Bài giảng Engineering electromagnetic - Chapter XIII: Plane Wave Reflection & Dispersion - Nguyễn Công Phương

Nguy ễn Công Ph ươ ng Engineering Electromagnetics Plane Wave Reflection & Dispersion Contents I. Introduction II. Vector Analysis III. Coulomb’s Law & Electric Field Intensity IV. Electric Flux Density, Gauss’ Law & Divergence V. Energy & Potential VI. Current & Conductors VII. Dielectrics & Capacitance VIII.Poisson’s & Laplace’s Equations IX. The Steady Magnetic Field X. Magnetic Forces & Inductance XI. Time – Varying Fields & Maxwell’s Equations XII. The Uniform Plane Wave XIII.Plane Wave Reflection & Dispersion XIV.Guided Waves & Radiation Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 2 Plane Wave Reflection & Dispersion 1. Reflection of Uniform Plane Waves at Normal Incidence 2. Standing Wave Ratio 3. Wave Reflection from Multiple Interfaces 4. Plane Wave Propagation in General Directions 5. Plane Wave Reflection at Oblique Incidence Angles 6. Wave Propagation in Dispersive Media Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 3 Reflection of Uniform Plane Waves at Normal Incidence (1) + + −α z EztEe( , )=1 cos(ω t − β z ) Region 1 x Region 2 x1 x 10 1 µ ε′ ε ′′ µ ε′ ε ′′ + + − 1, 1 , 2 2, 2 , 2 = jk1 z Exs1 E x 10 e + + +1 + − E, H = jk1 z 1 1 Hys1 E x 10 e η Incident wave + + 1 E2, H 2 + + − = jk2 z Exs2 E x 20 e − − Transmitted wave E1, H 1 +1 + − jk z H= E e 2 Reflected wave ys2η x 20 2 z += + →+ = + z = 0 Boundary c.: Exs1 E xs 2 Ex10 E x 20 z=0 z = 0 + + →η = η (unreasonable) + + E E 1 2 Boundary c.: H= H →x10 = x 20 xs1= xs 2 = η η − − z0 z 0 = jk1 z 1 2 Exs1 E x 10 e −1 − jk z H= − E e 1 ys1η x 10 1 Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 4 Reflection of Uniform Plane Waves at Normal Incidence (2) = = Region 1 x Region 2 Exs1 E xs 2 ( z 0) + − + →E + E = E µ, ε′ , ε ′′ µ, ε′ , ε ′′ →+=+ − + = x10 x 10 x 20 1 1 2 2 2 2 Exs1 E xs 1 E xs 2 ( z 0) + + = = + − + E, H Hys1 H ys 2 ( z 0) E E E 1 1 →x10 − x 10 = x 20 Incident wave + + →+=+ − + = η η η E2, H 2 Hys1 H ys 1 H ys 2 ( z 0) 1 1 2 − − E, H Transmitted wave +−η + η − 1 1 →+=EE2 E − 2 E xx10 10η x 10 η x 10 Reflected wave 1 1 z − + η− η z = 0 →E = E 2 1 x10 x 10 η+ η 2 1 − E η− η →Γ=x10 = 2 1 + η + η+ η →=τ Ex20 =2 2 =+Γ Ex10 2 1 + 1 E η+ η ++ − = + x10 1 2 Ex10 E x 10 E x 20 Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 5 Reflection of Uniform Plane Waves at Normal Incidence (3) − η− η + η Γ =Ex10 = 2 1 τ =Ex20 =2 2 =+Γ Region 1 x Region 2 + + 1 µ ε′ ε ′′ µ ε′ ε ′′ η+ η η+ η 1, 1 , 2 2, 2 , 2 Ex10 2 1 Ex10 1 2 + + Region 1 is dielectric, region 2 is conductor: E1, H 1 Incident wave + + jωµ + E, H η =2 = 0 →τ = 0 →E = 0 2 2 2 σ+ ωε ' x20 2j 2 − − Transmitted wave E1, H 1 Γ = − 1 →+ = − − Ex10 E x 10 Reflected wave +−+− β + β =+=jz1 − jz 1 z Exs1 E xs 1 E xs 1 Ee x 10 Ee x 10 z = 0 = + β Dielectric: jk10 j 1 − β β + + →=jz1 − jz 1 =− β Exs1( e