# Semiconductor Laser Linewidth Theory Revisited
## 8 Impact of a Passive Section, Chirp Reduction Factor and the Fabry-Perot Case
### Linewidth of a Laser Consisting of a Gain Chip Subject to Feedback from an External Cavity  
We consider a laser consisting of one active section for \(z\in[0,I]\) with length \(l\) and an arbitrary number of passive sections (including an external cavity) for \(z\in[l,L]\) with total length \(L-l\). The active section is of the Fabry-Perot (FP) type having no Bragg grating (\(\kappa=0\) for \(z\in[0,l]\)). The passive sections may contain Bragg gratings but no excess carriers (\(N=0\), \(\partial\beta/\partial N=0\), \(D_{\rm sp}=0\), \(R_{\rm st}=0\) for \(z\in[l,L]\)). The reference plane is located just inside the active section so that a possible finite reflectivity at the interface between active and passive sections belongs to the passive sections. Furthermore, we neglect spatial hole burning (i.e., \(\partial N/\partial z=0\)), nonlinear gain and index \((\partial\beta/\partial|f|^{2}=0,\,\partial R_{\rm st}/\partial|f|^{2}=R_{ \rm st}/|f|^{2})\) and gain dispersion (\(\beta_{D}=0,\,\partial\beta/\partial\omega=n_{\rm g}/c\)). See Figure 2 for an illustration of the device. The functions \(\Phi^{\pm}(z)\) as solution of (9) are defined on the whole cavity \(z\in[0,L]\). In the active section they are given by  
\[\Phi^{\pm}(z)|_{\rm active}=\Phi^{\pm}(l){\rm e}^{\mp i(\Delta\beta+\frac{n_{ \rm g}}{c}\Omega)(z-l)} \tag{89}\]  
and hence  
\[\Phi^{+}(z)\[PERSON]^{-}(z)|_{\rm active}=\[PERSON]^{+}(l)\Phi^{-}(l) \tag{90}\]holds. Furthermore we define the left and right reflectivities  
\[r^{+}=\frac{\[PERSON]^{-}(l)}{\[PERSON]^{+}(l)}\quad\text{and}\quad r^{-}=\frac{\[PERSON]^{+}(l) }{\[PERSON]^{-}(l)}\equiv r_{0}\text{e}^{-2 il\left(\Delta\overline{\beta}+\frac{n_{ \text{g}}}{\overline{\varepsilon}}\Omega\right)}, \tag{91}\]  
where we exploited the fact that the \(\Phi^{\pm}(z)\) are continuous at \(z=l\). The derivative of \(r^{+}\) can be expressed as (see Appendix B)  
\[\frac{\partial\text{ln}(r^{+})}{\partial\omega}=-2i\frac{\int_{l}^{L}\[PERSON]^{+} \[PERSON]^{-}\frac{\partial\beta}{\partial\omega}\,dz}{\[PERSON]^{+}(l)\[PERSON]^{-}(l)}. \tag{92}\]  
Therefore,  
\[ \begin{split}(\[PERSON],\frac{\partial\beta}{\partial\omega}\[PERSON])& =2\int_{0}^{l}\[PERSON]^{+}\[PERSON]^{-}\frac{\partial\beta}{\partial \omega}\,dz+2\int_{l}^{L}\[PERSON]^{+}\[PERSON]^{-}\frac{\partial\beta}{\partial\omega} \,dz\\ &=2\[PERSON]^{+}(l)\[PERSON]^{-}(l)\left[l\frac{\partial\beta}{\partial \omega}\right]_{\text{active}}+\frac{i}{2}\frac{\partial\text{ln}(r^{+})}{ \partial\omega}\right]\\ &=2l\[PERSON]^{+}(l)\[PERSON]^{-}(l)\frac{\partial\beta}{\partial\omega} \bigg{|}_{\text{active}}\chi\end{split}  \tag{93}\]  
where we introduced the parameter  
\[\chi=1+\frac{i\frac{\partial\text{ln}(r^{+})}{\partial\omega}}{2l\frac{ \partial\beta}{\partial\omega}}=1+i\frac{c}{2 ln_{\text{g}}}\frac{\partial \text{ln}(r^{+})}{\partial\omega} \tag{94}\]  
as in [14]. We omit \(|_{\text{active}}\) in what follows. Furthermore  
\[ \begin{split}(\Phi,\tau_{\text{d}}\frac{\partial\beta}{ \partial N}\frac{\partial R_{\text{st}}}{\partial|f|^{2}}\Phi)& =\frac{2}{|f|^{2}}\int_{0}^{l}\Phi^{+}\Phi^{-}\tau_{\text{d}} \frac{\partial\beta}{\partial N}R_{\text{st}}\,dz\\ &=\frac{2}{|f|^{2}}\Phi^{+}(l)\Phi^{-}(l)\frac{\partial\beta}{ \partial N}\int_{0}^{l}\tau_{\text{d}}R_{\text{st}}\,dz\end{split}  \tag{95}\]  
holds. Then from (75), (78) and (79)  
\[\alpha_{\text{H,eff}}=-\frac{\text{Re}\left(\frac{\frac{\partial\beta}{ \partial N}\chi^{*}}{\frac{\partial\beta}{\partial\omega}\chi^{*}}\right)}{ \text{Im}\left(\frac{\frac{\partial\beta}{\partial N}\chi^{*}}{\frac{\partial \beta}{\partial\omega}\chi^{*}}\right)}=-\frac{\text{Re}\left(\frac{\partial \beta}{\partial N}\chi^{*}\right)}{\text{Im}\left(\frac{\partial\beta}{ \partial N}\chi^{*}\right)}=\frac{\text{Re}(\alpha_{\text{H}}\chi+i\chi)}{ \text{Im}(\alpha_{\text{H}}\chi+i\chi)} \tag{96}\]  
and  
\[1+a_{\text{H,eff}}^{2}=(1+a_{\text{H}}^{2})\frac{|\chi|^{2}}{\text{Im}^{2}( \alpha_{\text{H}}\chi+i\chi)} \tag{97}\]  
follow.