Page 511 - Engineering Electromagnetics, 8th Edition

CHAPTER 13 Guided Waves 493

where
k 1u = κ 1 a x + βa z (127)
and
k 1d =−κ 1 a x + βa z (128)
The second term in (126) may either add to or subtract from the first term, since either
case would result in a symmetric intensity distribution in the x direction. We expect
this because the guide is symmetric. Now, using r = xa x + za z , (126) becomes

E y1s = E 0 [e jκ 1 x + e − jκ 1 x ]e − jβz = 2E 0 cos(κ 1 x)e − jβz (129)

for the choice of the plus sign in (126), and

E y1s = E 0 [e jκ 1 x − e − jκ 1 x ]e − jβz = 2 jE 0 sin(κ 1 x)e − jβz (130)

if the minus sign is chosen. Because κ 1 = n 1 k 0 cos θ 1 ,we see that larger values of κ 1
imply smaller values of θ 1 at a given frequency. In addition, larger κ 1 values result
in a greater number of spatial oscillations of the electric field over the transverse
dimension, as (129) and (130) show. We found similar behavior in the parallel-plate
guide. In the slab waveguide, as with the parallel-plate guide, we associate higher-
order modes with increasing values of κ 1 . 6
In the regions above and below the slab, waves propagate according to wave-
vectors k 2u and k 2d as shown in Figure 13.20. Above the slab, for example (x > d/2),
the TE electric field will be of the form

e
E y2s = E 02 e − jk 2 ·r = E 02 e − jκ 2 x − jβz (131)
However, κ 2 = n 2 k 0 cos θ 2 , where cos θ 2 ,given in (123), is imaginary. We may
therefore write
κ 2 =− jγ 2 (132)
where γ 2 is real and is given by (using 123)
1/2
2
n 1 2
γ 2 = jκ 2 = jn 2 k 0 cos θ 2 = jn 2 k 0 (− j) sin θ 1 − 1 (133)
n 2
Equation (131) now becomes
d
e
E y2s = E 02 e −γ 2 (x−d/2) − jβz x > (134)
2

6 It would be appropriate to add the mode number subscript, m,to κ 1 ,κ 2 ,β, and θ 1 , because, as was true
with the metal guides, we will obtain discrete values of these quantities. To keep notation simple, the m
subscript is suppressed, and we will assume it to be understood. Again, subscripts 1 and 2 in this section
indicate, respectively, the slab and surrounding regions, and have nothing to do with mode number.