Page 93 - Fiber Optic Communications Fund

74 Fiber Optic Communications

P (dBm)= 10 log G + 10 log F + 10 log P
10 in
10
out
10
= G(dB)− F(dB)+ P (dBm). (2.190)
in
Note that any loss (such as fiber attenuation) in the system is subtracted from the input power in dBm units
and any gain (such as amplifier gain) is added.

2.7.3 Fiber Dispersion
A medium is said to be dispersive if the group speed of light changes with the frequency of the optical wave.
For example, a pulse p with carrier frequency f = 193 THz and inverse group speed (f )= 5 μs/km is
1 1 1 1
delayed by
ΔT = (f )L = 50 μs (2.191)
1 1
1
after propagating through a 10-km fiber. Consider another pulse p with a different carrier frequency
2
f = f +Δf. If the fiber is not dispersive ( = 0), the inverse group speed (f ) is the same as (f ) and,
2 1 2 2 2 1 1
therefore, the pulse p is delayed by the same amount
2
ΔT = (f )L = 50 μs. (2.192)
2 1 2
In this case, the differential delay between the pulses is zero. Next, consider a dispersive fiber with
2
= 100 ps ∕km at 193 THz. By definition,
2
2
d 0 d 1 1 d 1
()= = =
2 2
d d 2 df
1 (f +Δf)− (f)
1 1
1
≃ (2.193)
2 Δf
or
(f )= (f )+ 2Δf . (2.194)
1 2 1 1 2
Let Δf = 1 THz. The pulse p is delayed by
2
ΔT = (f )L = (f )L + 2Δf L (2.195)
2 1 2 1 1 2
= 50.00628 μs. (2.196)

The differential delay between the pulses is given by
ΔT =ΔT −ΔT = 2Δf L (2.197)
1
2
2
= 6.28 ns. (2.198)
In other words, pulse p arrives at the fiber output later than pulse p by 6.28 ns. Instead of finding the deriva-
2
1
tive of with respect to frequency as in Eq. (2.193), we could define its derivative with respect to wavelength,
1
d 1
D = , (2.199)
d