ECE 107: Electromagnetism

Transcription

1 ECE 107: Electromagnetsm Set 8: Plane waves Instructor: Prof. Vtaly Lomakn Department of Electrcal and Computer Engneerng Unversty of Calforna, San Dego, CA

2 Wave equaton Source-free lossless Maxwell s equatons Apply curl Helmholtz equaton (HE) E = jωµ H εe = 0 H = jωε E µ H = 0 E = jωµ H 2 E + ω 2 µεe = 0 ( E ) 2 E 0 jωε E 2 E + k 2 E = 0 2 H + k 2 H = 0 k 2 = ω 2 µε wavenumber In Cartesan coordnates ( 2 + k 2 ) Ex,y,z = 0 2

3 Plane waves n free space (1) Radaton from a unform surface current x = y = 0 E = E x ˆx, H = H y ŷ Maxwell s eqs. reduce to TL equatons! d E x dz = jωµ H y d H y dz = jωε E x Helmholtz equatons reduce to d 2 Ex dz 2 Solutons 2 H y + k2 Ex = 0, d dz 2 + k2 H y = 0 E x = E + x0e jkz E, H y = + x0 η η = µ ε, E = ηj 2 x0 s0 e jkz J J s ˆ = x s z= 0 Js0 y x z 3

4 Plane waves n free space (2) Plane waves solutons are smlar to TL solutons! Frequency doman expressons E(z) = ˆx H(z) = ŷ Tme doman expressons Parameters Phase velocty ( E + e jkz + E x0 e jkz ) x0 E + x0 η e jkz E x0 η e jkz ( ) E(t,z) = ˆx Ex0 + cos(ωt kz +φ + )+ Ex0 + cos(ωt + kz +φ ) H(t,z) = ŷ E x0 η + cos(ωt kz +φ+ ) E x0 η cos(ωt + kz +φ ) vp = 1 µε = ω k (1 µ 0ε0 = 2 10 ) Wavelength λ = 2π k = vp f Characterstc (ntrnsc) mpedance η µ ε η µ ε π 8 = ( = =120 Ω)

5 Plane waves n free space (3) Equvalence between plane waves and TL waves Replace V E, I H,Z 0 η The resultng waves wll have the same behavor E x ẑ H y = 1 η E x I = 1 Z 0 V V 5

6 Plane waves n free space (4) Relaton between and propagatng drecton E & H Let kˆ = xˆsnθcosϕ+ yˆsnθsnϕ+ zˆcosθ be the propagaton drecton of a plane wave wth θ & ϕ beng angles General solutons Feld-drecton relatons E = E 0 e jk ˆk r, jk ˆk r H = H 0 e H 0 = 1 η ˆk E 0 E 0 = η ˆk H 0 or H = 1 η ˆk E E = η ˆk H Plane waves are TEM waves E ˆk, H ˆk, E H E, H, ˆk form a rght-handed trple Plane waves satsfy Maxwell s and Helmholtz equatons 6

7 Plane waves n free space (5) Examples ˆk = ẑ, Ex = E x + e jkz H y = E x η + ˆk = ẑ, E y = E y e jkz H x = E y η Any combnaton of the two above Parallel polarzaton (TM feld) ˆk = ˆxsnθ + ẑcosθ, H = ŷ H y + e jk(xsnθ+zcosθ ) E = η H y + ( ˆxcosθ ẑsnθ)e jk(xsnθ+zcosθ ) Perpendcular polarzaton (TE feld) ˆk = ˆxsnθ + ẑcosθ, E = ŷ E y + e jk(xsnθ+zcosθ ) H = E y + η ( ˆxcosθ + ẑsnθ)e jk(xsnθ+zcosθ ) + E = ˆxE x kˆ = zˆ Hˆ = yˆ H y + + E = ŷe y + H = ˆxH x ˆx E θ + H = ŷh y ˆx + E = ŷe y θ H ˆk ẑ ˆk ẑ 7

8 Plane waves n free space (6) Plane waves n lossy meda σ ε εc = ε j = ε + jε ω wave equaton 2 E γ 2 E = 0 Consder ˆk = ẑ, E = ˆx E x Skn depth Delectrcs vs. conductors ε ε 1 good delectrc (large δ s ) ε ε 1 good conductor (small δ s ) 8

