Despite the set of rate equations formulate the instantaneous population of the upper and lower levels, the population inversion of laser transition, gain and saturation properties, however those equations suffer a drawback. To my best knowledge, the rate equations should be rewritten to discriminate the coherence degree of the laser beam (that arises from stimulated emission and laser oscillation in the resonator) from amplified spontaneous emission (ASE).
I suggest to consider an expression for coherent component of the beam as a "complex degree of partial coherence" in both terms of spontaneous and stimulated emission rates where it is equal to unity for perfect coherence, less than unity for partial coherence and zero for non-coherent beams. It may be an approach to discriminate the coherent light from non-coherent one.
It is essential to define expressions to discriminate Coherence components(arises from stimulated emission) of the light from ASE( incoherent light due to spontaneous emmision)) in laser resonators , otherwise there is no difference of the corresponding notations in rate equations.
ASE is noticeable in high gain lasers such as dye ,excimer,FEL and nitrogen and fiber lasers. Er doped silica and Yb doped silca fiber lasers exhibit strong ASE to limit the inversion population and gain.
Even lasers based purely on ASE, e.g. single-pass plasma-based soft X-ray lasers, evince pretty good degree of coherence, albeit no resonator is present.
ASE is not laser but is accompanying laser beam. ASE has wider spectral band and is emitted in bigger solid angle whereas its degree of coherence is much smaller than laser beam. In soft X-ray laser the diffraction limit divergence is relatively lower than conventional lasers due to shorter wavelength, however ASE still exhibits wider spectral bandwidth.
ASE and parasitic lasing are significant in Er:silica and Yb:silica fiber lasers too.
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