Oblique incidence: Interface between dielectric media

Transcription

1 lecrmagnec Felds Oblque ncdence: Inerface beween delecrc meda Cnsder a planar nerface beween w delecrc meda. A plane wave s ncden a an angle frm medum. The nerface plane defnes he bundary beween he meda. The plane f ncdence cnans he prpagan vecr and s bh perpendcular he nerface plane and he phase planes f he wave. Phase plane β θ Plane f ncdence θ r Medum = r µ = µ r µ Inerface plane θ Medum = r µ = µ r µ Amangawa, 006 Dgal Maesr Seres 65

2 lecrmagnec Felds There are w elemenary renans (plarzans) fr he elecrmagnec felds: Perpendcular Plarzan The elecrc feld s perpendcular he plane f ncdence and he magnec feld s parallel he plane f ncdence. The felds are cnfgured as n he Transverse lecrc (T) mdes. Parallel Plarzan The magnec feld s perpendcular he plane f ncdence and he elecrc feld s parallel he plane f ncdence. The felds are cnfgured as n he Transverse Magnec (TM) mdes. Any plane wave wh general feld renan can be baned by superpsn f w waves wh perpendcular and parallel plarzan. Amangawa, 006 Dgal Maesr Seres 66

3 lecrmagnec Felds Incden wave β x Medum = r Perpendcular (T) plarzan β z β θ θ r Refleced wave r r βr µ = µ r µ y z Medum = r µ = µ r µ x θ β Transmed wave Amangawa, 006 Dgal Maesr Seres 67

4 lecrmagnec Felds The elecrc feld phasrs fr he perpendcular plarzan, wh reference he sysem f crdnaes n he fgure, are gven by r = = = jβ x jβ z x z e ˆ y jβ x jβ z rx rz e ˆ yr jβ x jβ z x z e ˆ y The prpagan vecr cmpnens n medum are expressed as β = β + β = β = ω µ x z r rx rz Amangawa, 006 Dgal Maesr Seres 68 y y y βx = βcsθ βz = βsnθ β = β + β = β β = β csθ β = β snθ rx r rz r

5 lecrmagnec Felds The prpagan vecr cmpnens n medum are expressed as β = β + β = β = ω µ x z β = β csθ β = β snθ x z The magnec feld cmpnens can be baned as β y = = snθ ˆ x + csθ ˆ z e ωµ η β r r yr r = = snθr ˆ x + csθr ˆ z e ωµ η β y = = snθ ˆ x + csθ ˆ z e ωµ η jβxx jβzz jβrxx jβrzz jβxx jβzz Amangawa, 006 Dgal Maesr Seres 69

6 lecrmagnec Felds Assumng ha he amplude f he ncden elecrc feld s gven, cmpleely specfy he prblem we need fnd he amplude f refleced and ransmed elecrc feld. The bundary cndn a he nerface (x = 0) saes ha he angenal elecrc feld mus be cnnuus. Because f he perpendcular plarzan, he angenal feld s als he al feld x = 0) j β zz j β rzz j zz y + yr = β y e e e The relan abve mus be vald fr any chce f z and we mus have (phase cnservan law) βz = βrz = βz The frs equaly ndcaes ha he refleced angle s he same as he ncden angle. β = β β snθ = β snθ θ = θ z rz r r Amangawa, 006 Dgal Maesr Seres 70

7 lecrmagnec Felds The secnd equaly prvdes he ransmed angle β = β β snθ = β snθ z z µ θ = sn snθ µ Snell's Law Snce we have als j zz j β rzz j β zz e = e = e he bundary cndn fr he elecrc feld becmes y + yr = y Amangawa, 006 Dgal Maesr Seres 7