eE ) x 10 j 2sin( 110 zE ) x → = + β ω EztEx1(,) 2 x 10 sin( 1 z )sin( t ) Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 6 Reflection of Uniform Plane Waves at Normal Incidence (4) − η− η + η Γ =Ex10 = 2 1 τ =Ex20 =2 2 =+Γ Region 1 x Region 2 + + 1 µ ε′ ε ′′ µ ε′ ε ′′ η+ η η+ η 1, 1 , 2 2, 2 , 2 Ex10 2 1 Ex10 1 2 + + Region 1 is dielectric, region 2 is conductor: E1, H 1 Incident wave + + + = β ω E2, H 2 EztEx1(,) 2 x 10 sin( 1 z )sin( t ) − − Transmitted wave =→β = π =±± E1, H 1 Ex10 1 zmm ( 0, 1, 2,...) Reflected wave π λ 2 1 z →zm =π →= zm z = 0 λ 2 1 x Conductor = − 3 λ = − λ = − 1 λ z 1 z 1 z 1 2 2 z = 0 z Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 7 Reflection of Uniform Plane Waves at Normal Incidence (5) − η− η + η Γ =Ex10 = 2 1 τ =Ex20 =2 2 =+Γ Region 1 x Region 2 + + 1 µ ε′ ε ′′ µ ε′ ε ′′ η+ η η+ η 1, 1 , 2 2, 2 , 2 Ex10 2 1 Ex10 1 2 + + Region 1 is dielectric, region 2 is conductor: E1, H 1 Incident wave + + =+ + − E, H Hys1 H ys 1 H ys 1 2 2 − − Transmitted wave + E1, H 1 + E H = xs 1 ys 1 η Reflected wave 1 z − z = 0 − E H = − xs 1 ys 1 η 1 + + Ex10 − jzβ jz β Ex10 →H =( e1 + e 1 ) →Hzt( , ) = 2 cos(β z )cos( ω t ) ys 1 η y1η 1 1 1 Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 8 Reflection of Uniform Plane Waves at Normal Incidence (6) − η− η + η Γ =Ex10 = 2 1 τ =Ex20 =2 2 =+Γ Region 1 x Region 2 + + 1 µ ε′ ε ′′ µ ε′ ε ′′ η+ η η+ η 1, 1 , 2 2, 2 , 2 Ex10 2 1 Ex10 1 2 + + Region 1 is dielectric, region 2 is dielectric: E1, H 1 Incident wave + + E2, H 2 η1 & η2 are positive real values, − − Transmitted wave α1 = α2 = 0 E1, H 1 Reflected wave z z = 0 Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 9 Reflection of Uniform Plane Waves at Normal Incidence (7) Ex. + = Given η1 = 100 Ω, η2 = 300 ΩE, x10 100 V/ m . Find the incident, reflected, and transmitted waves. Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 10 Reflection of Uniform Plane Waves at Normal Incidence (8) + +1 ++ 1 + Eˆ Region 2 S=Re[ EHˆ ] = Re[ E x10 ] Region 1 x 1,av xy 1010 x 10 η µ ε′ ε ′′ µ ε′ ε ′′ 2 2 ˆ1 1, 1 , 2 2, 2 , 2 1 1  + 2 = Re   E + + η x10 E1, H 1 2 ˆ1  Γˆ ˆ + Incident wave + + −=−1 −−ˆ =Γ 1 + Ex10 E2, H 2 S1,avRe[ EH xy 1010 ] Re[ E x 10 ] η − − 2 2 ˆ1 E, H Transmitted wave   2 1 1 =1 1 + Γ 2 Re   Ex10 Reflected wave 2 ηˆ  1 z →− = Γ 2 + z = 0 S1,av S 1, av + +++1 1 +τˆEˆ 11  + 2 SEH=Re[ˆ ] = Re[τ Ex10 ] = Re E τ 2 2,avxy 2020 x 10η η  x 10 2 2ˆ2 2  ˆ 2  2 Re[1/ηˆ ] + ηηη+ ˆ + +2 + =2τ2S = 122 τ 2 S →S =(1 −Γ ) S η1,av ηηη+ 1, av 2,av 1, av Re[1/ˆ1 ] 211 ˆ Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 11 Plane Wave Reflection & Dispersion 1. Reflection of Uniform Plane Waves at Normal Incidence 2. Standing Wave Ratio 3. Wave Reflection from Multiple Interfaces 4. Plane Wave Propagation in General Directions 5. Plane Wave Reflection at Oblique Incidence Angles 6. Wave Propagation in Dispersive Media Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 12 Standing Wave Ratio (1) +−+− jzβ + jz β x Region 2 E=+= EEEe1 +Γ Ee 1 Dielectric xs1 x 1 x 1 x 10 x 10 η η 1 2 η− η ϕ Γ=2 1 =Γ e j η+ η + + 2 1 E1, H 1 −jzβ jz( β + ϕ ) + → =1 +Γ 1 Incident wave + + Eexs1( e) E x 10 E2, H 2 = + Γ + − − Transmitted wave Exs1,max(1 ) E x 10 E1, H 1 →−β = β ++ ϕ π =±± Reflected wave 1z 1 z2 mm ( 0, 1, 2,...) 