# Semiconductor Laser Linewidth Theory Revisited
## Appendix B Calculation of Derivative of \(\mathbf{r^{+}}\)  
To derive (92) we have to calculate  
\[\partial_{\omega}\ln(r^{+})=\frac{1}{r^{+}}\partial_{\omega}r^{+}=\frac{1}{ \Phi^{-}}\partial_{\omega}\Phi^{-}-\frac{1}{\Phi^{+}}\partial_{\omega}\Phi^{+}, \tag{11}\]  
where we abbreviated \(\partial/\partial\omega\equiv\partial_{\omega}\). The coupled-wave Equation (9) together with their derivatives with respect to frequency can be written as  
\[\begin{split}\frac{\partial}{\partial z}\[PERSON]^{+}(z)& =-i\hat{\beta}(\Omega)\[PERSON]^{+}(z)-i\kappa\[PERSON]^{-}(z),\\ -\frac{\partial}{\partial z}\[PERSON]^{-}(z)&=-i\hat{ \beta}(\Omega)\Phi^{-}(z)-i\kappa\[PERSON]^{+}(z),\\ \frac{\partial}{\partial z}\partial_{\omega}\[PERSON]^{+}(z)& =-i\hat{\beta}(\Omega)\partial_{\omega}\[PERSON]^{+}(z)-i\kappa \partial_{\omega}\[PERSON]^{-}(z)-i\partial_{\omega}\beta(\Omega)\[PERSON]^{+}(z),\\ -\frac{\partial}{\partial z}\partial_{\omega}\[PERSON]^{-}(z)& =-i\hat{\beta}(\Omega)\partial_{\omega}\[PERSON]^{-}(z)-i\kappa \partial_{\omega}\[PERSON]^{+}(z)-i\partial_{\omega}\beta(\Omega)\[PERSON]^{-}(z),\end{split} \tag{12}\]  
where \(\hat{\beta}(\Omega)=\Delta\overline{\beta}(z)+\beta_{\mathcal{D}}(\Omega)+n_ {\mathrm{g}}\Omega/c\) and \(\partial_{\omega}\beta(\Omega)\) is defined in (15). By multiplying the left- and right-hand sides of these equations by \(\partial_{\omega}\Phi^{-},\partial_{\omega}\[PERSON]^{+},\)\(-\Phi^{-},\) and \(-\Phi^{+}\), respectively, we get  
\[\begin{split}\partial_{\omega}\Phi^{-}\frac{\partial}{\partial z }\Phi^{+}&=-i\hat{\beta}(\Omega)\[PERSON]^{+}\partial_{\omega}\Phi^{- }-i\kappa\Phi^{-}\partial_{\omega}\[PERSON]^{-},\\ -\partial_{\omega}\[PERSON]^{+}\frac{\partial}{\partial z}\[PERSON]^{-}& =-i\hat{\beta}(\Omega)\[PERSON]^{-}\partial_{\omega}\[PERSON]^{+}-i\kappa \[PERSON]^{+}\partial_{\omega}\[PERSON]^{+},\\ -\[PERSON]^{-}\frac{\partial}{\partial z}\partial_{\omega}\[PERSON]^{+}& =i\hat{\beta}(\Omega)\[PERSON]^{-}\partial_{\omega}\[PERSON]^{+}+i\kappa \[PERSON]^{-}\partial_{\omega}\[PERSON]^{-}+i\partial_{\omega}\beta(\Omega)\[PERSON]^{-} \[PERSON]^{+},\\ \[PERSON]^{+}\frac{\partial}{\partial z}\partial_{\omega}\[PERSON]^{-}& =i\hat{\beta}(\Omega)\[PERSON]^{+}\partial_{\omega}\[PERSON]^{-}+i\kappa \[PERSON]^{+}\partial_{\omega}\[PERSON]^{+}+i\partial_{\omega}\beta(\Omega)\[PERSON]^{+} \[PERSON]^{-}.\end{split} \tag{13}\]  
Adding all these equations implies  
\[\frac{\partial}{\partial z}\big{(}\[PERSON]^{+}\partial_{\omega}\[PERSON]^{-}-\[PERSON]^{-} \partial_{\omega}\[PERSON]^{+}\big{)}=2i\partial_{\omega}\beta(\Omega)\[PERSON]^{+} \[PERSON]^{-}. \tag{14}\]An integration over the passive sections corresponding to the coordinate interval \([l,L]\) yields  
\[\big{[}\[PERSON]^{+}\partial_{\omega}\[PERSON]^{-}-\[PERSON]^{-}\partial_{\omega}\[PERSON]^{+}\big{]} \big{|}_{l}^{L}=2i\int_{l}^{L}\partial_{\omega}\beta(\Omega)\[PERSON]^{+}\,\[PERSON]^{-}dz. \tag{101}\]  
Due to the reflecting boundary condition, the expression \([\[PERSON]^{+}\,\partial_{\omega}\[PERSON]^{-}-\[PERSON]^{-}\partial_{\omega}\[PERSON]^{+}]\) vanishes at \(z=L\) and  
\[-\[PERSON]^{+}(l)\,\partial_{\omega}\[PERSON]^{-}(l)+\[PERSON]^{-}(l)\,\partial_{\omega}\[PERSON]^ {+}(l)=2i\int_{l}^{L}\partial_{\omega}\beta(z,\Omega)\[PERSON]^{+}(z)\[PERSON]^{-}(z)dz \tag{102}\]  
follows. Dividing by \(-\[PERSON]^{+}(l)\Phi^{-}(l)\) finally yields  
\[\partial_{\omega}\log(r^{+})=-2i\frac{\int_{l}^{L}\partial_{\omega}\beta(z, \Omega)\[PERSON]^{+}(z)\Phi^{-}(z)dz}{\Phi^{+}(l)\Phi^{-}(l)}. \tag{103}\]