8 lecrmagnec Felds The angenal magnec feld mus als be cnnuus a he nerface. Ths apples n ur case he z cmpnens z + zr = z y yr y csθ csθ = csθ η η η η csθ y y r = y η csθ Slun f he sysem f bundary equans gves yr ηcsθ cs η θ Γ = = η csθ + η csθ y y η csθ τ ( ) = = η csθ + η csθ y Reflecn ceffcen Transmssn ceffcen Amangawa, 006 Dgal Maesr Seres 7

9 lecrmagnec Felds Fr he magnec feld, we can defne he reflecn ceffcen as In erms f elecrc feld, he magnec feld cmpnens are zr Γ ( ) = = yr zr = csθ = r cs η y z = csθ = cs η The reflecn ceffcen fr he magnec feld s hen z η csθ η csθ Γ ( ) = = Γ = η θ η θ yr y cs + cs r θ θ Amangawa, 006 Dgal Maesr Seres 73

10 lecrmagnec Felds The ransmssn ceffcen s defned as τ ( ) = The magnec feld cmpnens are The ransmssn ceffcen fr he magnec feld s hen τ = = η η csθ ( ) = = = η η θ η θ η η y y τ cs + cs Amangawa, 006 Dgal Maesr Seres 74

11 lecrmagnec Felds Parallel (TM) plarzan Incden wave β x Medum = r β z β µ = µ r µ z θ θ r Refleced wave r r βr Medum = r µ = µ r µ y x θ β Transmed wave Amangawa, 006 Dgal Maesr Seres 75

12 lecrmagnec Felds The magnec feld phasrs fr he parallel plarzan are gven by r = = = jβ x jβ z x z e ˆ y jβ x jβ z rx rz e ˆ yr jβ x jβ z x z e ˆ y and he elecrc feld cmpnens can be baned as β = = η snθ ˆ csθ ˆ e ω r r r ω ω y x z β = = η snθ ˆ + csθ ˆ e yr r x r z β = = η snθ ˆ csθ ˆ e y x z y y y jβxx jβzz jβrxx jβrzz jβxx jβzz Amangawa, 006 Dgal Maesr Seres 76

13 lecrmagnec Felds Als fr parallel plarzan ne can verfy ha he same relanshps beween angles apply, as fund earler fr he perpendcular plarzan, ncludng Snell s law θ = θ r θ µ = sn snθ µ We have agan w bundary cndns a he nerface. One cndn s fr cnnuy f he angenal magnec feld y + yr = y Amangawa, 006 Dgal Maesr Seres 77

14 lecrmagnec Felds A secnd cndn s fr cnnuy f he angenal elecrc feld + = z η csθ + η csθ = η csθ y yr y η csθ = y yr y η csθ Frm he equans prvded by he bundary cndns we ban he reflecn and ransmssn ceffcens fr he magnec feld f a wave wh parallel plarzan as zr yr ηcsθ cs η θ Γ = = η csθ + η csθ z y τ ( ) y η csθ = = η csθ + η csθ y Amangawa, 006 Dgal Maesr Seres 78

15 The reflecn ceffcen fr he elecrc feld s defned as zr Γ ( ) = = z r lecrmagnec Felds The angenal cmpnens f he elecrc feld can be expressed n erms f magnec feld as = csθ = η csθ zr r yr = csθ = η csθ z y The reflecn ceffcen fr he elecrc feld s η csθ η csθ Γ ( ) = = = Γ = η θ η θ zr yr z y cs + cs Amangawa, 006 Dgal Maesr Seres 79

16 lecrmagnec Felds The ransmssn ceffcen fr he elecrc feld s defned as The elecrc feld cmpnens are gven by The ransmssn ceffcen fr he elecrc feld becmes η y η τ = = = τ ( ) η η τ ( ) = y = η = η y y η ηcsθ ηcsθ = = η η csθ + η csθ η csθ + η csθ Amangawa, 006 Dgal Maesr Seres 80