1 z →z =−(ϕ + 2 m π ) z = 0 max β 2 1 = − Γ + Exs1,min(1 ) E x 10 1 →−βzz = β +++ ϕπ2 mm π ( =±± 0, 1, 2,...) →z =−[ϕ + (2 m + 1) π ] 1 1 min β 2 1 Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 13 Standing Wave Ratio (2) −jzβ jz( β + ϕ ) + 1 1 =1 + Γ 1 z= −(ϕ + 2 m π ) z=−[ϕ + (2 m + 1) π ] Eexs1( e) E x 10 max β min β 2 1 2 1 E λ / 2 xs 1 + Γ + (1 ) Ex10 − Γ + (1 ) Ex10 z ϕ+ π ϕ+ π ϕ+ π ϕ − 6 − 4 − 2 2β ϕ+ π 2β ϕ+ π 2β ϕ+ π 2β − 5 − 3 − 2β 2β 2β Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 14 Standing Wave Ratio (3) −jzβ jz( β + ϕ ) + =1 + Γ 1 Eexs1( e) E x 10 + − ϕ− jzβ ϕ jz β ϕ =j/21 + Γ j /2 1 j /2 Eeex10 ( eee) + − ϕ− jzβ ϕ jz β ϕ =j/21 + Γ j /2 1 j /2 Eeex10 ( eee) +−ϕ−β +− ϕ − β +Γj/2jz1 −Γ j /2 jz1 ()Eeex10() Eee x 10 +−jzβ + − ϕ − jz β ϕ jz β ϕ = −Γ1 +Γj/21 + j /2 1 j /2 Ex10(1 ) e E x 10 ( ee eee) +− β + ϕ = −Γj1 z +Γ j / 2 β + ϕ Ex10(1) e 2 Ee x 10 cos( 1 z / 2) → =−Γ+ωβ −+Γ + βϕ + ωϕ + Eztx1( , )( 1) E x 10 cos( tzE 1 ) 2 x 101 cos( z / 2)cos( t / 2) Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 15 Standing Wave Ratio (4) =−Γ+ωβ −+Γ + βϕ + ωϕ + Eztx1( , )( 1) E x 10 cos( tzE 1 ) 2 x 101 cos( z / 2)cos( t / 2) = + Γ Exs 1,max 1 = − Γ Exs 1,min 1 E 1+ Γ s =xs 1,max = − Γ Exs 1,min 1 Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 16 Plane Wave Reflection & Dispersion 1. Reflection of Uniform Plane Waves at Normal Incidence 2. Standing Wave Ratio 3. Wave Reflection from Multiple Interfaces 4. Plane Wave Propagation in General Directions 5. Plane Wave Reflection at Oblique Incidence Angles 6. Wave Propagation in Dispersive Media Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 17 Wave Reflection from Multiple Interfaces (1) x η η η The steady – state has 5 waves: 1 2 3 • Incident wave in region 1 • Reflected wave in region 1 Incident energy • Transmitted wave in region 3 ηin z • 2 opposite waves in region 2 – l 0 +− β − β + − =jz2 + jz 2 β = ω ε Exsx220 Ee Ee x 20where 2 r 2 c,& Ex20 E x 20 are complex +− β − β =jz2 + jz 2 Hys2 He y 20 He y 20 η− η Γ = 3 2 23 η+ η 3 2 −= Γ + Ex20 23 E x 20 ⋮ + − + + E − EΓ E H = x20 H =−x20 =− 23 x 20 y20 η y20 η η 2 2 2 Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 18 Wave Reflection from Multiple Interfaces (2) x +− β − β =jz2 + jz 2 η η η Exs2 Ee x 20 Ee x 20 1 2 3 +− β − β =jz2 + jz 2 Hys2 He y 20 He y 20 +− β − β jz2+ jz 2 η =Exs2 = Ee x 20 Ee x 20 Define w(z ) +− β − β Incident energy jz2+ jz 2 H ys 2 Hey20 He y 20 + + ηin z − ++E − Γ E E=Γ EH, =x20 , H =− 23 x 20 – l 0 x20 23 xy 20 20η