Chapter 4 Reflection and Transmission of Waves

Transkript

1 4- Chapter 4 Reflecton and Transmsson of Waves Dr. Stuart Long

2 4- Boundary Condtons ^ n H H 3 H 4 w H l y (fg. 4.)

3 4-3 Boundary Condtons n ^ H H 3 H4 w H l y Tae ˆ component of H J+ jω D (fg. 4.) H y H J + y jωd H4 -H3 H-H J + l w jωd H3-H4 H - H Jw + jωd + w l

4 4-4 Boundary Condtons for H Feld H 3 n ^ H H4 w H l y We can defne a surface current densty J s lm w Jw Amp Tang the lmt as w we obtan m J s (fg. 4.) H - H J w + jωd + H -H 3 4 l w H3-H4 H - H J s + jωd + l H - H J s

5 4-5 Boundary Condtons for H Feld H 3 n ^ H H4 w H l y (fg. 4.) The dscontnuty n tangental magnetc feld s equal to the surface current. ˆ ( H - H ) J n s

6 4-6 Boundary Condtons for Feld A smlar dervaton for jω B yelds The tangental electrc feld s contnuous across the boundary n ˆ ( - ) (no magnetc current source)

7 4-7 Boundary Condtons We can now deduce that: The surface current densty J s only ests on a "perfect" conductor. So f both meda have fnte conductvtes then and H are both contnuous. tan tan Snce the - feld cannot est nsde a "perfect" conductor, then the tangental -feld s ero on the surface (.e -feld s normal to perfect conductor surface.)

8 Boundary Condtons n ^ A 4-8 for D Feld w Recall that the total flu out of a volume s equal to the total charge n the volume ( D ρ ) v. Tae lm w flu D out of volume (D - D ) Area n n total charge n the volume ρ Are a The boundary condton for the D - feld s then gven by: s ρ Surface s Coul charge densty m D n -Dn ρ s The dscontnuty of the normal D-feld s equal to ρ s

9 Boundary Condtons n ^ A 4-9 for B Feld w Smlarly, from B The normal B-feld s contnuous across the boundary B -B n n

10 4- Boundary Condtons Summary Non-Perfect Conductors, H, B, D tan tan norm norm Are all contnuous Dt t t ε D t Bt Ht H t μ B t D D ε ε n n n n B B μ H μ H n n n n ε μ D B B D H ε μ H ε μ

11 4- Boundary Condtons Summary Perfect Conductors H B D ^n H εμ D tan norm tan B H norm nˆ D ρs ; D n ρs n ρs ε σ,h,b,d nˆ H J ; H J ; B μ J s t s t s

12 4- Boundary Condtons Concepts - tan s contnuous -H tan s dscontnuous by J s -B norm s contnuous - J s and ρ s est only on perfect conductor - All felds n a perfect conductor - D ε -D norm s dscontnuous by ρ s - B μ H H B I J s

13 4-3 Unform Plane Wave Propagatng n an Arbtrary Drecton Consder a unform plane wave propagatng n + ˆ and + ˆ drecton, and wth the electrc feld n the yˆ drecton. H r Surfaces of constant phase - j - j ˆ y e [4.7] (fg. 4.5)

14 4-4 Unform Plane Wave Propagatng n an Arbtrary Drecton We can defne ˆ + ˆ " Wave Vector " [4.8] r ˆ + ˆ " Poston Vector " [4.9] H r Surfaces of constant phase (fg. 4.5) yˆ e j r [4.] Stll a unform plane wave wth planes of constant phase perpendcular to

15 4-5 H H H y y - - ˆ + ˆ jωμ jωμ j j j ( ) ˆ j j e ( je ) - ˆ ( ) + jωμ [ ˆ + ˆ ] [ ˆ + ˆ ] e e ωμ ωμ ( j + j ) ( j + j ) H r Surfaces of constant phase H [ ˆ ˆ + ] e η ( j + j ) (fg. 4.5) Once agan we see that H s perpendcular to both and

16 4-6 Plane Wave Impngng on a Delectrc Interface r t r r t t θ r θ t θ ε ε r ε ε ε r ε μ ε μ ε (fg. 4.6)

17 4-7 Plane Wave Impngng on a Delectrc Interface Reflected Wave r [4.3] - j + j ˆ r r yr e r t Transmtted Wave t -j -j ˆ t t yt e [4.5] r r t t θ r θ t θ [4.] Incdent Wave yˆ e -j -j μ ε μ ε (fg. 4.6) Where R Reflecton Coeffcent T Transmsson Coeffcent

