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Unformatted text preview: 19 Total internal reflection (TIR) and evanes cent waves TE reflection: k i k r k t 2 1 x z H i k 1 cos 1 k 1 sin 1 k 2 sin 2 Medium 1 Medium 2 1 E yr E yi = 2 cos 1 1 cos 2 2 cos 1 + 1 cos 2 E yt E yi = 2 2 cos 1 2 cos 1 + 1 cos 2 TM reflection: k i k r k t 2 1 x z E i k 1 cos 1 k 1 sin 1 k 2 sin 2 Medium 1 Medium 2 1 E r E i = 2 cos 2 1 cos 1 2 cos 2 + 1 cos 1 E t E i = 2 2 cos 1 2 cos 2 + 1 cos 1 . Consider a TE or TMpolarized wave (or a superposition) incident on an interface at x = 0 surface as depicted in the margin at an incidence angle 1 . Independent of the polarization of the incident wave, the angle of trans mitted wave 2 can be found using Snells law k 1 s in 1 = k 2 s in 2 1 r 1 r s in 1 = 2 r 2 r s in 2 assuming lossless media on either side of the interface, where Refractive index: n = c v p = r r r o and r o are the relative permeability and permittivity, respectively, of the prop agation media. Moreover, r r = o o = c v p n above can be referred to as the refractive index of the propagation medium. Snells law, expressed in terms of refractive index, n 1 s in 1 = n 2 s in 2 s in 2 = n 1 n 2 s in 1 1 shows that for a given 1 , the corresponding s in 2 can be in excess of 1 when n 1 > n 2 , that is, for propagation from a high refractive index (optically thick) material such as glass into a lower refractive index (optically thin) material such as air. For example : if n 1 n 2 = 1 . 5 and 1 = 45 , then s in 2 = n 1 n 2 s in 1 = 1 . 5 s in 45 = 1 . 5 2 1 . 5 1 . 41 > 1 . But, s in 2 in excess of 1 cannot be solved for 2 as if it were a regular angle 1 describing the elevation of vector k t above the xaxis. In general when n 1 > n 2 and the incidence angle Critical angle: c = s in 1 n 2 n 1 1 > s in 1 n 2 n 1 = s in 1 2 2 1 1 c we will have s in 2 in excess of 1 and cos 2 = 1 s in 2 2 purely imaginary. in such situations use s in 2 and cos 2 = 1 s in 2 2 directly in the expressions for k t , , and as illustrated below. 1 Nor if sin 2 is complex valued because medium 2 is lossy and we need to use 2 r = 2 o + 2 j o in Snells law (as we already did in Lecture 18)....
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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.
 Fall '08
 Staff
 Electromagnet

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