y 20 η 2 2 − jzβ jz β 2+ Γ 2 →η = η e23 e w(z ) 2 − β β jz2− Γ jz 2 e23 e η− η ϕ Γ=3 2 ,ej =+ cosϕ j sin ϕ 23 η+ η 3 2 (ηη+ )(cos βzjz − sin βηη ) +− ( )(cos β zjz + sin β ) →η(z ) = η × 32 2 232 2 2 w 2 ηη+ β − βηη −− β + β (32 )(cos 2zjz sin 232 )( )(cos 2 zjz sin 2 ) η β− η β =η 3cos 2z j 2 sin 2 z 2 η β− η β 2cos 2z j 3 sin 2 z Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 19 Wave Reflection from Multiple Interfaces (3) x η η ++ − = =− η1 2 3 Exs1 E xs 1 E xs 2 ( zl ) →+ + − = =− Ex10 E x 10 Ezl xs 2 ( ) ++ − = =− Incident energy Hys1 H ys 1 H ys 2 ( zl ) ηin z + − = − →Ex10 − E x 10 = Ezl xs 2 ( ) – l 0 η η η − 1 1 w(l ) − η− η →=Γ=Ex10 in 1 η = η + , where in w =− E η+ η z l η β+ η β x10 in 1 →η = η 3cos 2l j 2 sin 2 l ηcos βz− j η sin β z in 2 ηcos βl+ j η sin β l η(z ) = η 3 2 2 2 2 2 3 2 w 2 η β− η β 2cos 2z j 3 sin 2 z η= η v 1: matched Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 20 Wave Reflection from Multiple Interfaces (4) x η1 η2 η3 η= η Assume:  3 1 β= π  2l m λ →l = m 2 Incident energy 2π 2 β = η 2 λ v z 2 – l 0 η β+ η β η= η 3cos 2l j 2 sin 2 l in 2 η β+ η β 2cos 2l j 3 sin 2 l →η = η in 3 Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 21 Wave Reflection from Multiple Interfaces (5) x η≠ η η1 η2 η3  3 1 Assume:  π β l=(2 m − 1)  2 2 λ →l =(2 m − 1) 2 Incident energy 2π 4 β = η 2 λ v z 2 – l 0 η β+ η β η= η 3cos 2l j 2 sin 2 l in 2 η β+ η β 2cos 2l j 3 sin 2 l η2 →η = 2 in η →η = η η 3 2 1 3 η= η Total transmission: v 1 Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 22 Wave Reflection from Multiple Interfaces (6) Ex. It is required to coat a glass surface with an appropriate dielectric layer to provide total transmission from air to the glass at a wavelength of 570 nm. The glass has dielectric constant, εr = 2.1. Find the required dielectric constant for the coating and its minimum thickness. µ η== η 0 =377 Ω 1 0 ε 0 µ µµ µ1 η 377 η ==0r = 0 == 1 =Ω260 3 ε εε εε ε 0r 0 r r 2.1 η= η η = × = Ω Total transmission: 2 1 3 377 260 313 2 η η  377  2 η = 1 →=ε 1  =  = 1.45 2 ε r2 η r2 2  313  λ 570 λ 473 λ =1 = = 473 nm →==l 2 =118nm = 0.118 µ m 2 µ ε × 2 r2 r 2 1 1.45 4 4 Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 23 Wave Reflection from Multiple Interfaces (7) x η1 η2 η3 η4 η β+ η β η= η 4cos 3lb j 3 sin 3 l b in, b 3 η β+ η β 3cos 3lb j 4 sin 3 l b Incident energy η β+ ηβ vb,cos 2l a j 22 sin l a η= η ηin, a ηin, b z in, a 2 ηβcosl+ j η sin β l 22a vb , 2 a – (la + lb) – lb 0 η− η la lb Γ = in, a 1 η+ η in, a 1 The reflected power fraction: |Γ|2 The fraction of the power transmitted into region 4: 1 – |Γ|2 Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 24 Plane Wave Reflection & Dispersion 1. Reflection of Uniform Plane Waves at Normal Incidence 2. Standing Wave Ratio 3. Wave Reflection from Multiple Interfaces 4. Plane Wave Propagation in General Directions 5. Plane Wave Reflection at