EEE 537 L5 Wave Propagation Across Interfaces

# EEE 537 L5 Wave Propagation Across Interfaces - Wave...

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ave Propagation Across Wave Propagation Across Interfaces EEE537 2010 Fall C.Z. Ning ECEE ASU 1

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Boundary Conditions ˆ ˆ r r s D D n σ = ) ( ˆ 2 1 r r medium 1 1 D r S Δ ∫∫ = S V dV S d D ρ r r r r n 0 ) ( 2 1 = B B n medium 2 2 D r 0 Δ h = S S d B 0 Normal component of D and B are continuous th i f h d it ( ) h s Δ = lim Using the integral form of the two curl equations, one can prove following boundary conditions: if there is no surface charge density medium 1 1 E r 1 H r W Δ = CS s d t B l d E r r r r 0 ) ( ˆ 2 1 = × E E n r r n ˆ medium 2 2 E r 2 H r 0 Δ h + = s d t D J l d H r r r r r s J H H n = × ) ( ˆ 2 1 r r n ˆ : Surface normal Tangential component of E and H are continuous if there is no surface current density EEE537 2010 Fall C.Z. Ning ECEE ASU 2
Wave Propagation Across Medium Interface: TE Waves ransverse electric Ε) ave: A C s ˆ n ˆ r Transverse electric (ΤΕ) wave: The electric field is in the z - direction. (s-polarization) α + = cos ˆ sin ˆ 1 1 1 k y k x k r r r r k i r k i r r r r + = 1 1 Ce Ae E z cos sin 1 1 1 k y k x k = I: B r k i = 2 Be E z β cos ˆ sin ˆ 2 2 2 k y k x k = r II: ( ) z E z E ˆ = r ( ) t i z z e E E ω = sin ' sin sin 2 1 1 Be Ce Ae x ik x ik x ik = + Tangential continuity at y=0 : Since this relation has to hold for all x, following two conditions must hold (understand?) ' = 2 2 2 n sin r n k = = = ε aw of EEE537 2010 Fall C.Z. Ning ECEE ASU sin sin 2 1 k k = 1 1 1 sin r n k Law of reflection Snell’s Law B C A = + x=0 in particular, 3

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Fresnel Formula for TE Waves • To determine the amplitudes, we need one more equation, provided by the 0 0 εμ μ = = k and H B continuity of tangential component of H ( ) 2 1 μ= × = E i H r r ω 0 Recall ? ω H has both x (tangential) and y (normal) components, we need only x-components } Ce Ae { e cos H cos cos sin 0 1 x 1 1 1 α ε y ik y ik x ik + = I: k β cos sin 0 2 x 2 2 Be e cos H y ik x ik = s s 2 1 II: 1 (2.33) B s cos C A or B cos ) C A ( cos 2 0 0 = = + Tangential continuity at y=0: EEE537 2010 Fall C.Z. Ning ECEE ASU cos 1 B C A = + 4
Fresnel Formula for TE Waves B ] cos sin cos sin 1 [ B ] cos cos 1 [ 2 1 2 α β ε + + = A B ] cos

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## This note was uploaded on 01/02/2012 for the course ECE 537 taught by Professor Czning during the Fall '10 term at ASU.

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EEE 537 L5 Wave Propagation Across Interfaces - Wave...

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