23. Fresnel Equations

Størrelse: px

Begynne med side:

Download "23. Fresnel Equations"

Christina Edvardsen
7 år siden
Visninger:

1 3. Fesel quaos M Waves a boudaes Fesel quaos: Refleco ad Tasmsso Coeffces Bewse s Agle Toal Ieal Refleco (TIR) vaesce Waves The Complex Refacve Idex Refleco fom Meals

2 We wll deve he Fesel equaos : efleco coeffce T cos s cos + s cos + s cos + s 1 : T asmsso coeffce cos cos + s cos cos + s asme d cde 1

3 M Waves a a Ieface Icde beam: o exp Refleced beam: o exp Tasmed beam: o exp ( k ω ) ( k ω ) ( k ω ) k k k k 1 0 k 1 0 k 0 1 o k k o k o T mode mode 1 1 Noe he defo of he posve -feld decos boh cases.

4 M Waves a a Ieface Icde beam: oexp Refleced beam: o exp Tasmed beam: o exp ( k ω ) ( k ω) ( k ω ) A he bouday bewee he wo meda ( he x y plae), all waves mus exs smulaeously, ad he ageal compoe mus be equal o boh sdes of he eface. Theefoe, fo all me ad fo all bouday pos o heeface, ) ) ) + ) o exp ( k ω) ) o exp + ( k ω) ) o exp ( k ω) 1 o k ˆ k o k o Assumg ha he wave ampludes ae cosa, he oly way ha hs ca be ue ove he ee eface ad fo all s f ( k ω) ( k ω) ( k ω) : Phase machg a he bouday :!

5 M Waves a a Ieface Phase machg codo: ( k ) ( k ) ( k ω ω ω ) A 0, hs esuls ω ω ω ω ω ω (Fequecy does o chage a he bouday!) A 0, hs esuls k k k k,, cosa he equao fo a plae pepedcula o k,, ad. k, k, ad k ae coplaa he plae of cdece. (Phases o he bouday does o chage!) 1 k k k k 0 k 1 0 k k Nomal k k k k k 1 0 k 1 0 k 0 x

6 M Waves a a Ieface A 0, k k k cosa k ι k 1 x Cosdeg he elao fo he cde ad efleced beams, k k k s k s k Sce he cde ad efleced beams ae he same medum, k ω k s s : law of efleco c k 0 k 1 0 Cosdeg he elao fo he cde ad asmed beams, k k k s k s k x Bu he cde ad asmed beams ae dffee meda, k ω c k ω c s s : law of efaco k Nomal k

7 Developme of he Fesel quaos Fom Maxwell ' s M feld heoy, we have he bouday codos a he eface fo he Tcase: T-case + B cos B cos B cos The above codos mply ha he ageal compoes of boh ad B ae equal o boh sdes of he eface. We have also assumed ha μ μ μ, as s ue fo mos delecc maeals. 0 -case Fo he mode : cos + cos cos B + B B

8 Developme of he Fesel quaos c Recall ha vb B B c T-case Le 1 efacve dex of cde medum efacve dex of efacg medum Fo he T mode : 1 + cos cos cos 1 1 -case Fo he mode : cos + cos cos

9 Developme of he Fesel quaos lmag fom each se of equaos ad solvg fo he efleco coeffce we oba : T-case T case : T cos cos + cos cos 1 case : whee 1 cos + cos cos + cos -case We kow ha s s s cos 1 s 1 s 1

10 Now we have deved he Fesel quaos Subsug we oba he Fesel equaos fo efleco coeffces : T case : case : T cos s cos + s cos + s cos + s 1 T-case Fo he asmsso coeffce : Tcase: T cos cos + s 1 case : cos cos + s -case T : + 1 : T T 1 These jus mea he bouday codos. Fo he T case : + Fo he mode : B + B B 1

11 Powe : Reflecace (R) ad Tasmace (T) The quaes ad ae aos of elecc feld ampludes. The aos R ad T ae he aos of efleced ad asmed powes, especvely, o he cde powe : R P P T P P Fom cosevao of eegy : P P + P 1 R + T A We ca expess he powe each of he felds ems of he poduc of a adace ad aea : P I A P I A P I A I A I A + I A I Acos I Acos + I Acos I cos I cos + I cos I cos cos ou ou ou ou ou Powe _ ao I cos cos Bu I ε 0c0 1ε 0c0 cos 1ε0c0 cos + ε0c0 cos 0 0 cos 0 cos R T cos 0 cos 0 0 cos 0 cos R T 0 cos 0 cos R * cos cos T * cos cos