9 Plane waves n free space (7) Power flow (1) Poyntng vector S s the power densty (power per unt area) It shows the strength AND drecton of power flow Power through a surface Average power densty Ths s analogous to TLs! 9

10 Plane waves n free space (8) Power flow (2) Example: Plane waves n a lossless medum 10

11 Plane waves n free space (9) Why are plane waves (PWs) mportant? PWs are smple solutons of Maxwell s equatons allowng learnng many mportant wave propertes Felds radated by antennas are local plane waves Plane waves allow canoncal soluton of several mportant problems, e.g. reflecton from an nterface Any feld/current can be represented (expanded) as an ntegral/summaton of a set of plane waves Plane wave representatons allow for solvng many mportant problems 11

12 Plane waves at boundares (1) Motvaton Problems nvolvng boundares are met n our every day lfe Boundares can lead to nterference problems and we need to know ther effects Boundares can be used to gude EM felds along them, e.g. fbers There are many applcatons of wave phenomena occurrng on boundares Problem of plane wave scatterng from a planar boundary can be solved analytcally 12

13 Plane waves at boundares (2) Normal ncdence (1) We have already establshed a close smlarty between plane wave and TL waves Ths smlarty can be extended to the problem of plane wave scatterng from an nterface 13

14 Plane waves at boundares (3) Normal ncdence (2) 14

15 Plane waves at boundares (4) Normal ncdence (3) From TL equvalence the reflecton r coeffcent E0 η2 η1 Γ= = E0 η2 + η1 r t From the boundary condton E0 + E0 = E0 and from the TL analogy, the transmsson coeffcent s gven by t E0 2η 2 1+Γ= τ τ = = E η + η For non-magnetc meda µ = µ = µ η0 η ε 0 1 ε2 2 ε1 η1 =, η2 = Γ=, τ = ε ε ε + ε ε + ε

16 Plane waves at boundares (5) Normal ncdence (4) All result for standng waves n TLs apply here as well! Standng wave rato For matched meda η1 = η2 Γ= 0, S = 1 For a PEC wall η 2 = 0 Γ= 1,S = Electrc feld maxma Electrc feld mnma 16

17 Plane waves at boundares (6) Normal ncdence (5): TL Analogy 17

18 Plane waves at boundares (7) Normal ncdence (6): Power flow Net average power flow n the frst medum Net average power flow n the second medum Power relatons S = S av1 av2 18

19 Plane waves at boundares (8) Oblque ncdence (1) What s dfferent? There are feld components normal and tangental to the nterface Reflecton and transmsson depend on the state of polarzaton of the ncdent feld A general polarzaton s wrtten as a sum of parallel (TM) and perpendcular (TE) polarzatons Reflectng propertes for the parallel and perpendcular polarzatons are studed separately Angle of transmsson s dfferent from angle of ncdence 19

20 Plane waves at boundares (9) Oblque ncdence (2): Snell s law Consder a plane wave at an ncdence angle E = E 0 e jk 1 (xsnθ +zcosθ ) = E0 e jk 1 xsnθ e jk 1 zcosθ The reflected and transmtted felds are E r = r jk E 0e 1 xsnθ r e jk 1 zcosθ r ; E t = t jk E0e 2 xsnθ t e jk 2 zcosθ t The boundary condtons at z=0 ( ) = ẑ E 0 t e jk 2 xsnθ t ẑ ( E + E r ) = ẑ E t ẑ E 0 e jk 1 xsnθ + E 0 r e jk 1 xsnθ r To satsfy the boundary condtons, the phase along the boundary has to be matched! jk xsnθ jk xsnθ e = e = e 1 1 r 2 jk xsnθ t θ 20

21 Plane waves at boundares (10) Oblque ncdence (3): Snell s law (cont d) From the phase matchng condton k snθ = k snθ = k snθ 1 1 r 2 t Defne ndces of refracton Snell s law (general form) n snθ = n snθ = n snθ 1 1 r 2 t Snell s law of reflecton θ = θ Snell s law of refracton snθt n u 1 snθ = n = u r p2 2 p1 n = k k, n = k k

