# Semiconductor Laser Linewidth Theory Revisited
## 7 Spectral Linewidth  
Collecting the results of the previous sections, we can now calculate the spectral linewidth. Rewriting (53) as  
\[\Delta\omega=\int_{-\infty}^{+\infty}\left\langle\frac{\partial\delta\varphi( t)}{\partial t}\frac{\partial\delta\varphi(t+\tau)}{\partial t}\right\rangle d\tau \tag{80}\]  
and inserting the equation of the phase fluctuations (77), we obtain  
\[\Delta\omega=\frac{\alpha_{\text{H,eff}}^{2}}{4(|f|^{2})^{2}}\int_{-\infty}^{ \infty}\left\langle F_{|f|^{2}}(t)F_{|f|^{2}}(t+\tau)\right\rangle d\tau+\int _{-\infty}^{\infty}\left\langle F_{\varphi}(t)F_{\varphi}(t+\tau)\right\rangle d\tau, \tag{81}\]  
where we have taken (63) into account. Inserting (61) and (62) results in the remarkably simple expression for the spectral linewidth  
\[\Delta\
u=\frac{\Delta\omega}{2\pi}=\frac{D_{f^{\prime}f}}{2\pi\langle|f|^{2 }\rangle}\Big{(}1+\alpha_{\text{H,eff}}^{2}\Big{)}. \tag{82}\]  
In the following, we will rewrite this expression in terms of the intra-cavity photon number  
\[I_{\text{ph}}=\frac{1}{\hbar\omega_{0}}\int_{0}^{L}P\ \text{Re}\left(\frac{ \partial\beta}{\partial\omega}\right)dz=\frac{P_{0}}{\hbar\omega_{0}}\frac{ \int_{0}^{L}\|\Phi\|^{2}\text{Re}\left(\frac{\partial\beta}{\partial\omega} \right)dz}{(1-|r_{0}|^{2})|\Phi^{-}(0)|^{2}} \tag{83}\]  
where  
\[P(z)=\langle|f|^{2}\rangle\|\Phi(z)\|^{2} \tag{84}\]  
is the stationary optical power inside the cavity and  
\[P_{0}=(1-|r_{0}|^{2})\langle|f|^{2}\rangle|\Phi^{-}(0)|^{2} \tag{85}\]is the outcoupled power at \(z=0\). Moreover, we define the rate of spontaneous emission into the lasing mode  
\[R_{\rm sp}=\frac{\int_{0}^{L}\|\Phi\|^{2}n_{\rm sp}\beta\,dz}{\int_{0}^{L}\|\Phi \|^{2}\,{\rm Re}\Big{(}\frac{\partial\beta}{\partial\omega}\Big{)}\,dz} \tag{86}\]  
and the longitudinal excess factor of spontaneous emission  
\[K=\frac{\Big{(}\int_{0}^{L}\|\Phi\|^{2}\,{\rm Re}\Big{(}\frac{\partial\beta}{ \partial\omega}\Big{)}\,dz\Big{)}^{2}}{\Big{|}\big{(}\Phi,\frac{\partial\beta} {\partial\omega}\Phi\big{)}\Big{|}^{2}}. \tag{87}\]  
Now, (82) can be written as  
\[\Delta\
u=\frac{KR_{\rm sp}}{4\pi I_{\rm ph}}\Big{(}1+\alpha_{\rm H,eff}^{2} \Big{)} \tag{88}\]  
which is the standard form found in the literature [14; 46].  
The mode profile \(\Phi(z)\) entering (83), (86), and (87) is obtained by solving (9), (36) and (68) self-consistently above threshold. Thus, \(\Phi\) is a mode of the active cavity affected by spatial hole burning. The expression for the \(K\)-factor generalizes the one given in [10] for non-uniform and complex-valued group velocity \((\partial\beta/\partial\omega)^{-1}\) and is in basic correspondence with [14; 50; 51], but fundamentally different from the modified \(K\)-factor introduced in [52].  
At an exceptional point, the mode is orthogonal to itself, i.e., \((\Phi,\frac{\partial\beta}{\partial\omega}\Phi)=0\), and the \(K\)-factor approaches infinity. However, one has to keep in mind that at an exceptional point the single-mode approximation used here fails. Instead, for the description of the dynamics in the vicinity of such a point one has use at least one eigenmode and the corresponding generalized eigenmode [10].