18 4-8 Plane Wave Impngng on a Delectrc Interface Incdent Wave Vector r t ˆ + ˆ r θ r r t t θ t Reflected Wave Vector r ˆ r ˆ r θ μ ε μ ε (fg. 4.6) Transmtted Wave Vecor t t ˆ + t ˆ t

19 4-9 Remember that tan s contnuous at the boundary () thus we have : j j j j j j e + yˆ e + y ˆ e ˆ r r t t y R T - j - j j e + R e r Te t [4.7] To be true for all values of r t [4.9] Phase matchng condton The tangental components of the three wave vectors are equal

20 4- r t r r t t ach wave satsfes the approprate Mawell equatons therefore the wave equatons become: I θ r θ μ ε μ ε θ t (fg. 4.6) + ω με + ω με r In medum + ω με t In medum

21 4- r t θ θ r θ t r t + ω μ ε [4.a] r r + ω μ ε [4.b] t + t ω μ ε [4.]

22 4- From geometry and understandng of the r t phase matchng condton we obtan : r θ r r t t θ t θ r snθ sn θr snθ snθ r snθ t I μ ε μ ε (fg. 4.6) t snθ t r Snell s Law (law of refracton) sn θ snθ t [4.5] θ r θ angle of ncdence s equal to angle of reflecton

23 4-3 Graphcal Representaton of Phase Matchng Condtons Radus r t Radus θ r θ t θ Note: wave bent toward normal (fg 4.7a) <

24 4-4 Graphcal Representaton of Phase Matchng Condtons Radus r t Radus θ r θ t θ Note: wave bent away from normal (fg 4.7b) >

25 4-5 Graphcal Representaton of Phase Matchng Condtons Radus r t Radus θ r θ t θ c θθ c θ t 9 > snθ

26 4-6 θ t - + t t t If θ > θ ; > < c t j whe re t ± α α - t yˆ T e α e j [4.6] t α (, yt,, ) yˆ Te cos( ωt ) In medum wave propagates n + drecton. Nonunform plane wave also called a surface wave ˆ

27 4-7 Crtcal Angle θ c If θ θ ; sn θ c c where sn θ c [4.7] Crtcal Angle θ c Angle of ncdence above whch total nternal reflecton occurs. It θ c can only occur when >.

28 4-8 Magntude of Reflected and Transmtted Waves - Depends on polaraton of Case I Case II Perpendcularly Polared Parallel Polared - The plane of ncdence s defned by the plane formed by the unt normal v vector normal to the boundary and the ncdent wave vector. ˆn

29 4-9 Magntude of Reflected and Transmtted Waves Case I: -feld Perpendcular to Plane of Incdence r θ Hr r t t H t H

30 4-3 Magntude of Reflected and Transmtted Waves Case I: -feld Perpendcular to Plane of Incdence The ncdent wave s gven by r θ r H r t t H t [4.] yˆ e j ( + ) H [4.] H ˆ ˆ ( + ) e ωμ j( + ) Note: ω με ε ωμ ωμ μ μ ε η

31 4-3 Magntude of Reflected and Transmtted Waves Case I: -feld Perpendcular to Plane of Incdence The reflected wave s gven by r θ r H r t t H t [4.3] r yˆ Re I j ( ) H [4.4] H r R ˆ ˆ I ( + + ) e ωμ j( )

32 4-3 Magntude of Reflected and Transmtted Waves Case I: -feld Perpendcular to Plane of Incdence The transmtted wave s gven by r θ r H r t t H t [4.5] t yˆ Te I j ( + ) t H [4.6] H t T ˆ ˆ I ( t + ) e ωμ j( + ) t

33 4-33 Quc revew snθ r t - cosθ t - - sn θ θ t Vector to ( ˆ ˆ) ˆ + ˆ ± - ω με ε ωμ ωμ μ μ ε η θ t t

34 Magntude of Reflected and Transmtted Waves r θ r H r t t H t 4-34 Case I: -feld Perpendcular to Plane of Incdence H At tan tan r t y y y y + e j ( ) j ( ) j( ) + Re I Te I + R I T I At tan tan r H H H H + H RI tti + ωμ ωμ ωμ H t NOT: At both tan and H tan must be contnuous

35 Magntude of Reflected and Transmtted Waves r θ r H r t t H t 4-35 Case I: -feld Perpendcular to Plane of Incdence H Usng the prevous equatons + R T I I R + ωμ ωμ I t I T ωμ we can fnd [4.] R I μ μ μ + μ t t Reflecton coeffcent for perpendcularly polared wave [4.3] T I μ μ + μ t Transmsson coeffcent for perpendcularly polared wave

