450fall09hw9_Sol - ECE 450 Fall 2009 Homework 9 Solutions...

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ECE 450 Fall 2009 Homework 9 Solutions Due: Thursday , Oct 29 th , 2009 1. In medium 1, the dielectric (glass), 2 · 2 ˆ ˆ ˆ which can be expressed in more general terms as 2 i x z j j i i o x z e e π + = = k r E E ± ± E e a. Since then from the 1 ˆ i i k k = k 2 2 x z j e π + term, one may intuitively see that that ˆ ˆ ˆ 2 i x z k + = which satisfies ˆ ˆ · i i k k 1 = and thus 1 2 k π = Or just equate ( ) 1 1 2 2 x z x z j j k x k z e π + + e , which yields: 2 2 1 1 1 1 x z k 1 and 2 2 2 2 2 x z k k k k π π = + π = = = ( ) 1 1 1 9 1 1 1 1 2 2 3 rad 10 300 MHz m p p r k v k c k v f 8 10 1.26 2.25 π ω λ π ω = = = = = × = ε ˆ × = b. Since you know e from the incident field, you can determine i θ graphically given that incidence angle is in the plane of incidence and is perpendicular to the direction or propagation. As a matter of fact, since is discernable from the given incident field, determining i E ± ˆ i k i θ becomes easier. (See figure below). From the figure, it is easy to see that 1 2 1 2 ˆ cos 45 ˆ sin x i i z i k k θ θ θ = = = ° = = 1
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1 0 0 , r μ Critical angle: Snell’s Law Æ 1 1 sin n 2 sin n 2 θ θ = , with 2 90 θ = ° , we find c θ as 2 2 1 1 1 1 2.25 1 sin 41.8 2.2 s n 5 i r c r c n n θ θ = = = = = ° ε ε Æ An evanescent wave will be produced in the half space x > 0 because the angle of incidence is greater than the critical angle. That is, 45 i 41.8 c θ θ = ° > = ° . This scenario leads to total internal reflection ( TIR ) of the incidence field impinging upon the interface. c. Starting from Snell’s Law: 1 2 2 1 2 1 2 2 2 2 2 1 2 1 2 1 2 2 sin sin Since, sin sin sin cos 1 sin cos 1 n n n n n n θ θ θ θ θ θ θ θ = = = = Since we are in a TIR condition, 2 2 1 1 2 2 sin 1 n n θ > , and thus we can write 0 0 , μ ε ε ε i E ± ˆ k i θ i θ ˆ x k ˆ k z 2
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2 2 2 1 1 2 1 1 2 2 2 2 2 sin cos 1 1 1 sin n n j j n n θ θ θ = = ± = ± α 2 2 2 sin 45 2.25 cos 1 0.354 cos 0.354 1 j j j θ θ = ± °− = ± = − Æ 2 θ is complex because as the wave travels from medium 1 to medium 2 in conjunction with the fact that 1 n n > i 2 c θ θ > 0.354 j , all of which leads to total internal reflection and thus the complex transmission
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