22 Plane waves at boundares (11) Oblque ncdence (4): Snell s law (cont d) Transmsson from and nto a dense medum Denser meda have larger n Transmsson nto a dense medum 2 1 t Incdence from a dense medum n2 < n1 θt > θ Total nternal reflecton (to be contnued ) π n sn 2 n θ = θ = snθ = 2 n > n θ < θ t 2 c n t 2 1 t θ = π n1 1 ( ) θ c = sn n2 n1 - crtcal angle θ > θc - total nternal reflecton (no feld can propagate n the second medum; further dscusson wll follow shortly) 22

23 Plane waves at boundares (13) Oblque ncdence (5): Perpendcular polarzaton Incdent feld E H = ŷe 0 E = ( ˆxcosθ + ẑsnθ ) 0 e jk 1 (xsnθ +zcosθ ) η 1 e jk 1 (xsnθ +zcosθ ) Reflected feld r E = ŷ r jk E 0e 1 (xsnθ r zcosθ r ) r E H = ( ˆxcosθr + ẑsnθ r ) r 0 e jk 1 (xsnθ r zcosθ r ) η 1 Transmtted feld E t H t t = ŷe 0 E = ( ˆxcosθ t + ẑsnθ t ) t 0 e jk 2 (xsnθ t +zcosθ t ) η 2 e jk 2 (xsnθ t +zcosθ t ) 23

24 Plane waves at boundares (14) Oblque ncdence (6): Perpendcular polarzaton Boundary condtons ẑ ( E ẑ ( H E y H x + E r ) = ẑ E t + H r ) = ẑ H t + E r y r + H x = E t y t = H x E 0 + E 0 r E 0 cosθ η 1 = E 0 t E + r 0 cosθ r η 1 Solve and obtan the coeffcents! E Γ = = E r 0 0 η2 η1 cosθt cosθ η2 η1 + cosθ cosθ t τ E = = 1+Γ = E t 0 0 = t E 0 cosθ t η 2 η2 2 cos θt η2 η1 + cosθ cosθ t 24

25 Plane waves at boundares (15) Oblque ncdence (7): Parallel polarzaton Incdent feld E = ( ˆxcosθ ẑsnθ ) E 0 E 0 H = ŷ e jk 1 (xsnθ +zcosθ ) η 1 Reflected feld Transmtted feld e jk 1 (xsnθ +zcosθ ) E r = ( ˆxcosθr + ẑsnθ r ) E 0 r e jk 1 (xsnθ r zcosθ r ) H r = ŷ E 0 r E η 1 e jk 1 (xsnθ r +zcosθ r ) t = ( ˆxcosθt ẑsnθ t ) E 0 The sgn s chosen to keep the same x component of the electrc feld! t jk e 2 (xsnθ t +zcosθ t ) t E 0 t H = ŷ e jk 2 (xsnθ t +zcosθ t ) η 2 25

26 Plane waves at boundares (16) Oblque ncdence (8): Parallel polarzaton Boundary condtons E x ẑ ( E + r E ) = ẑ t E ẑ ( H + r H ) = ẑ t H H y + E r x = E x t + H r y = H y t E 0 cosθ + E r 0 cosθr = E t 0 cosθ r E 0 η 1 Solve and obtan the coeffcents! E 0 r η 1 = t E 0 η 2 E η θ η cosθ Γ = = E η θ η cosθ r 0 2cos t cos t + 1 τ = E t 0 E 0 = (1+ Γ ) cosθ cosθ t = 2η 2 cosθ t cosθ η 2 cosθ t + η 1 cosθ cosθ t 26

27 Plane waves at boundares (16) Oblque ncdence (9): TL equvalence Observatons Boundary condtons are gven n terms of tangental components Tangental components depend on the ncdence angle and polarzaton Equvalence frst or second medum Defne the characterstc mpedance va the tangental components η 1,2 = E tan η 1,2 = Replace Z01 & Z02 n a TL juncton by η 1& η 2 for the perpendcular and η & η for the parallel polarzatons Dfferences: e,t,t H tan,t E tan,t H tan = =,t E y,t H x 1 2 jk1x sn θ,t E x = E,t H = 0 y,t E 0 = η 1,2,t H 0 cosθ,t cosθ,t,t cosθ,t,t H 0 = η 1,2 cosθ,t phase propagaton dentcal for ALL felds E,t,t E = ŷe,t y ˆx θ,t H = ŷ,t H y ˆx θ H ˆk ẑ ˆk ẑ 27

