Page 130 - Fiber Optic Communications Fund

Lasers 111

transition, while it increases due to absorption and external pumping. The net growth rate of the population
density of state 2 is
dN 2
= R pump + R abs + R stim + R spont + R . (3.70)
nr
dt
Here, R abs , R stim , and R spont are given by Eqs. (3.2), (3.3), and (3.4), respectively. R pump refers to the pumping
rate, which is the rate at which the population density of state 2 grows due to an external pump. A specific
pumping scheme is discussed in Section 3.8.
An atom in state 2 could drop down to state 1 by releasing the energy difference as translational, vibra-
tional, or rotational energies of the atom or nearby atoms/molecules. This is known as non-radiative transition,
since no photon is emitted as the atom makes transition from state 2 to state 1 and R represents the rate of
nr
non-radiative transition from state 2 to state 1. It is given by
R = CN , (3.71)
nr 2
where C is a constant similar to the Einstein coefficient A.
Using Eqs. (3.2), (3.3), (3.4), and (3.71) in Eq. (3.70), we find
dN 2
= R pump + BuN − BuN −(A + C)N . (3.72)
2
1
2
dt
The population density of the ground state increases due to stimulated emission, spontaneous emission, and
non-radiative transition, while it decreases due to absorption. The rate of change of the population density of
the ground state is
dN 1
= R stim + R spont + R + R abs
nr
dt
= BuN +(A + C)N − BuN . (3.73)
2
2
1
Next, let us consider the growth rate of photons. Let N ph be the photon density. When an atom makes a
transition from the excited state to the ground state due to stimulated emission, it emits a photon. If there are
R stim transitions per unit time per unit volume, the growth rate of photon density is also R stim . The photon
density in a laser cavity increases due to stimulated emission and spontaneous emission, while it decreases
due to absorption and loss in the cavity. The growth rate of photon density is given by

dN
ph
= R stim + R spont + R abs + R loss . (3.74)
dt
Here, R loss refers to the loss rate of photons due to internal loss and mirror loss in the cavity. Since the energy
of a photon is ℏ, the mean number of photons present in the electromagnetic radiation of energy E is
E
n ph = . (3.75)
ℏ
The photon density N is the mean number of photons per unit volume and the energy density u is the energy
ph
per unit volume. Therefore, they are related by
n ph E u
N ph = = = . (3.76)
V ℏV ℏ
In Section 3.3, we developed an expression for the time rate of change of the energy density u in the presence
of stimulated emission and loss as
du
=(G − cav )u. (3.77)
dt