Page 164 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Basic Description of Light
2
,
(
----- ct –(
Ex t) = E exp – σ 2 k x) cos ( ω t – k x) (4.51)
0
0
0
2
and is illustrated in Figure 4.2. The center of this wave packet travels at
c
the speed of light to the right, and has been defined so that its spatial
extension and its wavevector extension fulfil the relation
 1 
(
∆x∆k = ----- σ ) = 1 (4.52)
 σ  k
k
so that, if the packet is spread in space, it will have a precise frequency,
and if it is concentrated in space, its spectrum will be spread. The energy
density of the wave packet is now finite, and can be analytically com-
puted as
ε 0 2 2 2
(
v = ----E exp – [ σ ct – x) ] (4.53)
0
k
E
2
This is simply a constant shape Gaussian that moves to the right with the
speed of light , see Figure 4.2. The Gaussian does not change its shape
c
x = ct 2
σ 2
k
E exp – ------ ct –( x) cos ( ω t – k x)
0
0
0
E 0 2

∆k = σ k
Figure 4.2. The general features of 1
∆x = ------
a Gaussian wave packet. σ k

because of the linear dispersion relation ω = ck . For example, an elec-
tronic wave packet, for ω k() ≠ const , would change its shape, because
ω k() is not linear.

Semiconductors for Micro and Nanosystem Technology 161