17 lecrmagnec Felds Cnsderable smplfcans are pssble fr he cmmn case f nnmagnec delecrc meda wh Frs f all, Snell s law becmes r, equvalenly µ = µ = µ µ sn sn sn θ = θ = snθ µ snθ snθ n = = ( n = ndex f refracn n ) Snell s law prvdes hen a useful recpe express he reflecn and ransmssn ceffcens nly wh angles, hus elmnang he explc dependence n medum mpedance. Amangawa, 006 Dgal Maesr Seres 8

18 lecrmagnec Felds Afer sme rgnmerc manpulans, we ban he fllwng able f smplfed ceffcens fr elecrc and magnec feld sn Γ ( ) = Γ ( ) = sn Γ an ( ) = Γ ( ) = an ( θ θ) ( θ + θ) ( θ θ) ( θ + θ ) τ snθ cs ( ) ( ) θ = τ = sn τ ( ) ( ) = τ = sn ( θ + θ ) snθ csθ ( θ + θ ) cs( θ θ ) Amangawa, 006 Dgal Maesr Seres 8

19 lecrmagnec Felds Pwer flw The me-average pwer flw nrmal he nerface mus be cnnuus. We can express hs as We defne he reflecn and ransmssn ceffcens fr he meaverage pwer as R T cs r cs = cs θ θ θ η η η ncden pwer refleced pwer = refleced pwer = = ncden pwer ransmed pwer = = ncden pwer r η csθ η csθ ransmed pwer Amangawa, 006 Dgal Maesr Seres 83

20 lecrmagnec Felds The fllwng cnversn frmulas relae pwer and elecrc feld ceffcens R =Γ ( ) T = τ η ( ) η csθ csθ Ne ha reflecn and ransmssn ceffcens fr he meaverage pwer are always real psve quanes. The fllwng pwer cnservan cndn s always verfed R + T = Snce he pwer flw nrmal he nerface s cnsdered, he resuls baned abve apply equally perpendcular and parallel plarzan. Amangawa, 006 Dgal Maesr Seres 84

21 lecrmagnec Felds Nn magnec perfec delecrc meda Case > β θ θ β r Medum = r Medum = r µ = µ Frm Snell s law µ = µ z x y θ β θ θ θ θ sn = sn > < Amangawa, 006 Dgal Maesr Seres 85

22 lecrmagnec Felds Snce θ < θ here s always a ransmed (refraced) beam. The ransmssn ceffcens are always psve τ snθ cs θ = τ = > 0 sn ( θ + θ ) snθ cs θ τ = τ = > 0 sn ( θ + θ ) cs( θ θ ) ransmed and ncden wave are n phase a he bundary. τ ( ) = y y y y z z Amangawa, 006 Dgal Maesr Seres 86

23 lecrmagnec Felds Perpendcular plarzan ( ) ( ) sn θ sn θ θ ( ) θ Γ = Γ = sn θ + θ sn θ + θ The reflecn ceffcen fr he elecrc feld s always negave and have always phase dfference f 80 y yr The reflecn ceffcen fr he magnec feld s always psve z and are always n phase zr Γ ( ) = yr y Γ ( ) = zr z y z yr zr Amangawa, 006 Dgal Maesr Seres 87

24 lecrmagnec Felds Parallel plarzan ( ) ( ) an θ an θ θ ( ) θ Γ = Γ = an θ + θ an θ + θ When When ( θ ) θ + θ < 90 an θ + > 0 z y and have phase dfference f 80 zr and are n phase yr and are n phase θ + θ > 90 an θ + θ < 0 z y zr and yr have phase dfference f 80 y z z y yr zr zr yr Amangawa, 006 Dgal Maesr Seres 88

25 lecrmagnec Felds When θ + θ = 90 an θ + θ he reflecn ceffcens vansh (TOTAL TRANSMISSION) Fr θ + θ = 90 θ = θ Frm Snell s law ( ) B and csθ = snθ B (Brewser angle) ( ) an θ an θ θ 0; θ Γ = Γ = 0 an θ + θ an θ + θ snθb snθb = = anθ B = θb = snθ csθ B an Amangawa, 006 Dgal Maesr Seres 89