Oblique Incidence Angles 6. Wave Propagation in Dispersive Media Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 25 Plane Wave Propagation in General Directions (1) Phase: k.r z λ v = − jk.r p EEs 0e = + kkxx a k zz a k r=x a + z a kz x z r (x, z) → = + k.r kxx kz z θ x −jkx( + kz ) → = x z EEs 0e kx λx   π π ω ω θ = kz λ =2 = 2 = = atan   vp k  k 2+ 2 k 2+ 2 x kx k z kx k z Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 26 Plane Wave Propagation in General Directions (2) Ex. Given a 50 MHz uniform wave, it has electric field amplitude 10 V/m. The medium is o lossless, εr = ε’r= 9.0; µr = 1.0. The wave propagates in the x, y plane at a 30 angle to the x axis, & is linearly polarized along z. Find the phasor expression of the electric field. 6 ω ε 2π × 50 × 10 9 − k ===ω µε r = 3.14 m 1 c 3× 10 8 =o + o k3.14cos30 ax 3.14sin30 a y = + rx ax y a y −−jkx( + k y ) − + =jk.r =x y = jxy(2.7 1.6 ) Es Ee0 Ee 0 10 e V/m Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 27 Plane Wave Reflection & Dispersion 1. Reflection of Uniform Plane Waves at Normal Incidence 2. Standing Wave Ratio 3. Wave Reflection from Multiple Interfaces 4. Plane Wave Propagation in General Directions 5. Plane Wave Reflection at Oblique Incidence Angles 6. Wave Propagation in Dispersive Media Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 28 Plane Wave Reflection at Oblique Incidence Angles (1) + − + − + E E − + H H − k1 10 10 k1 k1 10 10 k1 θ θ′ θ θ′ 1 1 1 1 + − + −
+
− +U U − Ez10 Ez10 H H z10 Ez10 E H10 10 E10 10 θ θ′ θ θ′ 1 1 1 1 η η 1 z 1 z η θ η θ 2 2 2 2 θ θ 2 E20 2 H20 H20
E20 U Ez20 H z20 x k2 x k2 p – polarization, TM s – polarization, TE Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 29 Plane Wave Reflection at Oblique Incidence Angles (2) + + + − − + + − jk .r k E10 E10 k EE= e 1 1 θ′ 1 s1 10 θ 1 − +1 − − − − jk .r
+
− = 1 Ez10 Ez10 H EE e H10 10 s1 10 ′ − θ θ = jk2 .r 1 1 EEs2 20 e η 1 z + η θ k=k (cosθ a + sin θ a ) 2 2 11 1x 1 z − θ E = −θ′ + θ ′ 2 20 k11k ( cos 1 ax sin 1 a z ) H20
=θ + θ Ez20 k22k (cos 2 ax sin 2 a z ) x k2 = + rx ax z a z p – polarization, TM ω ε n ω ω ε n ω k =r1 = 1 k =r2 = 2 1 c c 2 c c Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 30 Plane Wave Reflection at Oblique Incidence Angles (3) + + + − − + + − jk .r k E10 E10 k EE= e 1 1 θ′ 1 s1 10 θ 1 − +1 − − − − jk .r
+
− = 1 Ez10 Ez10 H EE e H10 10 s1 10 ′ − θ θ = jk2 .r 1 1 EEs2 20 e η 1 z + +−jk+ .r + − jkxz( cosθ + sin θ ) η θ EEe=1 = Ecos θ e 1 1 1 2 2 zs1 z 10 10 1 − −−− − −θ' − θ ' θ E =jk1 .r = θ ' jkxz1( cos 1 sin 1 ) 2 20 EEezs1 z 10 E 10cos 1 e H
− −θ + θ 20 E =jk2 .r = θ jk2( x cos 2 z sin 2 ) z20 EEezs2 z 20 E 20cos 2 e + − x k2 + = = EEEzs1 zs 1 zs 2 (at x 0) p – polarization, TM +−θ − − θ′ − θ →θjk11 zsin + θ′ jk 11 z sin = θ jk 22 z sin Ee10cos 1 Ee 10 cos 1 Ee 20 cos 2 θ′ = θ →kzsinθ = kz sin θ′ = kz sin θ →  1 1 1 11 12 2 θ= θ →θ = θ k1sin 1 k 2 sin 2 n1sin 12 n sin 2 Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 31 Plane Wave Reflection at Oblique Incidence Angles (4) + + − − k E E k θ′ = θ 1 10 θ′ 10 1 1 1 θ 1 +1 − θ= θ
