# Semiconductor Laser Linewidth Theory Revisited
## 3 Equation of the Field Amplitude
Hence we have to calculate the PSD  
\[\frac{\partial|f|^{2}}{\partial t}+2\,\mathrm{Im}(\Omega)|f|^{2}-2\,\mathrm{ Im}\,\frac{(\Phi,\Delta M\Phi)}{(\Phi,\frac{\partial\beta}{\partial\omega}\Phi)}|f|^{2 }=2\,\mathrm{Re}(F_{f}f^{*}). \tag{24}\]0  
with  
\[\frac{\partial|f|^{2}}{\partial t}+2\,\mathrm{Im}(\Omega)|f|^{2}-2\,\mathrm{ Im}\,\frac{(\Phi,\Delta M\Phi)}{(\Phi,\frac{\partial\beta}{\partial\omega}\Phi)}|f|^{2 }=2\,\mathrm{Re}(F_{f}f^{*}). \tag{24}\]1  
Now we employ here a first approximation: We neglect in (42) the intensity fluctuations because above threshold they are damped, in contrast to the phase fluctuations, which can been seen by comparing (37) and (38), see the discussion in [45]. Therefore, we obtain  
\[\frac{\partial|f|^{2}}{\partial t}+2\,\mathrm{Im}(\Omega)|f|^{2}-2\,\mathrm{ Im}\,\frac{(\Phi,\Delta M\Phi)}{(\Phi,\frac{\partial\beta}{\partial\omega}\Phi)}|f|^{2 }=2\,\mathrm{Re}(F_{f}f^{*}). \tag{24}\]2  
with  
\[\frac{\partial|f|^{2}}{\partial t}+2\,\mathrm{Im}(\Omega)|f|^{2}-2\,\mathrm{ Im}\,\frac{(\Phi,\Delta M\Phi)}{(\Phi,\frac{\partial\beta}{\partial\omega}\Phi)}|f|^{2 }=2\,\mathrm{Re}(F_{f}f^{*}). \tag{24}\]3  
and \(\Omega=\mathrm{Re}(\Omega)\). The random variable \(\Delta_{\tau}\varphi\) is assumed to be Gaussian distributed [46], such that it holds  
\[\frac{\partial|f|^{2}}{\partial t}+2\,\mathrm{Im}(\Omega)|f|^{2}-2\,\mathrm{ Im}\,\frac{(\Phi,\Delta M\Phi)}{(\Phi,\frac{\partial\beta}{\partial\omega}\Phi)}|f|^{2 }=2\,\mathrm{Re}(F_{f}f^{*}). \tag{24}\]4  
The variance of the phase fluctuations is  
\[ \begin{split}\mathrm{Var}(\Delta_{\tau}\varphi)&= \langle(\Delta_{\tau}\varphi-\langle\Delta_{\tau}\varphi\rangle)^{2}\rangle \\ &=\langle(\Delta_{\tau}\varphi)^{2}\rangle\\ &=\langle(\delta\varphi(t+\tau))^{2}\rangle-2\langle\delta \varphi(t)\delta\varphi(t+\tau)\rangle+\langle(\delta\varphi(t))^{2}\rangle \\ &=2\Big{[}\langle(\delta\varphi(0))^{2}\rangle-\langle\delta \varphi(0)\delta\varphi(\tau)\rangle\Big{]},\end{split}  \tag{46}\]  
where we exploited \(\langle\Delta_{\tau}\varphi\rangle=0\) and the stationarity property. We introduce the PSD of phase fluctuations which is even in the frequency (\(S_{\delta\varphi}(\tilde{\omega})=S_{\delta\varphi}(-\tilde{\omega})\))  
\[S_{\delta\varphi}(\tilde{\omega})=\int_{-\infty}^{+\infty}\langle\delta \varphi(t)\delta\varphi(t+\tau)\rangle\mathrm{e}^{-i\tilde{\omega}\tau}\,d\tau \tag{47}\]  
and the PSD of optical frequency fluctuations \(\delta\omega(\tilde{\omega})=i\tilde{\omega}\delta\varphi(\tilde{\omega})\)  
\[S_{\delta\omega}(\tilde{\omega})=\int_{-\infty}^{+\infty}\langle\delta \omega(t)\delta\omega(t+\tau)\rangle\mathrm{e}^{-i\tilde{\omega}\tau}\,d\tau =\tilde{\omega}^{2}S_{\delta\varphi}(\tilde{\omega}). \tag{48}\]  
With these spectral densities, the variance of the phase fluctuations can be written as  
\[ \begin{split}\mathrm{Var}(\Delta_{\tau}\varphi)&= \frac{1}{\pi}\int_{-\infty}^{\infty}S_{\delta\varphi}(\tilde{\omega})\Big{(}1 -\mathrm{e}^{i\tilde{\omega}\tau}\Big{)}\,d\tilde{\omega}\\ &=\frac{1}{\pi}\int_{-\infty}^{\infty}S_{\delta\varphi}(\tilde{ \omega})\left(2\sin^{2}\!\left(\frac{\tilde{\omega}\tau}{2}\right)-i\sin( \tilde{\omega}\tau)\right)d\tilde{\omega}\\ &=\frac{2}{\pi}\int_{-\infty}^{\infty}S_{\delta\varphi}(\tilde{ \omega})\sin^{2}\!\left(\frac{\tilde{\omega}\tau}{2}\right)d\tilde{\omega}\\ &=\frac{\tau^{2}}{2\pi}\int_{-\infty}^{\infty}S_{\delta\omega}( \tilde{\omega})\left(\frac{\sin\left(\frac{\tilde{\omega}\tau}{2}\right)}{ \frac{\tilde{\omega}\tau}{2}}\right)^{2}d\tilde{\omega}.\end{split}