WAVES IN MEDIA Radio Communications SatelliteIonosphere (plasma) Refraction, moist or dense air Reflection Blue sky, red sunset Troposcatter Cloud, Rain Optical Fibers Optoelectronics on chips θ Devices Linear Circular polarization L8-1----------+ + + + + P ρp WAVES IN MEDIA – Constitutive Relations Vacuum: D =εoE ∇i D =ρ fE ρf = free charge density + + Dielectric D =ε =ε E E + P+ Materials: o+ ∇ε E =ρ +ρ + i o f p polarization charge density ∇iP =−ρ p P = “Polarization Vector” Magnetic Materials: ( ) ∇ = =μ =μ =μ + = i o o B0 B H in vacuum B H H M B M “Magnetization Vector” + e e B + + e e L8-2TYPES OF MEDIA Properties are function of: Designation: Field direction Anisotropic D = ε = μ E, B H Position Inhomogeneous Time: ≠ f(t) Stationary ≠ f(history) Amnesic Frequency Dispersive E or H Non-linear Temperature Temperature dependent Pressure Compressive L8-3L8-4 ANISOTROPIC DIELECTRICS x x y zxx xy xz y x y zyx yy yz z x y zzx zy zz D E D E E E D E E E D E E E = ε = ε +ε +ε = ε +ε +ε = ε +ε +ε x y z 0 0 Let 0 0 0 0 ⎡ ⎤ε ⎢ ⎥ε= ε⎢ ⎥ ⎢ ⎥ε⎣ ⎦ x,y,z are “principal axes” EY 0 EX Dy = εyEy Dx = εxEx y x D E ε≠ε ≠εx y zNote: When , D // E iff E // x,y, or zˆˆ ˆ Real , Lossless medium εμ⇒MAKING ANISOTROPIC MATERIALS eff eff AC d ε = ε = ε d/2 d + -V+Q -Q A [m2] εoεo A “Uniaxial Medium” εx = εy = εo “ordinary” εz = εe “extraordinary” εo > εe z x y εo < εe Atom or molecule ( ) eff A2C d 2 ε≅ ε ≅ ε ( ) o eff o AC d2 2 ε≅ ε ≅ε ε >> εo ε ε C = Q/V A L8-5L8-6 WAVES IN UNIAXIAL MEDIA fE j B D 0 H j D B 0 ∇× = − ω ∇ =ρ = ∇× = ω ∇ = i i Derive wave equation: ( ) ( ) 2 2E E E j H E∇× ∇× =∇ ∇ −∇ =− ωμ∇× =ω με i ∇ =iE0Can show (can also test final solution) Therefore: [ ]∂ ∂ ∂ + + + + +ω με = ∂ ∂ ∂ ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ 2 2 2 2 x y z2 2 2 xE yE zE E 0ˆ ˆ ˆ x y z Assume = 0 (assume UPW in z direction) Yields 3 decoupled equations (x,y,z components) Assume: ε ε= ε ε ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ e 00 0 0 0 0 = = εD E==σ = 0, μ = μoL8-7 BIREFRINGENT MEDIA Decoupled wave equations: ⎡ ⎤∂⎢ ⎥+ω με = = ω με ⎢ ⎥∂⎣ ⎦ ⎡ ⎤∂⎢ ⎥+ω με = = ω με ⎢ ⎥∂⎣ ⎦ e e e o 2 2 x2 2 2 y2 E 0 , k z E 0 , k z e2(k ) o2(k ) (x-polarization equation) (y-polarization equation) Thus x- and y-polarized waves propagate independently at different velocities If ve < vo then ve → “slow-axis velocity” − − ω ⎧ = με⎪∝ = ⎨ ⎪ = με⎩ e e eejk z j( / v )z x o v 1E e e where v 1 Solutions: “extraordinary” velocity “ordinary” velocityL8-8 BIREFRINGENT MEDIA Example: “Quarter-wave plate” Δφ φ − φ = − π π e o e o(k k )d = /2 = − φ − φ = + e oj j2 o What pol.? ˆ ˆE E (xe ye ) ejk d −ojk d −Say, “slow axis” 45° X Z Y d π/2 3π/2 0π Δφ Linear pol. LHC RHC ( )= +1 oE E x yˆ ˆInput: ⇒ 45o linear polarization Output: “Half-wave plate” E2 E1 Output:MIT OpenCourseWare http://ocw.mit.edu 6.013 Electromagnetics and Applications Spring 2009 For information about citing these materials or our Terms of Use, visit:
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