Page 46 - Fiber Optic Communications Fund

Electromagnetics and Optics 27

Amplitude

Δω Δω Δω Δω
– ω 0 – – ω 0 – ω 0 + ω 0 – ω 0 ω 0 + Frequency
2 2 2 2

Figure 1.24 The spectrum when two monochromatic waves are superposed.

Let the electric field intensity of these waves be
[ ]
E = cos ( −Δ∕2)t −(k −Δk∕2)z , (1.163)
0
1
[ ]
E = cos ( +Δ∕2)t −(k +Δk∕2)z . (1.164)
2 0
Using the formula
( ) ( )
C + D C − D
cos C + cos D = 2 cos cos ,
2 2
Eq. (1.162) can be written as
E = 2 cos (Δt −Δkz) cos ( t − k z). (1.165)
0
0
⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏟
field envelope carrier
Eq. (1.165) represents the modulation of an optical carrier of frequency by a sinusoid of frequency Δ.
0
Fig. 1.25 shows the total electric field intensity at z = 0. The broken line shows the field envelope and the
solid line shows rapid oscillations due to the optical carrier. We have seen before that
0
ph =
k 0
is the velocity of the carrier. It is called the phase velocity. Similarly, from Eq. (1.165), the speed with which
the envelope moves is given by
Δ
= (1.166)
g
Δk

Envelope
2

0
t

Figure 1.25 Superposition of two monochromatic electromagnetic waves. The broken lines and solid lines show the
field envelope and optical carrier, respectively.