Page 310 - Electromagnetics

Figure 4.23: Timing diagram for multiple reflections from a conductor-backed dielectric
slab.

n
n
the nth reflected wave amplitude is multiplied byan additional (−1) and (− 1 ) and is
time-delayed by an additional 2nτ.
It is important to understand that the time delay 2τ is not just the propagation time
for the wave to travel through the slab. To properlydescribe the timing between the
initially-reflected wave and the waves that reflect from the conductor we must consider
the field over identical observation planes as shown in Figure 4.23. For example, consider
the observation plane designated P-P intersecting the first “exit point” on interface 1.
To arrive at this plane the initially-reflected wave takes the path labeled B, arriving at
a time
D sin θ i
v 0
after the time of initial reflection, where v 0 = c is the velocityin region 0. To arrive at
this same plane the wave that penetrates the surface takes the path labeled A, arriving
at a time

2 1
v 1 cos θ t
where v 1 is the wave velocityin region 1 and θ t is the transmission angle. Noting that
D = 2 1 tan θ t , the time delaybetween the arrival of the two waves at the plane P-P is

2 1 D sin θ i 2 1 sin θ t sin θ i
T = − = 1 − .
v 1 cos θ t v 0 v 1 cos θ t v 0 /v 1
BySnell’s law of refraction (4.297) we can write

v 0 sin θ i
= ,
v 1 sin θ t
which, upon substitution, gives

1 cos θ t
T = 2 .
v 1
This is exactlythe time delay 2τ.

© 2001 by CRC Press LLC