12 3-. xeal ad Ieal Refleco T cos s cos + s cos + s cos + s T, > 0 T, > 0 xeal Refleco [ / > 1] 1 1 T, 1 ( ) / > 1 s 0 > ae always eal T, < T If > 0 he hee ae o phase chages afe efleco. T, o If < 0 he hee ae always π ( 180 ) phase chages. T, e π T, T, T, Noe fo he case: 1 ( ) 0 whe a p p Bewse s agle (o, polazg agle) (No efleco of mode)

13 Ieal Refleco [ / < 1] 1 T, > 0 T cos s cos + s cos + s cos + s TIR ego 1 > 1 ( ) o ( ) ( ) T, ( ) If s < 0, 1, / < 1 s > 0,, s < 0 If s > 0, ae always eal ( ) If s 0, 1 T, T, T, 1 T, s ( / ) c 1 T, φ φ T, T, e e BUT ccal agle T, ae complex! φ (- π ~ + π) phase chage may occu afe efleco T, < 0 If > 0 he hee ae o phase chages afe efleco. o If < 0 he hee ae π ( 180 ) phase chages. case : 0 1 ( p a ) Noe Bewse's agle fo he Toal eal efleco (TIR) whe > c

14 Devao of Bewse s Agle Bewse's agle p ( fo polazg agle): c ( p ) cosp + s p cos + s p p 0 p p cos s 4 p p R cos + s 4 p ( 1) cos p s p 0 p T eal efleco exeal efleco p a 1 Fo 1.50, p Bewse s agle : a p : > 1 o < 1 xeal & Ieal eflecos, bu -polazao oly Ccal agle : s c : < 1 T & polazaos, bu Ieal efleco oly

15 Toal Ieal Refleco (TIR) Ieal efleco < : 1 1 R c R 1, oal eal efleco( R), 1 Fo c s called TI 1 ad R * 1 fo boh (T ad ) cases. sacomplexumbe eal efleco T cos s cos + s cos + s cos + s Complex value

16 3-3. Phase chages o efleco Phase shf afe xeal Refleco T, s always a eal umbe fo exeal efleco, T, > 0 xeal efleco > T, 0 he he phase shf s 0 fo > 0, T, ad he phase shf s 180( π ) fo < 0. T, T, < T xeal Refleco T xeal Refleco Fo T case, π phase shf fo all cde agles Fo case, π phase shf fo p

17 Phase shf afe Ieal Refleco Ieal efleco > 0 fo < s T c s complex TIR ego whee > T 1 φt e e T T φ T c Complex value I TIR ego > 0 fo < a < 0 fo < c s complex TIR ego whee > φ e e φ c TIR TIR Fo T case, o phase shf fo < c φ T () phase shf fo > c Fo case, o phase shf fo c

18 Phase shfs o oal Ieal Refleco fo boh T- ad -cases Whe ( TIR case) he s complex ad fo boh he T ad cases has he fom : c a b cosα sα e sα b aα φ α a+ b cosα + sα e cosα a φ s he phase shf o oal eal efleco( TIR). α α φ e e + α T case : T cos s cos + s a cos b s 1 φ s T aα a cos φ T a s cos : > c Ieal efleco TIR (Complex ) Asmla aalyss fo he case gves : φ 1 s π a cos : > c

19 Theefoe, T, afe TIR s.. Ieal efleco Fo TIR case ( cde > c ) T, T cos s cos + s cos + s cos + s Complex value φ φ T π a a 1 1 s s cos cos φ T,

20 Summay of Phase Shfs o Ieal Refleco o ' 0 c cos T 1 o 0 o 0 c < φ φ T Δφ

21 Fesel Rhomb 3π o Noe φ φt ea 53 whe Afe wo cosequeve TIRs, 3π φ φt π Δ φ φ φt Quae wave eade φ φ T Δφ Lealy polazed lgh (45 o ) Cculaly Polazed lgh

22 Quae-wave eadao afe TIR π o Noe φ φt ea 69 whe??? π Δ φ φ φt Quae wave eade φ φ T Δφ Lealy polazed lgh (45 o ) Cculaly Polazed lgh

