θ + o cos θ e jk 1 sn θ o cos θ e jk 2 sn θ Tangenal ˆn ( + ) ˆn 0 0 o e jk 1 sn θ o e jk 1 sn θ o e jk 2 sn θ Phase machng agan poduces Snell s Laws. Then, applcaon of he bounday condons poduces o cos θ cos θ o Γ cos θ + cos θ o o 2 cos θ o τ cos θ + cos θ o 62
1.3 Toal Inenal Reflecon Suppose we have he case whee k 1 > k 2 (geneally ɛ 1 > ɛ 2 ). Then, ou k-veco dagam looks lke: θ k k 2 1 k 1 > k 2 If θ θ ges oo lage, we have he case whee k 2 sn θ can be lage enough o equal k 1 sn θ. When he ncdence angle θ s such ha we call θ θ c he ccal angle k 1 sn θ k 2 (so θ 90 ) In hs case, he ansmed wave avels paallel o he neface so ha he consan phase planes ae paallel o he y- plane. Noce ha f θ > θ c : k 2 sn θ k 1 sn θ sn θ k 1 sn θ > 1 k 2 A eal angle θ canno sasfy hs equaon. In fac, θ becomes a comple numbe. Rahe han deemnng θ, we can deemne sn θ and cos θ. sn θ s deemned by Snell s Law as above, whch can be ewen sn θ k 1 sn θ β k 2 k 2 Then ( ) 2 (k1 ) 2 cos θ 1 sn 2 k1 θ 1 sn 2 θ ±j sn 2 θ 1 ±j α k 2 k 2 k 2 We wll chose he negave oo fo easons o be seen below. The popagaon em n he agumen of he eponenal fo he ansmed wave becomes k 2 ( sn θ + cos θ ) k 2 ( β k 2 j α k 2 ) β jα Theefoe, we can we he ansmed feld as ŷ oe jk 2( sn θ + cos θ ) ŷ oe α e jβ The wave decays n, bu phase pogesson (popagaon) occus n. Noe ha a θ θ c, cos θ 0 so α 0 (no decay). We call such a wave an evanescen wave. The eflecon coeffcen fo hs case s Γ / cos θ / cos θ j/α / cos θ A / cos θ + / cos θ j /α + / cos θ A Γ A 1 A 63
The same s ue fo Γ. Theefoe, no eal powe s ansfeed acoss he neface. All powe s efleced. We heefoe call hs oal nenal eflecon. We can also we: ( ˆ cos θ + ẑ sn θ ) o e jk 2( sn θ + cos θ ) (ˆj sn 2 θ 1 + ẑ sn θ ) o e α e jβ S oe α jβ [ ] o e e α e jβ ẑj sn 2 θ 1 + ˆ sn θ o 2 [ˆ e 2α sn θ + ẑj sn 2 θ 1 Ths ndcaes ha he powe avelng n he ẑ decon s eacve (soed enegy). The eal powe s: S av 1 o 2 e 2α ˆ sn θ 2 ] whch flows n he ˆ decon. Noe ha hs powe flow smply epesens a powe flow elaed o phase pogesson n he decon. I akes no powe fom he ncden wave o susan hs. phase planes 1.4 Toal Tansmsson (Bewse s Angle) Suppose we wan Γ 0 (no eflecon) Γ / cos θ / cos θ / cos θ + / cos θ 0 cos θ cos θ µ 2 [ 1 sn 2 ] θ ɛ 2 We can hen solve hs o oban µ 1 ɛ 1 [ 1 sn 2 θ ] µ 1 ɛ 1 sn θ [ 1 ( k1 k 2 1 (µ 1 ɛ 2 /µ 2 ɛ 1 ) 1 (µ 1 /µ 2 ) 2 ) 2 sn 2 θ ] µ 1 ɛ 1 [ 1 µ ] 1ɛ 1 sn 2 θ µ 2 ɛ 2 If µ 1 µ 2, sn θ whch s mpossble. So, we can have oal ansmsson fo he pependcula polaaon n non-magnec meda. Fo he paallel polaaon: Γ cos θ cos θ cos θ + cos θ 0 cos θ cos θ 64
The soluon fo hs (followng an analogous pocedue) s: 1 (ɛ 1 µ 2 /ɛ 2 µ 1 ) sn θ 1 (ɛ 1 /ɛ 2 ) 2 ee, f µ 1 µ 2 : sn θ B cos θ B 1 ɛ 1 /ɛ 2 1 (ɛ 1 /ɛ 2 ) 2 1 1 + ɛ 1 /ɛ 2 1 ɛ 1 /ɛ 2 1 1 + ɛ 1 /ɛ 2 1 + ɛ 1 /ɛ 2 We call θ B Bewse s Angle. an θ B sn θ B ɛ2 cos θ B ɛ 1 ( ) θ B an 1 ɛ2 ɛ 1 To eplan Bewse s angle physcally, we have o evs he pemvy. Recall ha maeals ae made of aoms whose chage wll shf slghly unde he nfluence of an elecc feld (he aoms o molecules wll polae, wh posve and negave chage sepaang a b o made a dpole). The pemvy ɛ epesens he suscepbly of he medum o hs algnmen. When a me-vayng feld hs he medum, hese dpoles oscllae. Ths chage moon epesens a cuen whch causes e-adaon of he elecomagnec feld. Ths eplans why a efleced feld ge s ceaed. We wll lae see ha dpoles do no adae along he pola as. The oscllang dpole n maeal 2 won adae n he decon of he eflecon. Theefoe, 0. I can be shown ha θ B + θ 90 θ θ + - θ Noe ha fo he pependcula polaaon, he dpole oscllaon s nomal o he page, so we wll always have adaon n he efleced decon. Theefoe, hee s no Bewse angle fo hs case. 65