+
− k1sin 12 k sin 2 Ez10 Ez10 H H10 10 +−jk zsinθ − − jk z sin θ ′ θ11+ θ ′ 11 = θ θ′ E10cos 1 e E 10 cos 1 e 1 1 η − θ z = θ jk2 z sin 2 1 E20cos 2 e η θ 2 2 →+θ + − θ = θ θ E10cos 1 E 10 cos 1 E 20 cos 2 2 E20 ++ − = = H20
HHH10 10 20 (at x 0) Ez20 k +θ − θ θ x 2 →E10cos 1 − E 10 cos 1 = E 20 cos 2 η η η p – polarization, TM 1p 1 p 2 p η= η θη = η θ where 1p 1cos 12 , p 2 cos 2 Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 32 Plane Wave Reflection at Oblique Incidence Angles (5) + + − − k E E k 1 10 θ′ 10 1 θ 1 + − +1 − Ecosθ+ E cos θ = E cos θ
+
− 10 1 10 1 20 2 H Ez10 Ez10 H 10 10 + − θ θ′ Ecosθ E cos θ E cos θ 1 1 10 1− 10 1 = 20 2 η η η η 1 z 1p 1 p 2 p η θ 2 2 θ E 2 20  − η− η E10 2p 1 p H20
Γ = = Ez20 p + η+ η k  E 2p 1 p x 2 →  10 2η p – polarization, TM τ =E20 = 2 p  p + η+ η  E10 2p 1 p Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 33 Plane Wave Reflection at Oblique Incidence Angles (6) + − + H H − − k1 10 10 k1 θ θ′ Ey10 η− η 1 1 Γ = = 2s 1 s U + − U − s + H Ez10 + η+ η E z10 E10 E 2s 1 s 10 y10 θ θ′ 1 1 E η η z τ =y20 = 2 2s 1 s + η θ E η+ η 2 2 y10 2s 1 s θ 2 H20 η E20 U η = 1 H z20 1s θ k cos 1 x 2 η s – polarization, TE η = 2 2s θ cos 2 Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 34 Plane Wave Reflection at Oblique Incidence Angles (7) Ex. 1 A uniform plane wave is incident from air onto glass at an angle of 30 o from the normal. Find the fraction of the incident power that is reflected and transmitted for (a) p – polarization, & (b) s – polarization. Given glass refractive index n2 = 1.45. sin30 o nsinθ= n sin θ →= θ asin = 20.2 o 1 12 2 2 1.45 η= η o =× =Ω 1p 1 cos30 377 0.866 326 µ µ µ µ η =1 =r1 0 = 0 1 ε εε ε η 1r 10 0 →1 = ε η η r2 1 µ µ µ µ 2 → = n η =2 =r2 0 = 0 η 2 2 = ε 2 ε εε εε n2r 2 η 377 2r 20 r 20 →=η 1 = =260 Ω 2 n2 1.45 →=η η θ =o =Ω 2p 2cos 2 260cos20.2 244 Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 35 Plane Wave Reflection at Oblique Incidence Angles (8) Ex. 1 A uniform plane wave is incident from air onto glass at an angle of 30 o from the normal. Find the fraction of the incident power that is reflected and transmitted for (a) p – polarization, & (b) s – polarization. Given glass refractive index n2 = 1.45. η=Ω η = Ω 1p326 , 2 p 244 η− η 244− 326 Γ=2p 1 p = =− 0.144 p η+ η + 2p 1 p 244 326 P 2 reflected =Γ =−2 = p ( 0.144) 0.021 Pincident Ptransmitted =−Γ2 =−−2 = 1p 1 (0.144) 0.979 Pincident Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 36 Plane Wave Reflection at Oblique Incidence Angles (9) Ex. 1 A uniform plane wave is incident from air onto glass at an angle of 30 o from the normal. Find the fraction of the incident power that is reflected and transmitted for (a) p – polarization, & (b) s – polarization. Given glass refractive index n2 = 1.45. η 1 377 + + − − η = = =Ω435 k E10 E10 k 1s θ o 1 θ′ 1 cos 1 cos30 θ 1 +1 − η
