# Semiconductor Laser Linewidth Theory Revisited
## 3 Equation of the Field Amplitude
For \(m=n\), the factor \(\partial\beta/\partial\omega\), which can be interpreted as an inverse complex-valued group velocity, is the exact derivative  
\[\frac{\partial\beta}{\partial\omega}=\frac{n_{\text{g}}}{c}+\frac{\partial \beta_{\mathcal{D}}}{\partial\omega}\Big{|}_{\Omega_{m}}. \tag{15}\]  
In the general case of \(m\
eq n\) this holds only approximately:  
\[\frac{\partial\beta}{\partial\omega}=\frac{n_{\text{g}}}{c}+\frac{\beta_{ \mathcal{D}}(\Omega_{m})-\beta_{\mathcal{D}}(\Omega_{n})}{\Omega_{m}-\Omega_{ n}}\approx\frac{n_{\text{g}}}{c}+\frac{\partial\beta_{\mathcal{D}}}{\partial \omega}\Big{|}_{\Omega_{m}}. \tag{16}\]  
A system of ordinary differential equations for the amplitudes \(f_{m}(t)\) can be obtained by inserting (8) into (2), using (9), multiplying from left with \([\Phi_{n}^{-},\Phi_{n}^{+}]\), and integrating along \(z\). In order to exploit the orthogonality relation (13) with the approximation (16), we have again to expand  
\[\beta_{\mathcal{D}}(\tilde{\omega})-\beta_{\mathcal{D}}(\Omega_{m})=\frac{ \partial\beta_{\mathcal{D}}}{\partial\omega}\Big{|}_{\Omega_{m}}(\tilde{ \omega}-\Omega_{m}), \tag{17}\]where \(\tilde{\omega}=\omega-\omega_{0}\). Note that this is consistent with the slowly varying amplitude approximation. Transforming back into the time domain yields  
\[\int[\beta_{\mathcal{D}}(\tilde{\omega})-\beta_{\mathcal{D}}(\Omega_{m})]f_{m}( \tilde{\omega})\mathrm{e}^{i\tilde{\omega}t}\,d\tilde{\omega}=\Big{[}\widehat{ \mathcal{D}}-\beta_{\mathcal{D}}(\Omega_{m})\Big{]}f_{m}(t) \tag{18}\]  
and  
\[ \begin{split}\frac{\partial\beta_{\mathcal{D}}}{\partial\omega} \Big{|}_{\Omega_{m}}\int(\tilde{\omega}-\Omega_{m})f_{m}(\tilde{\omega}) \mathrm{e}^{i\tilde{\omega}t}\,d\tilde{\omega}&=-\frac{ \partial\beta_{\mathcal{D}}}{\partial\omega}\Big{|}_{\Omega_{m}}\bigg{[}\Omega _{m}f_{m}(t)+i\frac{\partial}{\partial t}\int f_{m}(\tilde{\omega})\mathrm{e} ^{i\tilde{\omega}t}\,d\tilde{\omega}\bigg{]}\\ &=-\frac{\partial\beta_{\mathcal{D}}}{\partial\omega}\Big{|}_{ \Omega_{m}}\bigg{[}\Omega_{m}f_{m}(t)+i\frac{\partial f_{m}}{\partial t} \bigg{]}.\end{split}  \tag{19}\]  
Therefore the amplitudes \(f_{m}\) fulfill  
\[(\Phi_{m},\frac{\partial\beta}{\partial\omega}\Phi_{m})\frac{\partial f_{m}} {\partial t}-i\Omega_{m}(\Phi_{m},\frac{\partial\beta}{\partial\omega}\Phi_{m })f_{m}+i\sum_{n}(\Phi_{m},\Delta M\Phi_{n})f_{n}=(\Phi_{m},F_{\mathrm{sp}}) \tag{20}\]  
with  
\[\Delta M=M-\overline{M}=(\beta-\overline{\beta}) \begin{bmatrix}1&0\\ 0&1\end{bmatrix} . \tag{21}\]  
In what follows, we employ the single mode approximation and drop the subscript \(m\). Then (20) becomes a single differential equation for the complex-valued amplitude \(f\) of the lasing mode  
\[\frac{\partial f}{\partial t}-i\Omega f+i\frac{(\Phi,\Delta M\Phi)}{(\Phi, \frac{\partial\beta}{\partial\omega}\Phi)}f=F_{f}. \tag{22}\]  
The new [PERSON] noise source entering (22) is given by  
\[F_{f}=\frac{(\Phi,F_{\mathrm{sp}})}{(\Phi,\frac{\partial\beta}{\partial\omega }\Phi)}. \tag{23}\]  
For an analysis above threshold, it is beneficial to derive equations for the modulus \(|f|\) and the phase \(\varphi\) of \(f\), because the fluctuations of \(|f|\) are damped but those of \(\varphi\) are not.