Page 546 - Fiber Optic Communications Fund

Appendix A

From Eq. (3.15), we find that the Einstein coefficients A and B are related by

A = ℏB, (A.1)
where
2 3
n 0
= . (A.2)
2 3
c
The spontaneous emission rate per unit volume is given by Eq. (3.4),
( )
dN 2
R =− = ℏBN . (A.3)
spont 2
dt
spont
In Eq. (A.3), the medium is assumed to be homogeneous with refractive index n and this emission rate takes
0
into account all the modes of the homogeneous medium in the frequency interval [, + d]. Typically,
amplifiers or lasers make use of single mode or multi-mode devices such as single/multi-mode fibers or
channel waveguides. In a single-mode fiber amplifier, the ASE coupled to a radiation mode escapes to the
cladding and does not contribute to the fiber amplifier output. Only the ASE coupled to the guided mode is of
practical interest. Therefore, we modify Eq. (A.3) such that the spontaneous emission rate corresponds to ASE
coupled to the guided mode. In fact, of Eq. (A.2) represents the number of modes of a homogeneous medium
per unit volume per unit frequency interval. To see that, consider an electromagnetic wave in a homogeneous
3
medium confined to a cube of volume L . The plane wave inside this cube is
= A cos(t − k x − k y − k z), (A.4)
x y z
with

= kc∕n , (A.5)
0
2 2 2 2
k = k + k + k . (A.6)
x y z
If L is infinite, k , k , and k can take arbitrary values satisfying Eq. (A.6). The propagation of the plane wave
x y z
is in the direction of k = k ̂x + k ̂y + k ̂z. Therefore, spontaneous emission occurs uniformly in all directions.
x y z

Fiber Optic Communications: Fundamentals and Applications, First Edition. Shiva Kumar and M. Jamal Deen.
© 2014 John Wiley & Sons, Ltd. Published 2014 by John Wiley & Sons, Ltd.