+
− 2 260 Ez10 Ez10 H η = = =Ω277 H10 10 2s θ o cos 2 cos20.2 θ θ′ 1 1 η− η 277 − 435 η Γ=2s 1 s = =− 0.222 1 z s η+ η + η θ 2s 1 s 277 435 2 2 P θ E reflected =Γ2 =−2 = 2 20 s ( 0.222) 0.049 Pincident H20
Ez20 Ptransmitted =−Γ2 =−−2 = k2 1s 1 (0.222) 0.951 x Pincident p – polarization, TM Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 37 Plane Wave Reflection at Oblique Incidence Angles (10) Total reflection: Γ2 =ΓΓ=ˆ 1 2 θ= − 2 θ n  cos2 1 sin 2 →θ = − 1 2 θ cos2 1  sin 1 nsinθ= n sin θ n2  1 12 2 →η = j η η= η θ 2p 2 p 2p 2cos 2 θ > n2 NÕu sin 1 n1 η= ηcos θ 1p 1 1 →η > 0 η > 1p 1 0 ηη− η − η η− η j21pp 1 p j 2 p Z →Γ=2p 1 p = =− =− →Γ Γˆ = 1 p η+ η ηη+ η + η ˆ p p 2p 1 p j21pp 1 p j 2 p Z →θ ≥ n2 →θ ≥ θ = n2 If sin 1 then total reflection 1 c asin n1 n1 Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 38 Plane Wave Reflection at Oblique Incidence Angles (11) Ex. 2 Compute n1 so that total reflection occurs at the back n1 surface. 45 o n2 = 1 Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 39 Plane Wave Reflection at Oblique Incidence Angles (12) Total transmission: Γ = 0 Γ = s 0 η− η →η = η Γ = 2s 1 s 2s 1 s s η+ η η 2s 1 s η = 1 η η 1s θ →2 = 1 cos 1 θ θ η cos2 cos 1 η = 2 nsinθ= n sin θ 2s θ 1 12 2 cos 2 −1 2  1 n  2 − →−η11  sin2 θη  =− 1 sin 2 θ  2 2n 11   1  2     2 n n1 2 2 →θ = θ = 2 Γ=→0η 1 −  sin θη =− 1 sin θ sin1 sin B p 2n 11 1 2+ 2 2  n1 n 2 Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 40 Plane Wave Reflection at Oblique Incidence Angles (13) Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 41 Plane Wave Reflection & Dispersion 1. Reflection of Uniform Plane Waves at Normal Incidence 2. Standing Wave Ratio 3. Wave Reflection from Multiple Interfaces 4. Plane Wave Propagation in General Directions 5. Plane Wave Reflection at Oblique Incidence Angles 6. Wave Propagation in Dispersive Media Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 42 Wave Propagation in Dispersive Media (1) Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 43 Wave Propagation in Dispersive Media (2) ω ω β() ω=k = ω µ ε ()() ω = n ω 0 c ωb −jzjtβ − ω − jzjt β − ω ω =a a + b b 0 Ec, net ( ztEe , ) 0 ( e e e ) ωa ∆=ωω − ω = ω − ω 0a b 0 β ∆=ββ − β = β − β 0a b 0 βa β0 βb − jβ z j ω t ∆β −∆ ω −∆ βω ∆ → =0 0 jzjt + jzjt Ec, net ( ztEe , ) 0 e( ee e e ) − β ω =j0 z j 0 t ∆−∆ω β 2Eee0 cos( t z ) → = = ∆−∆ω β ωβ − Eztnet(,) Re[ E cnet,0 ]2 E cos( tt )cos( 00 tt ) Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 44 Wave Propagation in Dispersive Media (3) = ∆−∆ω β ωβ − ω EztEnet ( , ) 20 cos( t t )cos( 0 tt 0 ) vp, sm = vg(ω0) ωb ω = 0 ω0 vp, carrier β vp, sb 0 ωa ∆ω β v = p, envelope ∆β βa β0 βb ∆ωd ω lim= = v (ω ) ∆ω → ∆β β g 0 0 d ω 0 Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 45