36 Magntude of Reflected and Transmtted Waves r θ r H r t t H t 4-36 Case I: -feld Perpendcular to Plane of Incdence H For nonmagnetc materals μ μ μ equ. 4.3 and 4.3 reduce to: R I + t t Reflecton coeffcent for perpendcularly polared wave T I + t Transmsson coeffcent for perpendcularly polared wave

37 4-37 Magntude of Reflected and Transmtted Waves Case II: -feld Parallel to Plane of Incdence H r t r r t H t θ θ t H (fg. 4.9)

38 4-38 Magntude of Reflected and Transmtted Waves Case II: -feld Parallel to Plane of Incdence r H r r t θ t t H t θ The ncdent wave s gven by H (fg. 4.9) [4.8] [4.9] H yˆ He j ( + ) H ˆ ˆ ( -) e ωε j( + ) Note: ω με ε η ωε ωε μ

39 4-39 Magntude of Reflected and Transmtted Waves Case II: -feld Parallel to Plane of Incdence r H r r t θ t t H t θ The reflected wave s gven by H (fg. 4.9) [4.3] H r yˆ R H e II j ( ) [4.3] r R ˆ ˆ II H ( - ) e ωε j( )

40 4-4 Magntude of Reflected and Transmtted Waves Case II: -feld Parallel to Plane of Incdence r H r r t θ t t H t θ The transmtted wave s gven by H (fg. 4.9) [4.3] [4.33] H t t yˆ T H e II j ( + ) T ˆ ˆ II H ( t -) e ωε t j( + ) t

41 4-4 Magntude of Reflected and Transmtted Waves Case II: -feld Parallel to Plane of Incdence r H r r t θ t t H t Usng Boundary Condtons as prevously done H θ (fg. 4.9) for Case I we obtan: [4.34] R II ε ε ε + ε t t Reflecton coeffcent for parallel polared wave [4.35] T II ε ε + ε t Transmsson coeffcent for parallel polared wave

42 4-4 Condtons for No Reflecton Total Transmsson R For non-magnetc delectrcs μ μ μ R I + t t Case I: For perpendcular polared R I t Snce we already now t, ths s only possble f ε ε, thus we fnd that for total transmsson to occur both meda must be the same ( no nterface at all)

43 4-43 Condtons for No ε Reflecton ε R II ε + ε t t Case II: For parallel polared R ε ε II t along wth the phase matchng condton we fnd that for total transmsson to occur the [4.36] the angle of ncdence must be θ b tan ε ε Brewster Angle or θ b Polaraton Angle Angle of ncdence at whch the wave s totally transmtted. It can est only when ncdent wave s parallel polared for nonmagnetc delectrcs.

44 4-44 Reflected Power as a Functon of Incdent Angle RI or RII θ b θ [Degrees] The materal s glass wth ε.5ε The Brewster angle s 56 (fg 4.)

45 4-45 Reflecton from a Perfect Conductor Perfect conductor σ r H r θ Oblque Incdence r Perfect Conductor r Normal Incdence H r r Perfect Conductor H H (fg 4.6) (fg 4.4a)

46 Reflecton from a Perfect Conductor r H r H θ r (fg 4.6) Perfect conductor r H H r r (fg 4.4a) 4-46 Perfect conductor Case I: For perpendcular polaraton ( j) for perfect conduc tor - t R I μ μt t μ + μ t t T I μ μ + μ t t

47 Reflecton from a Perfect Conductor r H r H θ r (fg 4.6) Perfect conductor r H H r r (fg 4.4a) 4-47 Perfect conductor Case II: For parallel polared for perfect conductor σ ε ε - j - j and t ε ω R II ε εt ε + ε + εt ε T II ε ε ε + εt ε

48 4-48 Normal Incdence of a Plane Wave on a Perfect Conductor r Hr r Perfect Conductor H (fg 4.4a)

49 Normal Incdence of a Plane Wave on a Perfect Conductor r H r r Perfect conductor 4-49 [4.44a] [4.44b] H ˆ e yˆ η j e j H (fg 4.4a) [4.45a] [4.45b] H r r ˆ e yˆ η e + + j j t t H

50 Normal Incdence of a Plane Wave on a Perfect Conductor r H r r Perfect conductor 4-5 H (fg 4.4a) Total felds n medum H j + j ( ) ˆ e e ˆ j sn j + j yˆ ( e + e ) yˆ cos η η [4.46a] [4.46b]

