# Semiconductor Laser Linewidth Theory Revisited
## Appendix C The Relation to Static Frequency Chirp  
Here we justify the naming of the factor \(F\) defined in (101). We start from the roundtrip condition  
\[r^{+}(\tilde{\omega},g)r^{-}(\tilde{\omega},g)=1 \tag{104}\]  
following from (91) which can be transformed into equations for the modulus  
\[\text{Re}\big{(}\text{ln}(r^{-}(\tilde{\omega},g))\big{)}+\text{Re}\big{(}\text {ln}(r^{+}(\tilde{\omega}))\big{)}=0 \tag{105}\]  
and the phase  
\[h(\tilde{\omega},g)\equiv\text{Im}\big{(}\text{ln}(r^{-}(\tilde{\omega},g)) \big{)}+\text{Im}\big{(}\text{ln}(r^{+}(\tilde{\omega}))\big{)}=2\pi m \tag{106}\]  
with \(\tilde{\omega}=\omega-\omega_{0}\) and \(m\in\mathbb{Z}\). As a result of the solution of (104) or, equivalently, (105) and (106), we obtain the real-valued frequency \(\Omega=\tilde{\omega}\) and the gain \(g\) of the steady lasing state. Let the modal index in the active section vary due to a fluctuation of some parameter such as carrier density or temperature. Equation (106) establishes a relation between the index fluctuation \(\delta\Delta n\) and the frequency fluctuation \(\delta\omega\),  
\[\delta h=\frac{\partial h}{\partial\omega}\delta\omega+\frac{\partial h}{ \partial n}\delta n=0, \tag{107}\]  
where \(\delta n\equiv\delta\Delta n\) and \(1/\partial n\equiv 1/\partial\Delta n\). The \(\omega\)-derivative of \(h\) is  
\[\frac{\partial h}{\partial\omega}=\text{Im}\bigg{(}\frac{\partial\,\text{ln} (r^{-})}{\partial\omega}\bigg{)}+\text{Im}\bigg{(}\frac{\partial\,\text{ln} (r^{+})}{\partial\omega}\bigg{)}+\text{Im}\bigg{(}\frac{\partial\,\text{ln}(r ^{-})}{\partial g}\bigg{)}\frac{\partial g}{\partial\omega}. \tag{108}\]  
The \(\omega\)-derivative of the lasing gain \(g\) can be determined from (105),  
\[\frac{\partial g}{\partial\omega}=-\frac{\text{Re}\bigg{(}\frac{\partial\, \text{ln}(r^{-})}{\partial\omega}\bigg{)}+\text{Re}\bigg{(}\frac{\partial\, \text{ln}(r^{+})}{\partial\omega}\bigg{)}}{\text{Re}\bigg{(}\frac{\partial\, \text{ln}(r^{-})}{\partial g}\bigg{)}}=-\frac{1}{l}\,\text{Re}\bigg{(}\frac{ \partial\,\text{ln}(r^{+})}{\partial\omega}\bigg{)} \tag{109}\]  
taking into account  
\[\frac{\partial\,\text{ln}(r^{-})}{\partial\omega}=-2 il\frac{n_{\text{g}}}{c} \tag{110}\]  
and  
\[\frac{\partial\,\text{ln}(r^{-})}{\partial g}=l(1+i\alpha_{\text{H}}), \tag{111}\]following from (91) and (11), such that  
\[\frac{\partial h}{\partial\omega}=-2l\frac{n_{\rm g}}{c}+\operatorname{Im}\left( \frac{\partial\ln(r^{+})}{\partial\omega}\right)-\alpha_{\rm H}\operatorname{ Re}\left(\frac{\partial\ln(r^{+})}{\partial\omega}\right) \tag{101}\]  
is obtained. Furthermore, (91) and (11) imply  
\[\frac{\partial\ln(r^{-})}{\partial n}=-2 il\frac{2\pi}{\lambda_{0}} \tag{102}\]  
and  
\[\text{Re}\big{(}\text{ln}(r^{-}(\tilde{\omega},g))\big{)}+\text{Re}\big{(}\text {ln}(r^{+}(\tilde{\omega}))\big{)}=0 \tag{105}\]0  
Therefore,  
\[\text{Re}\big{(}\text{ln}(r^{-}(\tilde{\omega},g))\big{)}+\text{Re}\big{(}\text {ln}(r^{+}(\tilde{\omega}))\big{)}=0 \tag{105}\]1  
and  
\[\text{Re}\big{(}\text{ln}(r^{-}(\tilde{\omega},g))\big{)}+\text{Re}\big{(}\text {ln}(r^{+}(\tilde{\omega}))\big{)}=0 \tag{105}\]2  
is gained. For the solitary laser with \(r^{+}(\tilde{\omega})\) taken at a fixed \(\tilde{\omega}=\Omega\), the same analysis results in  
\[\text{Re}\big{(}\text{ln}(r^{-}(\tilde{\omega},g))\big{)}+\text{Re}\big{(}\text {ln}(r^{+}(\tilde{\omega}))\big{)}=0 \tag{105}\]3  
and  
\[\text{Re}\big{(}\text{ln}(r^{-}(\tilde{\omega},g))\big{)}+\text{Re}\big{(}\text {ln}(r^{+}(\tilde{\omega}))\big{)}=0 \tag{105}\]4  
Therefore in a laser with a passive section or an external cavity, fluctuations of the frequency \(\delta\omega\) due to a fluctuation of some parameter in the active section are reduced (or enhanced) by the factor  
\[\text{Re}\big{(}\text{ln}(r^{-}(\tilde{\omega},g))\big{)}+\text{Re}\big{(}\text {ln}(r^{+}(\tilde{\omega}))\big{)}=0 \tag{105}\]5  
compared to a the solitary laser, as stated in, e.g., [16; 17].