26 lecrmagnec Felds Case < β θ θ β r Medum = r Medum = r µ = µ Frm Snell s law µ = µ z x y θ β snθ = snθ < θ > θ Amangawa, 006 Dgal Maesr Seres 90

27 lecrmagnec Felds Fr angles f ncdence such ha we have fr perpendcular plarzan ( ) ( ) ( ) sn θ sn θ θ θ Γ = = > 0 sn θ + θ sn θ + θ y and are always n phase yr ( ) sn θ sn θ θ θ Γ = = < 0 sn θ + θ sn θ + θ z and have always phase dfference f 80 zr snθ < y z zr yr Amangawa, 006 Dgal Maesr Seres 9

28 lecrmagnec Felds Fr parallel plarzan an ( ) an Γ ( ) = = an + an + ( ) θ θ θ θ θ θ θ θ ( ) an θ θ an θ θ Γ ( ) = = an θ + θ an θ + θ When ( + ) When θ + θ < 90 an θ θ > 0 and are n phase z z y zr and have phase dfference f 80 yr ( θ ) θ + θ > 90 an θ + z y and zr have phase dfference f 80 and are n phase y yr < z 0 y zr yr yr zr Amangawa, 006 Dgal Maesr Seres 9

29 lecrmagnec Felds When θ + θ = 90 an θ + θ Als n hs case he reflecn ceffcens vansh and we have TOTAL TRANSMISSION ( ) ( ) an θ an θ θ 0; θ Γ = Γ = 0 an θ + θ an θ + θ Tal ransmssn ccurs agan, fr parallel plarzan nly, a he Brewser angle θ = θ = B an Amangawa, 006 Dgal Maesr Seres 93

30 lecrmagnec Felds When snθ = snθ = snθ = θ = 90 we have a lm cndn fr TOTAL RFLCTION, vald fr bh plarzans. Ths parcular angle f ncdence s called crcal angle θ = θ = c sn θ c θ = 90 Amangawa, 006 Dgal Maesr Seres 94

31 lecrmagnec Felds Fr angles f ncdence beynd he crcal angle θ snθ = snθ > c > θ cs θ = magnary ±, chse" " csθ = sn θ = sn θ j The negave sgn s seleced, n rder ban he prper wave vecr n medum, as shwn laer. Amangawa, 006 Dgal Maesr Seres 95

32 The reflecn and ransmssn ceffcens becme cmplex Γ ( ) = Γ ( ) = Γ ( ) = Γ ( ) = τ csθ + j sn θ csθ j sn θ cs j sn csθ j sn θ csθ j sn θ csθ ( ) = τ ( ) = θ θ / csθ τ ( ) = τ ( ) = csθ j sn θ lecrmagnec Felds Amangawa, 006 Dgal Maesr Seres 96

33 lecrmagnec Felds If we cnsder he ceffcens fr me-average pwer flw, we have, fr bh plarzans R = Γ Γ = * T = R = 0 Ths means ha ncden and refleced waves carry he same meaverage pwer, and n pwer s ransmed medum. Bu hs des n mean ha he feld dsappears n medum. The nsananeus pwer ha eners medum s evenually refleced back medum. The elecrc feld phasr f he ransmed wave has he frm jβx x jβz z jβcsθ x jβsnθ z = e e = e e Amangawa, 006 Dgal Maesr Seres 97

34 The wave vecr cmpnens are β θ ω µ θ cs = j sn lecrmagnec Felds = jω µ θ = jβ θ = jα sn sn β θ ω µ θ β θ β sn = sn = sn = z The feld n medum crrespnds a surface wave, mvng alng he z drecn and expnenally decayng (evanescen) alng he x drecn j( jα x) jβz z α j x βz z e = e e = e Amangawa, 006 Dgal Maesr Seres 98

