Wave Propagation in Linear Media | re-examined

3.8 Turn-on e ects in a wave guide
Hx
4
2
0
-2
10
-4
0
7.5
2.5
X
5
5
T
2.5
0
7.5
Figure 3.22: Evolution of the longitudinal component (3.96) of the magnetic eld for aTE01 wave.
The parameters are = 0:8 and Y =1=4.
The longitudinal magnetic component is best calculated from the condition that the magnetic
eld be source-free, @Hx=@x = ,@Hy=@y, which yields after some simple manipulations
h Hx
,X
= cos Y e
H0
p 1, 2 1
p cos T ,
1 , 2
, 2 Z 1 cos ( X) cos T p 1+ 2
1+ 2 , 2
(3.96)
i
d :
0
We know already how the transverse components behave, so we need not reproduce the results
of the last section. The longitudinal component of the magnetic eld, however, is new. A
numerical evaluation is shown in g. 3.22 . We see that the wave front ofHx, too, propagates
with c. Furthermore, at t = x=c, the longitudinal component vanishes. Thus the front
of the wave travelling along the wave guide after turn-on consists exclusively of transverse
components, i. e. the mode acts exactly like a TEM mode at this instant. This becomes even
more impressive if we look at the magnetic eld lines of the wave ( g. 3.23). The example
demonstrates clearly that even waves with eld components in propagation direction do not
propagate faster than light.
Remark (Optical cuto frequencies) According to (3.86), the cuto frequency of a
wave guide is inversely proportional to its width or | in the case of a cylindrical guide
| its diameter. This imposes an inconvenient limit on the miniaturisation of integrated
optical devices. There is, however, a way to circumvent the problem. Takahara et al. [99]
proposed to use metallic wave guides at optical frequencies. At these high frequencies, the
metal actually exhibits a negative dielectric constant, so that alow-pass TM mode can
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