23 3-5. vaesce Waves a a Ieface Icde beam: oexp k Refleced beam: o exp k Tasmed beam: exp Fo he asmed beam : exp k ( ω ) o ( ω ) ( ω ) ( k ω ) o ) ) ) ) k k s x + k cos z xx + zz ( ) ( ) ( s cos ) k x + z Bu, cos 1 s 1 s Whe s > ( oal eal efleco), he : s cos 1 a puely magay umbe

24 vaesce Waves a a Ieface Fo he asmed beam wh a TIR codo we ca we he phase faco as : ( s > ), s k k x + z s 1 kx s 0 exp ω exp z ( α ) Defg he coeffce α : α k s π s 1 1 λ We ca we he asmed waveas: 1 > z h x kx s 0 exp ω exp ( α z) 1 The evaesce wave amplude wll decay apdly as peeaes o he lowe efacve dex medum. Noe ha he cde ad efleco waves fom a sadg wave x deco Peeao deph: 1 1 o h e α λ s π 1

25 Fusaed TIR T p faco of esy asmed acoss gap d Zhu e al., Vaable Tasmsso Oupu Couple ad Tue fo Rg Lase Sysems, Appl. Op. 4, (1985). d/λ

26 Fusaed Toal Ieal Reflecace Pell-Boca psm d Zhu e al., Vaable Tasmsso Oupu Couple ad Tue fo Rg Lase Sysems, Appl. Op. 4, (1985). d 1 ~ λ: chagg he eflecace Roao: chagg he wavelegh esoa a B

27 3-6. Complex Refacve Idex Fo a maeal wh coducvy( σ ) : σ % 1+ R + ε0 ω I % 1+ σ + R I R I ε0 ω Solvg fo he eal ad magay compoes we oba : σ R I 1 R I ε ω 0 σ ε σ 4 σ I 1 I I 0 I ε0ω ε0ω R I 0 ω Fom he quadac soluo we oba : σ σ 1± ε 0 ω ε0 ω I I We eed o ake he posve oo because s a eal umbe. I

28 Complex Refacve Idex Subsug ou expesso fo he complex efacve dex back o ou expesso fo he elecc feld we oba 0 exp ( k ω) ω 0 exp ( ) ( ˆ R + I uk ) ω c R I ω 0 exp ω ( uˆ k ) exp c c ( uˆ ) k The fs expoeal em s oscllaoy. The M wave popagaes wh a velocy of R / c. The secod expoeal has a eal agume (abso bed).

29 Complex Refacve Idex R I ω 0 exp ω ( uˆ ) exp ( ˆ k uk ) c c The secod em leads o absopo of he beam meals due o ducg a cue he medum. Ths causes he adace o decease as he wave popagaes hough he medum. * * ( ˆ Iω uk ) I 00 exp c ˆ Iω ( uk ) I I0 exp I0 exp α ( uˆ k ) c The absopo coeffce s defed Iω 4π : α c λ I

30 3-7. Refleco fom Meals Refleco fom meals s aalyzed by subsug he complex efacve dex % hefesel T case : T cos % s cos + % s equaos: case: % cos + % s % cos + % s Reflecace Subsug % + R I we oba : T case : ( ) ( ) ( ) ( ) cos s + cos + s + R I R I R I R I case : ( R I ) ( R ) I cos ( R I s ) ( R I) ( R I ) + ( R I) cos + ( R I s ) + ( R I) + + +

31 Refleco fom Meals a omal cdece ( 0) A omal cdece, 0 : A omal cdece (fom Hech, page 113) T % % + % cos s 1 cos + % s 1 % cos + % s 1 % % % 1 + % cos + s ( R I ) ( ) R The powe eflecace R s gve by I vsble λ R * ( R 1) ( + 1) ( ) ( ) I + I ( ) ( ) 1 R I 1 R + I 1 R + R + I 1+ R I 1+ R + I 1 + R + R + I R R +

Like dokumenter

Lecture 19. Non-Normal Incidence of Waves at Interfaces

Lecture 19. Non-Normal Incidence of Waves at Interfaces Lu 9 No-Nomal Id of Wavs a Ifas I hs lu you wll la: Wha happs wh wavs s a fa bw wo dff mda omg a a agl Rflo ad asmsso of wavs a fas Applao of -fld ad H-fld bouday odos Toal al flo Bws s agl C 303 Fall