# Semiconductor Laser Linewidth Theory Revisited
## 3 Equation of the Field Amplitude
For the modulus squared \(|f|^{2}\) (called intensity in the rest of the paper) it follows  
\[\frac{\partial|f|^{2}}{\partial t}+2\,\mathrm{Im}(\Omega)|f|^{2}-2\,\mathrm{ Im}\,\frac{(\Phi,\Delta M\Phi)}{(\Phi,\frac{\partial\beta}{\partial\omega}\Phi)}|f|^{2 }=2\,\mathrm{Re}(F_{f}f^{*}). \tag{24}\]  
An equation for the phase can be determined from  
\[\mathrm{Im}\bigg{(}\frac{\partial f}{\partial t}f^{*}\bigg{)}-\mathrm{Re}( \Omega)|f|^{2}+\mathrm{Re}\,\frac{(\Phi,\Delta M\Phi)}{(\Phi,\frac{\partial \beta}{\partial\omega}\Phi)}|f|^{2}=\mathrm{Im}(F_{f}f^{*}) \tag{25}\]  
and  
\[\frac{\partial f}{\partial t}=\mathrm{e}^{i\varphi}\frac{\partial|f|}{ \partial t}+if\frac{\partial\varphi}{\partial t}\quad\Longrightarrow\quad \mathrm{Im}\bigg{(}\frac{\partial f}{\partial t}f^{*}\bigg{)}=|f|^{2}\frac{ \partial\varphi}{\partial t}. \tag{26}\]  
The stochastic term on the right-hand side of (24) describes intensity noise, which does not have a vanishing expectation value anymore: \(\langle F_{f}f^{*}\rangle\
eq 0\). It is therefore convenient to rewrite the term by distilling out the expectation value and introducing two new real-valued Langevin forces \(F_{|f|^{2}}\) and \(F_{\varphi}\) with zero mean as  
\[2F_{f}f^{*}=2\langle F_{f}f^{*}\rangle+F_{|f|^{2}}+2i|f|^{2}F_{\varphi}. \tag{27}\]It will be shown below that \(\langle F_{f}f^{*}\rangle\) is real-valued, see (66). Then, the final equations for the modulus squared and phase of \(f\) read  
\[\frac{\partial|f|^{2}}{\partial t}=-2\,\text{Im}(\Omega)|f|^{2}+2\,\text{Im} \,\frac{(\Phi,\Delta M\Phi)}{(\Phi,\frac{\partial\beta}{\partial\omega}\Phi)}|f |^{2}+2\langle F_{f}f^{*}\rangle+F_{|f|^{2}} \tag{28}\]  
and  
\[\frac{\partial\varphi}{\partial t}=\text{Re}(\Omega)-\text{Re}\,\frac{(\Phi, \Delta M\Phi)}{(\Phi,\frac{\partial\beta}{\partial\omega}\Phi)}+F_{\varphi}. \tag{29}\]  
Now we consider small fluctuations  
\[N(z,t) =\langle N(z)\rangle+\delta N(z,t), \tag{30}\] \[\varphi(t) =\langle\varphi(t)\rangle+\delta\varphi(t),\] (31) \[F_{f}=\frac{(\Phi,F_{\mathrm{sp}})}{(\Phi,\frac{\partial\beta}{\partial\omega }\Phi)}. \tag{23}\]0  
and expand  
\[F_{f}=\frac{(\Phi,F_{\mathrm{sp}})}{(\Phi,\frac{\partial\beta}{\partial\omega }\Phi)}. \tag{23}\]1  
around the mean values  
\[F_{f}=\frac{(\Phi,F_{\mathrm{sp}})}{(\Phi,\frac{\partial\beta}{\partial\omega }\Phi)}. \tag{23}\]2 \[F_{f}=\frac{(\Phi,F_{\mathrm{sp}})}{(\Phi,\frac{\partial\beta}{\partial\omega }\Phi)}. \tag{23}\]3  
The mean intensity \(\langle|f|^{2}\rangle\) is required to satisfy  
\[F_{f}=\frac{(\Phi,F_{\mathrm{sp}})}{(\Phi,\frac{\partial\beta}{\partial\omega }\Phi)}. \tag{23}\]4  
Due to the extreme smallness of the spontaneous emission \(\langle F_{f}f^{*}\rangle\) going into the lasing mode, (35) can be replaced by  
\[F_{f}=\frac{(\Phi,F_{\mathrm{sp}})}{(\Phi,\frac{\partial\beta}{\partial\omega }\Phi)}. \tag{23}\]5  
above threshold \((\langle|f|^{2}\rangle>0)\), which will be exploited in what follows (i.e., \(\Omega=\text{Re}(\Omega)\)). The intensity and phase fluctuations fulfill the equations  
\[F_{f}=\frac{(\Phi,F_{\mathrm{sp}})}{(\Phi,\frac{\partial\beta}{\partial\omega }\Phi)}. \tag{23}\]6  
and  
\[F_{f}=\frac{(\Phi,F_{\mathrm{sp}})}{(\Phi,\frac{\partial\beta}{\partial\omega }\Phi)}. \tag{23}\]7  
which are the basis for the derivation of the linewidth formula in the following sections. **4. Lorentzian Line Shape**  
The power spectral density (PSD) of the optical field at \(z=0\) can be calculated utilizing the Wiener-Khinchin theorem as [44]  
\[F_{f}=\frac{(\Phi,F_{\mathrm{sp}})}{(\Phi,\frac{\partial\beta}{\partial\omega }\Phi)}. \tag{23}\]8  
with  
\[F_{f}=\frac{(\Phi,F_{\mathrm{sp}})}{(\Phi,\frac{\partial\beta}{\partial\omega }\Phi)}. \tag{23}\]9We do not address here technical issues regarding the non-existence of Fourier integrals of fluctuating quantities here and refer to the discussion in [44].