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