Page 548 - Fiber Optic Communications Fund

Appendix A 529

k y

k x

k z

k
k+dk

Figure A.1 Number of modes in the volume of a spherical shell enclosed between two spheres with radii k and k + dk.

For each mode defined by (k ̂x + k ̂y + k ̂z), there could be two polarizations (see Section 1.11). Therefore,
y
z
x
each mode can be considered as two polarization modes. The total number of modes per unit volume per unit
frequency interval, taking into account two polarization modes, is then
2dn dn dn z N
y
x
= = m (A.15)
3
L d L 3
2 3
n
= 0 , (A.16)
2 3
c
where N d is the number of modes in the frequency interval [, + d]. Eq. (A.16) is valid only for a
m
homogeneous medium. In the case of an optical fiber, the general expression Eq. (A.15) should be used. From
Eq. (A.3), the photon gain rate per unit volume due to spontaneous emission is
dN ph N ℏBN 2
( )
m
R = = ℏBN = , (A.17)
spont 2 3
dt L
spont
where N is the photon density. The spontaneous emission occurs over all the spatial and polarization modes
ph
of an optical fiber, and Eq. (A.17) represents the total spontaneous emission rate over all the modes. However,
all the modes do not contribute to the spontaneous emission at the amplifier output. In a single-mode fiber,
only the spontaneous emission coupled to the guided mode is of interest. For a single mode fiber with a single
polarization mode, N = 1 and Eq. (A.17) becomes
m
( )
dN ph
R = = AN , (A.18)
spont 2
dt
spont
where
3
A = ℏB∕L . (A.19)