28 Plane waves at boundares (17) Oblque ncdence (10): TL equvalence (cont d) Unfed expressons Partcular cases Perpendcular polarzaton Parallel polarzaton Γ, = η, 2 η, 1 η, 2 + η, 1 η 1,2 = η 1,2 cosθ,t Γ = η 2 cosθ t η 1 cosθ η 2 cosθ t + η 1 cosθ τ = 2η 2 cosθ t η 2 cosθ t + η 1 cosθ η 1,2 = η 1,2 cosθ,t Γ = η 2 cosθ t η 1 cosθ η 2 cosθ t + η 1 cosθ τ = 2η 2 cosθ t η 2 cosθ t + η 1 cosθ cosθ cosθ t 28

29 Plane waves at boundares (17) Oblque ncdence (11): Propertes Coeffcents depend on the polarzaton (for a general polarzaton, the feld s wrtten as a sum of two polarzatons wth dfferent propertes) The coeffcent depends on the angle of ncdence The coeffcents do not depend on the frequency for lossless case Dependng on the angle, the coeffcents can be real or COMPLEX even n the lossless case! 29

30 Plane waves at boundares (18) Oblque ncdence (12): Brewster angle Consder the case of matched mpedance η η Ths s obtaned under the Brewster angle η2 η 1 µε : sn 1 2 µ 2ε = θ 1 B = cosθ cosθ 2 1 ( µ µ ) = η Γ = = η Γ = ( ) t 2 1 for non-magnetc materals µ 1= µ 2 = µ 0, θb, does not exst! 1 εµ ( ε µ ) : η cosθt = η1 cosθ snθb = 2 1 ( ε1 ε2) 1 for non-magnetc materals θb = tan ε2 ε1 30

31 Plane waves at boundares (19) Oblque ncdence (13): Total nternal reflecton Incdence from a denser medum (e.g. from water to ar) Crtcal angle θ = θ = n n θ = π θ > θ cos θ =± j cos θ c t t η 2 2 jφ e, Γ = Γ = 1 1 ( ) n > n 1 2 sn 2 c 2 1 t & η are purely reactve (loads)!!! Γ = Γ = j e φ φ = φ ( θ ), φ = φ ( θ ) φ φ - phase shft at the nterface - total nternal reflecton!!! - angular dependence!!! - dependence on polarzaton!!! Γ φ 31

32 Plane waves at boundares (20) Total external reflecton

33 Plane waves at boundares (10) Crtcal angle c 2 1 t 1 ( n n ) θ = θ = sn θ = π 2 For q >q c total nternal reflecton G =1. Does ths mean E=0 n n 2 (z>0)? Evanescent waves ~ t t jk2 ( xsn θ t + z cosθt ) = 0 E where yˆ E e n1 snθt = snθ n 2 > 1 so for q >q c sn q t >1 2 2 n 1 2 cosθ 1 sn sn 1 t = ± θt = ± j θt = ± j sn θ 1 n2

34 Plane waves at boundares (10) Evanescent waves (2) k so n 2 2 = k1, k1 = n1 2π λ jk 2 zcosθ t = 2π n 2 1 sn 2 2 θ n 2 z = αz λn 1 α Choose for dampng ~ E yˆ E t t jk1x sn θ αz = 0 e Evanescent wave perssts over a dstance d=1/a n n 2. e d A demonstraton of evanescent wave s by placng a second prsm next to a prsm where the scatterng angle s > q c. A wave wll be transmtted that depends exponentally on the gap d between the two prsms. q >q c

35 Plane waves at boundares (21) Oblque ncdence (14): The phenomenon of total nternal reflecton s used to make fbers! 35

36 Plane waves at boundares (22) Oblque ncdence (15): Summary 36