51 Normal Incdence of a Plane Wave on a Perfect Conductor r H r r Perfect conductor 4-5 The nstantaneous values for and H are gven by H (fg 4.4a) jω t ( t, ) Re e ˆ snsnωt [4.48] H t He yˆ ωt η jω t (, ) Re cos cos [4.49] The surface current at the boundary s gven by ˆ ˆ J S n H -( ) H [4.47] η ˆ

52 4-5 Normal Incdence of a Plane Wave on a Perfect Conductor Standng-wave pattern of the feld π π fg(4.4b)

53 4-53 Normal Incdence of a Plane Wave on a Perfect Conductor Standng-wave pattern of the H feld H η 3π π fg(4.4c)

54 4-54 Oblque Incdence of a Perpendcularly Polared Plane Wave on a Perfect Conductor r H r r Perfect Conductor θ H (fg 4.6) Note: where: r ˆ + ˆ ˆ -ˆ snθ cosθ

55 4-55 Oblque Incdence of a Perpendcularly Polared Plane Wave on a Perfect Conductor [4.5a] [4.5b] H H r r yˆ e yˆ yˆ e j j η r (-yˆ ) e j + j η j j e r H r H θ j + j r Perfect conductor (fg 4.6) Note: R I t H t

56 4-56 Oblque Incdence of a Perpendcularly Polared Plane Wave on a Perfect Conductor r H r r Perfect conductor θ H (fg 4.6) Total felds n medum ( ) ( θ ) yˆ - j sn cos e j sn θ H { ˆ cos cos( cosθ ) η + ˆ jsnθ sn ( cosθ ) } - e - θ j sn θ

57 4-57 Oblque Incdence of a Perpendcularly Polared Plane Wave on a Perfect Conductor Standng-wave pattern of the y feld y -λ -λ cosθ cosθ fg(4.7)

58 4-58 Oblque Incdence of a Perpendcularly Polared Plane Wave on a Perfect Conductor Standng-wave pattern of the H feld H cosθ η

59 4-59 Oblque Incdence of a Perpendcularly Polared Plane Wave on a Perfect Conductor Standng-wave pattern of the H feld H snθ η

60 4-6 Oblque Incdence of a Perpendcularly Polared Plane Wave on a Perfect Conductor, H are ero at because of the boundary condtons. tan norm The dstance between nulls s λ λ. cosθ Note: when θ (normal ncdence) H H ; ; dstance between nulls η λ

61 4-6 Power Conservaton The tme-average Poyntng power densty can be found by: S Re H ( ) η η Respectvely, the ncdent, reflected and transmtted tme-average Poyntng power denstes can be found by: S I S r η η ( R ) ( T ) ; ; S t I η Where the component of each s gven by: S S cos θ ; S S cos θ r r r ; S S cosθ t t t

62 ample f [ ] MH V m ω μ ε ω μ ε π π m π f v μ ε 4-63 μ μ μ ε ε 4ε r r H r θ H t t H t θ θ 3 r π 4π ω μ ε ω μ ε εr 4 m 3 3

63 ample Cont. f [ ] MH V m μ ε 4-64 μ μ μ ε ε 4ε r r H r θ H t t H t θ θ 3 r snθ snθ t snθt snθ sn 3.6 θ t sn

64 ample Cont. snθ.5 r t cosθ.866 r μ ε 4-65 μ μ μ ε ε 4ε r r H r θ H t t H t θ θ 3 r t - 4 sn θ.94 you can chec by notng that t cosθt 4 cosθt cos4.5.94

65 ample Cont. R R I I μ μ μ + μ + t t t t.38 μ ε 4-66 μ μ μ ε ε 4ε r r H r θ H t t H t θ θ 3 r T I ( ) μ.866 μ + μ t t.68 Note: T I + R.68 + (-. 38) I

66 ample Cont. R T I I S η μ ε η η 4-67 μ μ μ ε ε 4ε r θ H r H r t t H t θ θ 3 r S r RI RI RI.459 η η η η S t T I.7639 η TI TI η η η

67 4-68 ample Cont. S ; S r.459 ; St.7639 η η η S S cos θ cos3.866 η η Sr Sr cos θr.459 cos3.64 η η St St cosθt.7639 cos η η

68 4-69 ample Cont. S.866 ; S r.64 ; St.7396 η η η η + η η ref. tran. nc. ( see p.99 for ) general proof

69 4-7 ample Cont. r θ r H r t t H t H In medum y total e ( + ) j ( e ) - j - - y total ( ) - j - j j e e -.38e y total - j - j e e -.38e j -.38e y total j cosθ

