Page 510 - Engineering Electromagnetics, 8th Edition

492 ENGINEERING ELECTROMAGNETICS

Figure 13.20 Plane wave geometry of a leaky wave in a
symmetric slab waveguide. For a guided mode, total reflection
occurs in the interior, and the x components of k 2u and k 2d are
imaginary.

Knowing that
η 2
η 2s = (121)
cos θ 2
and
η 2p = η 2 cos θ 2 (122)
the requirement is that cos θ 2 be zero or imaginary, where, from Eq. (75), Section 12.6,
2
1/2

2 2
1/2 n 1
cos θ 2 = 1 − sin θ 2 = 1 − sin θ 1 (123)
n 2
As a result, we require that
θ 1 ≥ θ c (124)
where the critical angle is defined through
n 2
sin θ c = (125)
n 1
Now, from the geometry of Figure 13.20, we can construct the field distribution
of a TE wave in the guide using plane wave superposition. In the slab region (−d/2 <
x < d/2), we have
d d
E y1s = E 0 e − jk 1u ·r ± E 0 e − jk 1d ·r − < x < (126)
2 2