Page 118 - Fiber Optic Communications Fund

Lasers 99

from collection of atoms is not perfectly monochromatic; this is because the emission can take place due to
transition from any of the sublevels. The interaction of electromagnetic waves over a range of frequencies
with the two-band system is described by a lineshape function l(). Out of N (or N ) atoms per unit volume,
2
1
N l()d (or N l()d) atoms interact with electromagnetic waves in the frequency interval [, + d].
1
2
The lineshape function is normalized such that
N l()d = N . (3.24)
∫ 2 2
In other words, N represents the total number of atoms per unit volume in band 2 and N l()d represents
2 2
the fraction of atoms per unit volume that could interact with radiation ranging from to + d. Therefore,
the total stimulation rate per unit volume is

R = Bu ()l()N d. (3.25)
stim ∫ s 2
Similarly, the absorption rate per unit volume is modified as

R abs = ∫ Bu ()l()N d. (3.26)
s
1
Let us consider a special case in which the radiation is a monochromatic wave of frequency . The energy
0
spectral density per unit volume of a monochromatic wave is an impulse function given by
u ()= u( − )∕(2). (3.27)
s
0
Since the energy spectral density is energy per unit frequency interval, its integration over frequency is energy.
Therefore,
u ()d = u ( − )d∕(2)= u. (3.28)
∫ s ∫ 0
Here, u is energy per unit volume or energy density. Substituting Eq. (3.27) in Eqs. (3.25) and (3.26), we find
R stim = Bul( )N , (3.29)
2
0
R = Bul( )N . (3.30)
abs 0 1
By letting Bl( ) → B, Eqs. (3.29) and (3.30) are the same as Eqs. (3.3) and (3.2), respectively, with the
0
exception that u is replaced by energy density u.
s

Example 3.1
In an atomic sytem, the spontaneous lifetime associated with 2 → 1 transition is 2 ns and the energy difference
between the levels is 2.4 × 10 −19 J. Calculate the Einstein A and B coefficients. Assume that the velocity of
8
light in the medium is 1.25 × 10 m∕s.
Solution:
From Eq. (3.23), we have
1 8 −1
A = = 5 × 10 s .
21
t
sp