Page 245 - Fiber Optic Communications Fund

226 Fiber Optic Communications

Using Eqs. (5.65) and (5.66) in Eq. (5.68), we obtain
∞
̃
2
N 1 = qI |H (f)| df
shot PC ∫ e
−∞
= 2qI B . (5.69)
PC e
Using Eq. (5.64), we find the variance of shot noise as
2 = N 1 = 2qI B . (5.70)
shot shot PC e
Eq. (5.70) is valid for arbitrary filter shapes if the effective bandwidth B is defined as
e
∞
1 2
̃
B = 2 ∫ −∞ |H (f)| df. (5.71)
e
e
When the dark current I is not negligible, Eq. (5.70) is modified as
d
2 = 2q(I + I )B . (5.72)
shot PC d e
For APD receivers, the variance of shot noise is given by (see Section 5.3.7) [62]
2
2 = 2qM F(RP + I )B , (5.73)
shot I d e
where M is the multiplication factor and F is the excess noise factor.

5.5.2 Thermal Noise

Electrons move randomly in a conductor. As the temperature increases, electrons move faster and therefore
the electron current increases. However, the mean value of the current is zero since, on average, there are as
many electrons moving in one direction as there are in the opposite direction. Because of the random motion
of electrons, the resulting current is noisy and is called “thermal noise” or “Johnson noise”.
In the presence of thermal noise, the current in the receiver circuit may be written as
I(t)= I + i (t), (5.74)
PC thermal
where I is the mean photocurrent (deterministic), and i (t) is the thermal noise current.
PC thermal
For low frequencies (f ≪ k T∕h), thermal noise can be regarded as white noise, i.e., its power spectral
B
density is constant. It is given by
(f)= 2k T∕R , (5.75)
thermal B L
where k is Boltzmann’s constant, R is the load resistance, and T is the absolute temperature. If B is the
B L e
effective bandwidth of the receiver, the noise variance can be calculated as before,
2 =< i 2 >= 4k TB ∕R . (5.76)
thermal thermal B e L
Eq. (5.76) does not include the noise sources in the amplifier circuit, such as that coming from resistors and
active elements. Eq. (5.76) can be modified to account for the noise sources within the amplifier as [24, 63]

2 = 4k TB F ∕R , (5.77)
B
L
e n
thermal
where F is the amplifier noise factor.
n