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

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Unformatted text preview: 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- 1 2 3 Consider the three-slab geometry depicted in the margin where a TEM wave field E i = x E i e- jk 1 z , accompanied by H i = y E i 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 = x E r e jk 1 z , accompanied by H r =- y E r 1 e jk 1 z , is set up in the same region, as well as E + = x E + e- jk 2 z , accompanied by H + = y E + 2 e- jk 2 z , and E- = x E- e jk 2 z , accompanied by H- =- y E- 2 e jk 2 z , 1 in the region- d < z < (region 2). Finally, in region z > , we will have E t = x E t e- jk 3 z , accompanied by H t = y E t 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- 1 2 3 E + + E- = E t and E +- E- 2 = E t 3 . These equations can be solved for E t and E- in terms of E + to obtain E t = 2 3 3 + 2 E + and E- = 3- 2 3 + 2 E + . 32 32 Note that we have defined a pair of coefficients representing the in- teraction at z = 0 interface: a transmission coefficient 32 and a 2 reflection coefficient 32 in terms of intrinsic impedances 3 and 2 in a manner analogous to similar relations seen in our studies of...
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This note was uploaded on 09/27/2011 for the course ECE 450 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.

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

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