Appendix A
From Eq. (3.15), we find that the Einstein coefficients 
and 
are related by
where
The spontaneous emission rate per unit volume is given by Eq. (3.4),
In Eq. (A.3), the medium is assumed to be homogeneous with refractive index 
and this emission rate takes into account all the modes of the homogeneous medium in the frequency interval 
. 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 ...
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