70 4-7 ample Cont. cosθ.866 snθ.5 r θ r H r t t H t In medum H H total ωμ e ( + ) R j( + e ) ωμ j I H e e +.38e total ωμ ( ) j j + j.866 H +.38e total η j cosθ.5 η j cosθ H.38 total e

71 4-7 ample Cont. In medum cosθ ε.94 t y r total.6e ( + ) - j t r r H r θ H t t H t.6 y total In medum H total ( ).94 (.6) tti j + t e ωμ ωμ H total..3 ; H η total η

72 4-73 ample Cont. j cosθ y -.38e total y total Frst mnmum occur when: y.38 cosθ π π cosθ.6 y π 3 π [ m] Standng-wave pattern of the y feld

73 4-74 ample Cont η H.38 total e j cosθ H total H H. η.54 η Standng-wave pattern of the H feld

74 4-75 ample Cont..5 η H.38e total j cosθ H total H.69 η H.3 η Standng-wave pattern of the H feld

75 4-76 ample Cont. Im Re + j cosθ.38e j j +.38e cosθ

76 4-77 ample μ μ ε ε H f 5 [ H] sea water μ ε μ 8 ε ˆ ω μ ε ˆ σ mho 4 m σ t t ˆ ω 8με j ˆ ω 8 ε σ t ˆ ω μ j ˆ ω note: t T I + t t

77 4-78 ample Cont. μ μ ε ε H In medum (sea water) H t t ˆ yˆ - -j T e I e I TI e I e R e η R - -j -jφ sea water μ ε σ μ 8 ε mho 4 m where S t η μ η ε e jφ ( T ) ( T ) I +jφ I t t e ˆ η η Re H Re Re η ˆ S η ˆ

78 4-79 ample Cont. μ μ ε ε H sea water μ ε μ 8 ε σ mho 4 m ω μ ε ε ω T I 5.4 t σ σ ω μ j ω 5 S S t η η η 5 Re Re 7. 5 η T I η η TI η

79 4-8 Drecton of Surface Currents J nˆ H S r H r θ r J (- ˆ ) H S H J ˆ S (- ) η ( θ) ˆ cosθcos cos e j snθ o J S y ˆ η cosθ e j snθ produces currents only n drecton ŷ H y wre grd

80 4-8 Wave Incdent on a Good Conductor Good conductor Re ( ) and σ ωε θ t t θ r t θ t ( j) ˆ ˆ t t ωμσ

81 ample 3 ( ) f 3 [ MH] η Prob. 4. parallel polared snθ snθ ε ε μ ε η t μ μ μ ε ε ε ε r H r θ H θ θ 45 r θ 3 t t r t θ H t t 4-8 R II cos sn ε θ ε θ ε θ ε θ cos + sn.78 T II ε cosθ ε θ ε sn θ cos +.78

82 ample 3 Cont. ( ) Prob. 4. H ( ˆ cosθ ˆ sn θ) e ωε j ( snθ + cos θ ) μ μ μ ε ε ε ε r H r H θ r t θ t θ θr 45 θ 3 t t H t 4-83 H η ( ˆ ˆ ) e j ( + ) r [ ˆ ˆ II j ( cos ) sn ) θ θ θ ] e R H ωε ( sn cos θ ) r H η R II ( ˆ ˆ ) e j ( )

83 ample 3 Cont. ( ) Prob. 4. μ μ μ ε ε ε ε r H r θ H θ θ 45 θ 3 t t r t θ H t t r 4-84 t T H ( ˆ sn ) e ˆ II θ ωε θ j( sn + ) t.536 ( 3 ˆ ˆ ) H η e j ( )

84 4-85 ample 3 Cont. ( Prob. 4.) H η π - π

85 ample 3 Cont. ( Prob. 4.) R.78 T.78 II S Re H ηh η S H ; II μ ε η η H H μ μ ε ε ε H r t r H θ r t θ t θ θr 45 θ 3 t H t 4-86 η η η η S r ( HRII ) H RII H RII H.5 TII η TII η H H H η η S t ( HTII ).83

86 ample 3 Cont. ( Prob. 4.) η S ; S r.5 ; St.83 H η H η H 4-87 S η η S cos θ H cos45 H.77 η η Sr Sr cos θr H.5 cos 45 H.36 η η St St cosθt H.83 cos3 H. 735

87 ample 3 Cont. ( Prob. 4.) η η η S H.77 ; Sr H. 36 ; St H η H η + H η H.36 ref..735 tran..77 nc.