Page 413 - Fiber Optic Communications Fund

394 Fiber Optic Communications

where we have used the frequency-shifting property. Using Parseval’s relation, the total energy is
∞ ∞
2 2
E = |q (t)| dt = |̃q (f)| df
tot ∫ in ∫ in
−∞ −∞
∞
∗
= ̃q (f)̃q (f)df
in
∫ in
−∞
∞ 5 5
∑ ∑ ∗
= ̃g (f − kΔf) ̃g (f − lΔf)df
∫ l
k
−∞ k=−5 l=−5
⎡ ⎤
∞ 5
⎢ ∑ 2 ∑ ∑ ∗ ⎥
= | ̃g (f − kΔf) + ̃g (f − kΔf)̃g (f − lΔf) df. (9.22)
|
∫ ⎢ | k | k l ⎥
−∞ ⎢k=−5 k l ⎥
k≠l
⎣ ⎦
Consider the second term on the right-hand side of Eq. (9.22). ̃g (f − kΔf) corresponds to the channel k
k
∗
centered at kΔf with a bandwidth of 50 GHz. Since Δf > 50 GHz, the overlap term ̃g (f − kΔf)̃g (f − lΔf)
k l
is zero when k ≠ l. Therefore, the total energy is
5 ∞
∑ 2
E = |̃g (f − kΔf)| df
tot ∫ k
k=−5 −∞
5 ∞
∑ 2
= |q (t)| dt. (9.23)
k
∫
k=−5 −∞
Thus, we see that the total energy is the sum of the energy of each channel. So, it follows that the total power
is 11 times the power per channel:
power per channel = 0dBm
= 10 0.1×0 mW = 1 mW; (9.24)
total power = 11 mW
= 10 log 11 dBm
10
= 10.413 dBm; (9.25)
total fiber loss = 0.2 × 50 = 10 dB; (9.26)
total power at the fiber output = 10.413 dBm − 10 dBm

= 0.413 dBm. (9.27)

9.3.1 WDM Components

Multiple wavelengths are combined using a multiplexer. The inverse operation of separating the wavelengths
of a combined signal is achieved using a demultiplexer. The photonic device used as a multiplexer can also
be used as a demultiplexer if the direction of propagation is reversed, because of the reciprocity property of
optical field propagation. The simplest example for a multiplexer/demultiplexer is a prism which separates