35 lecrmagnec Felds The surface wave mves parallel he surface, wh a phase velcy equal he apparen phase velcy alng z f he ncden wave n medum v v v v p p = = pz > p snθ Fr he surface wave, planes f cnsan amplude are parallel and planes f cnsan phase are nrmal he nerface. These planes d n cncde, herefre he surface wave s a nnransverse wave. θ > θ c Cnsan amplude planes Cnsan phase planes x Amangawa, 006 Dgal Maesr Seres 99

36 lecrmagnec Felds If yu cnsder a beam ncden n he nerface, s fund ha he pwer s ally refleced bu afer penerang fr sme dsance n medum. The refleced beam emerges dsplaced by a dsance D (called Gs-änchen shf, dscvered n 947) θ > θ c D Frm expermens, he dsplacemen s fund be D π π 0.5 = 0.5 α β sn θ Amangawa, 006 Dgal Maesr Seres 00

37 lecrmagnec Felds xamples: θ B =? Medum ar µ = µ = Medum θ B = an waer µ = µ { 80 = mcrwaves.8 pcal θ θ B B = an mcrwaves = an pcal Amangawa, 006 Dgal Maesr Seres 0

38 lecrmagnec Felds A he Brewser angle θ + θ = θ + θ = 90 B 6.38 mcrwaves θ 36.7 pcal Verfcan wh Snell s law snθ = snθ B snθ 6.38 mcrwaves sn B θ = 36.7 pcal Amangawa, 006 Dgal Maesr Seres 0

39 lecrmagnec Felds Medum θ B =? waer µ = µ { 80 = mcrwaves.8 pcal Medum θ B = an ar µ = µ = θ θ B B = an mcrwaves = an pcal Amangawa, 006 Dgal Maesr Seres 03

40 lecrmagnec Felds A he Brewser angle θ + θ = θ + θ = 90 B 83.6 mcrwaves θ 53.3 pcal Verfcan wh Snell s law snθ = snθ B snθ 83.6 mcrwaves sn B θ = 53.3 pcal Amangawa, 006 Dgal Maesr Seres 04

41 lecrmagnec Felds Medum θ c =? ar µ = µ = θ c = sn Medum waer µ = µ { 80 = mcrwaves.8 pcal The al reflecn angle des n exs snce > Amangawa, 006 Dgal Maesr Seres 05

42 lecrmagnec Felds Medum θ c =? waer µ = µ { 80 = mcrwaves.8 pcal θ c = sn Medum ar µ = µ = θ θ c c = sn mcrwaves = sn pcal Amangawa, 006 Dgal Maesr Seres 06

43 lecrmagnec Felds θ = 60 Medum µ = µ = 4 Medum ar µ = µ Cnsder a perpendcularly plarzed wave. Fnd he Brewser angle and he crcal angle: θ θ B c = = an = an = sn = sn = 30 4 Amangawa, 006 Dgal Maesr Seres 07

44 lecrmagnec Felds Fnd he cmpnens f he ncden prpagan vecr and f he x-cmpnen f he ransmed prpagan vecr n erms f β = ω µ β β cs 4 cs 60 x = β θ = ω µ = = β 3 β = β snθ = ω µ 4 sn 60 = β = 3β z x = z = z = z = 3 β β β β β β β β βx = ± β 3β = j β = jα chse " " Amangawa, 006 Dgal Maesr Seres 08

45 lecrmagnec Felds In he secnd medum, fnd he dsance a whch he feld srengh s /e f ha a he nerface d = = α β Wha s he value f he magnude f he reflecn ceffcen a he nerface? The reflecn ceffcen s a cmplex quany when he ncden angle exceeds he crcal angle. Because f al reflecn we knw ha mus be snce he me-average pwer reflecn ceffcen s Γ ( ) = R = Γ ( ) = Amangawa, 006 Dgal Maesr Seres 09