# Semiconductor Laser Linewidth Theory Revisited
## 10 Numerical Results for a DBR Laser
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Symbol** & **Unit** & **Value** & **First Used** \\ \hline reference wavelength & \(\lambda_{0}\) & m & \(1.064\cdot 10^{-6}\) & \\ front facet reflectivity & \(|r_{0}|^{2}\) & & 0.3 & (4) \\ rear facet reflectivity & \(|r_{L}|^{2}\) & & 0 & (4) \\ internal optical loss (both sections) & \(\alpha\) & m\({}^{-1}\) & 60 & (3c) \\ group index (both sections) & \(n_{\text{g}}\) & & 3.9 & (2) \\ built-in index detuning (both sections) & \(\Delta n_{0}\) & & 0 & (126) \\ \hline \hline \multicolumn{5}{c}{Active section} \\ \hline length & \(l\) & m & \(1\cdot 10^{-3}\) & \\ differential gain & \(g^{\prime}\) & m\({}^{2}\) & \(18.62\cdot 10^{-22}\) & (122) \\ \(\alpha\)-factor & \(\tilde{\alpha}_{\text{H}}\) & & 1 & (79) \\ transparency carrier density & \(N_{\text{tr}}\) & m\({}^{-3}\) & \(1.7\cdot 10^{24}\) & (122) \\ self-heating induced index tuning & \(v_{s}\) & A\({}^{-1}\) & \(9.16\cdot 10^{-3}\) & (126) \\ gain clamping density & \(N_{\text{cl}}\) & m\({}^{-3}\) & \(0.8\cdot 10^{24}\) & (122) \\ index clamping density & \(N_{\text{cl,i}}\) & m\({}^{-3}\) & \(0.01\cdot 10^{24}\) & (125) \\ gain saturation power & \(P_{\text{sat}}\) & W & 5.866 & (124) \\ dispersion peak amplitude & \(g_{D}\) & m\({}^{-1}\) & 50 & (A3) \\ dispersion peak frequency detuning & \(\omega_{D}\) & rad s\({}^{-1}\) & 0 & (A3) \\ dispersion HWHM & \(\gamma_{D}\) & rad s\({}^{-1}\) & \(83.25\cdot 10^{12}\) & (A3) \\ thickness of active region & \(d\) & m & \(4\cdot 10^{-6}\) & (5) \\ width of active region & \(W\) & m & \(10\cdot 10^{-9}\) & (6) \\ series resistance & \(R_{s}\) & \(\Omega\) & 1 & (7) \\ Fermi voltage derivative & \(\frac{dU_{\text{F}}}{dN}\) & Vm\({}^{3}\) & \(3\cdot 10^{-26}\) & (128) \\ defect recombination coefficient & \(A\) & s\({}^{-1}\) & \(4\cdot 10^{8}\) & (127) \\ bimolecular recombination coefficient & \(B\) & m\({}^{3}\) s\({}^{-1}\) & \(1\cdot 10^{-16}\) & (127) \\ Auger recombination coefficient & \(C\) & m\({}^{6}\) s\({}^{-1}\) & \(4\cdot 10^{-42}\) & (127) \\ injection current & \(I\) & A & \([0,300]\cdot 10^{-3}\) & (7) \\ \hline \multicolumn{5}{c}{Passive section} \\ \hline length & \(L-l\) & m & \(3\cdot 10^{-3}\) & \\ coupling coefficient & \(\kappa\) & m\({}^{-1}\) & 200 & (10) \\ cross-heating induced index tuning & \(v_{c}\) & A\({}^{-1}\) & \(1.83\cdot 10^{-3}\) & (126) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the simulated DBR laser. and the refractive index model with square-root-like dependency on the carrier density  
\[\Delta n_{N}=-\frac{\lambda_{0}\bar{\kappa}_{\rm H}g^{\prime}N_{\rm tr}}{2\pi} \left(\sqrt{\frac{\max(N,N_{\rm cl,i})}{N_{\rm tr}}}-1\right). \tag{125}\]  
Note, that at the transparency carrier density \(N_{\rm tr}\), the fixed parameter \(\bar{\kappa}_{\rm H}\) agrees with the \(\alpha\)-factor \(\alpha_{\rm H}\) from (79), whereas \(\Delta n_{N}\) vanishes. The \(\alpha\)-factor correspondingly increases for \(N>N_{\rm tr}\). The value of \(\bar{\kappa}_{\rm H}=1\) was chosen in agreement with simulations [58] and is in basic agreement with measurements, where \(\alpha\)-factors between 1 and 2 for a highly-strained InGaAs quantum well were obtained, depending on the detuning [62]. Other changes of the refractive index, including self-heating-induced self- and cross-heating contributions, are modeled by [30]  
\[\Delta n(z)=\Delta n_{0}(z)+\Delta n_{T}(z),\qquad\Delta n_{T}(z)= \begin{cases} v_{s}I,&z\in[0,l]\\ v_{c}I,&z\in[l,L]\end{cases} , \tag{126}\]  
where \(I\) is the injection current into the active (gain) section. We employ the commonly used cubic model for the rate of non-radiative and spontaneous recombination of the carriers in the active section  
\[R(N)=AN+BN^{2}+CN^{3}, \tag{127}\]  
is approximated by describing Shockley-Read-Hall recombination, direct band-to-band recombination and Auger recombination.