# Semiconductor Laser Linewidth Theory Revisited
## 8 Impact of a Passive Section, Chirp Reduction Factor and the Fabry-Perot Case
### Linewidth of the Fabry-Perot Laser Cavity
Second, in [1] the sub-threshold case is considered resulting in a further factor of 2, which can be seen as follows: Below threshold the spectral density can be directly calculated from (22) for \(\Delta M=0\) by Fourier transformation via  
\[\bar{R}_{\rm sp}(N)=\beta_{\rm sp}VR_{\rm rad}(N) \tag{106}\]4  
with the result  
\[\bar{R}_{\rm sp}(N)=\beta_{\rm sp}VR_{\rm rad}(N) \tag{106}\]5  
where we have used (35) and (66). This is a Lorentzian with the FWHM  
\[\bar{R}_{\rm sp}(N)=\beta_{\rm sp}VR_{\rm rad}(N) \tag{106}\]6  
which is a factor of 2 larger than (82) (for \(\alpha_{\text{H,eff}}=0\)). See [6] for a more thorough discussion. **9. Population Inversion Factor**  
The population inversion factor (sometimes misleadingly called'spontaneous emission factor') introduced in (58) and used in (86) and (113) is given by [28]  
\[\bar{R}_{\rm sp}(N)=\beta_{\rm sp}VR_{\rm rad}(N) \tag{106}\]7  
where \(U_{\text{F}}\) is the Fermi voltage (spacing of quasi-Fermi potentials of holes and electrons), \(q\) the elementary charge, \(k_{\text{B}}\) the Boltzmann constant and \(T\) the temperature. It has a singularity at the transparency density \(N_{\text{tr}}\), where \(\hbar\omega_{0}=U_{\text{F}}(N_{\text{tr}})\) and \(g(N_{\text{tr}})=0\). Therefore, the replacement  
\[\bar{R}_{\rm sp}(N)=\beta_{\rm sp}VR_{\rm rad}(N) \tag{106}\]8  
with constant \(n_{\text{sp}}\) often employed is critical because for high-\(Q\)-cavities with low \(\alpha\) and \(\alpha_{\text{out}}\) the spontaneous emission is underestimated. We calculated the modal gain, the spontaneous emission into the lasing mode and the inversion factor employing a numerical simulation based on a \(8\times 8\)\(\mathbf{k}\cdot\mathbf{p}\) band structure calculation and a free carrier theory for the optical response functions with phenomenological corrections for many-body effects such as band-gap renormalization, Coulomb enhancement and transition broadening [58]. The results for a 5 nm thick InGaAs quantum well emitting around 1064 nm embedded into an AlGaAs-based waveguide structure are compared with analytical models in \[\bar{R}_{\rm sp}(N)=\beta_{\rm sp}VR_{\rm rad}(N) \tag{106}\]9 The increase of the absorption (negative gain) at small increasing carrier densities observed in the simulation is caused by band-gap narrowing, shifting the absorption edge to longer wavelengths [59]. The gain at a fixed wavelength is modeled as  
\[\beta_{\rm sp}=\frac{c}{n_{\rm g}}\frac{n_{\rm sp}(N_{\rm th})g(N_{\rm th})}{ VR_{\rm rad}(N_{\rm th})}, \tag{107}\]0  
and the modal spontaneous emission as  
\[\beta_{\rm sp}=\frac{c}{n_{\rm g}}\frac{n_{\rm sp}(N_{\rm th})g(N_{\rm th})}{ VR_{\rm rad}(N_{\rm th})}, \tag{107}\]1  
with differential gain \(g^{\prime}=19\cdot 10^{-22}\) m\({}^{2}\), transparency density \(N_{\rm tr}=1.7\cdot 10^{24}\) m\({}^{-3}\) and gain clamping density \(N_{\rm cl}=7\cdot 10^{23}\) m\({}^{-3}\). Equation (123) exhibits the correct asymptotic behavior \(r_{\rm sp}\propto N^{2}\) for \(N\ll N_{\rm tr}\) and \(r_{\rm sp}=g\) for \(N\rightarrow\infty\). The agreement of \(r_{\rm sp}\) between simulation and model is remarkably good without the need of introducing any new parameters. The singularity of the inversion factor at \(N=N_{\rm tr}\) is clearly visible in \[\beta_{\rm sp}=\frac{c}{n_{\rm g}}\frac{n_{\rm sp}(N_{\rm th})g(N_{\rm th})}{ VR_{\rm rad}(N_{\rm th})}, \tag{107}\]2 If a constant value of \(n_{\rm sp}=2\) is used, the spontaneous emission is considerably under- or over-estimated (depending on the respective carrier density).