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350lect24 - 24 Evanescent waves and tunneling In this...

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24 Evanescent waves and tunneling In this lecture we will explore the tunneling phenomenon associated with evanescent waves established within finite-width regions. The multi-slab tunneling result to be derived in this lecture will: 1. Enhance our qualitative understanding of the frustrated-FIR ex- ample shown back in Lecture 19, 2. Illustrate a methodology based on transmission line analo- gies to be used in forthcoming lectures on waveguides. Region 1 Region 2 Region 3 z x - d ˜ E i ˜ E r ˜ E + ˜ E t ˜ E - η 1 η 2 η 3 Consider the three-slab geometry depicted in the margin where a TEM wave field ˜ E i = ˆ x E i e - jk 1 z , accompanied by ˜ H i = ˆ y E i η 1 e - jk 1 z , is incident from the left in the region z < - d (region 1). As a response a reflected wave ˜ E r = ˆ x E r e jk 1 z , accompanied by ˜ H r = - ˆ y E r η 1 e jk 1 z , is set up in the same region, as well as ˜ E + = ˆ x E + e - jk 2 z , accompanied by ˜ H + = ˆ y E + η 2 e - jk 2 z , and ˜ E - = ˆ x E - e jk 2 z , accompanied by ˜ H - = - ˆ y E - η 2 e jk 2 z , 1
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in the region - d < z < 0 (region 2). Finally, in region z > 0 , we will have ˜ E t = ˆ x E t e - jk 3 z , accompanied by ˜ H t = ˆ y E t η 3 e - jk 3 z . Our aim is to determine the amplitudes E t , E + , E - , E r in terms of E i using tangential boundary conditions at z = - d and z = 0 . We are in particular interested in the ratio of the transmitted power in region 3 to the incident power in region 1 as a function of slab width d as well as the refractive indices n 1 , n 2 , and n 3 , includ- ing the case when n 2 is purely imaginary, the case corresponding to region 2 being in evanescent mode. Starting with the boundary at z = 0 , the continuity of tangential ˜ E and ˜ H across the boundary requires that Region 1 Region 2 Region 3 z x - d ˜ E i ˜ E r ˜ E + ˜ E t ˜ E - η 1 η 2 η 3 E + + E - = E t and E + - E - η 2 = E t η 3 . These equations can be solved for E t and E - in terms of E + to obtain E t = 2 η 3 η 3 + η 2 E + and E - = η 3 - η 2 η 3 + η 2 E + . τ 32 Γ 32 Note that we have defined a pair of coe ffi cients representing the in- teraction at z = 0 interface: a transmission coe ffi cient τ 32 and a 2
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reflection coe ffi cient Γ 32 in terms of intrinsic impedances η 3 and η 2 in a manner analogous to similar relations seen in our studies of trans- mission line (TL) systems (in ECE 329).
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